Optimal. Leaf size=585 \[ -\frac {405 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {810 a^3 d \sqrt {a+b x^3}}{1729 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {54 a^2 \sqrt {a+b x^3} \left (1729 c x+935 d x^2\right )}{323323}+\frac {54\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (1729 \sqrt [3]{b} c-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{323323 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {30 a \left (a+b x^3\right )^{3/2} \left (247 c x+187 d x^2\right )}{46189}+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right ) \]
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Rubi [A] time = 0.46, antiderivative size = 585, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1852, 1853, 1878, 218, 1877} \[ \frac {54\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (1729 \sqrt [3]{b} c-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{323323 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {810 a^3 d \sqrt {a+b x^3}}{1729 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {405 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 \sqrt {a+b x^3} \left (1729 c x+935 d x^2\right )}{323323}+\frac {30 a \left (a+b x^3\right )^{3/2} \left (247 c x+187 d x^2\right )}{46189}+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right ) \]
Antiderivative was successfully verified.
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Rule 218
Rule 1852
Rule 1853
Rule 1877
Rule 1878
Rubi steps
\begin {align*} \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx &=\int (c+d x) \left (a+b x^3\right )^{5/2} \, dx\\ &=\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}+\frac {1}{2} (15 a) \int \left (\frac {2 c}{17}+\frac {2 d x}{19}\right ) \left (a+b x^3\right )^{3/2} \, dx\\ &=\frac {30 a \left (247 c x+187 d x^2\right ) \left (a+b x^3\right )^{3/2}}{46189}+\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}+\frac {1}{4} \left (135 a^2\right ) \int \left (\frac {4 c}{187}+\frac {4 d x}{247}\right ) \sqrt {a+b x^3} \, dx\\ &=\frac {54 a^2 \left (1729 c x+935 d x^2\right ) \sqrt {a+b x^3}}{323323}+\frac {30 a \left (247 c x+187 d x^2\right ) \left (a+b x^3\right )^{3/2}}{46189}+\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}+\frac {1}{8} \left (405 a^3\right ) \int \frac {\frac {8 c}{935}+\frac {8 d x}{1729}}{\sqrt {a+b x^3}} \, dx\\ &=\frac {54 a^2 \left (1729 c x+935 d x^2\right ) \sqrt {a+b x^3}}{323323}+\frac {30 a \left (247 c x+187 d x^2\right ) \left (a+b x^3\right )^{3/2}}{46189}+\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}+\frac {\left (405 a^3 d\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{1729 \sqrt [3]{b}}+\frac {\left (81 a^3 \left (1729 c-\frac {935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{323323}\\ &=\frac {810 a^3 d \sqrt {a+b x^3}}{1729 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {54 a^2 \left (1729 c x+935 d x^2\right ) \sqrt {a+b x^3}}{323323}+\frac {30 a \left (247 c x+187 d x^2\right ) \left (a+b x^3\right )^{3/2}}{46189}+\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}-\frac {405 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} d \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 )}{1729 b^{2/3} \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 {54\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (1729 \sqrt [3]{b} c-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) \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 )}{323323 b^{2/3} \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 = 0.05, size = 78, normalized size = 0.13 \[ \frac {a^2 x \sqrt {a+b x^3} \left (2 c \, _2F_1\left (-\frac {5}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+d x \, _2F_1\left (-\frac {5}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )\right )}{2 \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} d x^{7} + b^{2} c x^{6} + 2 \, a b d x^{4} + 2 \, a b c x^{3} + a^{2} d x + a^{2} c\right )} \sqrt {b x^{3} + a}, x\right ) \]
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 \[ \int {\left (b d x^{4} + b c x^{3} + a d x + a c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1618, normalized size = 2.77 \[ \text {result too large to display} \]
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 \[ \int {\left (b d x^{4} + b c x^{3} + a d x + a c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (b\,x^3+a\right )}^{3/2}\,\left (b\,d\,x^4+b\,c\,x^3+a\,d\,x+a\,c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.80, size = 265, normalized size = 0.45 \[ \frac {a^{\frac {5}{2}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {5}{2}} d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {2 a^{\frac {3}{2}} b c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {2 a^{\frac {3}{2}} b d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} b^{2} c x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt {a} b^{2} d x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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