Optimal. Leaf size=98 \[ \frac {3 a x^2}{5 \sqrt [4]{a+b x^4}}+\frac {1}{5} x^2 \left (a+b x^4\right )^{3/4}-\frac {3 a^{3/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {281, 201, 235,
233, 202} \begin {gather*} -\frac {3 a^{3/2} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a+b x^4}}+\frac {1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac {3 a x^2}{5 \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 202
Rule 233
Rule 235
Rule 281
Rubi steps
\begin {align*} \int x \left (a+b x^4\right )^{3/4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \left (a+b x^2\right )^{3/4} \, dx,x,x^2\right )\\ &=\frac {1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac {1}{10} (3 a) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac {\left (3 a \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx,x,x^2\right )}{10 \sqrt [4]{a+b x^4}}\\ &=\frac {3 a x^2}{5 \sqrt [4]{a+b x^4}}+\frac {1}{5} x^2 \left (a+b x^4\right )^{3/4}-\frac {\left (3 a \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{10 \sqrt [4]{a+b x^4}}\\ &=\frac {3 a x^2}{5 \sqrt [4]{a+b x^4}}+\frac {1}{5} x^2 \left (a+b x^4\right )^{3/4}-\frac {3 a^{3/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a+b x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 7.62, size = 51, normalized size = 0.52 \begin {gather*} \frac {x^2 \left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^4}{a}\right )}{2 \left (1+\frac {b x^4}{a}\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x \left (b \,x^{4}+a \right )^{\frac {3}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.07, size = 13, normalized size = 0.13 \begin {gather*} {\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac {3}{4}} x, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.47, size = 29, normalized size = 0.30 \begin {gather*} \frac {a^{\frac {3}{4}} x^{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\left (b\,x^4+a\right )}^{3/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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