Optimal. Leaf size=356 \[ \frac {2 x \left (d+e x^3\right )^{3/2} \left (391 a e^2-46 b d e+16 c d^2\right )}{4301 e^2}+\frac {18 d x \sqrt {d+e x^3} \left (391 a e^2-46 b d e+16 c d^2\right )}{21505 e^2}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} d^2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (391 a e^2-46 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt {3}\right )}{21505 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}-\frac {2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e} \]
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Rubi [A] time = 0.31, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1411, 388, 195, 218} \[ \frac {2 x \left (d+e x^3\right )^{3/2} \left (391 a e^2-46 b d e+16 c d^2\right )}{4301 e^2}+\frac {18 d x \sqrt {d+e x^3} \left (391 a e^2-46 b d e+16 c d^2\right )}{21505 e^2}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} d^2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (391 a e^2-46 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt {3}\right )}{21505 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}-\frac {2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e} \]
Antiderivative was successfully verified.
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Rule 195
Rule 218
Rule 388
Rule 1411
Rubi steps
\begin {align*} \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx &=\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac {2 \int \left (d+e x^3\right )^{3/2} \left (\frac {23 a e}{2}-\left (4 c d-\frac {23 b e}{2}\right ) x^3\right ) \, dx}{23 e}\\ &=-\frac {2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}-\frac {1}{391} \left (-391 a-\frac {2 d (8 c d-23 b e)}{e^2}\right ) \int \left (d+e x^3\right )^{3/2} \, dx\\ &=\frac {2 \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{4301}-\frac {2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac {\left (9 d \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right )\right ) \int \sqrt {d+e x^3} \, dx}{4301}\\ &=\frac {18 d \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right ) x \sqrt {d+e x^3}}{21505}+\frac {2 \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{4301}-\frac {2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac {\left (27 d^2 \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right )\right ) \int \frac {1}{\sqrt {d+e x^3}} \, dx}{21505}\\ &=\frac {18 d \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right ) x \sqrt {d+e x^3}}{21505}+\frac {2 \left (391 a+\frac {2 d (8 c d-23 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{4301}-\frac {2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} d^2 \left (16 c d^2-46 b d e+391 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right )|-7-4 \sqrt {3}\right )}{21505 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 101, normalized size = 0.28 \[ \frac {x \sqrt {d+e x^3} \left (\frac {\, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {e x^3}{d}\right ) \left (23 d e (17 a e-2 b d)+16 c d^3\right )}{\sqrt {\frac {e x^3}{d}+1}}-2 \left (d+e x^3\right )^2 \left (-23 b e+8 c d-17 c e x^3\right )\right )}{391 e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c e x^{9} + {\left (c d + b e\right )} x^{6} + {\left (b d + a e\right )} x^{3} + a d\right )} \sqrt {e x^{3} + d}, x\right ) \]
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 \[ \int {\left (c x^{6} + b x^{3} + a\right )} {\left (e x^{3} + d\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1010, normalized size = 2.84 \[ \text {result too large to display} \]
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 \[ \int {\left (c x^{6} + b x^{3} + a\right )} {\left (e x^{3} + d\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (e\,x^3+d\right )}^{3/2}\,\left (c\,x^6+b\,x^3+a\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.92, size = 257, normalized size = 0.72 \[ \frac {a d^{\frac {3}{2}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a \sqrt {d} e x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {b d^{\frac {3}{2}} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {b \sqrt {d} e x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {c d^{\frac {3}{2}} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {c \sqrt {d} e x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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