Optimal. Leaf size=82 \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} (2 b c-a d) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 b \left (a+b x^3\right )^{2/3}}+\frac {d x \sqrt [3]{a+b x^3}}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {388, 246, 245} \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} (2 b c-a d) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 b \left (a+b x^3\right )^{2/3}}+\frac {d x \sqrt [3]{a+b x^3}}{2 b} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 388
Rubi steps
\begin {align*} \int \frac {c+d x^3}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac {d x \sqrt [3]{a+b x^3}}{2 b}-\frac {(-2 b c+a d) \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx}{2 b}\\ &=\frac {d x \sqrt [3]{a+b x^3}}{2 b}-\frac {\left ((-2 b c+a d) \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{2 b \left (a+b x^3\right )^{2/3}}\\ &=\frac {d x \sqrt [3]{a+b x^3}}{2 b}+\frac {(2 b c-a d) x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 b \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 73, normalized size = 0.89 \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} (2 b c-a d) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+d x \left (a+b x^3\right )}{2 b \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}}, x\right ) \]
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 \[ \int \frac {d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {d \,x^{3}+c}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {d\,x^3+c}{{\left (b\,x^3+a\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.07, size = 78, normalized size = 0.95 \[ \frac {c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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