3.2391 \(\int \frac {(d+e x)^5}{(a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=389 \[ -\frac {8 (d+e x)^2 \left (8 a^2 c e^3-x (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-2 b c d \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {e \sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (-16 a^2 e^2-25 a b d e+b^2 d^2\right )+2 c e x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )+10 b^2 c e^3 (10 a e+3 b d)-16 c^3 d^2 e (7 b d-16 a e)-15 b^4 e^4+64 c^4 d^4\right )}{3 c^3 \left (b^2-4 a c\right )^2}-\frac {2 (d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{7/2}} \]

[Out]

-2/3*(e*x+d)^4*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2)+5/2*e^4*(-b*e+2*c*d)*arctanh(1/2*(2
*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)-8/3*(e*x+d)^2*(8*a^2*c*e^3-2*b*c*d*(3*a*e^2+c*d^2)+b^2*(-a*e^3+3*
c*d^2*e)-(-b*e+2*c*d)*(2*c^2*d^2-b^2*e^2-2*c*e*(-3*a*e+b*d))*x)/c/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/2)-1/3*e*(64
*c^4*d^4-15*b^4*e^4-16*c^3*d^2*e*(-16*a*e+7*b*d)+10*b^2*c*e^3*(10*a*e+3*b*d)+8*c^2*e^2*(-16*a^2*e^2-25*a*b*d*e
+b^2*d^2)+2*c*e*(-b*e+2*c*d)*(8*c^2*d^2-5*b^2*e^2-4*c*e*(-7*a*e+2*b*d))*x)*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2
)^2

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Rubi [A]  time = 0.47, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {738, 818, 779, 621, 206} \[ -\frac {e \sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (-16 a^2 e^2-25 a b d e+b^2 d^2\right )+2 c e x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )+10 b^2 c e^3 (10 a e+3 b d)-16 c^3 d^2 e (7 b d-16 a e)-15 b^4 e^4+64 c^4 d^4\right )}{3 c^3 \left (b^2-4 a c\right )^2}-\frac {8 (d+e x)^2 \left (8 a^2 c e^3-x (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-2 b c d \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d + e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - (8*(d + e*x)^2*(8
*a^2*c*e^3 - 2*b*c*d*(c*d^2 + 3*a*e^2) + b^2*(3*c*d^2*e - a*e^3) - (2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*
(b*d - 3*a*e))*x))/(3*c*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) - (e*(64*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(7
*b*d - 16*a*e) + 10*b^2*c*e^3*(3*b*d + 10*a*e) + 8*c^2*e^2*(b^2*d^2 - 25*a*b*d*e - 16*a^2*e^2) + 2*c*e*(2*c*d
- b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2*b*d - 7*a*e))*x)*Sqrt[a + b*x + c*x^2])/(3*c^3*(b^2 - 4*a*c)^2) + (5*
e^4*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(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 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), 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))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[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 {(d+e x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {(d+e x)^3 \left (2 \left (2 c d^2-3 b d e+4 a e^2\right )-2 e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2 (d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {8 (d+e x)^2 \left (8 a^2 c e^3-2 b c d \left (c d^2+3 a e^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {4 \int \frac {(d+e x) \left (e \left (b^3 d e^2+8 a c e \left (c d^2-4 a e^2\right )+4 b c d \left (2 c d^2+5 a e^2\right )-b^2 \left (14 c d^2 e-4 a e^3\right )\right )+e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{3 c \left (b^2-4 a c\right )^2}\\ &=-\frac {2 (d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {8 (d+e x)^2 \left (8 a^2 c e^3-2 b c d \left (c d^2+3 a e^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {e \left (64 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (7 b d-16 a e)+10 b^2 c e^3 (3 b d+10 a e)+8 c^2 e^2 \left (b^2 d^2-25 a b d e-16 a^2 e^2\right )+2 c e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}+\frac {\left (5 e^4 (2 c d-b e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c^3}\\ &=-\frac {2 (d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {8 (d+e x)^2 \left (8 a^2 c e^3-2 b c d \left (c d^2+3 a e^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {e \left (64 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (7 b d-16 a e)+10 b^2 c e^3 (3 b d+10 a e)+8 c^2 e^2 \left (b^2 d^2-25 a b d e-16 a^2 e^2\right )+2 c e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}+\frac {\left (5 e^4 (2 c d-b e)\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 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {8 (d+e x)^2 \left (8 a^2 c e^3-2 b c d \left (c d^2+3 a e^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {e \left (64 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (7 b d-16 a e)+10 b^2 c e^3 (3 b d+10 a e)+8 c^2 e^2 \left (b^2 d^2-25 a b d e-16 a^2 e^2\right )+2 c e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}+\frac {5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 1.81, size = 566, normalized size = 1.46 \[ \frac {b^4 e^4 \left (15 a^2 e-30 a c x (2 d+3 e x)+c^2 x^3 (3 e x-40 d)\right )-2 b^3 c \left (15 a^2 e^4 (d+7 e x)+2 a c e^4 x^2 (37 e x-45 d)+c^2 d^2 \left (d^3+15 d^2 e x-30 d e^2 x^2-10 e^3 x^3\right )\right )+4 b^2 c \left (-25 a^3 e^5+3 a^2 c e^4 x (35 d+4 e x)+a c^2 e \left (-5 d^4+60 d^3 e x-30 d^2 e^2 x^2+70 d e^3 x^3-6 e^4 x^4\right )+c^3 d^3 x \left (3 d^2-30 d e x+10 e^2 x^2\right )\right )+8 b c^2 \left (a^3 e^4 (25 d+39 e x)+4 a^2 c e^2 \left (5 d^3-15 d^2 e x+8 e^3 x^3\right )+3 a c^2 d^2 \left (d^3-5 d^2 e x+10 d e^2 x^2-10 e^3 x^3\right )+2 c^3 d^4 x^2 (3 d-5 e x)\right )+16 c^2 \left (8 a^4 e^5+a^3 c e^3 \left (-20 d^2-15 d e x+12 e^2 x^2\right )+a^2 c^2 e \left (-5 d^4-30 d^2 e^2 x^2-20 d e^3 x^3+3 e^4 x^4\right )+a c^3 d^3 x \left (3 d^2+10 e^2 x^2\right )+2 c^4 d^5 x^3\right )+10 b^5 e^4 x (3 a e+c x (2 e x-3 d))+15 b^6 e^5 x^2}{3 c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}+\frac {5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{2 c^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*b^6*e^5*x^2 + 10*b^5*e^4*x*(3*a*e + c*x*(-3*d + 2*e*x)) + b^4*e^4*(15*a^2*e + c^2*x^3*(-40*d + 3*e*x) - 30
*a*c*x*(2*d + 3*e*x)) - 2*b^3*c*(15*a^2*e^4*(d + 7*e*x) + 2*a*c*e^4*x^2*(-45*d + 37*e*x) + c^2*d^2*(d^3 + 15*d
^2*e*x - 30*d*e^2*x^2 - 10*e^3*x^3)) + 8*b*c^2*(2*c^3*d^4*x^2*(3*d - 5*e*x) + a^3*e^4*(25*d + 39*e*x) + 3*a*c^
2*d^2*(d^3 - 5*d^2*e*x + 10*d*e^2*x^2 - 10*e^3*x^3) + 4*a^2*c*e^2*(5*d^3 - 15*d^2*e*x + 8*e^3*x^3)) + 4*b^2*c*
(-25*a^3*e^5 + 3*a^2*c*e^4*x*(35*d + 4*e*x) + c^3*d^3*x*(3*d^2 - 30*d*e*x + 10*e^2*x^2) + a*c^2*e*(-5*d^4 + 60
*d^3*e*x - 30*d^2*e^2*x^2 + 70*d*e^3*x^3 - 6*e^4*x^4)) + 16*c^2*(8*a^4*e^5 + 2*c^4*d^5*x^3 + a*c^3*d^3*x*(3*d^
2 + 10*e^2*x^2) + a^3*c*e^3*(-20*d^2 - 15*d*e*x + 12*e^2*x^2) + a^2*c^2*e*(-5*d^4 - 30*d^2*e^2*x^2 - 20*d*e^3*
x^3 + 3*e^4*x^4)))/(3*c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) + (5*e^4*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(2*c^(7/2))

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fricas [B]  time = 3.33, size = 2249, normalized size = 5.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(15*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^5 + (2
*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)*x^4 + 2*(2*(b^5*c^2
- 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 - (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^3 + (2*(b^6*c - 6*a*b^4*c^
2 + 32*a^3*c^4)*d*e^4 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c
^3)*d*e^4 - (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^5)*x)*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*(160*a^2*b*c^4*d^3*e^2 - 320*a^3*c^4*d^2*e^3 + 3*(b^4*c^3 - 8*a
*b^2*c^4 + 16*a^2*c^5)*e^5*x^4 - 2*(b^3*c^4 - 12*a*b*c^5)*d^5 - 20*(a*b^2*c^4 + 4*a^2*c^5)*d^4*e - 10*(3*a^2*b
^3*c^2 - 20*a^3*b*c^3)*d*e^4 + (15*a^2*b^4*c - 100*a^3*b^2*c^2 + 128*a^4*c^3)*e^5 + 4*(8*c^7*d^5 - 20*b*c^6*d^
4*e + 10*(b^2*c^5 + 4*a*c^6)*d^3*e^2 + 5*(b^3*c^4 - 12*a*b*c^5)*d^2*e^3 - 10*(b^4*c^3 - 7*a*b^2*c^4 + 8*a^2*c^
5)*d*e^4 + (5*b^5*c^2 - 37*a*b^3*c^3 + 64*a^2*b*c^4)*e^5)*x^3 + 3*(16*b*c^6*d^5 - 40*b^2*c^5*d^4*e + 20*(b^3*c
^4 + 4*a*b*c^5)*d^3*e^2 - 40*(a*b^2*c^4 + 4*a^2*c^5)*d^2*e^3 - 10*(b^5*c^2 - 6*a*b^3*c^3)*d*e^4 + (5*b^6*c - 3
0*a*b^4*c^2 + 16*a^2*b^2*c^3 + 64*a^3*c^4)*e^5)*x^2 + 6*(40*a*b^2*c^4*d^3*e^2 - 80*a^2*b*c^4*d^2*e^3 + 2*(b^2*
c^5 + 4*a*c^6)*d^5 - 5*(b^3*c^4 + 4*a*b*c^5)*d^4*e - 10*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 4*a^3*c^4)*d*e^4 + (5*a*b
^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^3)*e^5)*x)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c^6
+ (b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*x^4 + 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*x^3 + (b^6*c^4 - 6*a*b^
4*c^5 + 32*a^3*c^7)*x^2 + 2*(a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*x), -1/6*(15*(2*(a^2*b^4*c - 8*a^3*b^2*
c^2 + 16*a^4*c^3)*d*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^5 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)
*d*e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)*x^4 + 2*(2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 -
(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d*e^4 - (b^7 - 6*a*b^
5*c + 32*a^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 - (a*b^6 - 8*a^2*b^4*c + 16
*a^3*b^2*c^2)*e^5)*x)*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*(160*a^2*b*c^4*d^3*e^2 - 320*a^3*c^4*d^2*e^3 + 3*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e^5*x^4 - 2*(b^3*c^4
- 12*a*b*c^5)*d^5 - 20*(a*b^2*c^4 + 4*a^2*c^5)*d^4*e - 10*(3*a^2*b^3*c^2 - 20*a^3*b*c^3)*d*e^4 + (15*a^2*b^4*
c - 100*a^3*b^2*c^2 + 128*a^4*c^3)*e^5 + 4*(8*c^7*d^5 - 20*b*c^6*d^4*e + 10*(b^2*c^5 + 4*a*c^6)*d^3*e^2 + 5*(b
^3*c^4 - 12*a*b*c^5)*d^2*e^3 - 10*(b^4*c^3 - 7*a*b^2*c^4 + 8*a^2*c^5)*d*e^4 + (5*b^5*c^2 - 37*a*b^3*c^3 + 64*a
^2*b*c^4)*e^5)*x^3 + 3*(16*b*c^6*d^5 - 40*b^2*c^5*d^4*e + 20*(b^3*c^4 + 4*a*b*c^5)*d^3*e^2 - 40*(a*b^2*c^4 + 4
*a^2*c^5)*d^2*e^3 - 10*(b^5*c^2 - 6*a*b^3*c^3)*d*e^4 + (5*b^6*c - 30*a*b^4*c^2 + 16*a^2*b^2*c^3 + 64*a^3*c^4)*
e^5)*x^2 + 6*(40*a*b^2*c^4*d^3*e^2 - 80*a^2*b*c^4*d^2*e^3 + 2*(b^2*c^5 + 4*a*c^6)*d^5 - 5*(b^3*c^4 + 4*a*b*c^5
)*d^4*e - 10*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 4*a^3*c^4)*d*e^4 + (5*a*b^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^3)*e^5)*
x)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c^6 + (b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*x^4
+ 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*x^3 + (b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*x^2 + 2*(a*b^5*c^4 - 8*
a^2*b^3*c^5 + 16*a^3*b*c^6)*x)]

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giac [B]  time = 0.36, size = 788, normalized size = 2.03 \[ \frac {{\left ({\left ({\left (\frac {3 \, {\left (b^{4} c^{2} e^{5} - 8 \, a b^{2} c^{3} e^{5} + 16 \, a^{2} c^{4} e^{5}\right )} x}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}} + \frac {4 \, {\left (8 \, c^{6} d^{5} - 20 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} + 40 \, a c^{5} d^{3} e^{2} + 5 \, b^{3} c^{3} d^{2} e^{3} - 60 \, a b c^{4} d^{2} e^{3} - 10 \, b^{4} c^{2} d e^{4} + 70 \, a b^{2} c^{3} d e^{4} - 80 \, a^{2} c^{4} d e^{4} + 5 \, b^{5} c e^{5} - 37 \, a b^{3} c^{2} e^{5} + 64 \, a^{2} b c^{3} e^{5}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x + \frac {3 \, {\left (16 \, b c^{5} d^{5} - 40 \, b^{2} c^{4} d^{4} e + 20 \, b^{3} c^{3} d^{3} e^{2} + 80 \, a b c^{4} d^{3} e^{2} - 40 \, a b^{2} c^{3} d^{2} e^{3} - 160 \, a^{2} c^{4} d^{2} e^{3} - 10 \, b^{5} c d e^{4} + 60 \, a b^{3} c^{2} d e^{4} + 5 \, b^{6} e^{5} - 30 \, a b^{4} c e^{5} + 16 \, a^{2} b^{2} c^{2} e^{5} + 64 \, a^{3} c^{3} e^{5}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x + \frac {6 \, {\left (2 \, b^{2} c^{4} d^{5} + 8 \, a c^{5} d^{5} - 5 \, b^{3} c^{3} d^{4} e - 20 \, a b c^{4} d^{4} e + 40 \, a b^{2} c^{3} d^{3} e^{2} - 80 \, a^{2} b c^{3} d^{2} e^{3} - 10 \, a b^{4} c d e^{4} + 70 \, a^{2} b^{2} c^{2} d e^{4} - 40 \, a^{3} c^{3} d e^{4} + 5 \, a b^{5} e^{5} - 35 \, a^{2} b^{3} c e^{5} + 52 \, a^{3} b c^{2} e^{5}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {2 \, b^{3} c^{3} d^{5} - 24 \, a b c^{4} d^{5} + 20 \, a b^{2} c^{3} d^{4} e + 80 \, a^{2} c^{4} d^{4} e - 160 \, a^{2} b c^{3} d^{3} e^{2} + 320 \, a^{3} c^{3} d^{2} e^{3} + 30 \, a^{2} b^{3} c d e^{4} - 200 \, a^{3} b c^{2} d e^{4} - 15 \, a^{2} b^{4} e^{5} + 100 \, a^{3} b^{2} c e^{5} - 128 \, a^{4} c^{2} e^{5}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (2 \, c d e^{4} - b e^{5}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((((3*(b^4*c^2*e^5 - 8*a*b^2*c^3*e^5 + 16*a^2*c^4*e^5)*x/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5) + 4*(8*c^6*d
^5 - 20*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 + 40*a*c^5*d^3*e^2 + 5*b^3*c^3*d^2*e^3 - 60*a*b*c^4*d^2*e^3 - 10*b^4*
c^2*d*e^4 + 70*a*b^2*c^3*d*e^4 - 80*a^2*c^4*d*e^4 + 5*b^5*c*e^5 - 37*a*b^3*c^2*e^5 + 64*a^2*b*c^3*e^5)/(b^4*c^
3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x + 3*(16*b*c^5*d^5 - 40*b^2*c^4*d^4*e + 20*b^3*c^3*d^3*e^2 + 80*a*b*c^4*d^3*e^
2 - 40*a*b^2*c^3*d^2*e^3 - 160*a^2*c^4*d^2*e^3 - 10*b^5*c*d*e^4 + 60*a*b^3*c^2*d*e^4 + 5*b^6*e^5 - 30*a*b^4*c*
e^5 + 16*a^2*b^2*c^2*e^5 + 64*a^3*c^3*e^5)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x + 6*(2*b^2*c^4*d^5 + 8*a*c^
5*d^5 - 5*b^3*c^3*d^4*e - 20*a*b*c^4*d^4*e + 40*a*b^2*c^3*d^3*e^2 - 80*a^2*b*c^3*d^2*e^3 - 10*a*b^4*c*d*e^4 +
70*a^2*b^2*c^2*d*e^4 - 40*a^3*c^3*d*e^4 + 5*a*b^5*e^5 - 35*a^2*b^3*c*e^5 + 52*a^3*b*c^2*e^5)/(b^4*c^3 - 8*a*b^
2*c^4 + 16*a^2*c^5))*x - (2*b^3*c^3*d^5 - 24*a*b*c^4*d^5 + 20*a*b^2*c^3*d^4*e + 80*a^2*c^4*d^4*e - 160*a^2*b*c
^3*d^3*e^2 + 320*a^3*c^3*d^2*e^3 + 30*a^2*b^3*c*d*e^4 - 200*a^3*b*c^2*d*e^4 - 15*a^2*b^4*e^5 + 100*a^3*b^2*c*e
^5 - 128*a^4*c^2*e^5)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))/(c*x^2 + b*x + a)^(3/2) - 5/2*(2*c*d*e^4 - b*e^5)*
log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)

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maple [B]  time = 0.09, size = 2395, normalized size = 6.16 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/3*d^2*e^3*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+5/6*d^3*e^2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+5/8
*d*e^4*b^2/c^3*x/(c*x^2+b*x+a)^(3/2)-5/48*d*e^4*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-5/6*d*e^4*b^5/c^3/(4*a
*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+5/3*d*e^4*b/c^3*a/(c*x^2+b*x+a)^(3/2)+5/2*d*e^4/c^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a
)^(1/2)-19/3*e^5*b^4/c^3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-5/2*e^5*b^3/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x
+e^5*a/c^3*b*x/(c*x^2+b*x+a)^(3/2)+2*e^5*a^2/c^3*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-5/3*d^4*e*b^2/c/(4*a*c-b^
2)/(c*x^2+b*x+a)^(3/2)-5/2*d^2*e^3*b/c^2*x/(c*x^2+b*x+a)^(3/2)+5/12*d^2*e^3*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^
(3/2)+20/3*d^3*e^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+80/3*d^3*e^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b-10/3
*d^4*e*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+20/3*d^3*e^2*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+20*d*e^4*b^2/c
*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-10*d^2*e^3*b/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+5/2*d*e^4*b^2/c^2*a/
(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+5*d*e^4/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/96*e^5*b^4/c^5
/(c*x^2+b*x+a)^(3/2)-5/4*e^5*b^2/c^4/(c*x^2+b*x+a)^(1/2)+8/3*e^5*a^2/c^3/(c*x^2+b*x+a)^(3/2)+e^5*x^4/c/(c*x^2+
b*x+a)^(3/2)-5/2*e^5*b/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+2/3*d^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(
3/2)*b-5/3*d^4*e/c/(c*x^2+b*x+a)^(3/2)+16*e^5*a^2/c^2*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+5/48*e^5*b^5/c^4/(
4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+5/6*e^5*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-19/24*e^5*b^4/c^4*a/(4*a*
c-b^2)/(c*x^2+b*x+a)^(3/2)+40/3*d^3*e^2*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+32*e^5*a^2/c*b/(4*a*c-b^2)^2/(
c*x^2+b*x+a)^(1/2)*x-19/12*e^5*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-38/3*e^5*b^3/c^2*a/(4*a*c-b^2)^2/(c
*x^2+b*x+a)^(1/2)*x+4*e^5*a^2/c^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-80/3*d^4*e*b*c/(4*a*c-b^2)^2/(c*x^2+b*x+
a)^(1/2)*x-5/24*d*e^4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-5/3*d*e^4*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^
(1/2)*x+5/4*d*e^4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+5/3*d^3*e^2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*
x+10/3*d^3*e^2*a/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b+160/3*d^3*e^2*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+10*
d*e^4*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+5*d*e^4/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+5/6*d^2*e^
3*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+20/3*d^2*e^3*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-5*d^2*e^3*b
^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-80*d^2*e^3*b*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-40*d^2*e^3*b^2/c*a
/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-5/4*e^5*b^2/c^3*x^2/(c*x^2+b*x+a)^(3/2)-5/16*e^5*b^3/c^4*x/(c*x^2+b*x+a)^(3
/2)+5/12*e^5*b^6/c^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-e^5*b^2/c^4*a/(c*x^2+b*x+a)^(3/2)+5/2*e^5*b/c^3*x/(c*x^
2+b*x+a)^(1/2)-5*d^3*e^2*x/c/(c*x^2+b*x+a)^(3/2)+5/2*d*e^4*b/c^2*x^2/(c*x^2+b*x+a)^(3/2)-10*d^2*e^3*x^2/c/(c*x
^2+b*x+a)^(3/2)+5/12*d^2*e^3*b^2/c^3/(c*x^2+b*x+a)^(3/2)+5/6*d^3*e^2*b/c^2/(c*x^2+b*x+a)^(3/2)-5/3*d*e^4*x^3/c
/(c*x^2+b*x+a)^(3/2)-20/3*d^2*e^3*a/c^2/(c*x^2+b*x+a)^(3/2)-5/4*e^5*b^4/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+5/
96*e^5*b^6/c^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+4*e^5*a/c^2*x^2/(c*x^2+b*x+a)^(3/2)+5/6*e^5*b/c^2*x^3/(c*x^2+b*
x+a)^(3/2)-5/48*d*e^4*b^3/c^4/(c*x^2+b*x+a)^(3/2)-5*d*e^4/c^2*x/(c*x^2+b*x+a)^(1/2)+5/2*d*e^4/c^3*b/(c*x^2+b*x
+a)^(1/2)-40/3*d^4*e*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+4/3*d^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*c*x+32/3*d^
5*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+16/3*d^5*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+b*x+a)^(5/2),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 \[ \int \frac {{\left (d+e\,x\right )}^5}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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