Optimal. Leaf size=60 \[ \frac {c x \sqrt {c+d x^3} F_1\left (\frac {1}{3};2,-\frac {3}{2};\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 \sqrt {\frac {d x^3}{c}+1}} \]
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Rubi [A] time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {430, 429} \[ \frac {c x \sqrt {c+d x^3} F_1\left (\frac {1}{3};2,-\frac {3}{2};\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 \sqrt {\frac {d x^3}{c}+1}} \]
Antiderivative was successfully verified.
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Rule 429
Rule 430
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx &=\frac {\left (c \sqrt {c+d x^3}\right ) \int \frac {\left (1+\frac {d x^3}{c}\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx}{\sqrt {1+\frac {d x^3}{c}}}\\ &=\frac {c x \sqrt {c+d x^3} F_1\left (\frac {1}{3};2,-\frac {3}{2};\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 \sqrt {1+\frac {d x^3}{c}}}\\ \end {align*}
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Mathematica [B] time = 0.43, size = 339, normalized size = 5.65 \[ \frac {x \left (\frac {a \left (24 x^3 \left (c+d x^3\right ) (b c-a d) \left (2 b c F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+a d F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )-64 a c \left (b c \left (3 c+d x^3\right )-a d^2 x^3\right ) F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )}{\left (a+b x^3\right ) \left (3 x^3 \left (2 b c F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+a d F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )-8 a c F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )}+d x^3 \sqrt {\frac {d x^3}{c}+1} (5 a d+b c) F_1\left (\frac {4}{3};\frac {1}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )}{24 a^2 b \sqrt {c+d x^3}} \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 801, normalized size = 13.35 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d\,x^3+c\right )}^{3/2}}{{\left (b\,x^3+a\right )}^2} \,d x \]
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 \[ \int \frac {\left (c + d x^{3}\right )^{\frac {3}{2}}}{\left (a + b x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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