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