Optimal. Leaf size=82 \[ \frac{3 a^2 \left (a-b x^4\right )^{5/4}}{5 b^4}-\frac{a^3 \sqrt [4]{a-b x^4}}{b^4}+\frac{\left (a-b x^4\right )^{13/4}}{13 b^4}-\frac{a \left (a-b x^4\right )^{9/4}}{3 b^4} \]
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Rubi [A] time = 0.0461735, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, 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 = 0.0236048, size = 51, normalized size = 0.62 \[ -\frac{\sqrt [4]{a-b x^4} \left (32 a^2 b x^4+128 a^3+20 a b^2 x^8+15 b^3 x^{12}\right )}{195 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 48, 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 = 1.34548, size = 92, normalized size = 1.12 \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 = 1.72709, size = 115, normalized size = 1.4 \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 = 9.44517, size = 94, normalized size = 1.15 \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 = 1.1808, size = 112, normalized size = 1.37 \begin{align*} -\frac{15 \,{\left (b x^{4} - a\right )}^{3}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} + 65 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{1}{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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