3.489 \(\int \frac{x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=609 \[ -\frac{x^{3/2}}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac{d x^{3/2}}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt [4]{b} (5 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}-\frac{\sqrt [4]{b} (5 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}-\frac{\sqrt [4]{d} (3 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} (b c-a d)^3}+\frac{\sqrt [4]{d} (3 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} (b c-a d)^3}-\frac{\sqrt [4]{b} (5 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}+\frac{\sqrt [4]{b} (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}+\frac{\sqrt [4]{d} (3 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} \sqrt [4]{c} (b c-a d)^3}-\frac{\sqrt [4]{d} (3 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} \sqrt [4]{c} (b c-a d)^3} \]

[Out]

-((d*x^(3/2))/((b*c - a*d)^2*(c + d*x^2))) - x^(3/2)/(2*(b*c - a*d)*(a + b*x^2)*
(c + d*x^2)) - (b^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(
1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + (b^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + (d^(1/4)*
(5*b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(1/4
)*(b*c - a*d)^3) - (d^(1/4)*(5*b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])
/c^(1/4)])/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^3) + (b^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt
[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*
d)^3) - (b^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^3) - (d^(1/4)*(5*b*c + 3*a*d)*Log[Sq
rt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(1/4)*(b*c -
a*d)^3) + (d^(1/4)*(5*b*c + 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
+ Sqrt[d]*x])/(8*Sqrt[2]*c^(1/4)*(b*c - a*d)^3)

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Rubi [A]  time = 1.65718, antiderivative size = 609, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{x^{3/2}}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac{d x^{3/2}}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt [4]{b} (5 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}-\frac{\sqrt [4]{b} (5 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}-\frac{\sqrt [4]{d} (3 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} (b c-a d)^3}+\frac{\sqrt [4]{d} (3 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} (b c-a d)^3}-\frac{\sqrt [4]{b} (5 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}+\frac{\sqrt [4]{b} (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}+\frac{\sqrt [4]{d} (3 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} \sqrt [4]{c} (b c-a d)^3}-\frac{\sqrt [4]{d} (3 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} \sqrt [4]{c} (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

-((d*x^(3/2))/((b*c - a*d)^2*(c + d*x^2))) - x^(3/2)/(2*(b*c - a*d)*(a + b*x^2)*
(c + d*x^2)) - (b^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(
1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + (b^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + (d^(1/4)*
(5*b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(1/4
)*(b*c - a*d)^3) - (d^(1/4)*(5*b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])
/c^(1/4)])/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)^3) + (b^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt
[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*
d)^3) - (b^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^3) - (d^(1/4)*(5*b*c + 3*a*d)*Log[Sq
rt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(1/4)*(b*c -
a*d)^3) + (d^(1/4)*(5*b*c + 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
+ Sqrt[d]*x])/(8*Sqrt[2]*c^(1/4)*(b*c - a*d)^3)

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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**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 2.8252, size = 583, normalized size = 0.96 \[ \frac{1}{16} \left (-\frac{8 b x^{3/2}}{\left (a+b x^2\right ) (b c-a d)^2}-\frac{8 d x^{3/2}}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{2} \sqrt [4]{b} (5 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a} (b c-a d)^3}+\frac{\sqrt{2} \sqrt [4]{b} (5 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a} (a d-b c)^3}+\frac{\sqrt{2} \sqrt [4]{d} (3 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{c} (a d-b c)^3}+\frac{\sqrt{2} \sqrt [4]{d} (3 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{c} (b c-a d)^3}+\frac{2 \sqrt{2} \sqrt [4]{b} (5 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} (a d-b c)^3}-\frac{2 \sqrt{2} \sqrt [4]{b} (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a} (a d-b c)^3}+\frac{2 \sqrt{2} \sqrt [4]{d} (3 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt [4]{c} (b c-a d)^3}-\frac{2 \sqrt{2} \sqrt [4]{d} (3 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt [4]{c} (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-8*b*x^(3/2))/((b*c - a*d)^2*(a + b*x^2)) - (8*d*x^(3/2))/((b*c - a*d)^2*(c +
d*x^2)) + (2*Sqrt[2]*b^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x]
)/a^(1/4)])/(a^(1/4)*(-(b*c) + a*d)^3) - (2*Sqrt[2]*b^(1/4)*(3*b*c + 5*a*d)*ArcT
an[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(-(b*c) + a*d)^3) + (2*Sqrt[
2]*d^(1/4)*(5*b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(1/
4)*(b*c - a*d)^3) - (2*Sqrt[2]*d^(1/4)*(5*b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/
4)*Sqrt[x])/c^(1/4)])/(c^(1/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(1/4)*(3*b*c + 5*a*d)
*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(b*c - a*d
)^3) + (Sqrt[2]*b^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sq
rt[x] + Sqrt[b]*x])/(a^(1/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(1/4)*(5*b*c + 3*a*d
)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(1/4)*(-(b*c) +
a*d)^3) + (Sqrt[2]*d^(1/4)*(5*b*c + 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4
)*Sqrt[x] + Sqrt[d]*x])/(c^(1/4)*(b*c - a*d)^3))/16

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Maple [A]  time = 0.03, size = 740, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*d^2/(a*d-b*c)^3*x^(3/2)/(d*x^2+c)*a+1/2*d/(a*d-b*c)^3*x^(3/2)/(d*x^2+c)*b*c
+5/16/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*b*c*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d
)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+5/8/(a*d-b*c)^3/(c/d)^(1/4
)*2^(1/2)*b*c*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+5/8/(a*d-b*c)^3/(c/d)^(1/4)*
2^(1/2)*b*c*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+3/16*d/(a*d-b*c)^3/(c/d)^(1/4)
*2^(1/2)*a*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)
*2^(1/2)+(c/d)^(1/2)))+3/8*d/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c
/d)^(1/4)*x^(1/2)+1)+3/8*d/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d
)^(1/4)*x^(1/2)-1)-1/2*b/(a*d-b*c)^3*x^(3/2)/(b*x^2+a)*a*d+1/2*b^2/(a*d-b*c)^3*x
^(3/2)/(b*x^2+a)*c-5/16/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*a*d*ln((x-(a/b)^(1/4)*x^
(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))-5/8/(a*d
-b*c)^3/(a/b)^(1/4)*2^(1/2)*a*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-5/8/(a*d-b
*c)^3/(a/b)^(1/4)*2^(1/2)*a*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-3/16*b/(a*d-
b*c)^3/(a/b)^(1/4)*2^(1/2)*c*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(
a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))-3/8*b/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*c
*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-3/8*b/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*c*a
rctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 24.3847, size = 7214, normalized size = 11.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^
2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(81*b^5*c^
4 + 540*a*b^4*c^3*d + 1350*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)
/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3
+ 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*
c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12
*a^12*b*c*d^11 + a^13*d^12))^(1/4)*arctan(-(a*b^9*c^9 - 9*a^2*b^8*c^8*d + 36*a^3
*b^7*c^7*d^2 - 84*a^4*b^6*c^6*d^3 + 126*a^5*b^5*c^5*d^4 - 126*a^6*b^4*c^4*d^5 +
84*a^7*b^3*c^3*d^6 - 36*a^8*b^2*c^2*d^7 + 9*a^9*b*c*d^8 - a^10*d^9)*(-(81*b^5*c^
4 + 540*a*b^4*c^3*d + 1350*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)
/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3
+ 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*
c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12
*a^12*b*c*d^11 + a^13*d^12))^(3/4)/((27*b^4*c^3 + 135*a*b^3*c^2*d + 225*a^2*b^2*
c*d^2 + 125*a^3*b*d^3)*sqrt(x) + sqrt((729*b^8*c^6 + 7290*a*b^7*c^5*d + 30375*a^
2*b^6*c^4*d^2 + 67500*a^3*b^5*c^3*d^3 + 84375*a^4*b^4*c^2*d^4 + 56250*a^5*b^3*c*
d^5 + 15625*a^6*b^2*d^6)*x - (81*a*b^11*c^10 + 54*a^2*b^10*c^9*d - 675*a^3*b^9*c
^8*d^2 - 120*a^4*b^8*c^7*d^3 + 2290*a^5*b^7*c^6*d^4 - 636*a^6*b^6*c^5*d^5 - 3534
*a^7*b^5*c^4*d^6 + 2440*a^8*b^4*c^3*d^7 + 1725*a^9*b^3*c^2*d^8 - 2250*a^10*b^2*c
*d^9 + 625*a^11*b*d^10)*sqrt(-(81*b^5*c^4 + 540*a*b^4*c^3*d + 1350*a^2*b^3*c^2*d
^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*
a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*
d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10
*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12))))) - 4*(a*b
^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4
+ (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(625*b^4*c^4*d + 1500*
a*b^3*c^3*d^2 + 1350*a^2*b^2*c^2*d^3 + 540*a^3*b*c*d^4 + 81*a^4*d^5)/(b^12*c^13
- 12*a*b^11*c^12*d + 66*a^2*b^10*c^11*d^2 - 220*a^3*b^9*c^10*d^3 + 495*a^4*b^8*c
^9*d^4 - 792*a^5*b^7*c^8*d^5 + 924*a^6*b^6*c^7*d^6 - 792*a^7*b^5*c^6*d^7 + 495*a
^8*b^4*c^5*d^8 - 220*a^9*b^3*c^4*d^9 + 66*a^10*b^2*c^3*d^10 - 12*a^11*b*c^2*d^11
+ a^12*c*d^12))^(1/4)*arctan(-(b^9*c^10 - 9*a*b^8*c^9*d + 36*a^2*b^7*c^8*d^2 -
84*a^3*b^6*c^7*d^3 + 126*a^4*b^5*c^6*d^4 - 126*a^5*b^4*c^5*d^5 + 84*a^6*b^3*c^4*
d^6 - 36*a^7*b^2*c^3*d^7 + 9*a^8*b*c^2*d^8 - a^9*c*d^9)*(-(625*b^4*c^4*d + 1500*
a*b^3*c^3*d^2 + 1350*a^2*b^2*c^2*d^3 + 540*a^3*b*c*d^4 + 81*a^4*d^5)/(b^12*c^13
- 12*a*b^11*c^12*d + 66*a^2*b^10*c^11*d^2 - 220*a^3*b^9*c^10*d^3 + 495*a^4*b^8*c
^9*d^4 - 792*a^5*b^7*c^8*d^5 + 924*a^6*b^6*c^7*d^6 - 792*a^7*b^5*c^6*d^7 + 495*a
^8*b^4*c^5*d^8 - 220*a^9*b^3*c^4*d^9 + 66*a^10*b^2*c^3*d^10 - 12*a^11*b*c^2*d^11
+ a^12*c*d^12))^(3/4)/((125*b^3*c^3*d + 225*a*b^2*c^2*d^2 + 135*a^2*b*c*d^3 + 2
7*a^3*d^4)*sqrt(x) + sqrt((15625*b^6*c^6*d^2 + 56250*a*b^5*c^5*d^3 + 84375*a^2*b
^4*c^4*d^4 + 67500*a^3*b^3*c^3*d^5 + 30375*a^4*b^2*c^2*d^6 + 7290*a^5*b*c*d^7 +
729*a^6*d^8)*x - (625*b^10*c^11*d - 2250*a*b^9*c^10*d^2 + 1725*a^2*b^8*c^9*d^3 +
2440*a^3*b^7*c^8*d^4 - 3534*a^4*b^6*c^7*d^5 - 636*a^5*b^5*c^6*d^6 + 2290*a^6*b^
4*c^5*d^7 - 120*a^7*b^3*c^4*d^8 - 675*a^8*b^2*c^3*d^9 + 54*a^9*b*c^2*d^10 + 81*a
^10*c*d^11)*sqrt(-(625*b^4*c^4*d + 1500*a*b^3*c^3*d^2 + 1350*a^2*b^2*c^2*d^3 + 5
40*a^3*b*c*d^4 + 81*a^4*d^5)/(b^12*c^13 - 12*a*b^11*c^12*d + 66*a^2*b^10*c^11*d^
2 - 220*a^3*b^9*c^10*d^3 + 495*a^4*b^8*c^9*d^4 - 792*a^5*b^7*c^8*d^5 + 924*a^6*b
^6*c^7*d^6 - 792*a^7*b^5*c^6*d^7 + 495*a^8*b^4*c^5*d^8 - 220*a^9*b^3*c^4*d^9 + 6
6*a^10*b^2*c^3*d^10 - 12*a^11*b*c^2*d^11 + a^12*c*d^12))))) - (a*b^2*c^3 - 2*a^2
*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 -
a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(81*b^5*c^4 + 540*a*b^4*c^3*d + 1350
*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)/(a*b^12*c^12 - 12*a^2*b^1
1*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 79
2*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*
d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12
))^(1/4)*log((a*b^9*c^9 - 9*a^2*b^8*c^8*d + 36*a^3*b^7*c^7*d^2 - 84*a^4*b^6*c^6*
d^3 + 126*a^5*b^5*c^5*d^4 - 126*a^6*b^4*c^4*d^5 + 84*a^7*b^3*c^3*d^6 - 36*a^8*b^
2*c^2*d^7 + 9*a^9*b*c*d^8 - a^10*d^9)*(-(81*b^5*c^4 + 540*a*b^4*c^3*d + 1350*a^2
*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)/(a*b^12*c^12 - 12*a^2*b^11*c^
11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^
6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8
- 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12))^(
3/4) + (27*b^4*c^3 + 135*a*b^3*c^2*d + 225*a^2*b^2*c*d^2 + 125*a^3*b*d^3)*sqrt(x
)) + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b
*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(81*b^5*c^4 +
540*a*b^4*c^3*d + 1350*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)/(a
*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 4
95*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5
*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^
12*b*c*d^11 + a^13*d^12))^(1/4)*log(-(a*b^9*c^9 - 9*a^2*b^8*c^8*d + 36*a^3*b^7*c
^7*d^2 - 84*a^4*b^6*c^6*d^3 + 126*a^5*b^5*c^5*d^4 - 126*a^6*b^4*c^4*d^5 + 84*a^7
*b^3*c^3*d^6 - 36*a^8*b^2*c^2*d^7 + 9*a^9*b*c*d^8 - a^10*d^9)*(-(81*b^5*c^4 + 54
0*a*b^4*c^3*d + 1350*a^2*b^3*c^2*d^2 + 1500*a^3*b^2*c*d^3 + 625*a^4*b*d^4)/(a*b^
12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*
a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^
7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*
b*c*d^11 + a^13*d^12))^(3/4) + (27*b^4*c^3 + 135*a*b^3*c^2*d + 225*a^2*b^2*c*d^2
+ 125*a^3*b*d^3)*sqrt(x)) + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d
- 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d
^3)*x^2)*(-(625*b^4*c^4*d + 1500*a*b^3*c^3*d^2 + 1350*a^2*b^2*c^2*d^3 + 540*a^3*
b*c*d^4 + 81*a^4*d^5)/(b^12*c^13 - 12*a*b^11*c^12*d + 66*a^2*b^10*c^11*d^2 - 220
*a^3*b^9*c^10*d^3 + 495*a^4*b^8*c^9*d^4 - 792*a^5*b^7*c^8*d^5 + 924*a^6*b^6*c^7*
d^6 - 792*a^7*b^5*c^6*d^7 + 495*a^8*b^4*c^5*d^8 - 220*a^9*b^3*c^4*d^9 + 66*a^10*
b^2*c^3*d^10 - 12*a^11*b*c^2*d^11 + a^12*c*d^12))^(1/4)*log((b^9*c^10 - 9*a*b^8*
c^9*d + 36*a^2*b^7*c^8*d^2 - 84*a^3*b^6*c^7*d^3 + 126*a^4*b^5*c^6*d^4 - 126*a^5*
b^4*c^5*d^5 + 84*a^6*b^3*c^4*d^6 - 36*a^7*b^2*c^3*d^7 + 9*a^8*b*c^2*d^8 - a^9*c*
d^9)*(-(625*b^4*c^4*d + 1500*a*b^3*c^3*d^2 + 1350*a^2*b^2*c^2*d^3 + 540*a^3*b*c*
d^4 + 81*a^4*d^5)/(b^12*c^13 - 12*a*b^11*c^12*d + 66*a^2*b^10*c^11*d^2 - 220*a^3
*b^9*c^10*d^3 + 495*a^4*b^8*c^9*d^4 - 792*a^5*b^7*c^8*d^5 + 924*a^6*b^6*c^7*d^6
- 792*a^7*b^5*c^6*d^7 + 495*a^8*b^4*c^5*d^8 - 220*a^9*b^3*c^4*d^9 + 66*a^10*b^2*
c^3*d^10 - 12*a^11*b*c^2*d^11 + a^12*c*d^12))^(3/4) + (125*b^3*c^3*d + 225*a*b^2
*c^2*d^2 + 135*a^2*b*c*d^3 + 27*a^3*d^4)*sqrt(x)) - (a*b^2*c^3 - 2*a^2*b*c^2*d +
a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*
d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(625*b^4*c^4*d + 1500*a*b^3*c^3*d^2 + 1350*a^2
*b^2*c^2*d^3 + 540*a^3*b*c*d^4 + 81*a^4*d^5)/(b^12*c^13 - 12*a*b^11*c^12*d + 66*
a^2*b^10*c^11*d^2 - 220*a^3*b^9*c^10*d^3 + 495*a^4*b^8*c^9*d^4 - 792*a^5*b^7*c^8
*d^5 + 924*a^6*b^6*c^7*d^6 - 792*a^7*b^5*c^6*d^7 + 495*a^8*b^4*c^5*d^8 - 220*a^9
*b^3*c^4*d^9 + 66*a^10*b^2*c^3*d^10 - 12*a^11*b*c^2*d^11 + a^12*c*d^12))^(1/4)*l
og(-(b^9*c^10 - 9*a*b^8*c^9*d + 36*a^2*b^7*c^8*d^2 - 84*a^3*b^6*c^7*d^3 + 126*a^
4*b^5*c^6*d^4 - 126*a^5*b^4*c^5*d^5 + 84*a^6*b^3*c^4*d^6 - 36*a^7*b^2*c^3*d^7 +
9*a^8*b*c^2*d^8 - a^9*c*d^9)*(-(625*b^4*c^4*d + 1500*a*b^3*c^3*d^2 + 1350*a^2*b^
2*c^2*d^3 + 540*a^3*b*c*d^4 + 81*a^4*d^5)/(b^12*c^13 - 12*a*b^11*c^12*d + 66*a^2
*b^10*c^11*d^2 - 220*a^3*b^9*c^10*d^3 + 495*a^4*b^8*c^9*d^4 - 792*a^5*b^7*c^8*d^
5 + 924*a^6*b^6*c^7*d^6 - 792*a^7*b^5*c^6*d^7 + 495*a^8*b^4*c^5*d^8 - 220*a^9*b^
3*c^4*d^9 + 66*a^10*b^2*c^3*d^10 - 12*a^11*b*c^2*d^11 + a^12*c*d^12))^(3/4) + (1
25*b^3*c^3*d + 225*a*b^2*c^2*d^2 + 135*a^2*b*c*d^3 + 27*a^3*d^4)*sqrt(x)) + 4*(2
*b*d*x^3 + (b*c + a*d)*x)*sqrt(x))/(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3
*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 +
a^3*d^3)*x^2)

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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**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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