Optimal. Leaf size=291 \[ \frac {a^4 f^{\frac {c}{a+b x}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{a+b x}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{a+b x}} (a+b x)^3}{b^5}-\frac {2 a^3 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^5}+\frac {a^2 c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{b^5}-\frac {a^4 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^5}+\frac {a^2 c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{b^5}+\frac {2 a^3 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{b^5}-\frac {a^2 c^3 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{b^5}-\frac {4 a c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^5}-\frac {c^5 \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right ) \log ^5(f)}{b^5} \]
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Rubi [A]
time = 0.22, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps
used = 13, 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^5 \log ^5(f) \text {Gamma}\left (-5,-\frac {c \log (f)}{a+b x}\right )}{b^5}-\frac {4 a c^4 \log ^4(f) \text {Gamma}\left (-4,-\frac {c \log (f)}{a+b x}\right )}{b^5}-\frac {a^4 c \log (f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}+\frac {a^4 (a+b x) f^{\frac {c}{a+b x}}}{b^5}+\frac {2 a^3 c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}-\frac {2 a^3 (a+b x)^2 f^{\frac {c}{a+b x}}}{b^5}-\frac {2 a^3 c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{b^5}-\frac {a^2 c^3 \log ^3(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}+\frac {a^2 c^2 \log ^2(f) (a+b x) f^{\frac {c}{a+b x}}}{b^5}+\frac {2 a^2 (a+b x)^3 f^{\frac {c}{a+b x}}}{b^5}+\frac {a^2 c \log (f) (a+b x)^2 f^{\frac {c}{a+b x}}}{b^5} \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^4 \, dx &=\int \left (\frac {a^4 f^{\frac {c}{a+b x}}}{b^4}-\frac {4 a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {6 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{b^4}-\frac {4 a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {f^{\frac {c}{a+b x}} (a+b x)^4}{b^4}\right ) \, dx\\ &=\frac {\int f^{\frac {c}{a+b x}} (a+b x)^4 \, dx}{b^4}-\frac {(4 a) \int f^{\frac {c}{a+b x}} (a+b x)^3 \, dx}{b^4}+\frac {\left (6 a^2\right ) \int f^{\frac {c}{a+b x}} (a+b x)^2 \, dx}{b^4}-\frac {\left (4 a^3\right ) \int f^{\frac {c}{a+b x}} (a+b x) \, dx}{b^4}+\frac {a^4 \int f^{\frac {c}{a+b x}} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{a+b x}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{a+b x}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{a+b x}} (a+b x)^3}{b^5}-\frac {4 a c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^5}-\frac {c^5 \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right ) \log ^5(f)}{b^5}+\frac {\left (2 a^2 c \log (f)\right ) \int f^{\frac {c}{a+b x}} (a+b x) \, dx}{b^4}-\frac {\left (2 a^3 c \log (f)\right ) \int f^{\frac {c}{a+b x}} \, dx}{b^4}+\frac {\left (a^4 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{a+b x}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{a+b x}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{a+b x}} (a+b x)^3}{b^5}-\frac {2 a^3 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^5}+\frac {a^2 c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{b^5}-\frac {a^4 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^5}-\frac {4 a c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^5}-\frac {c^5 \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right ) \log ^5(f)}{b^5}+\frac {\left (a^2 c^2 \log ^2(f)\right ) \int f^{\frac {c}{a+b x}} \, dx}{b^4}-\frac {\left (2 a^3 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{a+b x}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{a+b x}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{a+b x}} (a+b x)^3}{b^5}-\frac {2 a^3 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^5}+\frac {a^2 c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{b^5}-\frac {a^4 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^5}+\frac {a^2 c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{b^5}+\frac {2 a^3 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{b^5}-\frac {4 a c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^5}-\frac {c^5 \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right ) \log ^5(f)}{b^5}+\frac {\left (a^2 c^3 \log ^3(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{a+b x}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{a+b x}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{a+b x}} (a+b x)^3}{b^5}-\frac {2 a^3 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^5}+\frac {a^2 c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{b^5}-\frac {a^4 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^5}+\frac {a^2 c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{b^5}+\frac {2 a^3 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{b^5}-\frac {a^2 c^3 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{b^5}-\frac {4 a c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^5}-\frac {c^5 \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right ) \log ^5(f)}{b^5}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 241, normalized size = 0.83 \begin {gather*} \frac {a f^{\frac {c}{a+b x}} \left (24 a^4-154 a^3 c \log (f)+102 a^2 c^2 \log ^2(f)-19 a c^3 \log ^3(f)+c^4 \log ^4(f)\right )}{120 b^5}+\frac {-c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f) \left (120 a^4-240 a^3 c \log (f)+120 a^2 c^2 \log ^2(f)-20 a c^3 \log ^3(f)+c^4 \log ^4(f)\right )+b f^{\frac {c}{a+b x}} x \left (24 b^4 x^4+2 c \left (-48 a^3+18 a^2 b x-8 a b^2 x^2+3 b^3 x^3\right ) \log (f)+2 c^2 \left (43 a^2-7 a b x+b^2 x^2\right ) \log ^2(f)+c^3 (-18 a+b x) \log ^3(f)+c^4 \log ^4(f)\right )}{120 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 517, normalized size = 1.78
method | result | size |
risch | \(-\frac {7 c^{2} \ln \left (f \right )^{2} f^{\frac {c}{b x +a}} a \,x^{2}}{60 b^{3}}+\frac {43 c^{2} \ln \left (f \right )^{2} f^{\frac {c}{b x +a}} a^{2} x}{60 b^{4}}-\frac {3 c^{3} \ln \left (f \right )^{3} f^{\frac {c}{b x +a}} a x}{20 b^{4}}+\frac {c \ln \left (f \right ) a^{4} \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )}{b^{5}}-\frac {77 c \ln \left (f \right ) f^{\frac {c}{b x +a}} a^{4}}{60 b^{5}}-\frac {2 c \ln \left (f \right ) f^{\frac {c}{b x +a}} a \,x^{3}}{15 b^{2}}+\frac {3 c \ln \left (f \right ) f^{\frac {c}{b x +a}} a^{2} x^{2}}{10 b^{3}}-\frac {4 c \ln \left (f \right ) f^{\frac {c}{b x +a}} a^{3} x}{5 b^{4}}+\frac {17 c^{2} \ln \left (f \right )^{2} f^{\frac {c}{b x +a}} a^{3}}{20 b^{5}}-\frac {19 c^{3} \ln \left (f \right )^{3} f^{\frac {c}{b x +a}} a^{2}}{120 b^{5}}+\frac {c^{4} \ln \left (f \right )^{4} f^{\frac {c}{b x +a}} a}{120 b^{5}}+\frac {f^{\frac {c}{b x +a}} x^{5}}{5}+\frac {c^{2} \ln \left (f \right )^{2} f^{\frac {c}{b x +a}} x^{3}}{60 b^{2}}+\frac {c^{3} \ln \left (f \right )^{3} f^{\frac {c}{b x +a}} x^{2}}{120 b^{3}}+\frac {c^{4} \ln \left (f \right )^{4} f^{\frac {c}{b x +a}} x}{120 b^{4}}+\frac {c^{3} \ln \left (f \right )^{3} a^{2} \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )}{b^{5}}+\frac {a^{5} f^{\frac {c}{b x +a}}}{5 b^{5}}-\frac {2 c^{2} \ln \left (f \right )^{2} a^{3} \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )}{b^{5}}-\frac {c^{4} \ln \left (f \right )^{4} a \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )}{6 b^{5}}+\frac {c \ln \left (f \right ) f^{\frac {c}{b x +a}} x^{4}}{20 b}+\frac {c^{5} \ln \left (f \right )^{5} \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )}{120 b^{5}}\) | \(517\) |
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 = 243, normalized size = 0.84 \begin {gather*} \frac {{\left (24 \, b^{5} x^{5} + 24 \, a^{5} + {\left (b c^{4} x + a c^{4}\right )} \log \left (f\right )^{4} + {\left (b^{2} c^{3} x^{2} - 18 \, a b c^{3} x - 19 \, a^{2} c^{3}\right )} \log \left (f\right )^{3} + 2 \, {\left (b^{3} c^{2} x^{3} - 7 \, a b^{2} c^{2} x^{2} + 43 \, a^{2} b c^{2} x + 51 \, a^{3} c^{2}\right )} \log \left (f\right )^{2} + 2 \, {\left (3 \, b^{4} c x^{4} - 8 \, a b^{3} c x^{3} + 18 \, a^{2} b^{2} c x^{2} - 48 \, a^{3} b c x - 77 \, a^{4} c\right )} \log \left (f\right )\right )} f^{\frac {c}{b x + a}} - {\left (c^{5} \log \left (f\right )^{5} - 20 \, a c^{4} \log \left (f\right )^{4} + 120 \, a^{2} c^{3} \log \left (f\right )^{3} - 240 \, a^{3} c^{2} \log \left (f\right )^{2} + 120 \, a^{4} c \log \left (f\right )\right )} {\rm Ei}\left (\frac {c \log \left (f\right )}{b x + a}\right )}{120 \, b^{5}} \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^{4}\, 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^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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