Optimal. Leaf size=125 \[ \frac{2 a^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a+b x^4}}-\frac{2 a^2 x^2}{15 b \sqrt [4]{a+b x^4}}+\frac{1}{9} x^6 \left (a+b x^4\right )^{3/4}+\frac{a x^2 \left (a+b x^4\right )^{3/4}}{15 b} \]
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Rubi [A] time = 0.171353, antiderivative size = 125, 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 \[ \frac{2 a^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a+b x^4}}-\frac{2 a^2 x^2}{15 b \sqrt [4]{a+b x^4}}+\frac{1}{9} x^6 \left (a+b x^4\right )^{3/4}+\frac{a x^2 \left (a+b x^4\right )^{3/4}}{15 b} \]
Antiderivative was successfully verified.
[In] Int[x^5*(a + b*x^4)^(3/4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{15 b} + \frac{a x^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{15 b} + \frac{x^{6} \left (a + b x^{4}\right )^{\frac{3}{4}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(b*x**4+a)**(3/4),x)
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Mathematica [C] time = 0.0612009, size = 80, normalized size = 0.64 \[ \frac{x^2 \left (-3 a^2 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+3 a^2+8 a b x^4+5 b^2 x^8\right )}{45 b \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5*(a + b*x^4)^(3/4),x]
[Out]
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Maple [F] time = 0.038, size = 0, normalized size = 0. \[ \int{x}^{5} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(b*x^4+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x^5,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{5}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x^5,x, algorithm="fricas")
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Sympy [A] time = 5.80497, size = 29, normalized size = 0.23 \[ \frac{a^{\frac{3}{4}} x^{6}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(b*x**4+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x^5,x, algorithm="giac")
[Out]