Optimal. Leaf size=269 \[ -\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {3 a^2 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{2 b^4}-\frac {a c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{2 b^4}+\frac {a^3 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^4}-\frac {a c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{2 b^4}-\frac {3 a^2 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{2 b^4}+\frac {a c^3 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{2 b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4} \]
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Rubi [A]
time = 0.19, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2258, 2237,
2241, 2245, 2250} \begin {gather*} \frac {c^4 \log ^4(f) \text {Gamma}\left (-4,-\frac {c \log (f)}{a+b x}\right )}{b^4}+\frac {a^3 c \log (f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^4}-\frac {a^3 (a+b x) f^{\frac {c}{a+b x}}}{b^4}-\frac {3 a^2 c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{2 b^4}+\frac {3 a^2 (a+b x)^2 f^{\frac {c}{a+b x}}}{2 b^4}+\frac {3 a^2 c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{2 b^4}+\frac {a c^3 \log ^3(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{2 b^4}-\frac {a c^2 \log ^2(f) (a+b x) f^{\frac {c}{a+b x}}}{2 b^4}-\frac {a (a+b x)^3 f^{\frac {c}{a+b x}}}{b^4}-\frac {a c \log (f) (a+b x)^2 f^{\frac {c}{a+b x}}}{2 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2237
Rule 2241
Rule 2245
Rule 2250
Rule 2258
Rubi steps
\begin {align*} \int f^{\frac {c}{a+b x}} x^3 \, dx &=\int \left (-\frac {a^3 f^{\frac {c}{a+b x}}}{b^3}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)}{b^3}-\frac {3 a f^{\frac {c}{a+b x}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{a+b x}} (a+b x)^3}{b^3}\right ) \, dx\\ &=\frac {\int f^{\frac {c}{a+b x}} (a+b x)^3 \, dx}{b^3}-\frac {(3 a) \int f^{\frac {c}{a+b x}} (a+b x)^2 \, dx}{b^3}+\frac {\left (3 a^2\right ) \int f^{\frac {c}{a+b x}} (a+b x) \, dx}{b^3}-\frac {a^3 \int f^{\frac {c}{a+b x}} \, dx}{b^3}\\ &=-\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4}-\frac {(a c \log (f)) \int f^{\frac {c}{a+b x}} (a+b x) \, dx}{b^3}+\frac {\left (3 a^2 c \log (f)\right ) \int f^{\frac {c}{a+b x}} \, dx}{2 b^3}-\frac {\left (a^3 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{b^3}\\ &=-\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {3 a^2 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{2 b^4}-\frac {a c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{2 b^4}+\frac {a^3 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4}-\frac {\left (a c^2 \log ^2(f)\right ) \int f^{\frac {c}{a+b x}} \, dx}{2 b^3}+\frac {\left (3 a^2 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{2 b^3}\\ &=-\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {3 a^2 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{2 b^4}-\frac {a c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{2 b^4}+\frac {a^3 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^4}-\frac {a c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{2 b^4}-\frac {3 a^2 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{2 b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4}-\frac {\left (a c^3 \log ^3(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{2 b^3}\\ &=-\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {3 a^2 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{2 b^4}-\frac {a c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{2 b^4}+\frac {a^3 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^4}-\frac {a c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{2 b^4}-\frac {3 a^2 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{2 b^4}+\frac {a c^3 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{2 b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 179, normalized size = 0.67 \begin {gather*} -\frac {a f^{\frac {c}{a+b x}} \left (6 a^3-26 a^2 c \log (f)+11 a c^2 \log ^2(f)-c^3 \log ^3(f)\right )}{24 b^4}+\frac {c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f) \left (24 a^3-36 a^2 c \log (f)+12 a c^2 \log ^2(f)-c^3 \log ^3(f)\right )+b f^{\frac {c}{a+b x}} x \left (6 b^3 x^3+2 c \left (9 a^2-3 a b x+b^2 x^2\right ) \log (f)+c^2 (-10 a+b x) \log ^2(f)+c^3 \log ^3(f)\right )}{24 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 359, normalized size = 1.33
method | result | size |
risch | \(-\frac {c^{3} \ln \left (f \right )^{3} a \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )}{2 b^{4}}+\frac {f^{\frac {c}{b x +a}} x^{4}}{4}+\frac {c^{2} \ln \left (f \right )^{2} f^{\frac {c}{b x +a}} x^{2}}{24 b^{2}}+\frac {c^{3} \ln \left (f \right )^{3} f^{\frac {c}{b x +a}} x}{24 b^{3}}+\frac {3 c^{2} \ln \left (f \right )^{2} a^{2} \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )}{2 b^{4}}-\frac {f^{\frac {c}{b x +a}} a^{4}}{4 b^{4}}-\frac {c \ln \left (f \right ) f^{\frac {c}{b x +a}} a \,x^{2}}{4 b^{2}}+\frac {3 c \ln \left (f \right ) f^{\frac {c}{b x +a}} a^{2} x}{4 b^{3}}-\frac {11 c^{2} \ln \left (f \right )^{2} f^{\frac {c}{b x +a}} a^{2}}{24 b^{4}}+\frac {c^{3} \ln \left (f \right )^{3} f^{\frac {c}{b x +a}} a}{24 b^{4}}+\frac {13 c \ln \left (f \right ) f^{\frac {c}{b x +a}} a^{3}}{12 b^{4}}-\frac {c \ln \left (f \right ) a^{3} \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )}{b^{4}}-\frac {5 c^{2} \ln \left (f \right )^{2} f^{\frac {c}{b x +a}} a x}{12 b^{3}}+\frac {c \ln \left (f \right ) f^{\frac {c}{b x +a}} x^{3}}{12 b}+\frac {c^{4} \ln \left (f \right )^{4} \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )}{24 b^{4}}\) | \(359\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.09, size = 171, normalized size = 0.64 \begin {gather*} \frac {{\left (6 \, b^{4} x^{4} - 6 \, a^{4} + {\left (b c^{3} x + a c^{3}\right )} \log \left (f\right )^{3} + {\left (b^{2} c^{2} x^{2} - 10 \, a b c^{2} x - 11 \, a^{2} c^{2}\right )} \log \left (f\right )^{2} + 2 \, {\left (b^{3} c x^{3} - 3 \, a b^{2} c x^{2} + 9 \, a^{2} b c x + 13 \, a^{3} c\right )} \log \left (f\right )\right )} f^{\frac {c}{b x + a}} - {\left (c^{4} \log \left (f\right )^{4} - 12 \, a c^{3} \log \left (f\right )^{3} + 36 \, a^{2} c^{2} \log \left (f\right )^{2} - 24 \, a^{3} c \log \left (f\right )\right )} {\rm Ei}\left (\frac {c \log \left (f\right )}{b x + a}\right )}{24 \, b^{4}} \end {gather*}
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 \begin {gather*} \int f^{\frac {c}{a + b x}} x^{3}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int f^{\frac {c}{a+b\,x}}\,x^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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