Optimal. Leaf size=111 \[ -\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^2}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{2 b^2}+\frac {a \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^2}-\frac {c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{2 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, 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 = {2258, 2237,
2242, 2235, 2245, 2241} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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 \, 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)) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 89, normalized size = 0.80 \begin {gather*} \frac {f^{\frac {c}{(a+b x)^2}} \left (-a^2+b^2 x^2\right )+2 a \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}-c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.05, size = 93, normalized size = 0.84
method | result | size |
risch | \(\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} x^{2}}{2}-\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} a^{2}}{2 b^{2}}+\frac {\ln \left (f \right ) c \expIntegral \left (1, -\frac {c \ln \left (f \right )}{\left (b x +a \right )^{2}}\right )}{2 b^{2}}+\frac {a \ln \left (f \right ) c \sqrt {\pi }\, \erf \left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right )}{b^{2} \sqrt {-c \ln \left (f \right )}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 107, normalized size = 0.96 \begin {gather*} -\frac {2 \, \sqrt {\pi } a b \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 ) + c {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) \log \left (f\right ) - {\left (b^{2} x^{2} - a^{2}\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{\frac {c}{\left (a + b x\right )^{2}}} x\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int f^{\frac {c}{{\left (a+b\,x\right )}^2}}\,x \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________