Optimal. Leaf size=64 \[ \frac {x^2 \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {2}{3};2,\frac {1}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 \sqrt {c+d x^3}} \]
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Rubi [A] time = 0.04, antiderivative size = 64, 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} \[ \frac {x^2 \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {2}{3};2,\frac {1}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 \sqrt {c+d x^3}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {d x^3}{c}} \int \frac {x}{\left (a+b x^3\right )^2 \sqrt {1+\frac {d x^3}{c}}} \, dx}{\sqrt {c+d x^3}}\\ &=\frac {x^2 \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {2}{3};2,\frac {1}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [B] time = 0.16, size = 172, normalized size = 2.69 \[ \frac {-b d x^5 \left (a+b x^3\right ) \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {5}{3};\frac {1}{2},1;\frac {8}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+5 x^2 \left (a+b x^3\right ) \sqrt {\frac {d x^3}{c}+1} (b c-3 a d) F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+10 a b x^2 \left (c+d x^3\right )}{30 a^2 \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)} \]
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 \[ \int \frac {x}{{\left (b x^{3} + a\right )}^{2} \sqrt {d x^{3} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 923, normalized size = 14.42 \[ \text {result too large to display} \]
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 \[ \int \frac {x}{{\left (b x^{3} + a\right )}^{2} \sqrt {d x^{3} + c}}\,{d x} \]
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 \[ \int \frac {x}{{\left (b\,x^3+a\right )}^2\,\sqrt {d\,x^3+c}} \,d x \]
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 \[ \int \frac {x}{\left (a + b x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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