Optimal. Leaf size=78 \[ \frac{3 a^2 \left (a+b x^4\right )^{5/4}}{5 b^4}-\frac{a^3 \sqrt [4]{a+b x^4}}{b^4}+\frac{\left (a+b x^4\right )^{13/4}}{13 b^4}-\frac{a \left (a+b x^4\right )^{9/4}}{3 b^4} \]
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Rubi [A] time = 0.0449799, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{3 a^2 \left (a+b x^4\right )^{5/4}}{5 b^4}-\frac{a^3 \sqrt [4]{a+b x^4}}{b^4}+\frac{\left (a+b x^4\right )^{13/4}}{13 b^4}-\frac{a \left (a+b x^4\right )^{9/4}}{3 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{15}}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{3/4}}+\frac{3 a^2 \sqrt [4]{a+b x}}{b^3}-\frac{3 a (a+b x)^{5/4}}{b^3}+\frac{(a+b x)^{9/4}}{b^3}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^3 \sqrt [4]{a+b x^4}}{b^4}+\frac{3 a^2 \left (a+b x^4\right )^{5/4}}{5 b^4}-\frac{a \left (a+b x^4\right )^{9/4}}{3 b^4}+\frac{\left (a+b x^4\right )^{13/4}}{13 b^4}\\ \end{align*}
Mathematica [A] time = 0.0222493, size = 50, normalized size = 0.64 \[ \frac{\sqrt [4]{a+b x^4} \left (32 a^2 b x^4-128 a^3-20 a b^2 x^8+15 b^3 x^{12}\right )}{195 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-15\,{b}^{3}{x}^{12}+20\,a{b}^{2}{x}^{8}-32\,{a}^{2}b{x}^{4}+128\,{a}^{3}}{195\,{b}^{4}}\sqrt [4]{b{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0279, size = 86, normalized size = 1.1 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{13}{4}}}{13 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}} a}{3 \, b^{4}} + \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{2}}{5 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46871, size = 112, normalized size = 1.44 \begin{align*} \frac{{\left (15 \, b^{3} x^{12} - 20 \, a b^{2} x^{8} + 32 \, a^{2} b x^{4} - 128 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{195 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.41193, size = 92, normalized size = 1.18 \begin{align*} \begin{cases} - \frac{128 a^{3} \sqrt [4]{a + b x^{4}}}{195 b^{4}} + \frac{32 a^{2} x^{4} \sqrt [4]{a + b x^{4}}}{195 b^{3}} - \frac{4 a x^{8} \sqrt [4]{a + b x^{4}}}{39 b^{2}} + \frac{x^{12} \sqrt [4]{a + b x^{4}}}{13 b} & \text{for}\: b \neq 0 \\\frac{x^{16}}{16 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10276, size = 77, normalized size = 0.99 \begin{align*} \frac{15 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} - 65 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a + 117 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{2} - 195 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3}}{195 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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