Optimal. Leaf size=62 \[ \frac {(a+b x) f^{\frac {c}{(a+b x)^2}}}{b}-\frac {\sqrt {\pi } \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2206, 2211, 2204} \[ \frac {(a+b x) f^{\frac {c}{(a+b x)^2}}}{b}-\frac {\sqrt {\pi } \sqrt {c} \sqrt {\log (f)} \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2206
Rule 2211
Rubi steps
\begin {align*} \int f^{\frac {c}{(a+b x)^2}} \, dx &=\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)}{b}+(2 c \log (f)) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx\\ &=\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)}{b}-\frac {(2 c \log (f)) \operatorname {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{b}\\ &=\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)}{b}-\frac {\sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 1.00 \[ \frac {(a+b x) f^{\frac {c}{(a+b x)^2}}}{b}-\frac {\sqrt {\pi } \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 68, normalized size = 1.10 \[ \frac {\sqrt {\pi } b \sqrt {-\frac {c \log \relax (f)}{b^{2}}} \operatorname {erf}\left (\frac {b \sqrt {-\frac {c \log \relax (f)}{b^{2}}}}{b x + a}\right ) + {\left (b x + a\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b} \]
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 \[ \int f^{\frac {c}{{\left (b x + a\right )}^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 65, normalized size = 1.05 \[ -\frac {\sqrt {\pi }\, c \erf \left (\frac {\sqrt {-c \ln \relax (f )}}{b x +a}\right ) \ln \relax (f )}{\sqrt {-c \ln \relax (f )}\, b}+x \,f^{\frac {c}{\left (b x +a \right )^{2}}}+\frac {a \,f^{\frac {c}{\left (b x +a \right )^{2}}}}{b} \]
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 \[ 2 \, b c \int \frac {f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} x}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\,{d x} \log \relax (f) + f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.64, size = 53, normalized size = 0.85 \[ \frac {f^{\frac {c}{{\left (a+b\,x\right )}^2}}\,\left (a+b\,x\right )}{b}-\frac {c\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\sqrt {c\,\ln \relax (f)}}{a+b\,x}\right )\,\ln \relax (f)}{b\,\sqrt {c\,\ln \relax (f)}} \]
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 \[ \int f^{\frac {c}{\left (a + b x\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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