Optimal. Leaf size=556 \[ -\frac {108 a^3 x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {54 a^2 x^5 \sqrt {a+b x^3}}{6175 b}+\frac {18}{475} a x^8 \sqrt {a+b x^3}+\frac {432 a^4 \sqrt {a+b x^3}}{8645 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}-\frac {216 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{13/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {144 \sqrt {2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {285, 327, 309,
224, 1891} \begin {gather*} \frac {144 \sqrt {2} 3^{3/4} a^{13/3} \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 )}{8645 b^{8/3} \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 {216 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{13/3} \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}} E\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 )}{8645 b^{8/3} \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 {432 a^4 \sqrt {a+b x^3}}{8645 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {108 a^3 x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {54 a^2 x^5 \sqrt {a+b x^3}}{6175 b}+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}+\frac {18}{475} a x^8 \sqrt {a+b x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 224
Rule 285
Rule 309
Rule 327
Rule 1891
Rubi steps
\begin {align*} \int x^7 \left (a+b x^3\right )^{3/2} \, dx &=\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}+\frac {1}{25} (9 a) \int x^7 \sqrt {a+b x^3} \, dx\\ &=\frac {18}{475} a x^8 \sqrt {a+b x^3}+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}+\frac {1}{475} \left (27 a^2\right ) \int \frac {x^7}{\sqrt {a+b x^3}} \, dx\\ &=\frac {54 a^2 x^5 \sqrt {a+b x^3}}{6175 b}+\frac {18}{475} a x^8 \sqrt {a+b x^3}+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}-\frac {\left (54 a^3\right ) \int \frac {x^4}{\sqrt {a+b x^3}} \, dx}{1235 b}\\ &=-\frac {108 a^3 x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {54 a^2 x^5 \sqrt {a+b x^3}}{6175 b}+\frac {18}{475} a x^8 \sqrt {a+b x^3}+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}+\frac {\left (216 a^4\right ) \int \frac {x}{\sqrt {a+b x^3}} \, dx}{8645 b^2}\\ &=-\frac {108 a^3 x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {54 a^2 x^5 \sqrt {a+b x^3}}{6175 b}+\frac {18}{475} a x^8 \sqrt {a+b x^3}+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}+\frac {\left (216 a^4\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{8645 b^{7/3}}+\frac {\left (216 \sqrt {2 \left (2-\sqrt {3}\right )} a^{13/3}\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{8645 b^{7/3}}\\ &=-\frac {108 a^3 x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {54 a^2 x^5 \sqrt {a+b x^3}}{6175 b}+\frac {18}{475} a x^8 \sqrt {a+b x^3}+\frac {432 a^4 \sqrt {a+b x^3}}{8645 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}-\frac {216 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{13/3} \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}} E\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 )}{8645 b^{8/3} \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 {144 \sqrt {2} 3^{3/4} a^{13/3} \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 )}{8645 b^{8/3} \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*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 5.92, size = 81, normalized size = 0.15 \begin {gather*} \frac {2 x^2 \sqrt {a+b x^3} \left (-\left (\left (10 a-19 b x^3\right ) \left (a+b x^3\right )^2\right )+\frac {10 a^3 \, _2F_1\left (-\frac {3}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{475 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.14, size = 509, normalized size = 0.92 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.07, size = 80, normalized size = 0.14 \begin {gather*} -\frac {2 \, {\left (1080 \, a^{4} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (1729 \, b^{4} x^{11} + 2548 \, a b^{3} x^{8} + 189 \, a^{2} b^{2} x^{5} - 270 \, a^{3} b x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{43225 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.59, size = 39, normalized size = 0.07 \begin {gather*} \frac {a^{\frac {3}{2}} x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^7\,{\left (b\,x^3+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________