Optimal. Leaf size=130 \[ \frac {4 \sqrt {\pi } b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F) F^{c \left (a-\frac {b d}{e}\right )} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{3 e^{5/2}}-\frac {4 b c \log (F) F^{c (a+b x)}}{3 e^2 \sqrt {d+e x}}-\frac {2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2177, 2180, 2204} \[ \frac {4 \sqrt {\pi } b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F) F^{c \left (a-\frac {b d}{e}\right )} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{3 e^{5/2}}-\frac {4 b c \log (F) F^{c (a+b x)}}{3 e^2 \sqrt {d+e x}}-\frac {2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2180
Rule 2204
Rubi steps
\begin {align*} \int \frac {F^{c (a+b x)}}{(d+e x)^{5/2}} \, dx &=-\frac {2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}}+\frac {(2 b c \log (F)) \int \frac {F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx}{3 e}\\ &=-\frac {2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}}-\frac {4 b c F^{c (a+b x)} \log (F)}{3 e^2 \sqrt {d+e x}}+\frac {\left (4 b^2 c^2 \log ^2(F)\right ) \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}} \, dx}{3 e^2}\\ &=-\frac {2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}}-\frac {4 b c F^{c (a+b x)} \log (F)}{3 e^2 \sqrt {d+e x}}+\frac {\left (8 b^2 c^2 \log ^2(F)\right ) \operatorname {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 )}{3 e^3}\\ &=-\frac {2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}}-\frac {4 b c F^{c (a+b x)} \log (F)}{3 e^2 \sqrt {d+e x}}+\frac {4 b^{3/2} c^{3/2} 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 ) \log ^{\frac {3}{2}}(F)}{3 e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 92, normalized size = 0.71 \[ -\frac {2 \left (F^{c (a+b x)} (2 b c \log (F) (d+e x)+e)+2 e F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {b c (d+e x) \log (F)}{e}\right )\right )}{3 e^2 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 140, normalized size = 1.08 \[ -\frac {2 \, {\left (\frac {2 \, \sqrt {\pi } {\left (b c e^{2} x^{2} + 2 \, b c d e x + b c d^{2}\right )} \sqrt {-\frac {b c \log \relax (F)}{e}} \operatorname {erf}\left (\sqrt {e x + d} \sqrt {-\frac {b c \log \relax (F)}{e}}\right ) \log \relax (F)}{F^{\frac {b c d - a c e}{e}}} + \sqrt {e x + d} {\left (2 \, {\left (b c e x + b c d\right )} \log \relax (F) + e\right )} F^{b c x + a c}\right )}}{3 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]
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 \[ \int \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {F^{\left (b x +a \right ) c}}{\left (e x +d \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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