Optimal. Leaf size=277 \[ \frac {2}{3 a x^5 \sqrt {a+b x^3}}-\frac {13 \sqrt {a+b x^3}}{15 a^2 x^5}+\frac {91 b \sqrt {a+b x^3}}{60 a^3 x^2}+\frac {91 \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{60 \sqrt [4]{3} a^3 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {296, 331, 224}
\begin {gather*} \frac {91 b \sqrt {a+b x^3}}{60 a^3 x^2}-\frac {13 \sqrt {a+b x^3}}{15 a^2 x^5}+\frac {91 \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{60 \sqrt [4]{3} a^3 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2}{3 a x^5 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 224
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx &=\frac {2}{3 a x^5 \sqrt {a+b x^3}}+\frac {13 \int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx}{3 a}\\ &=\frac {2}{3 a x^5 \sqrt {a+b x^3}}-\frac {13 \sqrt {a+b x^3}}{15 a^2 x^5}-\frac {(91 b) \int \frac {1}{x^3 \sqrt {a+b x^3}} \, dx}{30 a^2}\\ &=\frac {2}{3 a x^5 \sqrt {a+b x^3}}-\frac {13 \sqrt {a+b x^3}}{15 a^2 x^5}+\frac {91 b \sqrt {a+b x^3}}{60 a^3 x^2}+\frac {\left (91 b^2\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{120 a^3}\\ &=\frac {2}{3 a x^5 \sqrt {a+b x^3}}-\frac {13 \sqrt {a+b x^3}}{15 a^2 x^5}+\frac {91 b \sqrt {a+b x^3}}{60 a^3 x^2}+\frac {91 \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{60 \sqrt [4]{3} a^3 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 54, normalized size = 0.19 \begin {gather*} -\frac {\sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (-\frac {5}{3},\frac {3}{2};-\frac {2}{3};-\frac {b x^3}{a}\right )}{5 a x^5 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 342, normalized size = 1.23 Too large to display
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.08, size = 80, normalized size = 0.29 \begin {gather*} \frac {91 \, {\left (b^{2} x^{8} + a b x^{5}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (91 \, b^{2} x^{6} + 39 \, a b x^{3} - 12 \, a^{2}\right )} \sqrt {b x^{3} + a}}{60 \, {\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.60, size = 44, normalized size = 0.16 \begin {gather*} \frac {\Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {3}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} x^{5} \Gamma \left (- \frac {2}{3}\right )} \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 = -0.00 \begin {gather*} \int \frac {1}{x^6\,{\left (b\,x^3+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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