Integrand size = 19, antiderivative size = 173 \[ \int F^{c (a+b x)} (d+e x)^{5/2} \, dx=-\frac {15 e^{5/2} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{8 b^{7/2} c^{7/2} \log ^{\frac {7}{2}}(F)}+\frac {15 e^2 F^{c (a+b x)} \sqrt {d+e x}}{4 b^3 c^3 \log ^3(F)}-\frac {5 e F^{c (a+b x)} (d+e x)^{3/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{5/2}}{b c \log (F)} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2207, 2211, 2235} \[ \int F^{c (a+b x)} (d+e x)^{5/2} \, dx=-\frac {15 \sqrt {\pi } e^{5/2} F^{c \left (a-\frac {b d}{e}\right )} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{8 b^{7/2} c^{7/2} \log ^{\frac {7}{2}}(F)}+\frac {15 e^2 \sqrt {d+e x} F^{c (a+b x)}}{4 b^3 c^3 \log ^3(F)}-\frac {5 e (d+e x)^{3/2} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac {(d+e x)^{5/2} F^{c (a+b x)}}{b c \log (F)} \]
[In]
[Out]
Rule 2207
Rule 2211
Rule 2235
Rubi steps \begin{align*} \text {integral}& = \frac {F^{c (a+b x)} (d+e x)^{5/2}}{b c \log (F)}-\frac {(5 e) \int F^{c (a+b x)} (d+e x)^{3/2} \, dx}{2 b c \log (F)} \\ & = -\frac {5 e F^{c (a+b x)} (d+e x)^{3/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{5/2}}{b c \log (F)}+\frac {\left (15 e^2\right ) \int F^{c (a+b x)} \sqrt {d+e x} \, dx}{4 b^2 c^2 \log ^2(F)} \\ & = \frac {15 e^2 F^{c (a+b x)} \sqrt {d+e x}}{4 b^3 c^3 \log ^3(F)}-\frac {5 e F^{c (a+b x)} (d+e x)^{3/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{5/2}}{b c \log (F)}-\frac {\left (15 e^3\right ) \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}} \, dx}{8 b^3 c^3 \log ^3(F)} \\ & = \frac {15 e^2 F^{c (a+b x)} \sqrt {d+e x}}{4 b^3 c^3 \log ^3(F)}-\frac {5 e F^{c (a+b x)} (d+e x)^{3/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{5/2}}{b c \log (F)}-\frac {\left (15 e^2\right ) \text {Subst}\left (\int F^{c \left (a-\frac {b d}{e}\right )+\frac {b c x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 b^3 c^3 \log ^3(F)} \\ & = -\frac {15 e^{5/2} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{8 b^{7/2} c^{7/2} \log ^{\frac {7}{2}}(F)}+\frac {15 e^2 F^{c (a+b x)} \sqrt {d+e x}}{4 b^3 c^3 \log ^3(F)}-\frac {5 e F^{c (a+b x)} (d+e x)^{3/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{5/2}}{b c \log (F)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.42 \[ \int F^{c (a+b x)} (d+e x)^{5/2} \, dx=\frac {e^2 F^{c \left (a-\frac {b d}{e}\right )} \sqrt {d+e x} \Gamma \left (\frac {7}{2},-\frac {b c (d+e x) \log (F)}{e}\right )}{b^3 c^3 \log ^3(F) \sqrt {-\frac {b c (d+e x) \log (F)}{e}}} \]
[In]
[Out]
\[\int F^{c \left (b x +a \right )} \left (e x +d \right )^{\frac {5}{2}}d x\]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.97 \[ \int F^{c (a+b x)} (d+e x)^{5/2} \, dx=\frac {\frac {15 \, \sqrt {\pi } \sqrt {-\frac {b c \log \left (F\right )}{e}} e^{3} \operatorname {erf}\left (\sqrt {e x + d} \sqrt {-\frac {b c \log \left (F\right )}{e}}\right )}{F^{\frac {b c d - a c e}{e}}} + 2 \, {\left (15 \, b c e^{2} \log \left (F\right ) + 4 \, {\left (b^{3} c^{3} e^{2} x^{2} + 2 \, b^{3} c^{3} d e x + b^{3} c^{3} d^{2}\right )} \log \left (F\right )^{3} - 10 \, {\left (b^{2} c^{2} e^{2} x + b^{2} c^{2} d e\right )} \log \left (F\right )^{2}\right )} \sqrt {e x + d} F^{b c x + a c}}{8 \, b^{4} c^{4} \log \left (F\right )^{4}} \]
[In]
[Out]
\[ \int F^{c (a+b x)} (d+e x)^{5/2} \, dx=\int F^{c \left (a + b x\right )} \left (d + e x\right )^{\frac {5}{2}}\, dx \]
[In]
[Out]
\[ \int F^{c (a+b x)} (d+e x)^{5/2} \, dx=\int { {\left (e x + d\right )}^{\frac {5}{2}} F^{{\left (b x + a\right )} c} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (141) = 282\).
Time = 0.43 (sec) , antiderivative size = 643, normalized size of antiderivative = 3.72 \[ \int F^{c (a+b x)} (d+e x)^{5/2} \, dx=-\frac {\frac {8 \, \sqrt {\pi } d^{3} e \operatorname {erf}\left (-\frac {\sqrt {-b c e \log \left (F\right )} \sqrt {e x + d}}{e}\right ) e^{\left (-\frac {b c d \log \left (F\right ) - a c e \log \left (F\right )}{e}\right )}}{\sqrt {-b c e \log \left (F\right )}} - 12 \, d^{2} {\left (\frac {\sqrt {\pi } {\left (2 \, b c d \log \left (F\right ) + e\right )} e \operatorname {erf}\left (-\frac {\sqrt {-b c e \log \left (F\right )} \sqrt {e x + d}}{e}\right ) e^{\left (-\frac {b c d \log \left (F\right ) - a c e \log \left (F\right )}{e}\right )}}{\sqrt {-b c e \log \left (F\right )} b c \log \left (F\right )} + \frac {2 \, \sqrt {e x + d} e e^{\left (\frac {{\left (e x + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )}{e}\right )}}{b c \log \left (F\right )}\right )} + 6 \, d {\left (\frac {\sqrt {\pi } {\left (4 \, b^{2} c^{2} d^{2} \log \left (F\right )^{2} + 4 \, b c d e \log \left (F\right ) + 3 \, e^{2}\right )} e \operatorname {erf}\left (-\frac {\sqrt {-b c e \log \left (F\right )} \sqrt {e x + d}}{e}\right ) e^{\left (-\frac {b c d \log \left (F\right ) - a c e \log \left (F\right )}{e}\right )}}{\sqrt {-b c e \log \left (F\right )} b^{2} c^{2} \log \left (F\right )^{2}} - \frac {2 \, {\left (2 \, {\left (e x + d\right )}^{\frac {3}{2}} b c e \log \left (F\right ) - 4 \, \sqrt {e x + d} b c d e \log \left (F\right ) - 3 \, \sqrt {e x + d} e^{2}\right )} e^{\left (\frac {{\left (e x + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )}{e}\right )}}{b^{2} c^{2} \log \left (F\right )^{2}}\right )} - \frac {\sqrt {\pi } {\left (8 \, b^{3} c^{3} d^{3} \log \left (F\right )^{3} + 12 \, b^{2} c^{2} d^{2} e \log \left (F\right )^{2} + 18 \, b c d e^{2} \log \left (F\right ) + 15 \, e^{3}\right )} e \operatorname {erf}\left (-\frac {\sqrt {-b c e \log \left (F\right )} \sqrt {e x + d}}{e}\right ) e^{\left (-\frac {b c d \log \left (F\right ) - a c e \log \left (F\right )}{e}\right )}}{\sqrt {-b c e \log \left (F\right )} b^{3} c^{3} \log \left (F\right )^{3}} - \frac {2 \, {\left (4 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{2} c^{2} e \log \left (F\right )^{2} - 12 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c^{2} d e \log \left (F\right )^{2} + 12 \, \sqrt {e x + d} b^{2} c^{2} d^{2} e \log \left (F\right )^{2} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} b c e^{2} \log \left (F\right ) + 18 \, \sqrt {e x + d} b c d e^{2} \log \left (F\right ) + 15 \, \sqrt {e x + d} e^{3}\right )} e^{\left (\frac {{\left (e x + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )}{e}\right )}}{b^{3} c^{3} \log \left (F\right )^{3}}}{8 \, e} \]
[In]
[Out]
Timed out. \[ \int F^{c (a+b x)} (d+e x)^{5/2} \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (d+e\,x\right )}^{5/2} \,d x \]
[In]
[Out]