3.503 \(\int \frac{1}{x^{7/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=881 \[ -\frac{3 (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{5/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 \left (24 b^4 c^4-32 a b^3 d c^3-32 a^2 b^2 d^2 c^2+87 a^3 b d^3 c-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+29 a b d c-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (d x^2+c\right )^2}-\frac{3 \left (24 b^3 c^3-32 a b^2 d c^2+87 a^2 b d^2 c-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}} \]

[Out]

(-3*(24*b^3*c^3 - 32*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 39*a^3*d^3))/(80*a^2*c^3*(b*
c - a*d)^3*x^(5/2)) + (3*(24*b^4*c^4 - 32*a*b^3*c^3*d - 32*a^2*b^2*c^2*d^2 + 87*
a^3*b*c*d^3 - 39*a^4*d^4))/(16*a^3*c^4*(b*c - a*d)^3*Sqrt[x]) + (d*(2*b*c + a*d)
)/(4*a*c*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^(5/2)*(a +
b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 29*a*b*c*d - 13*a^2*d^2))/(16*a*c^2*(b*c
- a*d)^3*x^(5/2)*(c + d*x^2)) - (3*b^(17/4)*(3*b*c - 7*a*d)*ArcTan[1 - (Sqrt[2]
*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*b^(17/4)*(3*
b*c - 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*
(b*c - a*d)^4) - (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 -
(Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^4) + (3*d^
(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt
[x])/c^(1/4)])/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^4) + (3*b^(17/4)*(3*b*c - 7*a*d)
*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)
*(b*c - a*d)^4) - (3*b^(17/4)*(3*b*c - 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1
/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*d^(13/4)*(119*
b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x
] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^4) - (3*d^(13/4)*(119*b^2*c^2 -
126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[
d]*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^4)

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Rubi [A]  time = 3.69103, antiderivative size = 881, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ -\frac{3 (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{5/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 \left (24 b^4 c^4-32 a b^3 d c^3-32 a^2 b^2 d^2 c^2+87 a^3 b d^3 c-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+29 a b d c-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (d x^2+c\right )^2}-\frac{3 \left (24 b^3 c^3-32 a b^2 d c^2+87 a^2 b d^2 c-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(-3*(24*b^3*c^3 - 32*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 39*a^3*d^3))/(80*a^2*c^3*(b*
c - a*d)^3*x^(5/2)) + (3*(24*b^4*c^4 - 32*a*b^3*c^3*d - 32*a^2*b^2*c^2*d^2 + 87*
a^3*b*c*d^3 - 39*a^4*d^4))/(16*a^3*c^4*(b*c - a*d)^3*Sqrt[x]) + (d*(2*b*c + a*d)
)/(4*a*c*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^(5/2)*(a +
b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 29*a*b*c*d - 13*a^2*d^2))/(16*a*c^2*(b*c
- a*d)^3*x^(5/2)*(c + d*x^2)) - (3*b^(17/4)*(3*b*c - 7*a*d)*ArcTan[1 - (Sqrt[2]
*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*b^(17/4)*(3*
b*c - 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*
(b*c - a*d)^4) - (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 -
(Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^4) + (3*d^
(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt
[x])/c^(1/4)])/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^4) + (3*b^(17/4)*(3*b*c - 7*a*d)
*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)
*(b*c - a*d)^4) - (3*b^(17/4)*(3*b*c - 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1
/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*d^(13/4)*(119*
b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x
] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^4) - (3*d^(13/4)*(119*b^2*c^2 -
126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[
d]*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^4)

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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(1/x**(7/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 4.51477, size = 729, normalized size = 0.83 \[ \frac{1}{640} \left (\frac{120 \sqrt{2} b^{17/4} (3 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4} (b c-a d)^4}+\frac{120 \sqrt{2} b^{17/4} (7 a d-3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4} (b c-a d)^4}+\frac{240 \sqrt{2} b^{17/4} (7 a d-3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{13/4} (b c-a d)^4}+\frac{240 \sqrt{2} b^{17/4} (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{13/4} (b c-a d)^4}-\frac{320 b^5 x^{3/2}}{a^3 \left (a+b x^2\right ) (a d-b c)^3}+\frac{1280 (3 a d+2 b c)}{a^3 c^4 \sqrt{x}}+\frac{15 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (b c-a d)^4}-\frac{15 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (b c-a d)^4}-\frac{30 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{17/4} (b c-a d)^4}+\frac{30 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{17/4} (b c-a d)^4}-\frac{256}{a^2 c^3 x^{5/2}}+\frac{40 d^4 x^{3/2} (37 b c-21 a d)}{c^4 \left (c+d x^2\right ) (b c-a d)^3}+\frac{160 d^4 x^{3/2}}{c^3 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(-256/(a^2*c^3*x^(5/2)) + (1280*(2*b*c + 3*a*d))/(a^3*c^4*Sqrt[x]) - (320*b^5*x^
(3/2))/(a^3*(-(b*c) + a*d)^3*(a + b*x^2)) + (160*d^4*x^(3/2))/(c^3*(b*c - a*d)^2
*(c + d*x^2)^2) + (40*d^4*(37*b*c - 21*a*d)*x^(3/2))/(c^4*(b*c - a*d)^3*(c + d*x
^2)) + (240*Sqrt[2]*b^(17/4)*(-3*b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x
])/a^(1/4)])/(a^(13/4)*(b*c - a*d)^4) + (240*Sqrt[2]*b^(17/4)*(3*b*c - 7*a*d)*Ar
cTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(13/4)*(b*c - a*d)^4) - (30*Sqrt
[2]*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4
)*Sqrt[x])/c^(1/4)])/(c^(17/4)*(b*c - a*d)^4) + (30*Sqrt[2]*d^(13/4)*(119*b^2*c^
2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^
(17/4)*(b*c - a*d)^4) + (120*Sqrt[2]*b^(17/4)*(3*b*c - 7*a*d)*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(13/4)*(b*c - a*d)^4) + (120*Sqrt[2
]*b^(17/4)*(-3*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt
[b]*x])/(a^(13/4)*(b*c - a*d)^4) + (15*Sqrt[2]*d^(13/4)*(119*b^2*c^2 - 126*a*b*c
*d + 39*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^
(17/4)*(b*c - a*d)^4) - (15*Sqrt[2]*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2
*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(17/4)*(b*c
- a*d)^4))/640

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Maple [A]  time = 0.046, size = 1170, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

-189/32*d^4/c^3/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*a*b*arctan(2^(1/2)/(c/d)^(1/4)*x
^(1/2)+1)-189/32*d^4/c^3/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*a*b*arctan(2^(1/2)/(c/d
)^(1/4)*x^(1/2)-1)+6/x^(1/2)/a^2/c^4*d+4/x^(1/2)/a^3/c^3*b+21/16*d^7/c^4/(a*d-b*
c)^4/(d*x^2+c)^2*x^(7/2)*a^2+37/16*d^5/c^2/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*b^2+2
5/16*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*a^2+41/16*d^4/c/(a*d-b*c)^4/(d*x^2+
c)^2*x^(3/2)*b^2-1/2*b^5/a^2/(a*d-b*c)^4*x^(3/2)/(b*x^2+a)*d-189/64*d^4/c^3/(a*d
-b*c)^4/(c/d)^(1/4)*2^(1/2)*a*b*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)))-2/5/a^2/c^3/x^(5/2)+117/128*d^5/c^4/
(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*a^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)))+117/64*d^5/c^4/(a*d-b*c)^4/(c/d)
^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+117/64*d^5/c^4/(a*d-b*c
)^4/(c/d)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+357/128*d^3/c^
2/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*b^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)))+357/64*d^3/c^2/(a*d-b*c)^4/(c/
d)^(1/4)*2^(1/2)*b^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+357/64*d^3/c^2/(a*d-b
*c)^4/(c/d)^(1/4)*2^(1/2)*b^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-21/16*b^4/a^
2/(a*d-b*c)^4/(a/b)^(1/4)*2^(1/2)*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)))-21/8*b^4/a^2/(a*d-b*c)^4/(a/b)^(
1/4)*2^(1/2)*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-21/8*b^4/a^2/(a*d-b*c)^4/(a
/b)^(1/4)*2^(1/2)*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+9/16*b^5/a^3/(a*d-b*c)
^4/(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)))+9/8*b^5/a^3/(a*d-b*c)^4/(a/b)^(1/4)*2^(1/2)
*c*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+9/8*b^5/a^3/(a*d-b*c)^4/(a/b)^(1/4)*2^(
1/2)*c*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+1/2*b^6/a^3/(a*d-b*c)^4*x^(3/2)/(b*
x^2+a)*c-29/8*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*a*b-33/8*d^5/c^2/(a*d-b*c)
^4/(d*x^2+c)^2*x^(3/2)*a*b

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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(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^(7/2)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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(1/x**(7/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.5731, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^(7/2)),x, algorithm="giac")

[Out]

Done