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

Optimal. Leaf size=121 \[ -\frac{b^{3/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a+b x^2}}+\frac{b^2 x}{2 a \sqrt [4]{a+b x^2}}-\frac{b \left (a+b x^2\right )^{3/4}}{2 a x}-\frac{\left (a+b x^2\right )^{3/4}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.114921, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^{3/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a+b x^2}}+\frac{b^2 x}{2 a \sqrt [4]{a+b x^2}}-\frac{b \left (a+b x^2\right )^{3/4}}{2 a x}-\frac{\left (a+b x^2\right )^{3/4}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(3/4)/x^4,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b^{2} \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{4} - \frac{\left (a + b x^{2}\right )^{\frac{3}{4}}}{3 x^{3}} + \frac{b^{2} x}{2 a \sqrt [4]{a + b x^{2}}} - \frac{b \left (a + b x^{2}\right )^{\frac{3}{4}}}{2 a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**2*Integral((a + b*x**2)**(-5/4), x)/4 - (a + b*x**2)**(3/4)/(3*x**3) + b**2*
x/(2*a*(a + b*x**2)**(1/4)) - b*(a + b*x**2)**(3/4)/(2*a*x)

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Mathematica [C]  time = 0.046641, size = 88, normalized size = 0.73 \[ \frac{b^2 x \sqrt [4]{\frac{a+b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{4 a \sqrt [4]{a+b x^2}}+\left (-\frac{b}{2 a x}-\frac{1}{3 x^3}\right ) \left (a+b x^2\right )^{3/4} \]

Antiderivative was successfully verified.

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

[Out]

(-1/(3*x^3) - b/(2*a*x))*(a + b*x^2)^(3/4) + (b^2*x*((a + b*x^2)/a)^(1/4)*Hyperg
eometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)])/(4*a*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/4)/x^4,x)

[Out]

int((b*x^2+a)^(3/4)/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(3/4)/x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^(3/4)/x^4, x)

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Sympy [A]  time = 4.03735, size = 34, normalized size = 0.28 \[ - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(3/4)*hyper((-3/2, -3/4), (-1/2,), b*x**2*exp_polar(I*pi)/a)/(3*x**3)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError