Optimal. Leaf size=291 \[ \frac {\sqrt {\pi } a^3 \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^4}-\frac {a^3 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^4}-\frac {3 a^2 c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{2 b^4}+\frac {3 a^2 (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{2 b^4}+\frac {2 \sqrt {\pi } a c^{3/2} \log ^{\frac {3}{2}}(f) \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^4}-\frac {c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{4 b^4}+\frac {(a+b x)^4 f^{\frac {c}{(a+b x)^2}}}{4 b^4}-\frac {a (a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{b^4}+\frac {c \log (f) (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{4 b^4}-\frac {2 a c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^4} \]
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Rubi [A] time = 0.30, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2226, 2206, 2211, 2204, 2214, 2210} \[ \frac {\sqrt {\pi } a^3 \sqrt {c} \sqrt {\log (f)} \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^4}-\frac {3 a^2 c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{2 b^4}+\frac {3 a^2 (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{2 b^4}-\frac {a^3 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^4}+\frac {2 \sqrt {\pi } a c^{3/2} \log ^{\frac {3}{2}}(f) \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^4}-\frac {c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{4 b^4}+\frac {(a+b x)^4 f^{\frac {c}{(a+b x)^2}}}{4 b^4}-\frac {a (a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{b^4}+\frac {c \log (f) (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{4 b^4}-\frac {2 a c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^4} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2206
Rule 2210
Rule 2211
Rule 2214
Rule 2226
Rubi steps
\begin {align*} \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx &=\int \left (-\frac {a^3 f^{\frac {c}{(a+b x)^2}}}{b^3}+\frac {3 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^3}-\frac {3 a f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^3}\right ) \, dx\\ &=\frac {\int f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \, dx}{b^3}-\frac {(3 a) \int f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \, dx}{b^3}+\frac {\left (3 a^2\right ) \int f^{\frac {c}{(a+b x)^2}} (a+b x) \, dx}{b^3}-\frac {a^3 \int f^{\frac {c}{(a+b x)^2}} \, dx}{b^3}\\ &=-\frac {a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{2 b^4}-\frac {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}{4 b^4}+\frac {(c \log (f)) \int f^{\frac {c}{(a+b x)^2}} (a+b x) \, dx}{2 b^3}-\frac {(2 a c \log (f)) \int f^{\frac {c}{(a+b x)^2}} \, dx}{b^3}+\frac {\left (3 a^2 c \log (f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{a+b x} \, dx}{b^3}-\frac {\left (2 a^3 c \log (f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b^3}\\ &=-\frac {a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{2 b^4}-\frac {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}{4 b^4}-\frac {2 a c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^4}+\frac {c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{4 b^4}-\frac {3 a^2 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{2 b^4}+\frac {\left (2 a^3 c \log (f)\right ) \operatorname {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{b^4}+\frac {\left (c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{a+b x} \, dx}{2 b^3}-\frac {\left (4 a c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b^3}\\ &=-\frac {a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{2 b^4}-\frac {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}{4 b^4}+\frac {a^3 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^4}-\frac {2 a c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^4}+\frac {c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{4 b^4}-\frac {3 a^2 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{2 b^4}-\frac {c^2 \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{4 b^4}+\frac {\left (4 a c^2 \log ^2(f)\right ) \operatorname {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{b^4}\\ &=-\frac {a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{2 b^4}-\frac {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}{4 b^4}+\frac {a^3 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^4}-\frac {2 a c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^4}+\frac {c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{4 b^4}-\frac {3 a^2 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{2 b^4}+\frac {2 a c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{b^4}-\frac {c^2 \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{4 b^4}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 148, normalized size = 0.51 \[ \frac {4 \sqrt {\pi } a \sqrt {c} \sqrt {\log (f)} \left (a^2+2 c \log (f)\right ) \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )-c \log (f) \left (6 a^2+c \log (f)\right ) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )+b x f^{\frac {c}{(a+b x)^2}} \left (-6 a c \log (f)+b^3 x^3+b c x \log (f)\right )}{4 b^4}-\frac {a^2 \left (a^2+7 c \log (f)\right ) f^{\frac {c}{(a+b x)^2}}}{4 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 156, normalized size = 0.54 \[ -\frac {4 \, \sqrt {\pi } {\left (a^{3} b + 2 \, a b c \log \relax (f)\right )} \sqrt {-\frac {c \log \relax (f)}{b^{2}}} \operatorname {erf}\left (\frac {b \sqrt {-\frac {c \log \relax (f)}{b^{2}}}}{b x + a}\right ) - {\left (b^{4} x^{4} - a^{4} + {\left (b^{2} c x^{2} - 6 \, a b c x - 7 \, a^{2} c\right )} \log \relax (f)\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + {\left (6 \, a^{2} c \log \relax (f) + c^{2} \log \relax (f)^{2}\right )} {\rm Ei}\left (\frac {c \log \relax (f)}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{4 \, b^{4}} \]
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 \[ \int f^{\frac {c}{{\left (b x + a\right )}^{2}}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 228, normalized size = 0.78 \[ \frac {x^{4} f^{\frac {c}{\left (b x +a \right )^{2}}}}{4}+\frac {c \,x^{2} f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )}{4 b^{2}}+\frac {\sqrt {\pi }\, a^{3} c \erf \left (\frac {\sqrt {-c \ln \relax (f )}}{b x +a}\right ) \ln \relax (f )}{\sqrt {-c \ln \relax (f )}\, b^{4}}-\frac {3 a c x \,f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )}{2 b^{3}}+\frac {2 \sqrt {\pi }\, a \,c^{2} \erf \left (\frac {\sqrt {-c \ln \relax (f )}}{b x +a}\right ) \ln \relax (f )^{2}}{\sqrt {-c \ln \relax (f )}\, b^{4}}-\frac {a^{4} f^{\frac {c}{\left (b x +a \right )^{2}}}}{4 b^{4}}-\frac {7 a^{2} c \,f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )}{4 b^{4}}+\frac {3 a^{2} c \Ei \left (1, -\frac {c \ln \relax (f )}{\left (b x +a \right )^{2}}\right ) \ln \relax (f )}{2 b^{4}}+\frac {c^{2} \Ei \left (1, -\frac {c \ln \relax (f )}{\left (b x +a \right )^{2}}\right ) \ln \relax (f )^{2}}{4 b^{4}} \]
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 \[ \frac {{\left (b^{3} x^{4} + b c x^{2} \log \relax (f) - 6 \, a c x \log \relax (f)\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \, b^{3}} + \int \frac {{\left (3 \, a^{4} c \log \relax (f) + {\left (6 \, a^{2} b^{2} c \log \relax (f) + b^{2} c^{2} \log \relax (f)^{2}\right )} x^{2} + 2 \, {\left (4 \, a^{3} b c \log \relax (f) - 3 \, a b c^{2} \log \relax (f)^{2}\right )} x\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{\frac {c}{{\left (a+b\,x\right )}^2}}\,x^3 \,d x \]
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 \[ \int f^{\frac {c}{\left (a + b x\right )^{2}}} x^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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