3.488 \(\int \frac{x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=624 \[ \frac{a \sqrt{x}}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{\sqrt{x} (a d+b c)}{2 b \left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt [4]{a} (3 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{b} (b c-a d)^3}-\frac{\sqrt [4]{a} (3 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{b} (b c-a d)^3}-\frac{\sqrt [4]{c} (5 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{d} (b c-a d)^3}+\frac{\sqrt [4]{c} (5 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{d} (b c-a d)^3}+\frac{\sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{b} (b c-a d)^3}-\frac{\sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{b} (b c-a d)^3}-\frac{\sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} \sqrt [4]{d} (b c-a d)^3}+\frac{\sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} \sqrt [4]{d} (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 1.60135, antiderivative size = 624, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{a \sqrt{x}}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{\sqrt{x} (a d+b c)}{2 b \left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt [4]{a} (3 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{b} (b c-a d)^3}-\frac{\sqrt [4]{a} (3 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{b} (b c-a d)^3}-\frac{\sqrt [4]{c} (5 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{d} (b c-a d)^3}+\frac{\sqrt [4]{c} (5 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{d} (b c-a d)^3}+\frac{\sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{b} (b c-a d)^3}-\frac{\sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{b} (b c-a d)^3}-\frac{\sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} \sqrt [4]{d} (b c-a d)^3}+\frac{\sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} \sqrt [4]{d} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

((b*c + a*d)*Sqrt[x])/(2*b*(b*c - a*d)^2*(c + d*x^2)) + (a*Sqrt[x])/(2*b*(b*c -
a*d)*(a + b*x^2)*(c + d*x^2)) + (a^(1/4)*(5*b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(1/4)*(b*c - a*d)^3) - (a^(1/4)*(5*b*c + 3*
a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(1/4)*(b*c - a*
d)^3) - (c^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/
(4*Sqrt[2]*d^(1/4)*(b*c - a*d)^3) + (c^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]
*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*d^(1/4)*(b*c - a*d)^3) + (a^(1/4)*(5*b*c
+ 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*
b^(1/4)*(b*c - a*d)^3) - (a^(1/4)*(5*b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(1/4)*(b*c - a*d)^3) - (c^(1/4)*(3*b*
c + 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2
]*d^(1/4)*(b*c - a*d)^3) + (c^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4
)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*d^(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**(7/2)/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 2.32611, size = 585, normalized size = 0.94 \[ \frac{1}{16} \left (\frac{8 a \sqrt{x}}{\left (a+b x^2\right ) (b c-a d)^2}+\frac{8 c \sqrt{x}}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{2} \sqrt [4]{a} (3 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b} (b c-a d)^3}-\frac{\sqrt{2} \sqrt [4]{a} (3 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b} (b c-a d)^3}+\frac{\sqrt{2} \sqrt [4]{c} (5 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{d} (a d-b c)^3}-\frac{\sqrt{2} \sqrt [4]{c} (5 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{d} (a d-b c)^3}+\frac{2 \sqrt{2} \sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b} (b c-a d)^3}-\frac{2 \sqrt{2} \sqrt [4]{a} (3 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b} (b c-a d)^3}+\frac{2 \sqrt{2} \sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt [4]{d} (a d-b c)^3}+\frac{2 \sqrt{2} \sqrt [4]{c} (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt [4]{d} (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]

[Out]

((8*a*Sqrt[x])/((b*c - a*d)^2*(a + b*x^2)) + (8*c*Sqrt[x])/((b*c - a*d)^2*(c + d
*x^2)) + (2*Sqrt[2]*a^(1/4)*(5*b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])
/a^(1/4)])/(b^(1/4)*(b*c - a*d)^3) - (2*Sqrt[2]*a^(1/4)*(5*b*c + 3*a*d)*ArcTan[1
+ (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(b^(1/4)*(b*c - a*d)^3) + (2*Sqrt[2]*c^(1
/4)*(3*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(d^(1/4)*(-(b
*c) + a*d)^3) + (2*Sqrt[2]*c^(1/4)*(3*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*S
qrt[x])/c^(1/4)])/(d^(1/4)*(b*c - a*d)^3) + (Sqrt[2]*a^(1/4)*(5*b*c + 3*a*d)*Log
[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(b^(1/4)*(b*c - a*d)^3)
- (Sqrt[2]*a^(1/4)*(5*b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x])/(b^(1/4)*(b*c - a*d)^3) + (Sqrt[2]*c^(1/4)*(3*b*c + 5*a*d)*Log[S
qrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(d^(1/4)*(-(b*c) + a*d)^3
) - (Sqrt[2]*c^(1/4)*(3*b*c + 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[
x] + Sqrt[d]*x])/(d^(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^(7/2)/(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

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

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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^(7/2)/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 27.9399, size = 5917, normalized size = 9.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/((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)*(-(625*a*b^4
*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*a^5*d^4)
/(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 4
95*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5
*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^1
1*b^2*c*d^11 + a^12*b*d^12))^(1/4)*arctan(-(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*
d^2 - a^3*d^3)*(-(625*a*b^4*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 54
0*a^4*b*c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2
- 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^
7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66
*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12))^(1/4)/((5*b*c + 3*a*d)*s
qrt(x) + sqrt((25*b^2*c^2 + 30*a*b*c*d + 9*a^2*d^2)*x + (b^6*c^6 - 6*a*b^5*c^5*d
+ 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5
+ a^6*d^6)*sqrt(-(625*a*b^4*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 54
0*a^4*b*c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2
- 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^
7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66
*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*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)*(-(81*b^4*c^5 + 540*a*b^3*c^4*d + 135
0*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^12*c^12*d - 12*a*b^11
*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 7
92*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4
*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^1
3))^(1/4)*arctan(-(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(-(81*b^4*
c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^
4)/(b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^
4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^
5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 -
12*a^11*b*c*d^12 + a^12*d^13))^(1/4)/((3*b*c + 5*a*d)*sqrt(x) + sqrt((9*b^2*c^2
+ 30*a*b*c*d + 25*a^2*d^2)*x + (b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 2
0*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(-(81*b^4*
c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^
4)/(b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^
4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^
5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 -
12*a^11*b*c*d^12 + a^12*d^13))))) + (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*a*b^4*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 5
40*a^4*b*c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^
2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b
^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 6
6*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12))^(1/4)*log((5*b*c + 3*a*
d)*sqrt(x) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(-(625*a*b^4*c^
4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*a^5*d^4)/(b
^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*
a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^
7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b
^2*c*d^11 + a^12*b*d^12))^(1/4)) - (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*a*b^4*c^4 + 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 54
0*a^4*b*c*d^3 + 81*a^5*d^4)/(b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2
- 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^
7*c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66
*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12*b*d^12))^(1/4)*log((5*b*c + 3*a*d
)*sqrt(x) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(-(625*a*b^4*c^4
+ 1500*a^2*b^3*c^3*d + 1350*a^3*b^2*c^2*d^2 + 540*a^4*b*c*d^3 + 81*a^5*d^4)/(b^
13*c^12 - 12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a
^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*a^7*b^6*c^5*d^7
+ 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^
2*c*d^11 + a^12*b*d^12))^(1/4)) - (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^4*c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*
b*c^2*d^3 + 625*a^4*c*d^4)/(b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*
d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*
b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 +
66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13))^(1/4)*log((3*b*c + 5*a*d)
*sqrt(x) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(-(81*b^4*c^5 + 5
40*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^1
2*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495
*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d
^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11
*b*c*d^12 + a^12*d^13))^(1/4)) + (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^4*c^5 + 540*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b
*c^2*d^3 + 625*a^4*c*d^4)/(b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d
^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b
^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 +
66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13))^(1/4)*log((3*b*c + 5*a*d)*
sqrt(x) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(-(81*b^4*c^5 + 54
0*a*b^3*c^4*d + 1350*a^2*b^2*c^3*d^2 + 1500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^12
*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*
a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^
8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*
b*c*d^12 + a^12*d^13))^(1/4)) - 4*((b*c + a*d)*x^2 + 2*a*c)*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**(7/2)/(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. \[ \int \frac{x^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="giac")

[Out]

integrate(x^(7/2)/((b*x^2 + a)^2*(d*x^2 + c)^2), x)