3.2481 \(\int \frac{(A+B x) (d+e x)^3}{(a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=397 \[ -\frac{2 \left (-x \left (2 b^2 c e \left (11 a B e^2+4 A c d e+3 B c d^2\right )-8 b c^2 \left (a A e^3+4 a B d e^2+3 A c d^2 e+B c d^3\right )+8 c^2 \left (2 A c d \left (a e^2+c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )+2 b^3 B c d e^2-3 b^4 B e^3\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+5 a B e \left (a e^2+c d^2\right )\right )+16 a c^2 e \left (a A e^2+3 a B d e+A c d^2\right )+b^3 (-B) \left (c d^2 e-3 a e^3\right )+4 b^2 c^2 d^2 (2 A e+B d)\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{B e^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]

[Out]

(2*(d + e*x)^2*(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))/
(3*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - (2*(4*b^2*c^2*d^2*(B*d + 2*A*e) + 16*a*c^2*e*(A*c*d^2 + 3*a*B*d*
e + a*A*e^2) - b^3*B*(c*d^2*e - 3*a*e^3) - 4*b*c*(5*a*B*e*(c*d^2 + a*e^2) + 2*A*c*d*(c*d^2 + 3*a*e^2)) - (2*b^
3*B*c*d*e^2 - 3*b^4*B*e^3 + 2*b^2*c*e*(3*B*c*d^2 + 4*A*c*d*e + 11*a*B*e^2) - 8*b*c^2*(B*c*d^3 + 3*A*c*d^2*e +
4*a*B*d*e^2 + a*A*e^3) + 8*c^2*(3*a*B*e*(c*d^2 - a*e^2) + 2*A*c*d*(c*d^2 + a*e^2)))*x))/(3*c^2*(b^2 - 4*a*c)^2
*Sqrt[a + b*x + c*x^2]) + (B*e^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(5/2)

________________________________________________________________________________________

Rubi [A]  time = 0.341134, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {818, 777, 621, 206} \[ -\frac{2 \left (-x \left (2 b^2 c e \left (11 a B e^2+4 A c d e+3 B c d^2\right )-8 b c^2 \left (a A e^3+4 a B d e^2+3 A c d^2 e+B c d^3\right )+8 c^2 \left (2 A c d \left (a e^2+c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )+2 b^3 B c d e^2-3 b^4 B e^3\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+5 a B e \left (a e^2+c d^2\right )\right )+16 a c^2 e \left (a A e^2+3 a B d e+A c d^2\right )+b^3 (-B) \left (c d^2 e-3 a e^3\right )+4 b^2 c^2 d^2 (2 A e+B d)\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{B e^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*(d + e*x)^2*(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))/
(3*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - (2*(4*b^2*c^2*d^2*(B*d + 2*A*e) + 16*a*c^2*e*(A*c*d^2 + 3*a*B*d*
e + a*A*e^2) - b^3*B*(c*d^2*e - 3*a*e^3) - 4*b*c*(5*a*B*e*(c*d^2 + a*e^2) + 2*A*c*d*(c*d^2 + 3*a*e^2)) - (2*b^
3*B*c*d*e^2 - 3*b^4*B*e^3 + 2*b^2*c*e*(3*B*c*d^2 + 4*A*c*d*e + 11*a*B*e^2) - 8*b*c^2*(B*c*d^3 + 3*A*c*d^2*e +
4*a*B*d*e^2 + a*A*e^3) + 8*c^2*(3*a*B*e*(c*d^2 - a*e^2) + 2*A*c*d*(c*d^2 + a*e^2)))*x))/(3*c^2*(b^2 - 4*a*c)^2
*Sqrt[a + b*x + c*x^2]) + (B*e^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(5/2)

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])

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 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 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])

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=\frac{2 (d+e x)^2 \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 )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{2 \int \frac{(d+e x) \left (\frac{1}{2} \left (-8 A c^2 d^2-2 b B e \left (\frac{b d}{2}-2 a e\right )+4 b c d (B d+2 A e)-4 a c e (3 B d+2 A e)\right )+\frac{3}{2} B \left (b^2-4 a c\right ) e^2 x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 c \left (b^2-4 a c\right )}\\ &=\frac{2 (d+e x)^2 \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 )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (4 b^2 c^2 d^2 (B d+2 A e)+16 a c^2 e \left (A c d^2+3 a B d e+a A e^2\right )-b^3 B \left (c d^2 e-3 a e^3\right )-4 b c \left (5 a B e \left (c d^2+a e^2\right )+2 A c d \left (c d^2+3 a e^2\right )\right )-\left (2 b^3 B c d e^2-3 b^4 B e^3+2 b^2 c e \left (3 B c d^2+4 A c d e+11 a B e^2\right )-8 b c^2 \left (B c d^3+3 A c d^2 e+4 a B d e^2+a A e^3\right )+8 c^2 \left (3 a B e \left (c d^2-a e^2\right )+2 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{\left (B e^3\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{c^2}\\ &=\frac{2 (d+e x)^2 \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 )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (4 b^2 c^2 d^2 (B d+2 A e)+16 a c^2 e \left (A c d^2+3 a B d e+a A e^2\right )-b^3 B \left (c d^2 e-3 a e^3\right )-4 b c \left (5 a B e \left (c d^2+a e^2\right )+2 A c d \left (c d^2+3 a e^2\right )\right )-\left (2 b^3 B c d e^2-3 b^4 B e^3+2 b^2 c e \left (3 B c d^2+4 A c d e+11 a B e^2\right )-8 b c^2 \left (B c d^3+3 A c d^2 e+4 a B d e^2+a A e^3\right )+8 c^2 \left (3 a B e \left (c d^2-a e^2\right )+2 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{\left (2 B e^3\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^2}\\ &=\frac{2 (d+e x)^2 \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 )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (4 b^2 c^2 d^2 (B d+2 A e)+16 a c^2 e \left (A c d^2+3 a B d e+a A e^2\right )-b^3 B \left (c d^2 e-3 a e^3\right )-4 b c \left (5 a B e \left (c d^2+a e^2\right )+2 A c d \left (c d^2+3 a e^2\right )\right )-\left (2 b^3 B c d e^2-3 b^4 B e^3+2 b^2 c e \left (3 B c d^2+4 A c d e+11 a B e^2\right )-8 b c^2 \left (B c d^3+3 A c d^2 e+4 a B d e^2+a A e^3\right )+8 c^2 \left (3 a B e \left (c d^2-a e^2\right )+2 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{B e^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 1.54027, size = 505, normalized size = 1.27 \[ \frac{B e^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{c^{5/2}}-\frac{2 \left (A c^2 \left (-12 b (d-e x) \left (2 a^2 e^2+a c (d-e x)^2+2 c^2 d^2 x^2\right )+8 \left (3 a^2 c e \left (d^2+e^2 x^2\right )+2 a^3 e^3-3 a c^2 d x \left (d^2+e^2 x^2\right )-2 c^3 d^3 x^3\right )+6 b^2 \left (d^2-6 d e x+e^2 x^2\right ) (a e-c d x)+b^3 \left (9 d^2 e x+d^3-9 d e^2 x^2-e^3 x^3\right )\right )+B \left (a^2 \left (-42 b^2 c e^3 x+3 b^3 e^3-24 b c^2 d e (d-3 e x)+8 c^3 \left (d^3+9 d e^2 x^2+4 e^3 x^3\right )\right )+4 a^3 c e^2 (6 c (2 d+e x)-5 b e)+2 a \left (b^2 c^2 \left (-18 d^2 e x+d^3+9 d e^2 x^2-14 e^3 x^3\right )-9 b^3 c e^3 x^2+3 b^4 e^3 x+6 b c^3 d x \left (d^2-3 d e x+3 e^2 x^2\right )-12 c^4 d^2 e x^3\right )+b x \left (3 b^2 c^2 d \left (d^2-3 d e x-e^2 x^2\right )+4 b^3 c e^3 x^2+3 b^4 e^3 x+6 b c^3 d^2 x (2 d-e x)+8 c^4 d^3 x^2\right )\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*(A*c^2*(6*b^2*(a*e - c*d*x)*(d^2 - 6*d*e*x + e^2*x^2) + b^3*(d^3 + 9*d^2*e*x - 9*d*e^2*x^2 - e^3*x^3) - 12
*b*(d - e*x)*(2*a^2*e^2 + 2*c^2*d^2*x^2 + a*c*(d - e*x)^2) + 8*(2*a^3*e^3 - 2*c^3*d^3*x^3 + 3*a^2*c*e*(d^2 + e
^2*x^2) - 3*a*c^2*d*x*(d^2 + e^2*x^2))) + B*(4*a^3*c*e^2*(-5*b*e + 6*c*(2*d + e*x)) + b*x*(3*b^4*e^3*x + 8*c^4
*d^3*x^2 + 4*b^3*c*e^3*x^2 + 6*b*c^3*d^2*x*(2*d - e*x) + 3*b^2*c^2*d*(d^2 - 3*d*e*x - e^2*x^2)) + 2*a*(3*b^4*e
^3*x - 9*b^3*c*e^3*x^2 - 12*c^4*d^2*e*x^3 + 6*b*c^3*d*x*(d^2 - 3*d*e*x + 3*e^2*x^2) + b^2*c^2*(d^3 - 18*d^2*e*
x + 9*d*e^2*x^2 - 14*e^3*x^3)) + a^2*(3*b^3*e^3 - 42*b^2*c*e^3*x - 24*b*c^2*d*e*(d - 3*e*x) + 8*c^3*(d^3 + 9*d
*e^2*x^2 + 4*e^3*x^3)))))/(3*c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) + (B*e^3*ArcTanh[(b + 2*c*x)/(2*Sqrt
[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2)

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 2443, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x)

[Out]

1/2*B*e^3/c^3*b/(c*x^2+b*x+a)^(1/2)-1/48*B*e^3*b^3/c^4/(c*x^2+b*x+a)^(3/2)-B*e^3/c^2*x/(c*x^2+b*x+a)^(1/2)-8/3
*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d^3-2/3*a/c^2/(c*x^2+b*x+a)^(3/2)*A*e^3-x^2/c/(c*x^2+b*x+a)^(3/2)*A*e
^3+1/24*b^2/c^3/(c*x^2+b*x+a)^(3/2)*A*e^3-1/3*B*e^3*x^3/c/(c*x^2+b*x+a)^(3/2)+2/3*A*d^3/(4*a*c-b^2)/(c*x^2+b*x
+a)^(3/2)*b-1/c/(c*x^2+b*x+a)^(3/2)*A*d^2*e-b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*d^2*e+1/24*b^4/c^3/(4*a*c-
b^2)/(c*x^2+b*x+a)^(3/2)*A*e^3+1/3*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*e^3-2*a/c^2/(c*x^2+b*x+a)^(3/2)
*B*d*e^2-1/4*b/c^2*x/(c*x^2+b*x+a)^(3/2)*A*e^3-3*x^2/c/(c*x^2+b*x+a)^(3/2)*B*d*e^2+1/2*B*e^3/c^3*b^3/(4*a*c-b^
2)/(c*x^2+b*x+a)^(1/2)+1/3*B*e^3*b/c^3*a/(c*x^2+b*x+a)^(3/2)-1/48*B*e^3*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2
)-1/6*B*e^3*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+1/2*B*e^3*b/c^2*x^2/(c*x^2+b*x+a)^(3/2)+1/8*B*e^3*b^2/c^
3*x/(c*x^2+b*x+a)^(3/2)+1/4*b/c^2/(c*x^2+b*x+a)^(3/2)*B*d^2*e+1/8*b^2/c^3/(c*x^2+b*x+a)^(3/2)*B*d*e^2-3/2*x/c/
(c*x^2+b*x+a)^(3/2)*B*d^2*e+1/4*b/c^2/(c*x^2+b*x+a)^(3/2)*A*d*e^2-8*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*d^
2*e+4/3*A*d^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*c-3/2*x/c/(c*x^2+b*x+a)^(3/2)*A*d*e^2+a/c/(4*a*c-b^2)/(c*x^2+b
*x+a)^(3/2)*b*B*d^2*e+16*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*d*e^2+16*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1
/2)*x*B*d^2*e+1/2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*d*e^2+1/2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*
B*d^2*e+a/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b*A*d*e^2+32/3*A*d^3*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+16/3*
A*d^3*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b-2/3*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^3-1/3*b^2/c/(4*a*c-b^2
)/(c*x^2+b*x+a)^(3/2)*B*d^3-3*b/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d*e^2-1/3/c/(c*x^2+b*x+a)^(3/2)*B*d^3+
B*e^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+4*B*e^3*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+
1/8*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d*e^2+B*e^3/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-1/24*B*e^3
*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-1/3*B*e^3*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+1/4*B*e^3*b^3
/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+2*B*e^3*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+2*a/(4*a*c-b^2)/(c*
x^2+b*x+a)^(3/2)*x*B*d^2*e+8*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b*A*d*e^2-16/3*b*c/(4*a*c-b^2)^2/(c*x^2+b*x+a
)^(1/2)*x*B*d^3+4*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d^2*e+2*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*
d*e^2+2*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d^2*e+2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*d*e^2-2*b/(4*a
*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*d^2*e+8*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b*B*d^2*e+4*b^2/(4*a*c-b^2)^2/(c*x
^2+b*x+a)^(1/2)*x*A*d*e^2+1/4*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*d*e^2+1/4*b^3/c^2/(4*a*c-b^2)/(c*x^2+b
*x+a)^(3/2)*B*d^2*e+1/2*B*e^3*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-24*b*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(
1/2)*x*B*d*e^2-12*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d*e^2-b/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*
e^3-3/2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d*e^2+1/4*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d*e^
2+2*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d*e^2-16*b*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*d^2*e+2/3*b
^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*e^3+b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d*e^2-3/4*b/c^2*x/(
c*x^2+b*x+a)^(3/2)*B*d*e^2+1/12*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*e^3-1/2*b^2/c^2*a/(4*a*c-b^2)/(c*x
^2+b*x+a)^(3/2)*A*e^3-8*b*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*e^3-4*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1
/2)*A*e^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 88.6694, size = 3903, normalized size = 9.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*((B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*e^3*x^4 + 2*(B*b^5*c - 8*B*a*b^3*c^2 + 16*B*a^2*b*c^3)*e^3
*x^3 + (B*b^6 - 6*B*a*b^4*c + 32*B*a^3*c^3)*e^3*x^2 + 2*(B*a*b^5 - 8*B*a^2*b^3*c + 16*B*a^3*b*c^2)*e^3*x + (B*
a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2)*e^3)*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*(24*(2*B*a^3 - A*a^2*b)*c^3*d*e^2 + (4*(2*B*a^2 - 3*A*a*b)*c^4 + (2*B*a*b^2 +
A*b^3)*c^3)*d^3 + 6*(4*A*a^2*c^4 - (4*B*a^2*b - A*a*b^2)*c^3)*d^2*e + (3*B*a^2*b^3*c - 20*B*a^3*b*c^2 + 16*A*
a^3*c^3)*e^3 + (8*(B*b*c^5 - 2*A*c^6)*d^3 - 6*(B*b^2*c^4 + 4*(B*a - A*b)*c^5)*d^2*e - 3*(B*b^3*c^3 + 8*A*a*c^5
- 2*(6*B*a*b - A*b^2)*c^4)*d*e^2 + (4*B*b^4*c^2 + 4*(8*B*a^2 + 3*A*a*b)*c^4 - (28*B*a*b^2 + A*b^3)*c^3)*e^3)*
x^3 + 3*(4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 3*(B*b^3*c^3 + 4*(B*a*b - A*b^2)*c^4)*d^2*e + 3*(4*(2*B*a^2 - A*a*b)*
c^4 + (2*B*a*b^2 - A*b^3)*c^3)*d*e^2 + (B*b^5*c - 6*B*a*b^3*c^2 + 2*A*a*b^2*c^3 + 8*A*a^2*c^4)*e^3)*x^2 + 3*(1
2*(2*B*a^2*b - A*a*b^2)*c^3*d*e^2 + (B*b^3*c^3 - 8*A*a*c^5 + 2*(2*B*a*b - A*b^2)*c^4)*d^3 + 3*(4*A*a*b*c^4 - (
4*B*a*b^2 - A*b^3)*c^3)*d^2*e + 2*(B*a*b^4*c - 7*B*a^2*b^2*c^2 + 4*(B*a^3 + A*a^2*b)*c^3)*e^3)*x)*sqrt(c*x^2 +
b*x + a))/(a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 -
8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 1
6*a^3*b*c^5)*x), -1/3*(3*((B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*e^3*x^4 + 2*(B*b^5*c - 8*B*a*b^3*c^2 + 16
*B*a^2*b*c^3)*e^3*x^3 + (B*b^6 - 6*B*a*b^4*c + 32*B*a^3*c^3)*e^3*x^2 + 2*(B*a*b^5 - 8*B*a^2*b^3*c + 16*B*a^3*b
*c^2)*e^3*x + (B*a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2)*e^3)*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*(24*(2*B*a^3 - A*a^2*b)*c^3*d*e^2 + (4*(2*B*a^2 - 3*A*a*b)*c^4 + (
2*B*a*b^2 + A*b^3)*c^3)*d^3 + 6*(4*A*a^2*c^4 - (4*B*a^2*b - A*a*b^2)*c^3)*d^2*e + (3*B*a^2*b^3*c - 20*B*a^3*b*
c^2 + 16*A*a^3*c^3)*e^3 + (8*(B*b*c^5 - 2*A*c^6)*d^3 - 6*(B*b^2*c^4 + 4*(B*a - A*b)*c^5)*d^2*e - 3*(B*b^3*c^3
+ 8*A*a*c^5 - 2*(6*B*a*b - A*b^2)*c^4)*d*e^2 + (4*B*b^4*c^2 + 4*(8*B*a^2 + 3*A*a*b)*c^4 - (28*B*a*b^2 + A*b^3)
*c^3)*e^3)*x^3 + 3*(4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 3*(B*b^3*c^3 + 4*(B*a*b - A*b^2)*c^4)*d^2*e + 3*(4*(2*B*a^
2 - A*a*b)*c^4 + (2*B*a*b^2 - A*b^3)*c^3)*d*e^2 + (B*b^5*c - 6*B*a*b^3*c^2 + 2*A*a*b^2*c^3 + 8*A*a^2*c^4)*e^3)
*x^2 + 3*(12*(2*B*a^2*b - A*a*b^2)*c^3*d*e^2 + (B*b^3*c^3 - 8*A*a*c^5 + 2*(2*B*a*b - A*b^2)*c^4)*d^3 + 3*(4*A*
a*b*c^4 - (4*B*a*b^2 - A*b^3)*c^3)*d^2*e + 2*(B*a*b^4*c - 7*B*a^2*b^2*c^2 + 4*(B*a^3 + A*a^2*b)*c^3)*e^3)*x)*s
qrt(c*x^2 + b*x + a))/(a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2
*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^2*
b^3*c^4 + 16*a^3*b*c^5)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.16554, size = 1067, normalized size = 2.69 \begin{align*} -\frac{B e^{3} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} - \frac{2 \,{\left ({\left ({\left (\frac{{\left (8 \, B b c^{4} d^{3} - 16 \, A c^{5} d^{3} - 6 \, B b^{2} c^{3} d^{2} e - 24 \, B a c^{4} d^{2} e + 24 \, A b c^{4} d^{2} e - 3 \, B b^{3} c^{2} d e^{2} + 36 \, B a b c^{3} d e^{2} - 6 \, A b^{2} c^{3} d e^{2} - 24 \, A a c^{4} d e^{2} + 4 \, B b^{4} c e^{3} - 28 \, B a b^{2} c^{2} e^{3} - A b^{3} c^{2} e^{3} + 32 \, B a^{2} c^{3} e^{3} + 12 \, A a b c^{3} e^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (4 \, B b^{2} c^{3} d^{3} - 8 \, A b c^{4} d^{3} - 3 \, B b^{3} c^{2} d^{2} e - 12 \, B a b c^{3} d^{2} e + 12 \, A b^{2} c^{3} d^{2} e + 6 \, B a b^{2} c^{2} d e^{2} - 3 \, A b^{3} c^{2} d e^{2} + 24 \, B a^{2} c^{3} d e^{2} - 12 \, A a b c^{3} d e^{2} + B b^{5} e^{3} - 6 \, B a b^{3} c e^{3} + 2 \, A a b^{2} c^{2} e^{3} + 8 \, A a^{2} c^{3} e^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (B b^{3} c^{2} d^{3} + 4 \, B a b c^{3} d^{3} - 2 \, A b^{2} c^{3} d^{3} - 8 \, A a c^{4} d^{3} - 12 \, B a b^{2} c^{2} d^{2} e + 3 \, A b^{3} c^{2} d^{2} e + 12 \, A a b c^{3} d^{2} e + 24 \, B a^{2} b c^{2} d e^{2} - 12 \, A a b^{2} c^{2} d e^{2} + 2 \, B a b^{4} e^{3} - 14 \, B a^{2} b^{2} c e^{3} + 8 \, B a^{3} c^{2} e^{3} + 8 \, A a^{2} b c^{2} e^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{2 \, B a b^{2} c^{2} d^{3} + A b^{3} c^{2} d^{3} + 8 \, B a^{2} c^{3} d^{3} - 12 \, A a b c^{3} d^{3} - 24 \, B a^{2} b c^{2} d^{2} e + 6 \, A a b^{2} c^{2} d^{2} e + 24 \, A a^{2} c^{3} d^{2} e + 48 \, B a^{3} c^{2} d e^{2} - 24 \, A a^{2} b c^{2} d e^{2} + 3 \, B a^{2} b^{3} e^{3} - 20 \, B a^{3} b c e^{3} + 16 \, A a^{3} c^{2} e^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

-B*e^3*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2) - 2/3*((((8*B*b*c^4*d^3 - 16*A*c^5
*d^3 - 6*B*b^2*c^3*d^2*e - 24*B*a*c^4*d^2*e + 24*A*b*c^4*d^2*e - 3*B*b^3*c^2*d*e^2 + 36*B*a*b*c^3*d*e^2 - 6*A*
b^2*c^3*d*e^2 - 24*A*a*c^4*d*e^2 + 4*B*b^4*c*e^3 - 28*B*a*b^2*c^2*e^3 - A*b^3*c^2*e^3 + 32*B*a^2*c^3*e^3 + 12*
A*a*b*c^3*e^3)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 3*(4*B*b^2*c^3*d^3 - 8*A*b*c^4*d^3 - 3*B*b^3*c^2*d^2*e
- 12*B*a*b*c^3*d^2*e + 12*A*b^2*c^3*d^2*e + 6*B*a*b^2*c^2*d*e^2 - 3*A*b^3*c^2*d*e^2 + 24*B*a^2*c^3*d*e^2 - 12
*A*a*b*c^3*d*e^2 + B*b^5*e^3 - 6*B*a*b^3*c*e^3 + 2*A*a*b^2*c^2*e^3 + 8*A*a^2*c^3*e^3)/(b^4*c^2 - 8*a*b^2*c^3 +
16*a^2*c^4))*x + 3*(B*b^3*c^2*d^3 + 4*B*a*b*c^3*d^3 - 2*A*b^2*c^3*d^3 - 8*A*a*c^4*d^3 - 12*B*a*b^2*c^2*d^2*e
+ 3*A*b^3*c^2*d^2*e + 12*A*a*b*c^3*d^2*e + 24*B*a^2*b*c^2*d*e^2 - 12*A*a*b^2*c^2*d*e^2 + 2*B*a*b^4*e^3 - 14*B*
a^2*b^2*c*e^3 + 8*B*a^3*c^2*e^3 + 8*A*a^2*b*c^2*e^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + (2*B*a*b^2*c^2*
d^3 + A*b^3*c^2*d^3 + 8*B*a^2*c^3*d^3 - 12*A*a*b*c^3*d^3 - 24*B*a^2*b*c^2*d^2*e + 6*A*a*b^2*c^2*d^2*e + 24*A*a
^2*c^3*d^2*e + 48*B*a^3*c^2*d*e^2 - 24*A*a^2*b*c^2*d*e^2 + 3*B*a^2*b^3*e^3 - 20*B*a^3*b*c*e^3 + 16*A*a^3*c^2*e
^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2)