Optimal. Leaf size=451 \[ \frac {7 a^2 c^{7/3} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} \operatorname {EllipticF}\left (\cos ^{-1}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{135 \sqrt [4]{3} b^2 \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}-\frac {14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac {(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac {2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.97, antiderivative size = 451, 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 = {279, 321, 329, 241, 225} \[ -\frac {14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac {7 a^2 c^{7/3} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{135 \sqrt [4]{3} b^2 \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac {(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac {2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 225
Rule 241
Rule 279
Rule 321
Rule 329
Rubi steps
\begin {align*} \int (c x)^{10/3} \sqrt [3]{a+b x^2} \, dx &=\frac {(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac {1}{15} (2 a) \int \frac {(c x)^{10/3}}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac {2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b}+\frac {(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}-\frac {\left (14 a^2 c^2\right ) \int \frac {(c x)^{4/3}}{\left (a+b x^2\right )^{2/3}} \, dx}{135 b}\\ &=-\frac {14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac {2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b}+\frac {(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac {\left (14 a^3 c^4\right ) \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )^{2/3}} \, dx}{405 b^2}\\ &=-\frac {14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac {2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b}+\frac {(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac {\left (14 a^3 c^3\right ) \operatorname {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^2}\\ &=-\frac {14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac {2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b}+\frac {(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac {\left (14 a^3 c^3\right ) \operatorname {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^2 \sqrt {\frac {a}{a+b x^2}} \sqrt {a+b x^2}}\\ &=-\frac {14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac {2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b}+\frac {(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac {7 a^2 c^{7/3} \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^2 \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*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 103, normalized size = 0.23 \[ \frac {c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (\sqrt [3]{\frac {b x^2}{a}+1} \left (-7 a^2+2 a b x^2+9 b^2 x^4\right )+7 a^2 \, _2F_1\left (-\frac {1}{3},\frac {1}{6};\frac {7}{6};-\frac {b x^2}{a}\right )\right )}{45 b^2 \sqrt [3]{\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {1}{3}} c^{3} x^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {10}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (c x \right )^{\frac {10}{3}} \left (b \,x^{2}+a \right )^{\frac {1}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {10}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,x\right )}^{10/3}\,{\left (b\,x^2+a\right )}^{1/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 127.90, size = 46, normalized size = 0.10 \[ \frac {\sqrt [3]{a} c^{\frac {10}{3}} x^{\frac {13}{3}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {19}{6}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________