Optimal. Leaf size=93 \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} (a d+b c) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a b \left (a+b x^3\right )^{2/3}}+\frac {x (b c-a d)}{2 a b \left (a+b x^3\right )^{2/3}} \]
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Rubi [A] time = 0.03, antiderivative size = 93, 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 = {385, 246, 245} \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} (a d+b c) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a b \left (a+b x^3\right )^{2/3}}+\frac {x (b c-a d)}{2 a b \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 385
Rubi steps
\begin {align*} \int \frac {c+d x^3}{\left (a+b x^3\right )^{5/3}} \, dx &=\frac {(b c-a d) x}{2 a b \left (a+b x^3\right )^{2/3}}+\frac {(b c+a d) \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx}{2 a b}\\ &=\frac {(b c-a d) x}{2 a b \left (a+b x^3\right )^{2/3}}+\frac {\left ((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 a b \left (a+b x^3\right )^{2/3}}\\ &=\frac {(b c-a d) x}{2 a b \left (a+b x^3\right )^{2/3}}+\frac {(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 a b \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 66, normalized size = 0.71 \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} (a d+b c) \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {4}{3};-\frac {b x^3}{a}\right )-a d x}{a b \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \]
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 \[ \int \frac {d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {d \,x^{3}+c}{\left (b \,x^{3}+a \right )^{\frac {5}{3}}}\, dx \]
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 \[ \int \frac {d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac {5}{3}}}\,{d x} \]
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 \[ \int \frac {d\,x^3+c}{{\left (b\,x^3+a\right )}^{5/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 16.36, size = 78, normalized size = 0.84 \[ \frac {c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {5}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {5}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{3}} \Gamma \left (\frac {7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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