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3.23.47 \(\int \frac {(A+B x) (d+e x)^5}{(a+b x+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=942 \[ \frac {B \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^5}{c^{7/2}}+\frac {2 \left (5 B e^3 \left (c d^2-3 a e^2\right ) b^5+4 B c^2 d^3 e^2 b^4-8 c e \left (16 A c^2 e d^3+B \left (11 c^2 d^4+7 a c e^2 d^2-20 a^2 e^4\right )\right ) b^3+32 c^3 d^2 \left (2 B c d^3+8 A c e d^2+17 a B e^2 d+16 a A e^3\right ) b^2-16 c^2 \left (8 A c d \left (c^2 d^4+6 a c e^2 d^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c e^2 d^2+33 a^2 e^4\right )\right ) b+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )+\left (-15 B e^5 b^6+10 B c d e^4 b^5+2 B c e^3 \left (3 c d^2+85 a e^2\right ) b^4+16 c^2 d e^2 \left (6 B c d^2+8 A c e d-7 a B e^2\right ) b^3-16 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c e^2 d^2+39 a^2 e^4\right )\right ) b^2+32 c^3 \left (4 A e \left (5 c^2 d^4+6 a c e^2 d^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c e^2 d^3+29 a^2 e^4 d\right )\right ) b-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c e^2 d^2-3 a^2 e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {c x^2+b x+a}}+\frac {2 (d+e x)^2 \left (B e \left (3 c d^2-5 a e^2\right ) b^3-4 c d \left (2 B c d^2+4 A c e d+a B e^2\right ) b^2+4 c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right ) b-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+\left (-5 B e^3 b^4+2 B c d e^2 b^3+2 c e \left (7 B c d^2+8 A c e d+19 a B e^2\right ) b^2-8 c^2 \left (2 B c d^3+6 A c e d^2+7 a B e^2 d+2 a A e^3\right ) b+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac {2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}} \]

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Rubi [A]  time = 0.96, antiderivative size = 942, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {818, 777, 621, 206} \begin {gather*} \frac {B \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^5}{c^{7/2}}+\frac {2 \left (5 B e^3 \left (c d^2-3 a e^2\right ) b^5+4 B c^2 d^3 e^2 b^4-8 c e \left (16 A c^2 e d^3+B \left (11 c^2 d^4+7 a c e^2 d^2-20 a^2 e^4\right )\right ) b^3+32 c^3 d^2 \left (2 B c d^3+8 A c e d^2+17 a B e^2 d+16 a A e^3\right ) b^2-16 c^2 \left (8 A c d \left (c^2 d^4+6 a c e^2 d^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c e^2 d^2+33 a^2 e^4\right )\right ) b+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )+\left (-15 B e^5 b^6+10 B c d e^4 b^5+2 B c e^3 \left (3 c d^2+85 a e^2\right ) b^4+16 c^2 d e^2 \left (6 B c d^2+8 A c e d-7 a B e^2\right ) b^3-16 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c e^2 d^2+39 a^2 e^4\right )\right ) b^2+32 c^3 \left (4 A e \left (5 c^2 d^4+6 a c e^2 d^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c e^2 d^3+29 a^2 e^4 d\right )\right ) b-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c e^2 d^2-3 a^2 e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {c x^2+b x+a}}+\frac {2 (d+e x)^2 \left (B e \left (3 c d^2-5 a e^2\right ) b^3-4 c d \left (2 B c d^2+4 A c e d+a B e^2\right ) b^2+4 c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right ) b-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+\left (-5 B e^3 b^4+2 B c d e^2 b^3+2 c e \left (7 B c d^2+8 A c e d+19 a B e^2\right ) b^2-8 c^2 \left (2 B c d^3+6 A c e d^2+7 a B e^2 d+2 a A e^3\right ) b+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac {2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^4*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/
(5*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) + (2*(d + e*x)^2*(b^3*B*e*(3*c*d^2 - 5*a*e^2) - 4*b^2*c*d*(2*B*c*d
^2 + 4*A*c*d*e + a*B*e^2) - 16*a*c^2*e*(5*a*B*d*e + 2*A*(c*d^2 + a*e^2)) + 4*b*c*(9*a*B*e*(c*d^2 + a*e^2) + 4*
A*c*d*(c*d^2 + 3*a*e^2)) + (2*b^3*B*c*d*e^2 - 5*b^4*B*e^3 + 2*b^2*c*e*(7*B*c*d^2 + 8*A*c*d*e + 19*a*B*e^2) - 8
*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 7*a*B*d*e^2 + 2*a*A*e^3) + 8*c^2*(5*a*B*e*(c*d^2 - a*e^2) + 4*A*c*d*(c*d^2 +
 a*e^2)))*x))/(15*c^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (2*(4*b^4*B*c^2*d^3*e^2 + 5*b^5*B*e^3*(c*d^2
- 3*a*e^2) + 32*b^2*c^3*d^2*(2*B*c*d^3 + 8*A*c*d^2*e + 17*a*B*d*e^2 + 16*a*A*e^3) + 64*a*c^3*e*(4*A*(c*d^2 + a
*e^2)^2 + 5*a*B*d*e*(c*d^2 + 4*a*e^2)) - 8*b^3*c*e*(16*A*c^2*d^3*e + B*(11*c^2*d^4 + 7*a*c*d^2*e^2 - 20*a^2*e^
4)) - 16*b*c^2*(8*A*c*d*(c^2*d^4 + 6*a*c*d^2*e^2 + 5*a^2*e^4) + a*B*e*(18*c^2*d^4 + 71*a*c*d^2*e^2 + 33*a^2*e^
4)) + (10*b^5*B*c*d*e^4 - 15*b^6*B*e^5 + 2*b^4*B*c*e^3*(3*c*d^2 + 85*a*e^2) + 16*b^3*c^2*d*e^2*(6*B*c*d^2 + 8*
A*c*d*e - 7*a*B*e^2) - 32*c^3*(8*A*c*d*(c*d^2 + a*e^2)^2 + 5*a*B*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 - 3*a^2*e^4)) -
16*b^2*c^2*e*(16*A*c*d*e*(2*c*d^2 + a*e^2) + B*(15*c^2*d^4 + 29*a*c*d^2*e^2 + 39*a^2*e^4)) + 32*b*c^3*(4*A*e*(
5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4) + B*(4*c^2*d^5 + 28*a*c*d^3*e^2 + 29*a^2*d*e^4)))*x))/(15*c^3*(b^2 - 4*a*
c)^3*Sqrt[a + b*x + c*x^2]) + (B*e^5*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(7/2)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=\frac {2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \int \frac {(d+e x)^3 \left (\frac {1}{2} \left (-16 A c^2 d^2-2 b B e \left (\frac {3 b d}{2}-4 a e\right )+8 b c d (B d+2 A e)-4 a c e (5 B d+4 A e)\right )+\frac {5}{2} B \left (b^2-4 a c\right ) e^2 x\right )}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 c \left (b^2-4 a c\right )}\\ &=\frac {2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 (d+e x)^2 \left (b^3 B e \left (3 c d^2-5 a e^2\right )-4 b^2 c d \left (2 B c d^2+4 A c d e+a B e^2\right )-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+4 b c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right )+\left (2 b^3 B c d e^2-5 b^4 B e^3+2 b^2 c e \left (7 B c d^2+8 A c d e+19 a B e^2\right )-8 b c^2 \left (2 B c d^3+6 A c d^2 e+7 a B d e^2+2 a A e^3\right )+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {4 \int \frac {(d+e x) \left (\frac {1}{4} \left (-5 b^4 B d e^3+8 b^2 c d e \left (11 B c d^2+16 A c d e+6 a B e^2\right )-4 b^3 B \left (c d^2 e^2-5 a e^4\right )+16 c^2 \left (8 A \left (c d^2+a e^2\right )^2+5 a B d e \left (2 c d^2+5 a e^2\right )\right )-16 b c \left (16 A c d e \left (c d^2+a e^2\right )+B \left (4 c^2 d^4+23 a c d^2 e^2+9 a^2 e^4\right )\right )\right )+\frac {15}{4} B \left (b^2-4 a c\right )^2 e^4 x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 c^2 \left (b^2-4 a c\right )^2}\\ &=\frac {2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 (d+e x)^2 \left (b^3 B e \left (3 c d^2-5 a e^2\right )-4 b^2 c d \left (2 B c d^2+4 A c d e+a B e^2\right )-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+4 b c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right )+\left (2 b^3 B c d e^2-5 b^4 B e^3+2 b^2 c e \left (7 B c d^2+8 A c d e+19 a B e^2\right )-8 b c^2 \left (2 B c d^3+6 A c d^2 e+7 a B d e^2+2 a A e^3\right )+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 b^4 B c^2 d^3 e^2+5 b^5 B e^3 \left (c d^2-3 a e^2\right )+32 b^2 c^3 d^2 \left (2 B c d^3+8 A c d^2 e+17 a B d e^2+16 a A e^3\right )+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )-8 b^3 c e \left (16 A c^2 d^3 e+B \left (11 c^2 d^4+7 a c d^2 e^2-20 a^2 e^4\right )\right )-16 b c^2 \left (8 A c d \left (c^2 d^4+6 a c d^2 e^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c d^2 e^2+33 a^2 e^4\right )\right )+\left (10 b^5 B c d e^4-15 b^6 B e^5+2 b^4 B c e^3 \left (3 c d^2+85 a e^2\right )+16 b^3 c^2 d e^2 \left (6 B c d^2+8 A c d e-7 a B e^2\right )-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c d^2 e^2-3 a^2 e^4\right )\right )-16 b^2 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c d^2 e^2+39 a^2 e^4\right )\right )+32 b c^3 \left (4 A e \left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c d^3 e^2+29 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {\left (B e^5\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{c^3}\\ &=\frac {2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 (d+e x)^2 \left (b^3 B e \left (3 c d^2-5 a e^2\right )-4 b^2 c d \left (2 B c d^2+4 A c d e+a B e^2\right )-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+4 b c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right )+\left (2 b^3 B c d e^2-5 b^4 B e^3+2 b^2 c e \left (7 B c d^2+8 A c d e+19 a B e^2\right )-8 b c^2 \left (2 B c d^3+6 A c d^2 e+7 a B d e^2+2 a A e^3\right )+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 b^4 B c^2 d^3 e^2+5 b^5 B e^3 \left (c d^2-3 a e^2\right )+32 b^2 c^3 d^2 \left (2 B c d^3+8 A c d^2 e+17 a B d e^2+16 a A e^3\right )+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )-8 b^3 c e \left (16 A c^2 d^3 e+B \left (11 c^2 d^4+7 a c d^2 e^2-20 a^2 e^4\right )\right )-16 b c^2 \left (8 A c d \left (c^2 d^4+6 a c d^2 e^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c d^2 e^2+33 a^2 e^4\right )\right )+\left (10 b^5 B c d e^4-15 b^6 B e^5+2 b^4 B c e^3 \left (3 c d^2+85 a e^2\right )+16 b^3 c^2 d e^2 \left (6 B c d^2+8 A c d e-7 a B e^2\right )-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c d^2 e^2-3 a^2 e^4\right )\right )-16 b^2 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c d^2 e^2+39 a^2 e^4\right )\right )+32 b c^3 \left (4 A e \left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c d^3 e^2+29 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {\left (2 B e^5\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{c^3}\\ &=\frac {2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 (d+e x)^2 \left (b^3 B e \left (3 c d^2-5 a e^2\right )-4 b^2 c d \left (2 B c d^2+4 A c d e+a B e^2\right )-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+4 b c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right )+\left (2 b^3 B c d e^2-5 b^4 B e^3+2 b^2 c e \left (7 B c d^2+8 A c d e+19 a B e^2\right )-8 b c^2 \left (2 B c d^3+6 A c d^2 e+7 a B d e^2+2 a A e^3\right )+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 b^4 B c^2 d^3 e^2+5 b^5 B e^3 \left (c d^2-3 a e^2\right )+32 b^2 c^3 d^2 \left (2 B c d^3+8 A c d^2 e+17 a B d e^2+16 a A e^3\right )+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )-8 b^3 c e \left (16 A c^2 d^3 e+B \left (11 c^2 d^4+7 a c d^2 e^2-20 a^2 e^4\right )\right )-16 b c^2 \left (8 A c d \left (c^2 d^4+6 a c d^2 e^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c d^2 e^2+33 a^2 e^4\right )\right )+\left (10 b^5 B c d e^4-15 b^6 B e^5+2 b^4 B c e^3 \left (3 c d^2+85 a e^2\right )+16 b^3 c^2 d e^2 \left (6 B c d^2+8 A c d e-7 a B e^2\right )-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c d^2 e^2-3 a^2 e^4\right )\right )-16 b^2 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c d^2 e^2+39 a^2 e^4\right )\right )+32 b c^3 \left (4 A e \left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c d^3 e^2+29 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {B e^5 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 9.19, size = 1608, normalized size = 1.71 \begin {gather*} \frac {\frac {2 \sqrt {c} \left (A \left (\left (3 d^5+25 e x d^4+150 e^2 x^2 d^3-150 e^3 x^3 d^2-25 e^4 x^4 d-3 e^5 x^5\right ) b^5+10 (a e-c d x) \left (d^4+20 e x d^3-90 e^2 x^2 d^2+20 e^3 x^3 d+e^4 x^4\right ) b^4+40 (d-e x) \left (2 a^2 \left (d^2-14 e x d+e^2 x^2\right ) e^2+2 c^2 d^2 x^2 \left (d^2-14 e x d+e^2 x^2\right )-a c (d-e x)^2 \left (d^2+18 e x d+e^2 x^2\right )\right ) b^3+80 \left (-2 a^3 \left (3 d^2-10 e x d+3 e^2 x^2\right ) e^3-3 a^2 c \left (d^4-10 e x d^3+10 e^2 x^2 d^2-10 e^3 x^3 d+e^4 x^4\right ) e+2 c^3 d^3 x^3 \left (3 d^2-10 e x d+3 e^2 x^2\right )+3 a c^2 d x \left (d^4-10 e x d^3+10 e^2 x^2 d^2-10 e^3 x^3 d+e^4 x^4\right )\right ) b^2+80 (d-e x) \left (8 a^4 e^4+4 a^3 c \left (3 d^2-2 e x d+3 e^2 x^2\right ) e^2+8 c^4 d^4 x^4+3 a^2 c^2 (d-e x)^4+4 a c^3 d^2 x^2 \left (3 d^2-2 e x d+3 e^2 x^2\right )\right ) b+32 \left (-8 a^5 e^5-20 a^4 c \left (d^2+e^2 x^2\right ) e^3-5 a^3 c^2 \left (3 d^4+10 e^2 x^2 d^2+3 e^4 x^4\right ) e+8 c^5 d^5 x^5+20 a c^4 d^3 x^3 \left (d^2+e^2 x^2\right )+5 a^2 c^3 d x \left (3 d^4+10 e^2 x^2 d^2+3 e^4 x^4\right )\right )\right ) c^3+B \left (-16 c^2 e^4 (80 c d-33 b e+30 c e x) a^5-80 c e^2 \left (2 \left (4 d^3+20 e^2 x^2 d+7 e^3 x^3\right ) c^3+b e \left (-16 d^2+40 e x d-3 e^2 x^2\right ) c^2-21 b^2 e^3 x c+2 b^3 e^3\right ) a^4-\left (32 \left (3 d^5+50 e^2 x^2 d^3+75 e^4 x^4 d+23 e^5 x^5\right ) c^5+80 b e \left (-6 d^4+20 e x d^3-40 e^2 x^2 d^2+60 e^3 x^3 d+5 e^4 x^4\right ) c^4+80 b^2 e^2 \left (6 d^3-40 e x d^2+30 e^2 x^2 d-27 e^3 x^3\right ) c^3-1400 b^3 e^5 x^2 c^2+490 b^4 e^5 x c-15 b^5 e^5\right ) a^3+\left (45 e^5 x b^6-465 c e^5 x^2 b^5-150 c^2 e^5 x^3 b^4+40 c^3 e \left (d^4-30 e x d^3+60 e^2 x^2 d^2-10 e^3 x^3 d+30 e^4 x^4\right ) b^3-48 c^4 \left (d^5-25 e x d^4+50 e^2 x^2 d^3-100 e^3 x^3 d^2+25 e^4 x^4 d-19 e^5 x^5\right ) b^2-240 c^5 d x \left (d^4-5 e x d^3+10 e^2 x^2 d^2-10 e^3 x^3 d+5 e^4 x^4\right ) b+160 c^6 d^2 e x^3 \left (5 d^2+6 e^2 x^2\right )\right ) a^2+\left (45 e^5 x^2 b^7-100 c e^5 x^3 b^6-375 c^2 e^5 x^4 b^5+2 c^3 \left (d^5+50 e x d^4-450 e^2 x^2 d^3+200 e^3 x^3 d^2+25 e^4 x^4 d-129 e^5 x^5\right ) b^4+40 c^4 d x \left (-3 d^4+25 e x d^3-50 e^2 x^2 d^2+30 e^3 x^3 d+5 e^4 x^4\right ) b^3+240 c^5 d^2 x^2 \left (-2 d^3+5 e x d^2-10 e^2 x^2 d+2 e^3 x^3\right ) b^2-160 c^6 d^3 x^3 \left (2 d^2-5 e x d+6 e^2 x^2\right ) b+320 c^7 d^4 e x^5\right ) a+b x \left (15 e^5 x^2 b^7+35 c e^5 x^3 b^6+23 c^2 e^5 x^4 b^5+5 c^3 d \left (d^4+15 e x d^3-30 e^2 x^2 d^2-10 e^3 x^3 d-3 e^4 x^4\right ) b^4-10 c^4 d^2 x \left (4 d^3-45 e x d^2+20 e^2 x^2 d+2 e^3 x^3\right ) b^3-40 c^5 d^3 x^2 \left (6 d^2-15 e x d+2 e^2 x^2\right ) b^2+80 c^6 d^4 x^3 (3 e x-4 d) b-128 c^7 d^5 x^4\right )\right )\right )}{(a+x (b+c x))^{5/2}}-15 B \left (b^2-4 a c\right )^3 e^5 \log \left (b+2 c x+2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{15 c^{7/2} \left (4 a c-b^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x]

[Out]

((2*Sqrt[c]*(A*c^3*(10*b^4*(a*e - c*d*x)*(d^4 + 20*d^3*e*x - 90*d^2*e^2*x^2 + 20*d*e^3*x^3 + e^4*x^4) + b^5*(3
*d^5 + 25*d^4*e*x + 150*d^3*e^2*x^2 - 150*d^2*e^3*x^3 - 25*d*e^4*x^4 - 3*e^5*x^5) + 40*b^3*(d - e*x)*(2*a^2*e^
2*(d^2 - 14*d*e*x + e^2*x^2) + 2*c^2*d^2*x^2*(d^2 - 14*d*e*x + e^2*x^2) - a*c*(d - e*x)^2*(d^2 + 18*d*e*x + e^
2*x^2)) + 80*b*(d - e*x)*(8*a^4*e^4 + 8*c^4*d^4*x^4 + 3*a^2*c^2*(d - e*x)^4 + 4*a^3*c*e^2*(3*d^2 - 2*d*e*x + 3
*e^2*x^2) + 4*a*c^3*d^2*x^2*(3*d^2 - 2*d*e*x + 3*e^2*x^2)) + 80*b^2*(-2*a^3*e^3*(3*d^2 - 10*d*e*x + 3*e^2*x^2)
 + 2*c^3*d^3*x^3*(3*d^2 - 10*d*e*x + 3*e^2*x^2) - 3*a^2*c*e*(d^4 - 10*d^3*e*x + 10*d^2*e^2*x^2 - 10*d*e^3*x^3
+ e^4*x^4) + 3*a*c^2*d*x*(d^4 - 10*d^3*e*x + 10*d^2*e^2*x^2 - 10*d*e^3*x^3 + e^4*x^4)) + 32*(-8*a^5*e^5 + 8*c^
5*d^5*x^5 - 20*a^4*c*e^3*(d^2 + e^2*x^2) + 20*a*c^4*d^3*x^3*(d^2 + e^2*x^2) - 5*a^3*c^2*e*(3*d^4 + 10*d^2*e^2*
x^2 + 3*e^4*x^4) + 5*a^2*c^3*d*x*(3*d^4 + 10*d^2*e^2*x^2 + 3*e^4*x^4))) + B*(-16*a^5*c^2*e^4*(80*c*d - 33*b*e
+ 30*c*e*x) - 80*a^4*c*e^2*(2*b^3*e^3 - 21*b^2*c*e^3*x + b*c^2*e*(-16*d^2 + 40*d*e*x - 3*e^2*x^2) + 2*c^3*(4*d
^3 + 20*d*e^2*x^2 + 7*e^3*x^3)) + b*x*(15*b^7*e^5*x^2 + 35*b^6*c*e^5*x^3 - 128*c^7*d^5*x^4 + 23*b^5*c^2*e^5*x^
4 + 80*b*c^6*d^4*x^3*(-4*d + 3*e*x) - 40*b^2*c^5*d^3*x^2*(6*d^2 - 15*d*e*x + 2*e^2*x^2) - 10*b^3*c^4*d^2*x*(4*
d^3 - 45*d^2*e*x + 20*d*e^2*x^2 + 2*e^3*x^3) + 5*b^4*c^3*d*(d^4 + 15*d^3*e*x - 30*d^2*e^2*x^2 - 10*d*e^3*x^3 -
 3*e^4*x^4)) + a*(45*b^7*e^5*x^2 - 100*b^6*c*e^5*x^3 - 375*b^5*c^2*e^5*x^4 + 320*c^7*d^4*e*x^5 - 160*b*c^6*d^3
*x^3*(2*d^2 - 5*d*e*x + 6*e^2*x^2) + 240*b^2*c^5*d^2*x^2*(-2*d^3 + 5*d^2*e*x - 10*d*e^2*x^2 + 2*e^3*x^3) + 40*
b^3*c^4*d*x*(-3*d^4 + 25*d^3*e*x - 50*d^2*e^2*x^2 + 30*d*e^3*x^3 + 5*e^4*x^4) + 2*b^4*c^3*(d^5 + 50*d^4*e*x -
450*d^3*e^2*x^2 + 200*d^2*e^3*x^3 + 25*d*e^4*x^4 - 129*e^5*x^5)) + a^2*(45*b^6*e^5*x - 465*b^5*c*e^5*x^2 - 150
*b^4*c^2*e^5*x^3 + 160*c^6*d^2*e*x^3*(5*d^2 + 6*e^2*x^2) - 240*b*c^5*d*x*(d^4 - 5*d^3*e*x + 10*d^2*e^2*x^2 - 1
0*d*e^3*x^3 + 5*e^4*x^4) + 40*b^3*c^3*e*(d^4 - 30*d^3*e*x + 60*d^2*e^2*x^2 - 10*d*e^3*x^3 + 30*e^4*x^4) - 48*b
^2*c^4*(d^5 - 25*d^4*e*x + 50*d^3*e^2*x^2 - 100*d^2*e^3*x^3 + 25*d*e^4*x^4 - 19*e^5*x^5)) - a^3*(-15*b^5*e^5 +
 490*b^4*c*e^5*x - 1400*b^3*c^2*e^5*x^2 + 80*b^2*c^3*e^2*(6*d^3 - 40*d^2*e*x + 30*d*e^2*x^2 - 27*e^3*x^3) + 80
*b*c^4*e*(-6*d^4 + 20*d^3*e*x - 40*d^2*e^2*x^2 + 60*d*e^3*x^3 + 5*e^4*x^4) + 32*c^5*(3*d^5 + 50*d^3*e^2*x^2 +
75*d*e^4*x^4 + 23*e^5*x^5)))))/(a + x*(b + c*x))^(5/2) - 15*B*(b^2 - 4*a*c)^3*e^5*Log[b + 2*c*x + 2*Sqrt[c]*Sq
rt[a + x*(b + c*x)]])/(15*c^(7/2)*(-b^2 + 4*a*c)^3)

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IntegrateAlgebraic [B]  time = 30.31, size = 2751, normalized size = 2.92 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*(3*A*b^5*c^3*d^5 + 2*a*b^4*B*c^3*d^5 - 40*a*A*b^3*c^4*d^5 - 48*a^2*b^2*B*c^4*d^5 + 240*a^2*A*b*c^5*d^5 - 96
*a^3*B*c^5*d^5 + 10*a*A*b^4*c^3*d^4*e + 40*a^2*b^3*B*c^3*d^4*e - 240*a^2*A*b^2*c^4*d^4*e + 480*a^3*b*B*c^4*d^4
*e - 480*a^3*A*c^5*d^4*e + 80*a^2*A*b^3*c^3*d^3*e^2 - 480*a^3*b^2*B*c^3*d^3*e^2 + 960*a^3*A*b*c^4*d^3*e^2 - 64
0*a^4*B*c^4*d^3*e^2 - 480*a^3*A*b^2*c^3*d^2*e^3 + 1280*a^4*b*B*c^3*d^2*e^3 - 640*a^4*A*c^4*d^2*e^3 + 640*a^4*A
*b*c^3*d*e^4 - 1280*a^5*B*c^3*d*e^4 + 15*a^3*b^5*B*e^5 - 160*a^4*b^3*B*c*e^5 + 528*a^5*b*B*c^2*e^5 - 256*a^5*A
*c^3*e^5 + 5*b^5*B*c^3*d^5*x - 10*A*b^4*c^4*d^5*x - 120*a*b^3*B*c^4*d^5*x + 240*a*A*b^2*c^5*d^5*x - 240*a^2*b*
B*c^5*d^5*x + 480*a^2*A*c^6*d^5*x + 25*A*b^5*c^3*d^4*e*x + 100*a*b^4*B*c^3*d^4*e*x - 600*a*A*b^3*c^4*d^4*e*x +
 1200*a^2*b^2*B*c^4*d^4*e*x - 1200*a^2*A*b*c^5*d^4*e*x + 200*a*A*b^4*c^3*d^3*e^2*x - 1200*a^2*b^3*B*c^3*d^3*e^
2*x + 2400*a^2*A*b^2*c^4*d^3*e^2*x - 1600*a^3*b*B*c^4*d^3*e^2*x - 1200*a^2*A*b^3*c^3*d^2*e^3*x + 3200*a^3*b^2*
B*c^3*d^2*e^3*x - 1600*a^3*A*b*c^4*d^2*e^3*x + 1600*a^3*A*b^2*c^3*d*e^4*x - 3200*a^4*b*B*c^3*d*e^4*x + 45*a^2*
b^6*B*e^5*x - 490*a^3*b^4*B*c*e^5*x + 1680*a^4*b^2*B*c^2*e^5*x - 640*a^4*A*b*c^3*e^5*x - 480*a^5*B*c^3*e^5*x -
 40*b^4*B*c^4*d^5*x^2 + 80*A*b^3*c^5*d^5*x^2 - 480*a*b^2*B*c^5*d^5*x^2 + 960*a*A*b*c^6*d^5*x^2 + 75*b^5*B*c^3*
d^4*e*x^2 - 200*A*b^4*c^4*d^4*e*x^2 + 1000*a*b^3*B*c^4*d^4*e*x^2 - 2400*a*A*b^2*c^5*d^4*e*x^2 + 1200*a^2*b*B*c
^5*d^4*e*x^2 + 150*A*b^5*c^3*d^3*e^2*x^2 - 900*a*b^4*B*c^3*d^3*e^2*x^2 + 2000*a*A*b^3*c^4*d^3*e^2*x^2 - 2400*a
^2*b^2*B*c^4*d^3*e^2*x^2 + 2400*a^2*A*b*c^5*d^3*e^2*x^2 - 1600*a^3*B*c^5*d^3*e^2*x^2 - 900*a*A*b^4*c^3*d^2*e^3
*x^2 + 2400*a^2*b^3*B*c^3*d^2*e^3*x^2 - 2400*a^2*A*b^2*c^4*d^2*e^3*x^2 + 3200*a^3*b*B*c^4*d^2*e^3*x^2 - 1600*a
^3*A*c^5*d^2*e^3*x^2 + 1200*a^2*A*b^3*c^3*d*e^4*x^2 - 2400*a^3*b^2*B*c^3*d*e^4*x^2 + 1600*a^3*A*b*c^4*d*e^4*x^
2 - 3200*a^4*B*c^4*d*e^4*x^2 + 45*a*b^7*B*e^5*x^2 - 465*a^2*b^5*B*c*e^5*x^2 + 1400*a^3*b^3*B*c^2*e^5*x^2 - 480
*a^3*A*b^2*c^3*e^5*x^2 + 240*a^4*b*B*c^3*e^5*x^2 - 640*a^4*A*c^4*e^5*x^2 - 240*b^3*B*c^5*d^5*x^3 + 480*A*b^2*c
^6*d^5*x^3 - 320*a*b*B*c^6*d^5*x^3 + 640*a*A*c^7*d^5*x^3 + 450*b^4*B*c^4*d^4*e*x^3 - 1200*A*b^3*c^5*d^4*e*x^3
+ 1200*a*b^2*B*c^5*d^4*e*x^3 - 1600*a*A*b*c^6*d^4*e*x^3 + 800*a^2*B*c^6*d^4*e*x^3 - 150*b^5*B*c^3*d^3*e^2*x^3
+ 900*A*b^4*c^4*d^3*e^2*x^3 - 2000*a*b^3*B*c^4*d^3*e^2*x^3 + 2400*a*A*b^2*c^5*d^3*e^2*x^3 - 2400*a^2*b*B*c^5*d
^3*e^2*x^3 + 1600*a^2*A*c^6*d^3*e^2*x^3 - 150*A*b^5*c^3*d^2*e^3*x^3 + 400*a*b^4*B*c^3*d^2*e^3*x^3 - 2000*a*A*b
^3*c^4*d^2*e^3*x^3 + 4800*a^2*b^2*B*c^4*d^2*e^3*x^3 - 2400*a^2*A*b*c^5*d^2*e^3*x^3 + 200*a*A*b^4*c^3*d*e^4*x^3
 - 400*a^2*b^3*B*c^3*d*e^4*x^3 + 2400*a^2*A*b^2*c^4*d*e^4*x^3 - 4800*a^3*b*B*c^4*d*e^4*x^3 + 15*b^8*B*e^5*x^3
- 100*a*b^6*B*c*e^5*x^3 - 150*a^2*b^4*B*c^2*e^5*x^3 - 80*a^2*A*b^3*c^3*e^5*x^3 + 2160*a^3*b^2*B*c^3*e^5*x^3 -
960*a^3*A*b*c^4*e^5*x^3 - 1120*a^4*B*c^4*e^5*x^3 - 320*b^2*B*c^6*d^5*x^4 + 640*A*b*c^7*d^5*x^4 + 600*b^3*B*c^5
*d^4*e*x^4 - 1600*A*b^2*c^6*d^4*e*x^4 + 800*a*b*B*c^6*d^4*e*x^4 - 200*b^4*B*c^4*d^3*e^2*x^4 + 1200*A*b^3*c^5*d
^3*e^2*x^4 - 2400*a*b^2*B*c^5*d^3*e^2*x^4 + 1600*a*A*b*c^6*d^3*e^2*x^4 - 50*b^5*B*c^3*d^2*e^3*x^4 - 200*A*b^4*
c^4*d^2*e^3*x^4 + 1200*a*b^3*B*c^4*d^2*e^3*x^4 - 2400*a*A*b^2*c^5*d^2*e^3*x^4 + 2400*a^2*b*B*c^5*d^2*e^3*x^4 -
 25*A*b^5*c^3*d*e^4*x^4 + 50*a*b^4*B*c^3*d*e^4*x^4 + 600*a*A*b^3*c^4*d*e^4*x^4 - 1200*a^2*b^2*B*c^4*d*e^4*x^4
+ 1200*a^2*A*b*c^5*d*e^4*x^4 - 2400*a^3*B*c^5*d*e^4*x^4 + 35*b^7*B*c*e^5*x^4 - 375*a*b^5*B*c^2*e^5*x^4 + 10*a*
A*b^4*c^3*e^5*x^4 + 1200*a^2*b^3*B*c^3*e^5*x^4 - 240*a^2*A*b^2*c^4*e^5*x^4 - 400*a^3*b*B*c^4*e^5*x^4 - 480*a^3
*A*c^5*e^5*x^4 - 128*b*B*c^7*d^5*x^5 + 256*A*c^8*d^5*x^5 + 240*b^2*B*c^6*d^4*e*x^5 - 640*A*b*c^7*d^4*e*x^5 + 3
20*a*B*c^7*d^4*e*x^5 - 80*b^3*B*c^5*d^3*e^2*x^5 + 480*A*b^2*c^6*d^3*e^2*x^5 - 960*a*b*B*c^6*d^3*e^2*x^5 + 640*
a*A*c^7*d^3*e^2*x^5 - 20*b^4*B*c^4*d^2*e^3*x^5 - 80*A*b^3*c^5*d^2*e^3*x^5 + 480*a*b^2*B*c^5*d^2*e^3*x^5 - 960*
a*A*b*c^6*d^2*e^3*x^5 + 960*a^2*B*c^6*d^2*e^3*x^5 - 15*b^5*B*c^3*d*e^4*x^5 - 10*A*b^4*c^4*d*e^4*x^5 + 200*a*b^
3*B*c^4*d*e^4*x^5 + 240*a*A*b^2*c^5*d*e^4*x^5 - 1200*a^2*b*B*c^5*d*e^4*x^5 + 480*a^2*A*c^6*d*e^4*x^5 + 23*b^6*
B*c^2*e^5*x^5 - 3*A*b^5*c^3*e^5*x^5 - 258*a*b^4*B*c^3*e^5*x^5 + 40*a*A*b^3*c^4*e^5*x^5 + 912*a^2*b^2*B*c^4*e^5
*x^5 - 240*a^2*A*b*c^5*e^5*x^5 - 736*a^3*B*c^5*e^5*x^5))/(15*c^3*(-b^2 + 4*a*c)^3*(a + b*x + c*x^2)^(5/2)) - (
B*e^5*Log[b*c^3 + 2*c^4*x - 2*c^(7/2)*Sqrt[a + b*x + c*x^2]])/c^(7/2)

________________________________________________________________________________________

fricas [B]  time = 80.94, size = 5253, normalized size = 5.58

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")

[Out]

[1/30*(15*((B*b^6*c^3 - 12*B*a*b^4*c^4 + 48*B*a^2*b^2*c^5 - 64*B*a^3*c^6)*e^5*x^6 + 3*(B*b^7*c^2 - 12*B*a*b^5*
c^3 + 48*B*a^2*b^3*c^4 - 64*B*a^3*b*c^5)*e^5*x^5 + 3*(B*b^8*c - 11*B*a*b^6*c^2 + 36*B*a^2*b^4*c^3 - 16*B*a^3*b
^2*c^4 - 64*B*a^4*c^5)*e^5*x^4 + (B*b^9 - 6*B*a*b^7*c - 24*B*a^2*b^5*c^2 + 224*B*a^3*b^3*c^3 - 384*B*a^4*b*c^4
)*e^5*x^3 + 3*(B*a*b^8 - 11*B*a^2*b^6*c + 36*B*a^3*b^4*c^2 - 16*B*a^4*b^2*c^3 - 64*B*a^5*c^4)*e^5*x^2 + 3*(B*a
^2*b^7 - 12*B*a^3*b^5*c + 48*B*a^4*b^3*c^2 - 64*B*a^5*b*c^3)*e^5*x + (B*a^3*b^6 - 12*B*a^4*b^4*c + 48*B*a^5*b^
2*c^2 - 64*B*a^6*c^3)*e^5)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c
) - 4*a*c) + 4*(640*(2*B*a^5 - A*a^4*b)*c^4*d*e^4 + (48*(2*B*a^3 - 5*A*a^2*b)*c^6 + 8*(6*B*a^2*b^2 + 5*A*a*b^3
)*c^5 - (2*B*a*b^4 + 3*A*b^5)*c^4)*d^5 + 10*(48*A*a^3*c^6 - 24*(2*B*a^3*b - A*a^2*b^2)*c^5 - (4*B*a^2*b^3 + A*
a*b^4)*c^4)*d^4*e + 80*(4*(2*B*a^4 - 3*A*a^3*b)*c^5 + (6*B*a^3*b^2 - A*a^2*b^3)*c^4)*d^3*e^2 + 160*(4*A*a^4*c^
5 - (8*B*a^4*b - 3*A*a^3*b^2)*c^4)*d^2*e^3 - (15*B*a^3*b^5*c - 160*B*a^4*b^3*c^2 + 528*B*a^5*b*c^3 - 256*A*a^5
*c^4)*e^5 + (128*(B*b*c^8 - 2*A*c^9)*d^5 - 80*(3*B*b^2*c^7 + 4*(B*a - 2*A*b)*c^8)*d^4*e + 80*(B*b^3*c^6 - 8*A*
a*c^8 + 6*(2*B*a*b - A*b^2)*c^7)*d^3*e^2 + 20*(B*b^4*c^5 - 48*(B*a^2 - A*a*b)*c^7 - 4*(6*B*a*b^2 - A*b^3)*c^6)
*d^2*e^3 + 5*(3*B*b^5*c^4 - 96*A*a^2*c^7 + 48*(5*B*a^2*b - A*a*b^2)*c^6 - 2*(20*B*a*b^3 - A*b^4)*c^5)*d*e^4 -
(23*B*b^6*c^3 - 16*(46*B*a^3 + 15*A*a^2*b)*c^6 + 8*(114*B*a^2*b^2 + 5*A*a*b^3)*c^5 - 3*(86*B*a*b^4 + A*b^5)*c^
4)*e^5)*x^5 + 5*(64*(B*b^2*c^7 - 2*A*b*c^8)*d^5 - 40*(3*B*b^3*c^6 + 4*(B*a*b - 2*A*b^2)*c^7)*d^4*e + 40*(B*b^4
*c^5 - 8*A*a*b*c^7 + 6*(2*B*a*b^2 - A*b^3)*c^6)*d^3*e^2 + 10*(B*b^5*c^4 - 48*(B*a^2*b - A*a*b^2)*c^6 - 4*(6*B*
a*b^3 - A*b^4)*c^5)*d^2*e^3 + 5*(48*(2*B*a^3 - A*a^2*b)*c^6 + 24*(2*B*a^2*b^2 - A*a*b^3)*c^5 - (2*B*a*b^4 - A*
b^5)*c^4)*d*e^4 - (7*B*b^7*c^2 - 75*B*a*b^5*c^3 - 96*A*a^3*c^6 - 16*(5*B*a^3*b + 3*A*a^2*b^2)*c^5 + 2*(120*B*a
^2*b^3 + A*a*b^4)*c^4)*e^5)*x^4 + 5*(16*(3*B*b^3*c^6 - 8*A*a*c^8 + 2*(2*B*a*b - 3*A*b^2)*c^7)*d^5 - 10*(9*B*b^
4*c^5 + 16*(B*a^2 - 2*A*a*b)*c^7 + 24*(B*a*b^2 - A*b^3)*c^6)*d^4*e + 10*(3*B*b^5*c^4 - 32*A*a^2*c^7 + 48*(B*a^
2*b - A*a*b^2)*c^6 + 2*(20*B*a*b^3 - 9*A*b^4)*c^5)*d^3*e^2 + 10*(48*A*a^2*b*c^6 - 8*(12*B*a^2*b^2 - 5*A*a*b^3)
*c^5 - (8*B*a*b^4 - 3*A*b^5)*c^4)*d^2*e^3 + 40*(12*(2*B*a^3*b - A*a^2*b^2)*c^5 + (2*B*a^2*b^3 - A*a*b^4)*c^4)*
d*e^4 - (3*B*b^8*c - 20*B*a*b^6*c^2 - 30*B*a^2*b^4*c^3 - 32*(7*B*a^4 + 6*A*a^3*b)*c^5 + 16*(27*B*a^3*b^2 - A*a
^2*b^3)*c^4)*e^5)*x^3 + 5*(8*(B*b^4*c^5 - 24*A*a*b*c^7 + 2*(6*B*a*b^2 - A*b^3)*c^6)*d^5 - 5*(3*B*b^5*c^4 + 48*
(B*a^2*b - 2*A*a*b^2)*c^6 + 8*(5*B*a*b^3 - A*b^4)*c^5)*d^4*e + 10*(16*(2*B*a^3 - 3*A*a^2*b)*c^6 + 8*(6*B*a^2*b
^2 - 5*A*a*b^3)*c^5 + 3*(6*B*a*b^4 - A*b^5)*c^4)*d^3*e^2 + 20*(16*A*a^3*c^6 - 8*(4*B*a^3*b - 3*A*a^2*b^2)*c^5
- 3*(8*B*a^2*b^3 - 3*A*a*b^4)*c^4)*d^2*e^3 + 80*(4*(2*B*a^4 - A*a^3*b)*c^5 + 3*(2*B*a^3*b^2 - A*a^2*b^3)*c^4)*
d*e^4 - (9*B*a*b^7*c - 93*B*a^2*b^5*c^2 + 280*B*a^3*b^3*c^3 - 128*A*a^4*c^5 + 48*(B*a^4*b - 2*A*a^3*b^2)*c^4)*
e^5)*x^2 + 5*(320*(2*B*a^4*b - A*a^3*b^2)*c^4*d*e^4 - (B*b^5*c^4 + 96*A*a^2*c^7 - 48*(B*a^2*b - A*a*b^2)*c^6 -
 2*(12*B*a*b^3 + A*b^4)*c^5)*d^5 + 5*(48*A*a^2*b*c^6 - 24*(2*B*a^2*b^2 - A*a*b^3)*c^5 - (4*B*a*b^4 + A*b^5)*c^
4)*d^4*e + 40*(4*(2*B*a^3*b - 3*A*a^2*b^2)*c^5 + (6*B*a^2*b^3 - A*a*b^4)*c^4)*d^3*e^2 + 80*(4*A*a^3*b*c^5 - (8
*B*a^3*b^2 - 3*A*a^2*b^3)*c^4)*d^2*e^3 - (9*B*a^2*b^6*c - 98*B*a^3*b^4*c^2 + 336*B*a^4*b^2*c^3 - 32*(3*B*a^5 +
 4*A*a^4*b)*c^4)*e^5)*x)*sqrt(c*x^2 + b*x + a))/(a^3*b^6*c^4 - 12*a^4*b^4*c^5 + 48*a^5*b^2*c^6 - 64*a^6*c^7 +
(b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9 - 64*a^3*c^10)*x^6 + 3*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*b^3*c^8 - 64*
a^3*b*c^9)*x^5 + 3*(b^8*c^5 - 11*a*b^6*c^6 + 36*a^2*b^4*c^7 - 16*a^3*b^2*c^8 - 64*a^4*c^9)*x^4 + (b^9*c^4 - 6*
a*b^7*c^5 - 24*a^2*b^5*c^6 + 224*a^3*b^3*c^7 - 384*a^4*b*c^8)*x^3 + 3*(a*b^8*c^4 - 11*a^2*b^6*c^5 + 36*a^3*b^4
*c^6 - 16*a^4*b^2*c^7 - 64*a^5*c^8)*x^2 + 3*(a^2*b^7*c^4 - 12*a^3*b^5*c^5 + 48*a^4*b^3*c^6 - 64*a^5*b*c^7)*x),
 -1/15*(15*((B*b^6*c^3 - 12*B*a*b^4*c^4 + 48*B*a^2*b^2*c^5 - 64*B*a^3*c^6)*e^5*x^6 + 3*(B*b^7*c^2 - 12*B*a*b^5
*c^3 + 48*B*a^2*b^3*c^4 - 64*B*a^3*b*c^5)*e^5*x^5 + 3*(B*b^8*c - 11*B*a*b^6*c^2 + 36*B*a^2*b^4*c^3 - 16*B*a^3*
b^2*c^4 - 64*B*a^4*c^5)*e^5*x^4 + (B*b^9 - 6*B*a*b^7*c - 24*B*a^2*b^5*c^2 + 224*B*a^3*b^3*c^3 - 384*B*a^4*b*c^
4)*e^5*x^3 + 3*(B*a*b^8 - 11*B*a^2*b^6*c + 36*B*a^3*b^4*c^2 - 16*B*a^4*b^2*c^3 - 64*B*a^5*c^4)*e^5*x^2 + 3*(B*
a^2*b^7 - 12*B*a^3*b^5*c + 48*B*a^4*b^3*c^2 - 64*B*a^5*b*c^3)*e^5*x + (B*a^3*b^6 - 12*B*a^4*b^4*c + 48*B*a^5*b
^2*c^2 - 64*B*a^6*c^3)*e^5)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x +
a*c)) - 2*(640*(2*B*a^5 - A*a^4*b)*c^4*d*e^4 + (48*(2*B*a^3 - 5*A*a^2*b)*c^6 + 8*(6*B*a^2*b^2 + 5*A*a*b^3)*c^5
 - (2*B*a*b^4 + 3*A*b^5)*c^4)*d^5 + 10*(48*A*a^3*c^6 - 24*(2*B*a^3*b - A*a^2*b^2)*c^5 - (4*B*a^2*b^3 + A*a*b^4
)*c^4)*d^4*e + 80*(4*(2*B*a^4 - 3*A*a^3*b)*c^5 + (6*B*a^3*b^2 - A*a^2*b^3)*c^4)*d^3*e^2 + 160*(4*A*a^4*c^5 - (
8*B*a^4*b - 3*A*a^3*b^2)*c^4)*d^2*e^3 - (15*B*a^3*b^5*c - 160*B*a^4*b^3*c^2 + 528*B*a^5*b*c^3 - 256*A*a^5*c^4)
*e^5 + (128*(B*b*c^8 - 2*A*c^9)*d^5 - 80*(3*B*b^2*c^7 + 4*(B*a - 2*A*b)*c^8)*d^4*e + 80*(B*b^3*c^6 - 8*A*a*c^8
 + 6*(2*B*a*b - A*b^2)*c^7)*d^3*e^2 + 20*(B*b^4*c^5 - 48*(B*a^2 - A*a*b)*c^7 - 4*(6*B*a*b^2 - A*b^3)*c^6)*d^2*
e^3 + 5*(3*B*b^5*c^4 - 96*A*a^2*c^7 + 48*(5*B*a^2*b - A*a*b^2)*c^6 - 2*(20*B*a*b^3 - A*b^4)*c^5)*d*e^4 - (23*B
*b^6*c^3 - 16*(46*B*a^3 + 15*A*a^2*b)*c^6 + 8*(114*B*a^2*b^2 + 5*A*a*b^3)*c^5 - 3*(86*B*a*b^4 + A*b^5)*c^4)*e^
5)*x^5 + 5*(64*(B*b^2*c^7 - 2*A*b*c^8)*d^5 - 40*(3*B*b^3*c^6 + 4*(B*a*b - 2*A*b^2)*c^7)*d^4*e + 40*(B*b^4*c^5
- 8*A*a*b*c^7 + 6*(2*B*a*b^2 - A*b^3)*c^6)*d^3*e^2 + 10*(B*b^5*c^4 - 48*(B*a^2*b - A*a*b^2)*c^6 - 4*(6*B*a*b^3
 - A*b^4)*c^5)*d^2*e^3 + 5*(48*(2*B*a^3 - A*a^2*b)*c^6 + 24*(2*B*a^2*b^2 - A*a*b^3)*c^5 - (2*B*a*b^4 - A*b^5)*
c^4)*d*e^4 - (7*B*b^7*c^2 - 75*B*a*b^5*c^3 - 96*A*a^3*c^6 - 16*(5*B*a^3*b + 3*A*a^2*b^2)*c^5 + 2*(120*B*a^2*b^
3 + A*a*b^4)*c^4)*e^5)*x^4 + 5*(16*(3*B*b^3*c^6 - 8*A*a*c^8 + 2*(2*B*a*b - 3*A*b^2)*c^7)*d^5 - 10*(9*B*b^4*c^5
 + 16*(B*a^2 - 2*A*a*b)*c^7 + 24*(B*a*b^2 - A*b^3)*c^6)*d^4*e + 10*(3*B*b^5*c^4 - 32*A*a^2*c^7 + 48*(B*a^2*b -
 A*a*b^2)*c^6 + 2*(20*B*a*b^3 - 9*A*b^4)*c^5)*d^3*e^2 + 10*(48*A*a^2*b*c^6 - 8*(12*B*a^2*b^2 - 5*A*a*b^3)*c^5
- (8*B*a*b^4 - 3*A*b^5)*c^4)*d^2*e^3 + 40*(12*(2*B*a^3*b - A*a^2*b^2)*c^5 + (2*B*a^2*b^3 - A*a*b^4)*c^4)*d*e^4
 - (3*B*b^8*c - 20*B*a*b^6*c^2 - 30*B*a^2*b^4*c^3 - 32*(7*B*a^4 + 6*A*a^3*b)*c^5 + 16*(27*B*a^3*b^2 - A*a^2*b^
3)*c^4)*e^5)*x^3 + 5*(8*(B*b^4*c^5 - 24*A*a*b*c^7 + 2*(6*B*a*b^2 - A*b^3)*c^6)*d^5 - 5*(3*B*b^5*c^4 + 48*(B*a^
2*b - 2*A*a*b^2)*c^6 + 8*(5*B*a*b^3 - A*b^4)*c^5)*d^4*e + 10*(16*(2*B*a^3 - 3*A*a^2*b)*c^6 + 8*(6*B*a^2*b^2 -
5*A*a*b^3)*c^5 + 3*(6*B*a*b^4 - A*b^5)*c^4)*d^3*e^2 + 20*(16*A*a^3*c^6 - 8*(4*B*a^3*b - 3*A*a^2*b^2)*c^5 - 3*(
8*B*a^2*b^3 - 3*A*a*b^4)*c^4)*d^2*e^3 + 80*(4*(2*B*a^4 - A*a^3*b)*c^5 + 3*(2*B*a^3*b^2 - A*a^2*b^3)*c^4)*d*e^4
 - (9*B*a*b^7*c - 93*B*a^2*b^5*c^2 + 280*B*a^3*b^3*c^3 - 128*A*a^4*c^5 + 48*(B*a^4*b - 2*A*a^3*b^2)*c^4)*e^5)*
x^2 + 5*(320*(2*B*a^4*b - A*a^3*b^2)*c^4*d*e^4 - (B*b^5*c^4 + 96*A*a^2*c^7 - 48*(B*a^2*b - A*a*b^2)*c^6 - 2*(1
2*B*a*b^3 + A*b^4)*c^5)*d^5 + 5*(48*A*a^2*b*c^6 - 24*(2*B*a^2*b^2 - A*a*b^3)*c^5 - (4*B*a*b^4 + A*b^5)*c^4)*d^
4*e + 40*(4*(2*B*a^3*b - 3*A*a^2*b^2)*c^5 + (6*B*a^2*b^3 - A*a*b^4)*c^4)*d^3*e^2 + 80*(4*A*a^3*b*c^5 - (8*B*a^
3*b^2 - 3*A*a^2*b^3)*c^4)*d^2*e^3 - (9*B*a^2*b^6*c - 98*B*a^3*b^4*c^2 + 336*B*a^4*b^2*c^3 - 32*(3*B*a^5 + 4*A*
a^4*b)*c^4)*e^5)*x)*sqrt(c*x^2 + b*x + a))/(a^3*b^6*c^4 - 12*a^4*b^4*c^5 + 48*a^5*b^2*c^6 - 64*a^6*c^7 + (b^6*
c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9 - 64*a^3*c^10)*x^6 + 3*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*b^3*c^8 - 64*a^3*b
*c^9)*x^5 + 3*(b^8*c^5 - 11*a*b^6*c^6 + 36*a^2*b^4*c^7 - 16*a^3*b^2*c^8 - 64*a^4*c^9)*x^4 + (b^9*c^4 - 6*a*b^7
*c^5 - 24*a^2*b^5*c^6 + 224*a^3*b^3*c^7 - 384*a^4*b*c^8)*x^3 + 3*(a*b^8*c^4 - 11*a^2*b^6*c^5 + 36*a^3*b^4*c^6
- 16*a^4*b^2*c^7 - 64*a^5*c^8)*x^2 + 3*(a^2*b^7*c^4 - 12*a^3*b^5*c^5 + 48*a^4*b^3*c^6 - 64*a^5*b*c^7)*x)]

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giac [B]  time = 0.84, size = 2525, normalized size = 2.68

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")

[Out]

-B*e^5*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2) + 2/15*((((((128*B*b*c^7*d^5 - 256
*A*c^8*d^5 - 240*B*b^2*c^6*d^4*e - 320*B*a*c^7*d^4*e + 640*A*b*c^7*d^4*e + 80*B*b^3*c^5*d^3*e^2 + 960*B*a*b*c^
6*d^3*e^2 - 480*A*b^2*c^6*d^3*e^2 - 640*A*a*c^7*d^3*e^2 + 20*B*b^4*c^4*d^2*e^3 - 480*B*a*b^2*c^5*d^2*e^3 + 80*
A*b^3*c^5*d^2*e^3 - 960*B*a^2*c^6*d^2*e^3 + 960*A*a*b*c^6*d^2*e^3 + 15*B*b^5*c^3*d*e^4 - 200*B*a*b^3*c^4*d*e^4
 + 10*A*b^4*c^4*d*e^4 + 1200*B*a^2*b*c^5*d*e^4 - 240*A*a*b^2*c^5*d*e^4 - 480*A*a^2*c^6*d*e^4 - 23*B*b^6*c^2*e^
5 + 258*B*a*b^4*c^3*e^5 + 3*A*b^5*c^3*e^5 - 912*B*a^2*b^2*c^4*e^5 - 40*A*a*b^3*c^4*e^5 + 736*B*a^3*c^5*e^5 + 2
40*A*a^2*b*c^5*e^5)*x/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6) + 5*(64*B*b^2*c^6*d^5 - 128*A*b*c
^7*d^5 - 120*B*b^3*c^5*d^4*e - 160*B*a*b*c^6*d^4*e + 320*A*b^2*c^6*d^4*e + 40*B*b^4*c^4*d^3*e^2 + 480*B*a*b^2*
c^5*d^3*e^2 - 240*A*b^3*c^5*d^3*e^2 - 320*A*a*b*c^6*d^3*e^2 + 10*B*b^5*c^3*d^2*e^3 - 240*B*a*b^3*c^4*d^2*e^3 +
 40*A*b^4*c^4*d^2*e^3 - 480*B*a^2*b*c^5*d^2*e^3 + 480*A*a*b^2*c^5*d^2*e^3 - 10*B*a*b^4*c^3*d*e^4 + 5*A*b^5*c^3
*d*e^4 + 240*B*a^2*b^2*c^4*d*e^4 - 120*A*a*b^3*c^4*d*e^4 + 480*B*a^3*c^5*d*e^4 - 240*A*a^2*b*c^5*d*e^4 - 7*B*b
^7*c*e^5 + 75*B*a*b^5*c^2*e^5 - 240*B*a^2*b^3*c^3*e^5 - 2*A*a*b^4*c^3*e^5 + 80*B*a^3*b*c^4*e^5 + 48*A*a^2*b^2*
c^4*e^5 + 96*A*a^3*c^5*e^5)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x + 5*(48*B*b^3*c^5*d^5 +
64*B*a*b*c^6*d^5 - 96*A*b^2*c^6*d^5 - 128*A*a*c^7*d^5 - 90*B*b^4*c^4*d^4*e - 240*B*a*b^2*c^5*d^4*e + 240*A*b^3
*c^5*d^4*e - 160*B*a^2*c^6*d^4*e + 320*A*a*b*c^6*d^4*e + 30*B*b^5*c^3*d^3*e^2 + 400*B*a*b^3*c^4*d^3*e^2 - 180*
A*b^4*c^4*d^3*e^2 + 480*B*a^2*b*c^5*d^3*e^2 - 480*A*a*b^2*c^5*d^3*e^2 - 320*A*a^2*c^6*d^3*e^2 - 80*B*a*b^4*c^3
*d^2*e^3 + 30*A*b^5*c^3*d^2*e^3 - 960*B*a^2*b^2*c^4*d^2*e^3 + 400*A*a*b^3*c^4*d^2*e^3 + 480*A*a^2*b*c^5*d^2*e^
3 + 80*B*a^2*b^3*c^3*d*e^4 - 40*A*a*b^4*c^3*d*e^4 + 960*B*a^3*b*c^4*d*e^4 - 480*A*a^2*b^2*c^4*d*e^4 - 3*B*b^8*
e^5 + 20*B*a*b^6*c*e^5 + 30*B*a^2*b^4*c^2*e^5 - 432*B*a^3*b^2*c^3*e^5 + 16*A*a^2*b^3*c^3*e^5 + 224*B*a^4*c^4*e
^5 + 192*A*a^3*b*c^4*e^5)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x + 5*(8*B*b^4*c^4*d^5 + 96*
B*a*b^2*c^5*d^5 - 16*A*b^3*c^5*d^5 - 192*A*a*b*c^6*d^5 - 15*B*b^5*c^3*d^4*e - 200*B*a*b^3*c^4*d^4*e + 40*A*b^4
*c^4*d^4*e - 240*B*a^2*b*c^5*d^4*e + 480*A*a*b^2*c^5*d^4*e + 180*B*a*b^4*c^3*d^3*e^2 - 30*A*b^5*c^3*d^3*e^2 +
480*B*a^2*b^2*c^4*d^3*e^2 - 400*A*a*b^3*c^4*d^3*e^2 + 320*B*a^3*c^5*d^3*e^2 - 480*A*a^2*b*c^5*d^3*e^2 - 480*B*
a^2*b^3*c^3*d^2*e^3 + 180*A*a*b^4*c^3*d^2*e^3 - 640*B*a^3*b*c^4*d^2*e^3 + 480*A*a^2*b^2*c^4*d^2*e^3 + 320*A*a^
3*c^5*d^2*e^3 + 480*B*a^3*b^2*c^3*d*e^4 - 240*A*a^2*b^3*c^3*d*e^4 + 640*B*a^4*c^4*d*e^4 - 320*A*a^3*b*c^4*d*e^
4 - 9*B*a*b^7*e^5 + 93*B*a^2*b^5*c*e^5 - 280*B*a^3*b^3*c^2*e^5 - 48*B*a^4*b*c^3*e^5 + 96*A*a^3*b^2*c^3*e^5 + 1
28*A*a^4*c^4*e^5)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x - 5*(B*b^5*c^3*d^5 - 24*B*a*b^3*c^
4*d^5 - 2*A*b^4*c^4*d^5 - 48*B*a^2*b*c^5*d^5 + 48*A*a*b^2*c^5*d^5 + 96*A*a^2*c^6*d^5 + 20*B*a*b^4*c^3*d^4*e +
5*A*b^5*c^3*d^4*e + 240*B*a^2*b^2*c^4*d^4*e - 120*A*a*b^3*c^4*d^4*e - 240*A*a^2*b*c^5*d^4*e - 240*B*a^2*b^3*c^
3*d^3*e^2 + 40*A*a*b^4*c^3*d^3*e^2 - 320*B*a^3*b*c^4*d^3*e^2 + 480*A*a^2*b^2*c^4*d^3*e^2 + 640*B*a^3*b^2*c^3*d
^2*e^3 - 240*A*a^2*b^3*c^3*d^2*e^3 - 320*A*a^3*b*c^4*d^2*e^3 - 640*B*a^4*b*c^3*d*e^4 + 320*A*a^3*b^2*c^3*d*e^4
 + 9*B*a^2*b^6*e^5 - 98*B*a^3*b^4*c*e^5 + 336*B*a^4*b^2*c^2*e^5 - 96*B*a^5*c^3*e^5 - 128*A*a^4*b*c^3*e^5)/(b^6
*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x - (2*B*a*b^4*c^3*d^5 + 3*A*b^5*c^3*d^5 - 48*B*a^2*b^2*c^
4*d^5 - 40*A*a*b^3*c^4*d^5 - 96*B*a^3*c^5*d^5 + 240*A*a^2*b*c^5*d^5 + 40*B*a^2*b^3*c^3*d^4*e + 10*A*a*b^4*c^3*
d^4*e + 480*B*a^3*b*c^4*d^4*e - 240*A*a^2*b^2*c^4*d^4*e - 480*A*a^3*c^5*d^4*e - 480*B*a^3*b^2*c^3*d^3*e^2 + 80
*A*a^2*b^3*c^3*d^3*e^2 - 640*B*a^4*c^4*d^3*e^2 + 960*A*a^3*b*c^4*d^3*e^2 + 1280*B*a^4*b*c^3*d^2*e^3 - 480*A*a^
3*b^2*c^3*d^2*e^3 - 640*A*a^4*c^4*d^2*e^3 - 1280*B*a^5*c^3*d*e^4 + 640*A*a^4*b*c^3*d*e^4 + 15*B*a^3*b^5*e^5 -
160*B*a^4*b^3*c*e^5 + 528*B*a^5*b*c^2*e^5 - 256*A*a^5*c^3*e^5)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a
^3*c^6))/(c*x^2 + b*x + a)^(5/2)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 7765, normalized size = 8.24 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^5}{{\left (c\,x^2+b\,x+a\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x)

[Out]

int(((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**5/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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