3.1024 \(\int \frac{x^{5/2} (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=459 \[ \frac{\left (-\frac{-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\sqrt{x} \left (a \left (20 a B c-12 A b c+b^2 B\right )-x \left (12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

[Out]

-(x^(3/2)*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(2*c*(b^2 - 4*a*c)*(a
+ b*x + c*x^2)^2) - (Sqrt[x]*(a*(b^2*B - 12*A*b*c + 20*a*B*c) - (b^3*B + 3*A*b^
2*c - 16*a*b*B*c + 12*a*A*c^2)*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + ((b
^3*B + 3*A*b^2*c - 16*a*b*B*c + 12*a*A*c^2 - (b^4*B + 3*A*b^3*c - 18*a*b^2*B*c +
36*a*A*b*c^2 - 40*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x]
)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^2*Sqrt[b - Sqrt
[b^2 - 4*a*c]]) + ((b^3*B + 3*A*b^2*c - 16*a*b*B*c + 12*a*A*c^2 + (b^4*B + 3*A*b
^3*c - 18*a*b^2*B*c + 36*a*A*b*c^2 - 40*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sq
rt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4
*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 9.18121, antiderivative size = 459, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{\left (-\frac{-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\sqrt{x} \left (a \left (20 a B c-12 A b c+b^2 B\right )-x \left (12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(x^(3/2)*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(2*c*(b^2 - 4*a*c)*(a
+ b*x + c*x^2)^2) - (Sqrt[x]*(a*(b^2*B - 12*A*b*c + 20*a*B*c) - (b^3*B + 3*A*b^
2*c - 16*a*b*B*c + 12*a*A*c^2)*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + ((b
^3*B + 3*A*b^2*c - 16*a*b*B*c + 12*a*A*c^2 - (b^4*B + 3*A*b^3*c - 18*a*b^2*B*c +
36*a*A*b*c^2 - 40*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x]
)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^2*Sqrt[b - Sqrt
[b^2 - 4*a*c]]) + ((b^3*B + 3*A*b^2*c - 16*a*b*B*c + 12*a*A*c^2 + (b^4*B + 3*A*b
^3*c - 18*a*b^2*B*c + 36*a*A*b*c^2 - 40*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sq
rt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4
*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x+A)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 4.39374, size = 539, normalized size = 1.17 \[ \frac{-\frac{4 \sqrt{x} \left (2 a^2 B c+a \left (b c (A+3 B x)-2 A c^2 x+b^2 (-B)\right )+b^2 x (A c-b B)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{2 \sqrt{x} \left (b^2 c (11 a B+3 A c x)+4 a b c^2 (A-4 B x)+12 a c^2 (A c x-3 a B)+b^3 c (2 A+B x)-2 b^4 B\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{\sqrt{2} \sqrt{c} \left (4 a c^2 \left (3 A \sqrt{b^2-4 a c}+10 a B\right )+3 b^2 c \left (A \sqrt{b^2-4 a c}+6 a B\right )-4 a b c \left (4 B \sqrt{b^2-4 a c}+9 A c\right )+b^3 \left (B \sqrt{b^2-4 a c}-3 A c\right )+b^4 (-B)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (4 a c^2 \left (3 A \sqrt{b^2-4 a c}-10 a B\right )+3 b^2 c \left (A \sqrt{b^2-4 a c}-6 a B\right )+4 a b c \left (9 A c-4 B \sqrt{b^2-4 a c}\right )+b^3 \left (B \sqrt{b^2-4 a c}+3 A c\right )+b^4 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{8 c^2} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[x]*(-2*b^4*B + 4*a*b*c^2*(A - 4*B*x) + b^3*c*(2*A + B*x) + 12*a*c^2*(-3
*a*B + A*c*x) + b^2*c*(11*a*B + 3*A*c*x)))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) -
(4*Sqrt[x]*(2*a^2*B*c + b^2*(-(b*B) + A*c)*x + a*(-(b^2*B) - 2*A*c^2*x + b*c*(A
+ 3*B*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (Sqrt[2]*Sqrt[c]*(-(b^4*B) +
3*b^2*c*(6*a*B + A*Sqrt[b^2 - 4*a*c]) + 4*a*c^2*(10*a*B + 3*A*Sqrt[b^2 - 4*a*c])
+ b^3*(-3*A*c + B*Sqrt[b^2 - 4*a*c]) - 4*a*b*c*(9*A*c + 4*B*Sqrt[b^2 - 4*a*c]))
*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(
5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(b^4*B + 3*b^2*c*(-6*a*B +
A*Sqrt[b^2 - 4*a*c]) + 4*a*c^2*(-10*a*B + 3*A*Sqrt[b^2 - 4*a*c]) + 4*a*b*c*(9*A*
c - 4*B*Sqrt[b^2 - 4*a*c]) + b^3*(3*A*c + B*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*
Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqr
t[b^2 - 4*a*c]]))/(8*c^2)

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Maple [B]  time = 0.176, size = 9253, normalized size = 20.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)/(c*x^2+b*x+a)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{{\left (12 \, A b c^{2} -{\left (b^{2} c + 20 \, a c^{2}\right )} B\right )} x^{\frac{9}{2}} + 3 \,{\left ({\left (7 \, b^{2} c - 4 \, a c^{2}\right )} A -{\left (b^{3} + 8 \, a b c\right )} B\right )} x^{\frac{7}{2}} +{\left ({\left (7 \, b^{3} + 8 \, a b c\right )} A -{\left (17 \, a b^{2} + 4 \, a^{2} c\right )} B\right )} x^{\frac{5}{2}} -{\left (12 \, B a^{2} b -{\left (5 \, a b^{2} + 4 \, a^{2} c\right )} A\right )} x^{\frac{3}{2}}}{4 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} + \int \frac{{\left (12 \, A b c -{\left (b^{2} + 20 \, a c\right )} B\right )} x^{\frac{3}{2}} - 3 \,{\left (12 \, B a b -{\left (5 \, b^{2} + 4 \, a c\right )} A\right )} \sqrt{x}}{8 \,{\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*((12*A*b*c^2 - (b^2*c + 20*a*c^2)*B)*x^(9/2) + 3*((7*b^2*c - 4*a*c^2)*A - (
b^3 + 8*a*b*c)*B)*x^(7/2) + ((7*b^3 + 8*a*b*c)*A - (17*a*b^2 + 4*a^2*c)*B)*x^(5/
2) - (12*B*a^2*b - (5*a*b^2 + 4*a^2*c)*A)*x^(3/2))/(a^2*b^4 - 8*a^3*b^2*c + 16*a
^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*
a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 1
6*a^3*b*c^2)*x) + integrate(1/8*((12*A*b*c - (b^2 + 20*a*c)*B)*x^(3/2) - 3*(12*B
*a*b - (5*b^2 + 4*a*c)*A)*sqrt(x))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c -
8*a*b^2*c^2 + 16*a^2*c^3)*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x), x)

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Fricas [A]  time = 6.74896, size = 9526, normalized size = 20.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4
+ 16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a
*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x)*sqrt(
-(B^2*b^7 - 240*(4*A*B*a^3 - 3*A^2*a^2*b)*c^4 + 120*(14*B^2*a^3*b - 16*A*B*a^2*b
^2 + 3*A^2*a*b^3)*c^3 + (280*B^2*a^2*b^3 - 60*A*B*a*b^4 + 9*A^2*b^5)*c^2 - (35*B
^2*a*b^5 - 6*A*B*b^6)*c + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b
^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((B^4*b^4 + 81*A^4*c^4 - 18*(25*A^
2*B^2*a - 6*A^3*B*b)*c^3 + (625*B^4*a^2 - 300*A*B^3*a*b + 54*A^2*B^2*b^2)*c^2 -
2*(25*B^4*a*b^2 - 6*A*B^3*b^3)*c)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 6
40*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 +
160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*log(1/2*s
qrt(1/2)*(B^3*b^10 - 2304*(5*A^2*B*a^4 - 3*A^3*a^3*b)*c^6 + 64*(500*B^3*a^5 - 42
0*A*B^2*a^4*b + 198*A^2*B*a^3*b^2 - 81*A^3*a^2*b^3)*c^5 - 16*(1480*B^3*a^4*b^2 -
1284*A*B^2*a^3*b^3 + 324*A^2*B*a^2*b^4 - 81*A^3*a*b^5)*c^4 + 4*(1424*B^3*a^3*b^
4 - 1332*A*B^2*a^2*b^5 + 234*A^2*B*a*b^6 - 27*A^3*b^7)*c^3 - (392*B^3*a^2*b^6 -
492*A*B^2*a*b^7 + 63*A^2*B*b^8)*c^2 - (17*B^3*a*b^8 + 6*A*B^2*b^9)*c - (B*b^13*c
^3 - 24576*A*a^6*c^10 + 4096*(13*B*a^6*b + 3*A*a^5*b^2)*c^9 - 1536*(44*B*a^5*b^3
- 5*A*a^4*b^4)*c^8 + 3840*(9*B*a^4*b^5 - 2*A*a^3*b^6)*c^7 - 160*(56*B*a^3*b^7 -
15*A*a^2*b^8)*c^6 + 48*(25*B*a^2*b^9 - 7*A*a*b^10)*c^5 - 18*(4*B*a*b^11 - A*b^1
2)*c^4)*sqrt((B^4*b^4 + 81*A^4*c^4 - 18*(25*A^2*B^2*a - 6*A^3*B*b)*c^3 + (625*B^
4*a^2 - 300*A*B^3*a*b + 54*A^2*B^2*b^2)*c^2 - 2*(25*B^4*a*b^2 - 6*A*B^3*b^3)*c)/
(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10
- 1024*a^5*c^11)))*sqrt(-(B^2*b^7 - 240*(4*A*B*a^3 - 3*A^2*a^2*b)*c^4 + 120*(14
*B^2*a^3*b - 16*A*B*a^2*b^2 + 3*A^2*a*b^3)*c^3 + (280*B^2*a^2*b^3 - 60*A*B*a*b^4
+ 9*A^2*b^5)*c^2 - (35*B^2*a*b^5 - 6*A*B*b^6)*c + (b^10*c^3 - 20*a*b^8*c^4 + 16
0*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((B^4*b^4
+ 81*A^4*c^4 - 18*(25*A^2*B^2*a - 6*A^3*B*b)*c^3 + (625*B^4*a^2 - 300*A*B^3*a*b
+ 54*A^2*B^2*b^2)*c^2 - 2*(25*B^4*a*b^2 - 6*A*B^3*b^3)*c)/(b^10*c^6 - 20*a*b^8*
c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(
b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 -
1024*a^5*c^8)) - (35*B^4*a*b^6 - 15*A*B^3*b^7 - 1296*A^4*a^2*c^5 + 648*(14*A^3*
B*a^2*b - 5*A^4*a*b^2)*c^4 + (10000*B^4*a^4 - 30000*A*B^3*a^3*b + 9936*A^2*B^2*a
^2*b^2 + 1080*A^3*B*a*b^3 - 405*A^4*b^4)*c^3 + 3*(5000*B^4*a^3*b^2 - 3864*A*B^3*
a^2*b^3 + 1080*A^2*B^2*a*b^4 - 135*A^3*B*b^5)*c^2 - 3*(497*B^4*a^2*b^4 - 315*A*B
^3*a*b^5 + 45*A^2*B^2*b^6)*c)*sqrt(x)) - sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 +
16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3
+ 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a
^2*b^3*c^2 + 16*a^3*b*c^3)*x)*sqrt(-(B^2*b^7 - 240*(4*A*B*a^3 - 3*A^2*a^2*b)*c^4
+ 120*(14*B^2*a^3*b - 16*A*B*a^2*b^2 + 3*A^2*a*b^3)*c^3 + (280*B^2*a^2*b^3 - 60
*A*B*a*b^4 + 9*A^2*b^5)*c^2 - (35*B^2*a*b^5 - 6*A*B*b^6)*c + (b^10*c^3 - 20*a*b^
8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqr
t((B^4*b^4 + 81*A^4*c^4 - 18*(25*A^2*B^2*a - 6*A^3*B*b)*c^3 + (625*B^4*a^2 - 300
*A*B^3*a*b + 54*A^2*B^2*b^2)*c^2 - 2*(25*B^4*a*b^2 - 6*A*B^3*b^3)*c)/(b^10*c^6 -
20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5
*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4
*b^2*c^7 - 1024*a^5*c^8))*log(-1/2*sqrt(1/2)*(B^3*b^10 - 2304*(5*A^2*B*a^4 - 3*A
^3*a^3*b)*c^6 + 64*(500*B^3*a^5 - 420*A*B^2*a^4*b + 198*A^2*B*a^3*b^2 - 81*A^3*a
^2*b^3)*c^5 - 16*(1480*B^3*a^4*b^2 - 1284*A*B^2*a^3*b^3 + 324*A^2*B*a^2*b^4 - 81
*A^3*a*b^5)*c^4 + 4*(1424*B^3*a^3*b^4 - 1332*A*B^2*a^2*b^5 + 234*A^2*B*a*b^6 - 2
7*A^3*b^7)*c^3 - (392*B^3*a^2*b^6 - 492*A*B^2*a*b^7 + 63*A^2*B*b^8)*c^2 - (17*B^
3*a*b^8 + 6*A*B^2*b^9)*c - (B*b^13*c^3 - 24576*A*a^6*c^10 + 4096*(13*B*a^6*b + 3
*A*a^5*b^2)*c^9 - 1536*(44*B*a^5*b^3 - 5*A*a^4*b^4)*c^8 + 3840*(9*B*a^4*b^5 - 2*
A*a^3*b^6)*c^7 - 160*(56*B*a^3*b^7 - 15*A*a^2*b^8)*c^6 + 48*(25*B*a^2*b^9 - 7*A*
a*b^10)*c^5 - 18*(4*B*a*b^11 - A*b^12)*c^4)*sqrt((B^4*b^4 + 81*A^4*c^4 - 18*(25*
A^2*B^2*a - 6*A^3*B*b)*c^3 + (625*B^4*a^2 - 300*A*B^3*a*b + 54*A^2*B^2*b^2)*c^2
- 2*(25*B^4*a*b^2 - 6*A*B^3*b^3)*c)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 -
640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))*sqrt(-(B^2*b^7 - 240*(4*
A*B*a^3 - 3*A^2*a^2*b)*c^4 + 120*(14*B^2*a^3*b - 16*A*B*a^2*b^2 + 3*A^2*a*b^3)*c
^3 + (280*B^2*a^2*b^3 - 60*A*B*a*b^4 + 9*A^2*b^5)*c^2 - (35*B^2*a*b^5 - 6*A*B*b^
6)*c + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b
^2*c^7 - 1024*a^5*c^8)*sqrt((B^4*b^4 + 81*A^4*c^4 - 18*(25*A^2*B^2*a - 6*A^3*B*b
)*c^3 + (625*B^4*a^2 - 300*A*B^3*a*b + 54*A^2*B^2*b^2)*c^2 - 2*(25*B^4*a*b^2 - 6
*A*B^3*b^3)*c)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 12
80*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 -
640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) - (35*B^4*a*b^6 - 15*A*B^3*b
^7 - 1296*A^4*a^2*c^5 + 648*(14*A^3*B*a^2*b - 5*A^4*a*b^2)*c^4 + (10000*B^4*a^4
- 30000*A*B^3*a^3*b + 9936*A^2*B^2*a^2*b^2 + 1080*A^3*B*a*b^3 - 405*A^4*b^4)*c^3
+ 3*(5000*B^4*a^3*b^2 - 3864*A*B^3*a^2*b^3 + 1080*A^2*B^2*a*b^4 - 135*A^3*B*b^5
)*c^2 - 3*(497*B^4*a^2*b^4 - 315*A*B^3*a*b^5 + 45*A^2*B^2*b^6)*c)*sqrt(x)) + sqr
t(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2
*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2
+ 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x)*sqrt(-(B^2*b^7
- 240*(4*A*B*a^3 - 3*A^2*a^2*b)*c^4 + 120*(14*B^2*a^3*b - 16*A*B*a^2*b^2 + 3*A^
2*a*b^3)*c^3 + (280*B^2*a^2*b^3 - 60*A*B*a*b^4 + 9*A^2*b^5)*c^2 - (35*B^2*a*b^5
- 6*A*B*b^6)*c - (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 +
1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((B^4*b^4 + 81*A^4*c^4 - 18*(25*A^2*B^2*a -
6*A^3*B*b)*c^3 + (625*B^4*a^2 - 300*A*B^3*a*b + 54*A^2*B^2*b^2)*c^2 - 2*(25*B^4
*a*b^2 - 6*A*B^3*b^3)*c)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^
4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*
b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*log(1/2*sqrt(1/2)*
(B^3*b^10 - 2304*(5*A^2*B*a^4 - 3*A^3*a^3*b)*c^6 + 64*(500*B^3*a^5 - 420*A*B^2*a
^4*b + 198*A^2*B*a^3*b^2 - 81*A^3*a^2*b^3)*c^5 - 16*(1480*B^3*a^4*b^2 - 1284*A*B
^2*a^3*b^3 + 324*A^2*B*a^2*b^4 - 81*A^3*a*b^5)*c^4 + 4*(1424*B^3*a^3*b^4 - 1332*
A*B^2*a^2*b^5 + 234*A^2*B*a*b^6 - 27*A^3*b^7)*c^3 - (392*B^3*a^2*b^6 - 492*A*B^2
*a*b^7 + 63*A^2*B*b^8)*c^2 - (17*B^3*a*b^8 + 6*A*B^2*b^9)*c + (B*b^13*c^3 - 2457
6*A*a^6*c^10 + 4096*(13*B*a^6*b + 3*A*a^5*b^2)*c^9 - 1536*(44*B*a^5*b^3 - 5*A*a^
4*b^4)*c^8 + 3840*(9*B*a^4*b^5 - 2*A*a^3*b^6)*c^7 - 160*(56*B*a^3*b^7 - 15*A*a^2
*b^8)*c^6 + 48*(25*B*a^2*b^9 - 7*A*a*b^10)*c^5 - 18*(4*B*a*b^11 - A*b^12)*c^4)*s
qrt((B^4*b^4 + 81*A^4*c^4 - 18*(25*A^2*B^2*a - 6*A^3*B*b)*c^3 + (625*B^4*a^2 - 3
00*A*B^3*a*b + 54*A^2*B^2*b^2)*c^2 - 2*(25*B^4*a*b^2 - 6*A*B^3*b^3)*c)/(b^10*c^6
- 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a
^5*c^11)))*sqrt(-(B^2*b^7 - 240*(4*A*B*a^3 - 3*A^2*a^2*b)*c^4 + 120*(14*B^2*a^3*
b - 16*A*B*a^2*b^2 + 3*A^2*a*b^3)*c^3 + (280*B^2*a^2*b^3 - 60*A*B*a*b^4 + 9*A^2*
b^5)*c^2 - (35*B^2*a*b^5 - 6*A*B*b^6)*c - (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6
*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((B^4*b^4 + 81*A^4
*c^4 - 18*(25*A^2*B^2*a - 6*A^3*B*b)*c^3 + (625*B^4*a^2 - 300*A*B^3*a*b + 54*A^2
*B^2*b^2)*c^2 - 2*(25*B^4*a*b^2 - 6*A*B^3*b^3)*c)/(b^10*c^6 - 20*a*b^8*c^7 + 160
*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3
- 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5
*c^8)) - (35*B^4*a*b^6 - 15*A*B^3*b^7 - 1296*A^4*a^2*c^5 + 648*(14*A^3*B*a^2*b -
5*A^4*a*b^2)*c^4 + (10000*B^4*a^4 - 30000*A*B^3*a^3*b + 9936*A^2*B^2*a^2*b^2 +
1080*A^3*B*a*b^3 - 405*A^4*b^4)*c^3 + 3*(5000*B^4*a^3*b^2 - 3864*A*B^3*a^2*b^3 +
1080*A^2*B^2*a*b^4 - 135*A^3*B*b^5)*c^2 - 3*(497*B^4*a^2*b^4 - 315*A*B^3*a*b^5
+ 45*A^2*B^2*b^6)*c)*sqrt(x)) - sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^
3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2
*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^
2 + 16*a^3*b*c^3)*x)*sqrt(-(B^2*b^7 - 240*(4*A*B*a^3 - 3*A^2*a^2*b)*c^4 + 120*(1
4*B^2*a^3*b - 16*A*B*a^2*b^2 + 3*A^2*a*b^3)*c^3 + (280*B^2*a^2*b^3 - 60*A*B*a*b^
4 + 9*A^2*b^5)*c^2 - (35*B^2*a*b^5 - 6*A*B*b^6)*c - (b^10*c^3 - 20*a*b^8*c^4 + 1
60*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((B^4*b^
4 + 81*A^4*c^4 - 18*(25*A^2*B^2*a - 6*A^3*B*b)*c^3 + (625*B^4*a^2 - 300*A*B^3*a*
b + 54*A^2*B^2*b^2)*c^2 - 2*(25*B^4*a*b^2 - 6*A*B^3*b^3)*c)/(b^10*c^6 - 20*a*b^8
*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/
(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7
- 1024*a^5*c^8))*log(-1/2*sqrt(1/2)*(B^3*b^10 - 2304*(5*A^2*B*a^4 - 3*A^3*a^3*b)
*c^6 + 64*(500*B^3*a^5 - 420*A*B^2*a^4*b + 198*A^2*B*a^3*b^2 - 81*A^3*a^2*b^3)*c
^5 - 16*(1480*B^3*a^4*b^2 - 1284*A*B^2*a^3*b^3 + 324*A^2*B*a^2*b^4 - 81*A^3*a*b^
5)*c^4 + 4*(1424*B^3*a^3*b^4 - 1332*A*B^2*a^2*b^5 + 234*A^2*B*a*b^6 - 27*A^3*b^7
)*c^3 - (392*B^3*a^2*b^6 - 492*A*B^2*a*b^7 + 63*A^2*B*b^8)*c^2 - (17*B^3*a*b^8 +
6*A*B^2*b^9)*c + (B*b^13*c^3 - 24576*A*a^6*c^10 + 4096*(13*B*a^6*b + 3*A*a^5*b^
2)*c^9 - 1536*(44*B*a^5*b^3 - 5*A*a^4*b^4)*c^8 + 3840*(9*B*a^4*b^5 - 2*A*a^3*b^6
)*c^7 - 160*(56*B*a^3*b^7 - 15*A*a^2*b^8)*c^6 + 48*(25*B*a^2*b^9 - 7*A*a*b^10)*c
^5 - 18*(4*B*a*b^11 - A*b^12)*c^4)*sqrt((B^4*b^4 + 81*A^4*c^4 - 18*(25*A^2*B^2*a
- 6*A^3*B*b)*c^3 + (625*B^4*a^2 - 300*A*B^3*a*b + 54*A^2*B^2*b^2)*c^2 - 2*(25*B
^4*a*b^2 - 6*A*B^3*b^3)*c)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*
b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))*sqrt(-(B^2*b^7 - 240*(4*A*B*a^3 -
3*A^2*a^2*b)*c^4 + 120*(14*B^2*a^3*b - 16*A*B*a^2*b^2 + 3*A^2*a*b^3)*c^3 + (280
*B^2*a^2*b^3 - 60*A*B*a*b^4 + 9*A^2*b^5)*c^2 - (35*B^2*a*b^5 - 6*A*B*b^6)*c - (b
^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 -
1024*a^5*c^8)*sqrt((B^4*b^4 + 81*A^4*c^4 - 18*(25*A^2*B^2*a - 6*A^3*B*b)*c^3 + (
625*B^4*a^2 - 300*A*B^3*a*b + 54*A^2*B^2*b^2)*c^2 - 2*(25*B^4*a*b^2 - 6*A*B^3*b^
3)*c)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^
2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b
^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) - (35*B^4*a*b^6 - 15*A*B^3*b^7 - 1296
*A^4*a^2*c^5 + 648*(14*A^3*B*a^2*b - 5*A^4*a*b^2)*c^4 + (10000*B^4*a^4 - 30000*A
*B^3*a^3*b + 9936*A^2*B^2*a^2*b^2 + 1080*A^3*B*a*b^3 - 405*A^4*b^4)*c^3 + 3*(500
0*B^4*a^3*b^2 - 3864*A*B^3*a^2*b^3 + 1080*A^2*B^2*a*b^4 - 135*A^3*B*b^5)*c^2 - 3
*(497*B^4*a^2*b^4 - 315*A*B^3*a*b^5 + 45*A^2*B^2*b^6)*c)*sqrt(x)) + 2*(B*a^2*b^2
- (B*b^3*c + 12*A*a*c^3 - (16*B*a*b - 3*A*b^2)*c^2)*x^3 + (B*b^4 + 4*(9*B*a^2 -
4*A*a*b)*c^2 + 5*(B*a*b^2 - A*b^3)*c)*x^2 + 4*(5*B*a^3 - 3*A*a^2*b)*c + (2*B*a*
b^3 + 4*A*a^2*c^2 + (28*B*a^2*b - 19*A*a*b^2)*c)*x)*sqrt(x))/(a^2*b^4*c - 8*a^3*
b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8
*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b
^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(B*x+A)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out