Optimal. Leaf size=44 \[ \frac {(a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2208} \[ \frac {(a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \text {Gamma}\left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b} \]
Antiderivative was successfully verified.
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Rule 2208
Rubi steps
\begin {align*} \int f^{\frac {c}{(a+b x)^3}} \, dx &=\frac {(a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 1.00 \[ \frac {(a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 94, normalized size = 2.14 \[ -\frac {b \left (-\frac {c \log \relax (f)}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \relax (f)}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - {\left (b x + a\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{b} \]
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 \[ \int f^{\frac {c}{{\left (b x + a\right )}^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int f^{\frac {c}{\left (b x +a \right )^{3}}}\, dx \]
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 \[ 3 \, b c \int \frac {f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}} x}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}\,{d x} \log \relax (f) + f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.97, size = 68, normalized size = 1.55 \[ \frac {\left (a+b\,x\right )\,\left (\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {c\,\ln \relax (f)}{{\left (a+b\,x\right )}^3}\right )}^{1/3}-\Gamma \left (\frac {2}{3},-\frac {c\,\ln \relax (f)}{{\left (a+b\,x\right )}^3}\right )\,{\left (-\frac {c\,\ln \relax (f)}{{\left (a+b\,x\right )}^3}\right )}^{1/3}+f^{\frac {c}{{\left (a+b\,x\right )}^3}}\right )}{b} \]
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 \[ \int f^{\frac {c}{\left (a + b x\right )^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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