3.1149 \(\int \frac{1}{x^3 \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{3 \sqrt{b} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 a^{3/2} \sqrt [4]{a+b x^4}}-\frac{1}{2 a x^2 \sqrt [4]{a+b x^4}} \]

[Out]

-1/(2*a*x^2*(a + b*x^4)^(1/4)) - (3*Sqrt[b]*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcT
an[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(2*a^(3/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.109789, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 \sqrt{b} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 a^{3/2} \sqrt [4]{a+b x^4}}-\frac{1}{2 a x^2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^4)^(5/4)),x]

[Out]

-1/(2*a*x^2*(a + b*x^4)^(1/4)) - (3*Sqrt[b]*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcT
an[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(2*a^(3/2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{1}{2 a x^{2} \sqrt [4]{a + b x^{4}}} - \frac{3 b x^{2}}{2 a^{2} \sqrt [4]{a + b x^{4}}} + \frac{3 b \int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{4 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**4+a)**(5/4),x)

[Out]

-1/(2*a*x**2*(a + b*x**4)**(1/4)) - 3*b*x**2/(2*a**2*(a + b*x**4)**(1/4)) + 3*b*
Integral((a + b*x**2)**(-1/4), (x, x**2))/(4*a**2)

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Mathematica [C]  time = 0.0536698, size = 71, normalized size = 0.87 \[ \frac{3 b x^4 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-2 \left (a+3 b x^4\right )}{4 a^2 x^2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^4)^(5/4)),x]

[Out]

(-2*(a + 3*b*x^4) + 3*b*x^4*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/
2, -((b*x^4)/a)])/(4*a^2*x^2*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^4+a)^(5/4),x)

[Out]

int(1/x^3/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^3),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{7} + a x^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^3),x, algorithm="fricas")

[Out]

integral(1/((b*x^7 + a*x^3)*(b*x^4 + a)^(1/4)), x)

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Sympy [A]  time = 3.97207, size = 31, normalized size = 0.38 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**4+a)**(5/4),x)

[Out]

-hyper((-1/2, 5/4), (1/2,), b*x**4*exp_polar(I*pi)/a)/(2*a**(5/4)*x**2)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^3),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^3), x)