Optimal. Leaf size=206 \[ \frac {a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^3}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{3 b^3}-\frac {a^2 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^3}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{3 b^3}+\frac {a c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^3}-\frac {2 c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{3 b^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 11, 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^2 \sqrt {c} \sqrt {\log (f)} \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^3}+\frac {a^2 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^3}-\frac {2 \sqrt {\pi } c^{3/2} \log ^{\frac {3}{2}}(f) \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{3 b^3}+\frac {a c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{b^3}+\frac {(a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{3 b^3}-\frac {a (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^3}+\frac {2 c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{3 b^3} \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^2 \, dx &=\int \left (\frac {a^2 f^{\frac {c}{(a+b x)^2}}}{b^2}-\frac {2 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}{b^2}\right ) \, dx\\ &=\frac {\int f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \, dx}{b^2}-\frac {(2 a) \int f^{\frac {c}{(a+b x)^2}} (a+b x) \, dx}{b^2}+\frac {a^2 \int f^{\frac {c}{(a+b x)^2}} \, dx}{b^2}\\ &=\frac {a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^3}-\frac {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}{3 b^3}+\frac {(2 c \log (f)) \int f^{\frac {c}{(a+b x)^2}} \, dx}{3 b^2}-\frac {(2 a c \log (f)) \int \frac {f^{\frac {c}{(a+b x)^2}}}{a+b x} \, dx}{b^2}+\frac {\left (2 a^2 c \log (f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b^2}\\ &=\frac {a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^3}-\frac {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}{3 b^3}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{3 b^3}+\frac {a c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^3}-\frac {\left (2 a^2 c \log (f)\right ) \text {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{b^3}+\frac {\left (4 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{3 b^2}\\ &=\frac {a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^3}-\frac {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}{3 b^3}-\frac {a^2 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^3}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{3 b^3}+\frac {a c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^3}-\frac {\left (4 c^2 \log ^2(f)\right ) \text {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{3 b^3}\\ &=\frac {a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^3}-\frac {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}{3 b^3}-\frac {a^2 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^3}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{3 b^3}+\frac {a c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^3}-\frac {2 c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{3 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 131, normalized size = 0.64 \begin {gather*} \frac {a f^{\frac {c}{(a+b x)^2}} \left (a^2+2 c \log (f)\right )}{3 b^3}+\frac {3 a c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)-\sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)} \left (3 a^2+2 c \log (f)\right )+b f^{\frac {c}{(a+b x)^2}} x \left (b^2 x^2+2 c \log (f)\right )}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 175, normalized size = 0.85
method | result | size |
risch | \(\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} x^{3}}{3}+\frac {a^{3} f^{\frac {c}{\left (b x +a \right )^{2}}}}{3 b^{3}}+\frac {2 \ln \left (f \right ) c \,f^{\frac {c}{\left (b x +a \right )^{2}}} x}{3 b^{2}}+\frac {2 \ln \left (f \right ) c \,f^{\frac {c}{\left (b x +a \right )^{2}}} a}{3 b^{3}}-\frac {2 \ln \left (f \right )^{2} c^{2} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right )}{3 b^{3} \sqrt {-c \ln \left (f \right )}}-\frac {a^{2} \ln \left (f \right ) c \sqrt {\pi }\, \erf \left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right )}{b^{3} \sqrt {-c \ln \left (f \right )}}-\frac {a \ln \left (f \right ) c \expIntegral \left (1, -\frac {c \ln \left (f \right )}{\left (b x +a \right )^{2}}\right )}{b^{3}}\) | \(175\) |
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 = 0.36, size = 128, normalized size = 0.62 \begin {gather*} \frac {3 \, a 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 ) + \sqrt {\pi } {\left (3 \, a^{2} b + 2 \, b c \log \left (f\right )\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 (b^{3} x^{3} + a^{3} + 2 \, {\left (b c x + a c\right )} \log \left (f\right )\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{3 \, b^{3}} \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^{2}\, 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^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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