Optimal. Leaf size=415 \[ -\frac {\sqrt {\pi } a^4 \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^5}+\frac {a^4 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^5}+\frac {2 a^3 c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{b^5}-\frac {2 a^3 (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^5}-\frac {4 \sqrt {\pi } a^2 c^{3/2} \log ^{\frac {3}{2}}(f) \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^5}+\frac {2 a^2 (a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{b^5}+\frac {4 a^2 c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^5}-\frac {4 \sqrt {\pi } c^{5/2} \log ^{\frac {5}{2}}(f) \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{15 b^5}+\frac {a c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{b^5}+\frac {4 c^2 \log ^2(f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{15 b^5}+\frac {(a+b x)^5 f^{\frac {c}{(a+b x)^2}}}{5 b^5}-\frac {a (a+b x)^4 f^{\frac {c}{(a+b x)^2}}}{b^5}+\frac {2 c \log (f) (a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{15 b^5}-\frac {a c \log (f) (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps used = 19, 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} \[ -\frac {4 \sqrt {\pi } a^2 c^{3/2} \log ^{\frac {3}{2}}(f) \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^5}-\frac {\sqrt {\pi } a^4 \sqrt {c} \sqrt {\log (f)} \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^5}+\frac {2 a^3 c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{b^5}+\frac {2 a^2 (a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{b^5}-\frac {2 a^3 (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^5}+\frac {a^4 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^5}+\frac {4 a^2 c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^5}-\frac {4 \sqrt {\pi } c^{5/2} \log ^{\frac {5}{2}}(f) \text {Erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{15 b^5}+\frac {a c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{b^5}+\frac {4 c^2 \log ^2(f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{15 b^5}+\frac {(a+b x)^5 f^{\frac {c}{(a+b x)^2}}}{5 b^5}-\frac {a (a+b x)^4 f^{\frac {c}{(a+b x)^2}}}{b^5}+\frac {2 c \log (f) (a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{15 b^5}-\frac {a c \log (f) (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
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^4 \, dx &=\int \left (\frac {a^4 f^{\frac {c}{(a+b x)^2}}}{b^4}-\frac {4 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^4}+\frac {6 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^4}-\frac {4 a f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^4}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^4}\right ) \, dx\\ &=\frac {\int f^{\frac {c}{(a+b x)^2}} (a+b x)^4 \, dx}{b^4}-\frac {(4 a) \int f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \, dx}{b^4}+\frac {\left (6 a^2\right ) \int f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \, dx}{b^4}-\frac {\left (4 a^3\right ) \int f^{\frac {c}{(a+b x)^2}} (a+b x) \, dx}{b^4}+\frac {a^4 \int f^{\frac {c}{(a+b x)^2}} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}+\frac {(2 c \log (f)) \int f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \, dx}{5 b^4}-\frac {(2 a c \log (f)) \int f^{\frac {c}{(a+b x)^2}} (a+b x) \, dx}{b^4}+\frac {\left (4 a^2 c \log (f)\right ) \int f^{\frac {c}{(a+b x)^2}} \, dx}{b^4}-\frac {\left (4 a^3 c \log (f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{a+b x} \, dx}{b^4}+\frac {\left (2 a^4 c \log (f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}+\frac {4 a^2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac {a c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac {2 a^3 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}-\frac {\left (2 a^4 c \log (f)\right ) \operatorname {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{b^5}+\frac {\left (4 c^2 \log ^2(f)\right ) \int f^{\frac {c}{(a+b x)^2}} \, dx}{15 b^4}-\frac {\left (2 a c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{a+b x} \, dx}{b^4}+\frac {\left (8 a^2 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}-\frac {a^4 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^5}+\frac {4 a^2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac {a c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac {2 a^3 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}+\frac {4 c^2 f^{\frac {c}{(a+b x)^2}} (a+b x) \log ^2(f)}{15 b^5}+\frac {a c^2 \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{b^5}-\frac {\left (8 a^2 c^2 \log ^2(f)\right ) \operatorname {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{b^5}+\frac {\left (8 c^3 \log ^3(f)\right ) \int \frac {f^{\frac {c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{15 b^4}\\ &=\frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}-\frac {a^4 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^5}+\frac {4 a^2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac {a c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac {2 a^3 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}-\frac {4 a^2 c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{b^5}+\frac {4 c^2 f^{\frac {c}{(a+b x)^2}} (a+b x) \log ^2(f)}{15 b^5}+\frac {a c^2 \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{b^5}-\frac {\left (8 c^3 \log ^3(f)\right ) \operatorname {Subst}\left (\int f^{c x^2} \, dx,x,\frac {1}{a+b x}\right )}{15 b^5}\\ &=\frac {a^4 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^5}{5 b^5}-\frac {a^4 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^5}+\frac {4 a^2 c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac {a c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac {2 c f^{\frac {c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac {2 a^3 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}-\frac {4 a^2 c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{b^5}+\frac {4 c^2 f^{\frac {c}{(a+b x)^2}} (a+b x) \log ^2(f)}{15 b^5}+\frac {a c^2 \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{b^5}-\frac {4 c^{5/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {5}{2}}(f)}{15 b^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 195, normalized size = 0.47 \[ \frac {a \left (3 a^4+47 a^2 c \log (f)+4 c^2 \log ^2(f)\right ) f^{\frac {c}{(a+b x)^2}}}{15 b^5}+\frac {b x f^{\frac {c}{(a+b x)^2}} \left (c \log (f) \left (36 a^2-9 a b x+2 b^2 x^2\right )+3 b^4 x^4+4 c^2 \log ^2(f)\right )+15 a c \log (f) \left (2 a^2+c \log (f)\right ) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^2}\right )-\sqrt {\pi } \sqrt {c} \sqrt {\log (f)} \left (15 a^4+60 a^2 c \log (f)+4 c^2 \log ^2(f)\right ) \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{15 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 201, normalized size = 0.48 \[ \frac {\sqrt {\pi } {\left (15 \, a^{4} b + 60 \, a^{2} b c \log \relax (f) + 4 \, b c^{2} \log \relax (f)^{2}\right )} \sqrt {-\frac {c \log \relax (f)}{b^{2}}} \operatorname {erf}\left (\frac {b \sqrt {-\frac {c \log \relax (f)}{b^{2}}}}{b x + a}\right ) + {\left (3 \, b^{5} x^{5} + 3 \, a^{5} + 4 \, {\left (b c^{2} x + a c^{2}\right )} \log \relax (f)^{2} + {\left (2 \, b^{3} c x^{3} - 9 \, a b^{2} c x^{2} + 36 \, a^{2} b c x + 47 \, a^{3} c\right )} \log \relax (f)\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 15 \, {\left (2 \, a^{3} c \log \relax (f) + a c^{2} \log \relax (f)^{2}\right )} {\rm Ei}\left (\frac {c \log \relax (f)}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{15 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\frac {c}{{\left (b x + a\right )}^{2}}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.11, size = 343, normalized size = 0.83 \[ \frac {x^{5} f^{\frac {c}{\left (b x +a \right )^{2}}}}{5}+\frac {2 c \,x^{3} f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )}{15 b^{2}}-\frac {3 a c \,x^{2} f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )}{5 b^{3}}-\frac {\sqrt {\pi }\, a^{4} c \erf \left (\frac {\sqrt {-c \ln \relax (f )}}{b x +a}\right ) \ln \relax (f )}{\sqrt {-c \ln \relax (f )}\, b^{5}}+\frac {12 a^{2} c x \,f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )}{5 b^{4}}-\frac {4 \sqrt {\pi }\, a^{2} c^{2} \erf \left (\frac {\sqrt {-c \ln \relax (f )}}{b x +a}\right ) \ln \relax (f )^{2}}{\sqrt {-c \ln \relax (f )}\, b^{5}}+\frac {4 c^{2} x \,f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )^{2}}{15 b^{4}}-\frac {4 \sqrt {\pi }\, c^{3} \erf \left (\frac {\sqrt {-c \ln \relax (f )}}{b x +a}\right ) \ln \relax (f )^{3}}{15 \sqrt {-c \ln \relax (f )}\, b^{5}}+\frac {a^{5} f^{\frac {c}{\left (b x +a \right )^{2}}}}{5 b^{5}}+\frac {47 a^{3} c \,f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )}{15 b^{5}}-\frac {2 a^{3} c \Ei \left (1, -\frac {c \ln \relax (f )}{\left (b x +a \right )^{2}}\right ) \ln \relax (f )}{b^{5}}+\frac {4 a \,c^{2} f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \relax (f )^{2}}{15 b^{5}}-\frac {a \,c^{2} \Ei \left (1, -\frac {c \ln \relax (f )}{\left (b x +a \right )^{2}}\right ) \ln \relax (f )^{2}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (3 \, b^{4} x^{5} + 2 \, b^{2} c x^{3} \log \relax (f) - 9 \, a b c x^{2} \log \relax (f) + 4 \, {\left (9 \, a^{2} c \log \relax (f) + c^{2} \log \relax (f)^{2}\right )} x\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{15 \, b^{4}} - \int \frac {2 \, {\left (18 \, a^{5} c \log \relax (f) + 2 \, a^{3} c^{2} \log \relax (f)^{2} + 15 \, {\left (2 \, a^{3} b^{2} c \log \relax (f) + a b^{2} c^{2} \log \relax (f)^{2}\right )} x^{2} + {\left (45 \, a^{4} b c \log \relax (f) - 30 \, a^{2} b c^{2} \log \relax (f)^{2} - 4 \, b c^{3} \log \relax (f)^{3}\right )} x\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{15 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{\frac {c}{{\left (a+b\,x\right )}^2}}\,x^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\frac {c}{\left (a + b x\right )^{2}}} x^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________