Optimal. Leaf size=99 \[ \frac{3 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{a} \sqrt [4]{a+b x^4}}-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5} \]
[Out]
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Rubi [A] time = 0.159725, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{3 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{a} \sqrt [4]{a+b x^4}}-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^(3/4)/x^6,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{3 b x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{10 \sqrt [4]{a + b x^{4}}} - \frac{3 b}{5 x \sqrt [4]{a + b x^{4}}} - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(3/4)/x**6,x)
[Out]
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Mathematica [C] time = 0.0480042, size = 83, normalized size = 0.84 \[ \frac{-a^2+2 b^2 x^8 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-4 a b x^4-3 b^2 x^8}{5 a x^5 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)^(3/4)/x^6,x]
[Out]
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Maple [F] time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(3/4)/x^6,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)/x^6,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)/x^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.28071, size = 31, normalized size = 0.31 \[ - \frac{b^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**(3/4)/x**6,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)/x^6,x, algorithm="giac")
[Out]