Optimal. Leaf size=336 \[ \frac {54 a^2 x \sqrt {a+b x^3} (23 A b-8 a B)}{21505 b^2}-\frac {36\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (23 A b-8 a B) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{21505 b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 x^4 \left (a+b x^3\right )^{3/2} (23 A b-8 a B)}{391 b}+\frac {18 a x^4 \sqrt {a+b x^3} (23 A b-8 a B)}{4301 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b} \]
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Rubi [A] time = 0.18, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {459, 279, 321, 218} \[ \frac {54 a^2 x \sqrt {a+b x^3} (23 A b-8 a B)}{21505 b^2}-\frac {36\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (23 A b-8 a B) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{21505 b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 x^4 \left (a+b x^3\right )^{3/2} (23 A b-8 a B)}{391 b}+\frac {18 a x^4 \sqrt {a+b x^3} (23 A b-8 a B)}{4301 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b} \]
Antiderivative was successfully verified.
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Rule 218
Rule 279
Rule 321
Rule 459
Rubi steps
\begin {align*} \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}-\frac {\left (2 \left (-\frac {23 A b}{2}+4 a B\right )\right ) \int x^3 \left (a+b x^3\right )^{3/2} \, dx}{23 b}\\ &=\frac {2 (23 A b-8 a B) x^4 \left (a+b x^3\right )^{3/2}}{391 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}+\frac {(9 a (23 A b-8 a B)) \int x^3 \sqrt {a+b x^3} \, dx}{391 b}\\ &=\frac {18 a (23 A b-8 a B) x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 (23 A b-8 a B) x^4 \left (a+b x^3\right )^{3/2}}{391 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}+\frac {\left (27 a^2 (23 A b-8 a B)\right ) \int \frac {x^3}{\sqrt {a+b x^3}} \, dx}{4301 b}\\ &=\frac {54 a^2 (23 A b-8 a B) x \sqrt {a+b x^3}}{21505 b^2}+\frac {18 a (23 A b-8 a B) x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 (23 A b-8 a B) x^4 \left (a+b x^3\right )^{3/2}}{391 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}-\frac {\left (54 a^3 (23 A b-8 a B)\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{21505 b^2}\\ &=\frac {54 a^2 (23 A b-8 a B) x \sqrt {a+b x^3}}{21505 b^2}+\frac {18 a (23 A b-8 a B) x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 (23 A b-8 a B) x^4 \left (a+b x^3\right )^{3/2}}{391 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}-\frac {36\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 (23 A b-8 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 )}{21505 b^{7/3} \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 = 0.17, size = 93, normalized size = 0.28 \[ \frac {2 x \sqrt {a+b x^3} \left (\frac {a^2 (8 a B-23 A b) \, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\sqrt {\frac {b x^3}{a}+1}}-\left (a+b x^3\right )^2 \left (8 a B-23 A b-17 b B x^3\right )\right )}{391 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b x^{9} + {\left (B a + A b\right )} x^{6} + A a x^{3}\right )} \sqrt {b x^{3} + a}, x\right ) \]
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 \[ \int {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 694, normalized size = 2.07 \[ \left (\frac {2 \sqrt {b \,x^{3}+a}\, b \,x^{7}}{17}+\frac {40 \sqrt {b \,x^{3}+a}\, a \,x^{4}}{187}+\frac {36 i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, a^{3} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) b}}\right )}{935 \sqrt {b \,x^{3}+a}\, b^{2}}+\frac {54 \sqrt {b \,x^{3}+a}\, a^{2} x}{935 b}\right ) A +\left (\frac {2 \sqrt {b \,x^{3}+a}\, b \,x^{10}}{23}+\frac {52 \sqrt {b \,x^{3}+a}\, a \,x^{7}}{391}+\frac {54 \sqrt {b \,x^{3}+a}\, a^{2} x^{4}}{4301 b}-\frac {288 i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, a^{4} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) b}}\right )}{21505 \sqrt {b \,x^{3}+a}\, b^{3}}-\frac {432 \sqrt {b \,x^{3}+a}\, a^{3} x}{21505 b^{2}}\right ) B \]
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 \[ \int {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{3}\,{d x} \]
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 \[ \int x^3\,\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.06, size = 172, normalized size = 0.51 \[ \frac {A a^{\frac {3}{2}} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {A \sqrt {a} b x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {B a^{\frac {3}{2}} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {B \sqrt {a} b x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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