Optimal. Leaf size=415 \[ \frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}-\frac {a^4 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^5}+\frac {4 a^2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac {a c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac {2 a^3 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}-\frac {4 a^2 c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{b^5}+\frac {4 c^2 f^{\frac {c}{(a+b x)^2}} (a+b x) \log ^2(f)}{15 b^5}+\frac {a c^2 \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{b^5}-\frac {4 c^{5/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {5}{2}}(f)}{15 b^5} \]
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Rubi [A]
time = 0.32, antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2258, 2237,
2242, 2235, 2245, 2241} \begin {gather*} -\frac {\sqrt {\pi } a^4 \sqrt {c} \sqrt {\log (f)} \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^5}+\frac {a^4 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^5}+\frac {2 a^3 c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{b^5}-\frac {2 a^3 (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^5}-\frac {4 \sqrt {\pi } a^2 c^{3/2} \log ^{\frac {3}{2}}(f) \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^5}+\frac {2 a^2 (a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{b^5}+\frac {4 a^2 c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^5}-\frac {4 \sqrt {\pi } c^{5/2} \log ^{\frac {5}{2}}(f) \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{15 b^5}+\frac {a c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{b^5}+\frac {4 c^2 \log ^2(f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{15 b^5}+\frac {(a+b x)^5 f^{\frac {c}{(a+b x)^2}}}{5 b^5}-\frac {a (a+b x)^4 f^{\frac {c}{(a+b x)^2}}}{b^5}+\frac {2 c \log (f) (a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{15 b^5}-\frac {a c \log (f) (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2237
Rule 2241
Rule 2242
Rule 2245
Rule 2258
Rubi steps
\begin {align*} \int f^{\frac {c}{(a+b x)^2}} x^4 \, dx &=\int \left (\frac {a^4 f^{\frac {c}{(a+b x)^2}}}{b^4}-\frac {4 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^4}+\frac {6 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^4}-\frac {4 a f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^4}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^4}\right ) \, dx\\ &=\frac {\int f^{\frac {c}{(a+b x)^2}} (a+b x)^4 \, dx}{b^4}-\frac {(4 a) \int f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \, dx}{b^4}+\frac {\left (6 a^2\right ) \int f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \, dx}{b^4}-\frac {\left (4 a^3\right ) \int f^{\frac {c}{(a+b x)^2}} (a+b x) \, dx}{b^4}+\frac {a^4 \int f^{\frac {c}{(a+b x)^2}} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}+\frac {(2 c \log (f)) \int f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \, dx}{5 b^4}-\frac {(2 a c \log (f)) \int f^{\frac {c}{(a+b x)^2}} (a+b x) \, dx}{b^4}+\frac {\left (4 a^2 c \log (f)\right ) \int f^{\frac {c}{(a+b x)^2}} \, dx}{b^4}-\frac {\left (4 a^3 c \log (f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{a+b x} \, dx}{b^4}+\frac {\left (2 a^4 c \log (f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}+\frac {4 a^2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac {a c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac {2 a^3 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}-\frac {\left (2 a^4 c \log (f)\right ) \text {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{b^5}+\frac {\left (4 c^2 \log ^2(f)\right ) \int f^{\frac {c}{(a+b x)^2}} \, dx}{15 b^4}-\frac {\left (2 a c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{a+b x} \, dx}{b^4}+\frac {\left (8 a^2 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}-\frac {a^4 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^5}+\frac {4 a^2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac {a c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac {2 a^3 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}+\frac {4 c^2 f^{\frac {c}{(a+b x)^2}} (a+b x) \log ^2(f)}{15 b^5}+\frac {a c^2 \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{b^5}-\frac {\left (8 a^2 c^2 \log ^2(f)\right ) \text {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{b^5}+\frac {\left (8 c^3 \log ^3(f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{15 b^4}\\ &=\frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}-\frac {a^4 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^5}+\frac {4 a^2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac {a c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac {2 a^3 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}-\frac {4 a^2 c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{b^5}+\frac {4 c^2 f^{\frac {c}{(a+b x)^2}} (a+b x) \log ^2(f)}{15 b^5}+\frac {a c^2 \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{b^5}-\frac {\left (8 c^3 \log ^3(f)\right ) \text {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{15 b^5}\\ &=\frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}-\frac {a^4 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^5}+\frac {4 a^2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac {a c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac {2 a^3 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}-\frac {4 a^2 c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{b^5}+\frac {4 c^2 f^{\frac {c}{(a+b x)^2}} (a+b x) \log ^2(f)}{15 b^5}+\frac {a c^2 \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{b^5}-\frac {4 c^{5/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {5}{2}}(f)}{15 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 195, normalized size = 0.47 \begin {gather*} \frac {a f^{\frac {c}{(a+b x)^2}} \left (3 a^4+47 a^2 c \log (f)+4 c^2 \log ^2(f)\right )}{15 b^5}+\frac {15 a c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f) \left (2 a^2+c \log (f)\right )-\sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)} \left (15 a^4+60 a^2 c \log (f)+4 c^2 \log ^2(f)\right )+b f^{\frac {c}{(a+b x)^2}} x \left (3 b^4 x^4+c \left (36 a^2-9 a b x+2 b^2 x^2\right ) \log (f)+4 c^2 \log ^2(f)\right )}{15 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 343, normalized size = 0.83
method | result | size |
risch | \(-\frac {3 \ln \left (f \right ) c \,f^{\frac {c}{\left (b x +a \right )^{2}}} a \,x^{2}}{5 b^{3}}+\frac {12 \ln \left (f \right ) c \,f^{\frac {c}{\left (b x +a \right )^{2}}} a^{2} x}{5 b^{4}}-\frac {a^{4} \ln \left (f \right ) c \sqrt {\pi }\, \erf \left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right )}{b^{5} \sqrt {-c \ln \left (f \right )}}-\frac {4 a^{2} \ln \left (f \right )^{2} c^{2} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right )}{b^{5} \sqrt {-c \ln \left (f \right )}}-\frac {2 a^{3} \ln \left (f \right ) c \expIntegral \left (1, -\frac {c \ln \left (f \right )}{\left (b x +a \right )^{2}}\right )}{b^{5}}-\frac {a \ln \left (f \right )^{2} c^{2} \expIntegral \left (1, -\frac {c \ln \left (f \right )}{\left (b x +a \right )^{2}}\right )}{b^{5}}+\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} x^{5}}{5}+\frac {a^{5} f^{\frac {c}{\left (b x +a \right )^{2}}}}{5 b^{5}}+\frac {2 \ln \left (f \right ) c \,f^{\frac {c}{\left (b x +a \right )^{2}}} x^{3}}{15 b^{2}}+\frac {4 \ln \left (f \right )^{2} c^{2} f^{\frac {c}{\left (b x +a \right )^{2}}} x}{15 b^{4}}-\frac {4 \ln \left (f \right )^{3} c^{3} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right )}{15 b^{5} \sqrt {-c \ln \left (f \right )}}+\frac {47 \ln \left (f \right ) c \,f^{\frac {c}{\left (b x +a \right )^{2}}} a^{3}}{15 b^{5}}+\frac {4 \ln \left (f \right )^{2} c^{2} f^{\frac {c}{\left (b x +a \right )^{2}}} a}{15 b^{5}}\) | \(343\) |
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 = 0.40, size = 201, normalized size = 0.48 \begin {gather*} \frac {\sqrt {\pi } {\left (15 \, a^{4} b + 60 \, a^{2} b c \log \left (f\right ) + 4 \, b c^{2} \log \left (f\right )^{2}\right )} \sqrt {-\frac {c \log \left (f\right )}{b^{2}}} \operatorname {erf}\left (\frac {b \sqrt {-\frac {c \log \left (f\right )}{b^{2}}}}{b x + a}\right ) + {\left (3 \, b^{5} x^{5} + 3 \, a^{5} + 4 \, {\left (b c^{2} x + a c^{2}\right )} \log \left (f\right )^{2} + {\left (2 \, b^{3} c x^{3} - 9 \, a b^{2} c x^{2} + 36 \, a^{2} b c x + 47 \, a^{3} c\right )} \log \left (f\right )\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 15 \, {\left (2 \, a^{3} c \log \left (f\right ) + a c^{2} \log \left (f\right )^{2}\right )} {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{15 \, 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}{\left (a + b x\right )^{2}}} 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}{{\left (a+b\,x\right )}^2}}\,x^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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