Optimal. Leaf size=448 \[ \frac {16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac {8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac {(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}-\frac {8 a^2 \sqrt [3]{c} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {c^{4/3}+\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{135 \sqrt [4]{3} b \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}} \]
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Rubi [A]
time = 0.54, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {285, 327, 335,
247, 231} \begin {gather*} -\frac {8 a^2 \sqrt [3]{c} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\text {ArcCos}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{135 \sqrt [4]{3} b \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac {16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac {(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}+\frac {8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 231
Rule 247
Rule 285
Rule 327
Rule 335
Rubi steps
\begin {align*} \int (c x)^{4/3} \left (a+b x^2\right )^{4/3} \, dx &=\frac {(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}+\frac {1}{15} (8 a) \int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx\\ &=\frac {8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac {(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}+\frac {1}{135} \left (16 a^2\right ) \int \frac {(c x)^{4/3}}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac {16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac {8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac {(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}-\frac {\left (16 a^3 c^2\right ) \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )^{2/3}} \, dx}{405 b}\\ &=\frac {16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac {8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac {(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}-\frac {\left (16 a^3 c\right ) \text {Subst}\left (\int \frac {1}{\left (a+\frac {b x^6}{c^2}\right )^{2/3}} \, dx,x,\sqrt [3]{c x}\right )}{135 b}\\ &=\frac {16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac {8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac {(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}-\frac {\left (16 a^3 c\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {b x^6}{c^2}}} \, dx,x,\frac {\sqrt [3]{c x}}{\sqrt [6]{a+b x^2}}\right )}{135 b \sqrt {\frac {a}{a+b x^2}} \sqrt {a+b x^2}}\\ &=\frac {16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac {8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac {(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}-\frac {8 a^2 \sqrt [3]{c} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {c^{4/3}+\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{135 \sqrt [4]{3} b \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^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 = 10.04, size = 89, normalized size = 0.20 \begin {gather*} \frac {c \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (\left (a+b x^2\right )^2 \sqrt [3]{1+\frac {b x^2}{a}}-a^2 \, _2F_1\left (-\frac {4}{3},\frac {1}{6};\frac {7}{6};-\frac {b x^2}{a}\right )\right )}{5 b \sqrt [3]{1+\frac {b x^2}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (c x \right )^{\frac {4}{3}} \left (b \,x^{2}+a \right )^{\frac {4}{3}}\, dx\]
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 = 9.02, size = 46, normalized size = 0.10 \begin {gather*} \frac {a^{\frac {4}{3}} c^{\frac {4}{3}} x^{\frac {7}{3}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{6}\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 {\left (c\,x\right )}^{4/3}\,{\left (b\,x^2+a\right )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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