Optimal. Leaf size=119 \[ -\frac {4 a^2 x}{15 b \sqrt [4]{a+b x^2}}+\frac {2 a x \left (a+b x^2\right )^{3/4}}{15 b}+\frac {2}{9} x^3 \left (a+b x^2\right )^{3/4}+\frac {4 a^{5/2} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {285, 327, 235,
233, 202} \begin {gather*} \frac {4 a^{5/2} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a+b x^2}}-\frac {4 a^2 x}{15 b \sqrt [4]{a+b x^2}}+\frac {2 a x \left (a+b x^2\right )^{3/4}}{15 b}+\frac {2}{9} x^3 \left (a+b x^2\right )^{3/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 233
Rule 235
Rule 285
Rule 327
Rubi steps
\begin {align*} \int x^2 \left (a+b x^2\right )^{3/4} \, dx &=\frac {2}{9} x^3 \left (a+b x^2\right )^{3/4}+\frac {1}{3} a \int \frac {x^2}{\sqrt [4]{a+b x^2}} \, dx\\ &=\frac {2 a x \left (a+b x^2\right )^{3/4}}{15 b}+\frac {2}{9} x^3 \left (a+b x^2\right )^{3/4}-\frac {\left (2 a^2\right ) \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{15 b}\\ &=\frac {2 a x \left (a+b x^2\right )^{3/4}}{15 b}+\frac {2}{9} x^3 \left (a+b x^2\right )^{3/4}-\frac {\left (2 a^2 \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{15 b \sqrt [4]{a+b x^2}}\\ &=-\frac {4 a^2 x}{15 b \sqrt [4]{a+b x^2}}+\frac {2 a x \left (a+b x^2\right )^{3/4}}{15 b}+\frac {2}{9} x^3 \left (a+b x^2\right )^{3/4}+\frac {\left (2 a^2 \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{15 b \sqrt [4]{a+b x^2}}\\ &=-\frac {4 a^2 x}{15 b \sqrt [4]{a+b x^2}}+\frac {2 a x \left (a+b x^2\right )^{3/4}}{15 b}+\frac {2}{9} x^3 \left (a+b x^2\right )^{3/4}+\frac {4 a^{5/2} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 8.09, size = 62, normalized size = 0.52 \begin {gather*} \frac {2 x \left (a+b x^2\right )^{3/4} \left (a+b x^2-\frac {a \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )}{\left (1+\frac {b x^2}{a}\right )^{3/4}}\right )}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{2} \left (b \,x^{2}+a \right )^{\frac {3}{4}}\, dx\]
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 = 0.00, 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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Sympy [C] Result contains complex when optimal does not.
time = 0.55, size = 29, normalized size = 0.24 \begin {gather*} \frac {a^{\frac {3}{4}} x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} \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^2\,{\left (b\,x^2+a\right )}^{3/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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