Optimal. Leaf size=111 \[ \frac {\sqrt {\pi } a \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^2}-\frac {c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{2 b^2}+\frac {(a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{2 b^2}-\frac {a (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^2} \]
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Rubi [A] time = 0.12, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2226, 2206, 2211, 2204, 2214, 2210} \[ \frac {\sqrt {\pi } a \sqrt {c} \sqrt {\log (f)} \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^2}-\frac {c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{2 b^2}+\frac {(a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{2 b^2}-\frac {a (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^2} \]
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 \, dx &=\int \left (-\frac {a f^{\frac {c}{(a+b x)^2}}}{b}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)}{b}\right ) \, dx\\ &=\frac {\int f^{\frac {c}{(a+b x)^2}} (a+b x) \, dx}{b}-\frac {a \int f^{\frac {c}{(a+b x)^2}} \, dx}{b}\\ &=-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^2}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{2 b^2}+\frac {(c \log (f)) \int \frac {f^{\frac {c}{(a+b x)^2}}}{a+b x} \, dx}{b}-\frac {(2 a c \log (f)) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b}\\ &=-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^2}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{2 b^2}-\frac {c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{2 b^2}+\frac {(2 a c \log (f)) \operatorname {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{b^2}\\ &=-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^2}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{2 b^2}+\frac {a \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^2}-\frac {c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 89, normalized size = 0.80 \[ \frac {\left (b^2 x^2-a^2\right ) f^{\frac {c}{(a+b x)^2}}+2 \sqrt {\pi } a \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )-c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 107, normalized size = 0.96 \[ -\frac {2 \, \sqrt {\pi } a b \sqrt {-\frac {c \log \relax (f)}{b^{2}}} \operatorname {erf}\left (\frac {b \sqrt {-\frac {c \log \relax (f)}{b^{2}}}}{b x + a}\right ) + c {\rm Ei}\left (\frac {c \log \relax (f)}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) \log \relax (f) - {\left (b^{2} x^{2} - a^{2}\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, b^{2}} \]
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 \[ \int f^{\frac {c}{{\left (b x + a\right )}^{2}}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 93, normalized size = 0.84 \[ \frac {x^{2} f^{\frac {c}{\left (b x +a \right )^{2}}}}{2}+\frac {\sqrt {\pi }\, a c \erf \left (\frac {\sqrt {-c \ln \relax (f )}}{b x +a}\right ) \ln \relax (f )}{\sqrt {-c \ln \relax (f )}\, b^{2}}-\frac {a^{2} f^{\frac {c}{\left (b x +a \right )^{2}}}}{2 b^{2}}+\frac {c \Ei \left (1, -\frac {c \ln \relax (f )}{\left (b x +a \right )^{2}}\right ) \ln \relax (f )}{2 b^{2}} \]
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 \[ b c \int \frac {f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} x^{2}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\,{d x} \log \relax (f) + \frac {1}{2} \, f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} x^{2} \]
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 \[ \int f^{\frac {c}{{\left (a+b\,x\right )}^2}}\,x \,d x \]
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 \[ \int f^{\frac {c}{\left (a + b x\right )^{2}}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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