Optimal. Leaf size=65 \[ \frac {c x^2 \sqrt {c+d x^3} F_1\left (\frac {2}{3};1,-\frac {3}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a \sqrt {\frac {d x^3}{c}+1}} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {511, 510} \[ \frac {c x^2 \sqrt {c+d x^3} F_1\left (\frac {2}{3};1,-\frac {3}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a \sqrt {\frac {d x^3}{c}+1}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {x \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx &=\frac {\left (c \sqrt {c+d x^3}\right ) \int \frac {x \left (1+\frac {d x^3}{c}\right )^{3/2}}{a+b x^3} \, dx}{\sqrt {1+\frac {d x^3}{c}}}\\ &=\frac {c x^2 \sqrt {c+d x^3} F_1\left (\frac {2}{3};1,-\frac {3}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a \sqrt {1+\frac {d x^3}{c}}}\\ \end {align*}
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Mathematica [B] time = 0.21, size = 149, normalized size = 2.29 \[ \frac {x^2 \left (2 d x^3 \sqrt {\frac {d x^3}{c}+1} (10 b c-7 a d) F_1\left (\frac {5}{3};\frac {1}{2},1;\frac {8}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+5 c \sqrt {\frac {d x^3}{c}+1} (7 b c-4 a d) F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+20 a d \left (c+d x^3\right )\right )}{70 a b \sqrt {c+d x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.37, size = 930, normalized size = 14.31 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,{\left (d\,x^3+c\right )}^{3/2}}{b\,x^3+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (c + d x^{3}\right )^{\frac {3}{2}}}{a + b x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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