Optimal. Leaf size=614 \[ -\frac {72 \sqrt {2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-2 a B) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {108 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-2 a B) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {216 a^3 \sqrt {a+b x^3} (5 A b-2 a B)}{8645 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {54 a^2 x^2 \sqrt {a+b x^3} (5 A b-2 a B)}{8645 b^2}+\frac {2 x^5 \left (a+b x^3\right )^{3/2} (5 A b-2 a B)}{95 b}+\frac {18 a x^5 \sqrt {a+b x^3} (5 A b-2 a B)}{1235 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b} \]
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Rubi [A] time = 0.37, antiderivative size = 614, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {459, 279, 321, 303, 218, 1877} \[ \frac {54 a^2 x^2 \sqrt {a+b x^3} (5 A b-2 a B)}{8645 b^2}-\frac {216 a^3 \sqrt {a+b x^3} (5 A b-2 a B)}{8645 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {72 \sqrt {2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-2 a B) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {108 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-2 a B) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 x^5 \left (a+b x^3\right )^{3/2} (5 A b-2 a B)}{95 b}+\frac {18 a x^5 \sqrt {a+b x^3} (5 A b-2 a B)}{1235 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b} \]
Antiderivative was successfully verified.
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Rule 218
Rule 279
Rule 303
Rule 321
Rule 459
Rule 1877
Rubi steps
\begin {align*} \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}-\frac {\left (2 \left (-\frac {25 A b}{2}+5 a B\right )\right ) \int x^4 \left (a+b x^3\right )^{3/2} \, dx}{25 b}\\ &=\frac {2 (5 A b-2 a B) x^5 \left (a+b x^3\right )^{3/2}}{95 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}+\frac {(9 a (5 A b-2 a B)) \int x^4 \sqrt {a+b x^3} \, dx}{95 b}\\ &=\frac {18 a (5 A b-2 a B) x^5 \sqrt {a+b x^3}}{1235 b}+\frac {2 (5 A b-2 a B) x^5 \left (a+b x^3\right )^{3/2}}{95 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}+\frac {\left (27 a^2 (5 A b-2 a B)\right ) \int \frac {x^4}{\sqrt {a+b x^3}} \, dx}{1235 b}\\ &=\frac {54 a^2 (5 A b-2 a B) x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {18 a (5 A b-2 a B) x^5 \sqrt {a+b x^3}}{1235 b}+\frac {2 (5 A b-2 a B) x^5 \left (a+b x^3\right )^{3/2}}{95 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}-\frac {\left (108 a^3 (5 A b-2 a B)\right ) \int \frac {x}{\sqrt {a+b x^3}} \, dx}{8645 b^2}\\ &=\frac {54 a^2 (5 A b-2 a B) x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {18 a (5 A b-2 a B) x^5 \sqrt {a+b x^3}}{1235 b}+\frac {2 (5 A b-2 a B) x^5 \left (a+b x^3\right )^{3/2}}{95 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}-\frac {\left (108 a^3 (5 A b-2 a B)\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 (108 \sqrt {2 \left (2-\sqrt {3}\right )} a^{10/3} (5 A b-2 a B)\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{8645 b^{7/3}}\\ &=\frac {54 a^2 (5 A b-2 a B) x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {18 a (5 A b-2 a B) x^5 \sqrt {a+b x^3}}{1235 b}-\frac {216 a^3 (5 A b-2 a B) \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 (5 A b-2 a B) x^5 \left (a+b x^3\right )^{3/2}}{95 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}+\frac {108 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} (5 A b-2 a B) \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 {72 \sqrt {2} 3^{3/4} a^{10/3} (5 A b-2 a B) \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*}
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Mathematica [C] time = 0.24, size = 96, normalized size = 0.16 \[ \frac {2 x^2 \sqrt {a+b x^3} \left (\frac {5 a^2 (2 a B-5 A b) \, _2F_1\left (-\frac {3}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{\sqrt {\frac {b x^3}{a}+1}}-\left (a+b x^3\right )^2 \left (10 a B-25 A b-19 b B x^3\right )\right )}{475 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b x^{10} + {\left (B a + A b\right )} x^{7} + A a x^{4}\right )} \sqrt {b x^{3} + a}, x\right ) \]
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 \[ \int {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1002, normalized size = 1.63 \[ \text {result too large to display} \]
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 \[ \int {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{4}\,{d x} \]
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 \[ \int x^4\,\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.80, size = 172, normalized size = 0.28 \[ \frac {A a^{\frac {3}{2}} x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {A \sqrt {a} b x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} + \frac {B a^{\frac {3}{2}} x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} + \frac {B \sqrt {a} b x^{11} \Gamma \left (\frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{3} \\ \frac {14}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {14}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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