3.2096 \(\int \frac{x^3}{\left (a+\frac{b}{x^4}\right )^{5/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{5 x^4 \sqrt{a+\frac{b}{x^4}}}{4 a^3}-\frac{5 x^4}{6 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x^4}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

[Out]

-x^4/(6*a*(a + b/x^4)^(3/2)) - (5*x^4)/(6*a^2*Sqrt[a + b/x^4]) + (5*Sqrt[a + b/x
^4]*x^4)/(4*a^3) - (5*b*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/(4*a^(7/2))

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Rubi [A]  time = 0.153784, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{5 x^4 \sqrt{a+\frac{b}{x^4}}}{4 a^3}-\frac{5 x^4}{6 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x^4}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x^4)^(5/2),x]

[Out]

-x^4/(6*a*(a + b/x^4)^(3/2)) - (5*x^4)/(6*a^2*Sqrt[a + b/x^4]) + (5*Sqrt[a + b/x
^4]*x^4)/(4*a^3) - (5*b*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/(4*a^(7/2))

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Rubi in Sympy [A]  time = 12.9809, size = 83, normalized size = 0.9 \[ - \frac{x^{4}}{6 a \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}} - \frac{5 x^{4}}{6 a^{2} \sqrt{a + \frac{b}{x^{4}}}} + \frac{5 x^{4} \sqrt{a + \frac{b}{x^{4}}}}{4 a^{3}} - \frac{5 b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a+b/x**4)**(5/2),x)

[Out]

-x**4/(6*a*(a + b/x**4)**(3/2)) - 5*x**4/(6*a**2*sqrt(a + b/x**4)) + 5*x**4*sqrt
(a + b/x**4)/(4*a**3) - 5*b*atanh(sqrt(a + b/x**4)/sqrt(a))/(4*a**(7/2))

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Mathematica [A]  time = 0.0869119, size = 101, normalized size = 1.1 \[ \frac{\sqrt{a} x^2 \left (3 a^2 x^8+20 a b x^4+15 b^2\right )-15 b \left (a x^4+b\right )^{3/2} \log \left (\sqrt{a} \sqrt{a x^4+b}+a x^2\right )}{12 a^{7/2} x^2 \sqrt{a+\frac{b}{x^4}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b/x^4)^(5/2),x]

[Out]

(Sqrt[a]*x^2*(15*b^2 + 20*a*b*x^4 + 3*a^2*x^8) - 15*b*(b + a*x^4)^(3/2)*Log[a*x^
2 + Sqrt[a]*Sqrt[b + a*x^4]])/(12*a^(7/2)*Sqrt[a + b/x^4]*x^2*(b + a*x^4))

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Maple [B]  time = 0.08, size = 282, normalized size = 3.1 \[{\frac{1}{12\,{x}^{10}} \left ( a{x}^{4}+b \right ) ^{{\frac{5}{2}}} \left ( 3\,\sqrt{a{x}^{4}+b}{a}^{15/2}{x}^{10}+6\,{a}^{13/2}b\sqrt{a{x}^{4}+b}{x}^{6}+14\,{a}^{13/2}\sqrt{-{\frac{ \left ( a{x}^{2}+\sqrt{-ab} \right ) \left ( -a{x}^{2}+\sqrt{-ab} \right ) }{a}}}{x}^{6}b-15\,\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{8}{a}^{7}b+3\,{a}^{11/2}{b}^{2}\sqrt{a{x}^{4}+b}{x}^{2}+12\,{a}^{11/2}{b}^{2}\sqrt{-{\frac{ \left ( a{x}^{2}+\sqrt{-ab} \right ) \left ( -a{x}^{2}+\sqrt{-ab} \right ) }{a}}}{x}^{2}-30\,{a}^{6}{b}^{2}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{4}-15\,{a}^{5}{b}^{3}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) \right ){a}^{-{\frac{13}{2}}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}} \left ( -a{x}^{2}+\sqrt{-ab} \right ) ^{-2} \left ( a{x}^{2}+\sqrt{-ab} \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^4)^(5/2),x)

[Out]

1/12*(a*x^4+b)^(5/2)/a^(13/2)*(3*(a*x^4+b)^(1/2)*a^(15/2)*x^10+6*a^(13/2)*b*(a*x
^4+b)^(1/2)*x^6+14*a^(13/2)*(-1/a*(a*x^2+(-a*b)^(1/2))*(-a*x^2+(-a*b)^(1/2)))^(1
/2)*x^6*b-15*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*x^8*a^7*b+3*a^(11/2)*b^2*(a*x^4+b)^
(1/2)*x^2+12*a^(11/2)*b^2*(-1/a*(a*x^2+(-a*b)^(1/2))*(-a*x^2+(-a*b)^(1/2)))^(1/2
)*x^2-30*a^6*b^2*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*x^4-15*a^5*b^3*ln(x^2*a^(1/2)+(
a*x^4+b)^(1/2)))/((a*x^4+b)/x^4)^(5/2)/x^10/(-a*x^2+(-a*b)^(1/2))^2/(a*x^2+(-a*b
)^(1/2))^2

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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^3/(a + b/x^4)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261846, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a^{2} b x^{8} + 2 \, a b^{2} x^{4} + b^{3}\right )} \sqrt{a} \log \left (2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right ) + 2 \,{\left (3 \, a^{3} x^{12} + 20 \, a^{2} b x^{8} + 15 \, a b^{2} x^{4}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{24 \,{\left (a^{6} x^{8} + 2 \, a^{5} b x^{4} + a^{4} b^{2}\right )}}, \frac{15 \,{\left (a^{2} b x^{8} + 2 \, a b^{2} x^{4} + b^{3}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) +{\left (3 \, a^{3} x^{12} + 20 \, a^{2} b x^{8} + 15 \, a b^{2} x^{4}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{12 \,{\left (a^{6} x^{8} + 2 \, a^{5} b x^{4} + a^{4} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x^4)^(5/2),x, algorithm="fricas")

[Out]

[1/24*(15*(a^2*b*x^8 + 2*a*b^2*x^4 + b^3)*sqrt(a)*log(2*a*x^4*sqrt((a*x^4 + b)/x
^4) - (2*a*x^4 + b)*sqrt(a)) + 2*(3*a^3*x^12 + 20*a^2*b*x^8 + 15*a*b^2*x^4)*sqrt
((a*x^4 + b)/x^4))/(a^6*x^8 + 2*a^5*b*x^4 + a^4*b^2), 1/12*(15*(a^2*b*x^8 + 2*a*
b^2*x^4 + b^3)*sqrt(-a)*arctan(sqrt(-a)/sqrt((a*x^4 + b)/x^4)) + (3*a^3*x^12 + 2
0*a^2*b*x^8 + 15*a*b^2*x^4)*sqrt((a*x^4 + b)/x^4))/(a^6*x^8 + 2*a^5*b*x^4 + a^4*
b^2)]

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Sympy [A]  time = 20.9482, size = 819, normalized size = 8.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x**4)**(5/2),x)

[Out]

6*a**17*x**16*sqrt(1 + b/(a*x**4))/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 7
2*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) + 46*a**16*b*x**12*sqrt(1 + b/(a*x**4
))/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2*x**4 + 24*a**(3
3/2)*b**3) + 15*a**16*b*x**12*log(b/(a*x**4))/(24*a**(39/2)*x**12 + 72*a**(37/2)
*b*x**8 + 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) - 30*a**16*b*x**12*log(sqr
t(1 + b/(a*x**4)) + 1)/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*
b**2*x**4 + 24*a**(33/2)*b**3) + 70*a**15*b**2*x**8*sqrt(1 + b/(a*x**4))/(24*a**
(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3)
+ 45*a**15*b**2*x**8*log(b/(a*x**4))/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8
+ 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) - 90*a**15*b**2*x**8*log(sqrt(1 +
b/(a*x**4)) + 1)/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2*x
**4 + 24*a**(33/2)*b**3) + 30*a**14*b**3*x**4*sqrt(1 + b/(a*x**4))/(24*a**(39/2)
*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) + 45*
a**14*b**3*x**4*log(b/(a*x**4))/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a
**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) - 90*a**14*b**3*x**4*log(sqrt(1 + b/(a*x
**4)) + 1)/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2*x**4 +
24*a**(33/2)*b**3) + 15*a**13*b**4*log(b/(a*x**4))/(24*a**(39/2)*x**12 + 72*a**(
37/2)*b*x**8 + 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) - 30*a**13*b**4*log(s
qrt(1 + b/(a*x**4)) + 1)/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2
)*b**2*x**4 + 24*a**(33/2)*b**3)

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GIAC/XCAS [A]  time = 0.279033, size = 151, normalized size = 1.64 \[ \frac{1}{12} \, b{\left (\frac{2 \,{\left (a + \frac{6 \,{\left (a x^{4} + b\right )}}{x^{4}}\right )} x^{4}}{{\left (a x^{4} + b\right )} a^{3} \sqrt{\frac{a x^{4} + b}{x^{4}}}} + \frac{15 \, \arctan \left (\frac{\sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} - \frac{3 \, \sqrt{\frac{a x^{4} + b}{x^{4}}}}{{\left (a - \frac{a x^{4} + b}{x^{4}}\right )} a^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x^4)^(5/2),x, algorithm="giac")

[Out]

1/12*b*(2*(a + 6*(a*x^4 + b)/x^4)*x^4/((a*x^4 + b)*a^3*sqrt((a*x^4 + b)/x^4)) +
15*arctan(sqrt((a*x^4 + b)/x^4)/sqrt(-a))/(sqrt(-a)*a^3) - 3*sqrt((a*x^4 + b)/x^
4)/((a - (a*x^4 + b)/x^4)*a^3))