Optimal. Leaf size=206 \[ -\frac {\sqrt {\pi } a^2 \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^3}+\frac {a^2 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^3}-\frac {2 \sqrt {\pi } c^{3/2} \log ^{\frac {3}{2}}(f) \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{3 b^3}+\frac {a c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{b^3}+\frac {(a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{3 b^3}-\frac {a (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^3}+\frac {2 c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{3 b^3} \]
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Rubi [A] time = 0.21, 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 = {2226, 2206, 2211, 2204, 2214, 2210} \[ -\frac {\sqrt {\pi } a^2 \sqrt {c} \sqrt {\log (f)} \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^3}+\frac {a^2 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^3}-\frac {2 \sqrt {\pi } c^{3/2} \log ^{\frac {3}{2}}(f) \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{3 b^3}+\frac {a c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{b^3}+\frac {(a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{3 b^3}-\frac {a (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^3}+\frac {2 c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{3 b^3} \]
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^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 ) \operatorname {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 ) \operatorname {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 = 0.12, size = 131, normalized size = 0.64 \[ \frac {a \left (a^2+2 c \log (f)\right ) f^{\frac {c}{(a+b x)^2}}}{3 b^3}+\frac {-\sqrt {\pi } \sqrt {c} \sqrt {\log (f)} \left (3 a^2+2 c \log (f)\right ) \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )+b x f^{\frac {c}{(a+b x)^2}} \left (b^2 x^2+2 c \log (f)\right )+3 a c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{3 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 128, normalized size = 0.62 \[ \frac {3 \, a c {\rm Ei}\left (\frac {c \log \relax (f)}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) \log \relax (f) + \sqrt {\pi } {\left (3 \, a^{2} b + 2 \, b c \log \relax (f)\right )} \sqrt {-\frac {c \log \relax (f)}{b^{2}}} \operatorname {erf}\left (\frac {b \sqrt {-\frac {c \log \relax (f)}{b^{2}}}}{b x + a}\right ) + {\left (b^{3} x^{3} + a^{3} + 2 \, {\left (b c x + a c\right )} \log \relax (f)\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{3 \, b^{3}} \]
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 \[ \int f^{\frac {c}{{\left (b x + a\right )}^{2}}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 175, normalized size = 0.85 \[ \frac {x^{3} f^{\frac {c}{\left (b x +a \right )^{2}}}}{3}-\frac {\sqrt {\pi }\, a^{2} c \erf \left (\frac {\sqrt {-c \ln \relax (f )}}{b x +a}\right ) \ln \relax (f )}{\sqrt {-c \ln \relax (f )}\, b^{3}}+\frac {2 c x \,f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )}{3 b^{2}}-\frac {2 \sqrt {\pi }\, c^{2} \erf \left (\frac {\sqrt {-c \ln \relax (f )}}{b x +a}\right ) \ln \relax (f )^{2}}{3 \sqrt {-c \ln \relax (f )}\, b^{3}}+\frac {a^{3} f^{\frac {c}{\left (b x +a \right )^{2}}}}{3 b^{3}}+\frac {2 a c \,f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )}{3 b^{3}}-\frac {a c \Ei \left (1, -\frac {c \ln \relax (f )}{\left (b x +a \right )^{2}}\right ) \ln \relax (f )}{b^{3}} \]
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 \[ \frac {{\left (b^{2} x^{3} + 2 \, c x \log \relax (f)\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{3 \, b^{2}} - \int \frac {2 \, {\left (3 \, a b^{2} c x^{2} \log \relax (f) + a^{3} c \log \relax (f) + {\left (3 \, a^{2} b c \log \relax (f) - 2 \, b c^{2} \log \relax (f)^{2}\right )} x\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{3 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}}\,{d x} \]
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 \[ \int f^{\frac {c}{{\left (a+b\,x\right )}^2}}\,x^2 \,d x \]
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 \[ \int f^{\frac {c}{\left (a + b x\right )^{2}}} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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