Integrand size = 13, antiderivative size = 92 \[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=-\frac {a (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^2}+\frac {(a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^2} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2258, 2239, 2250} \[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=\frac {(a+b x)^2 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^2}-\frac {a (a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^2} \]
[In]
[Out]
Rule 2239
Rule 2250
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a f^{\frac {c}{(a+b x)^3}}}{b}+\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)}{b}\right ) \, dx \\ & = \frac {\int f^{\frac {c}{(a+b x)^3}} (a+b x) \, dx}{b}-\frac {a \int f^{\frac {c}{(a+b x)^3}} \, dx}{b} \\ & = -\frac {a (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^2}+\frac {(a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93 \[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=\frac {(a+b x) \left (-a \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}+(a+b x) \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}\right )}{3 b^2} \]
[In]
[Out]
\[\int f^{\frac {c}{\left (b x +a \right )^{3}}} x d x\]
[In]
[Out]
none
Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.67 \[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=-\frac {b^{2} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - 2 \, a b \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - {\left (b^{2} x^{2} - a^{2}\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{2 \, b^{2}} \]
[In]
[Out]
\[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=\int f^{\frac {c}{\left (a + b x\right )^{3}}} x\, dx \]
[In]
[Out]
\[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x \,d x } \]
[In]
[Out]
\[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x \,d x } \]
[In]
[Out]
Timed out. \[ \int f^{\frac {c}{(a+b x)^3}} x \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^3}}\,x \,d x \]
[In]
[Out]