Optimal. Leaf size=184 \[ -\frac {a^3 (a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac {a^2 (a+b x)^2 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{b^4}+\frac {a c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^3}\right )}{b^4}-\frac {a (a+b x)^3 f^{\frac {c}{(a+b x)^3}}}{b^4}+\frac {(a+b x)^4 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^4} \]
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Rubi [A] time = 0.15, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2226, 2208, 2218, 2214, 2210} \[ \frac {a^2 (a+b x)^2 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3} \text {Gamma}\left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{b^4}-\frac {a^3 (a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \text {Gamma}\left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac {(a+b x)^4 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3} \text {Gamma}\left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac {a c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^3}\right )}{b^4}-\frac {a (a+b x)^3 f^{\frac {c}{(a+b x)^3}}}{b^4} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 2210
Rule 2214
Rule 2218
Rule 2226
Rubi steps
\begin {align*} \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx &=\int \left (-\frac {a^3 f^{\frac {c}{(a+b x)^3}}}{b^3}+\frac {3 a^2 f^{\frac {c}{(a+b x)^3}} (a+b x)}{b^3}-\frac {3 a f^{\frac {c}{(a+b x)^3}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^3}\right ) \, dx\\ &=\frac {\int f^{\frac {c}{(a+b x)^3}} (a+b x)^3 \, dx}{b^3}-\frac {(3 a) \int f^{\frac {c}{(a+b x)^3}} (a+b x)^2 \, dx}{b^3}+\frac {\left (3 a^2\right ) \int f^{\frac {c}{(a+b x)^3}} (a+b x) \, dx}{b^3}-\frac {a^3 \int f^{\frac {c}{(a+b x)^3}} \, dx}{b^3}\\ &=-\frac {a f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^4}-\frac {a^3 (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^4}+\frac {a^2 (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{b^4}+\frac {(a+b x)^4 \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^4}-\frac {(3 a c \log (f)) \int \frac {f^{\frac {c}{(a+b x)^3}}}{a+b x} \, dx}{b^3}\\ &=-\frac {a f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^4}+\frac {a c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{b^4}-\frac {a^3 (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^4}+\frac {a^2 (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{b^4}+\frac {(a+b x)^4 \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^4}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 167, normalized size = 0.91 \[ \frac {3 a c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^3}\right )-(a+b x) \left (a^3 \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )+3 a (a+b x) \left ((a+b x) f^{\frac {c}{(a+b x)^3}}-a \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right )\right )+c \log (f) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right )\right )}{3 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 221, normalized size = 1.20 \[ -\frac {6 \, a^{2} b^{2} \left (-\frac {c \log \relax (f)}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \relax (f)}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - 4 \, a c {\rm Ei}\left (\frac {c \log \relax (f)}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \relax (f) - {\left (4 \, a^{3} b - 3 \, b c \log \relax (f)\right )} \left (-\frac {c \log \relax (f)}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \relax (f)}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - {\left (b^{4} x^{4} - a^{4} + 3 \, {\left (b c x + a c\right )} \log \relax (f)\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \, b^{4}} \]
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 \[ \int f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int x^{3} f^{\frac {c}{\left (b x +a \right )^{3}}}\, dx \]
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 \[ \frac {{\left (b^{3} x^{4} + 3 \, c x \log \relax (f)\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \, b^{3}} - \int \frac {3 \, {\left (4 \, a b^{3} c x^{3} \log \relax (f) + 6 \, a^{2} b^{2} c x^{2} \log \relax (f) + a^{4} c \log \relax (f) + {\left (4 \, a^{3} b c \log \relax (f) - 3 \, b c^{2} \log \relax (f)^{2}\right )} x\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\,{d x} \]
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 \[ \int f^{\frac {c}{{\left (a+b\,x\right )}^3}}\,x^3 \,d x \]
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 \[ \int f^{\frac {c}{\left (a + b x\right )^{3}}} x^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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