Optimal. Leaf size=138 \[ \frac {3 \sqrt {\pi } e^{3/2} F^{c \left (a-\frac {b d}{e}\right )} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{4 b^{5/2} c^{5/2} \log ^{\frac {5}{2}}(F)}-\frac {3 e \sqrt {d+e x} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac {(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)} \]
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Rubi [A] time = 0.12, antiderivative size = 138, 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 = {2176, 2180, 2204} \[ \frac {3 \sqrt {\pi } e^{3/2} F^{c \left (a-\frac {b d}{e}\right )} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{4 b^{5/2} c^{5/2} \log ^{\frac {5}{2}}(F)}-\frac {3 e \sqrt {d+e x} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac {(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2180
Rule 2204
Rubi steps
\begin {align*} \int F^{c (a+b x)} (d+e x)^{3/2} \, dx &=\frac {F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}-\frac {(3 e) \int F^{c (a+b x)} \sqrt {d+e x} \, dx}{2 b c \log (F)}\\ &=-\frac {3 e F^{c (a+b x)} \sqrt {d+e x}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}+\frac {\left (3 e^2\right ) \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}} \, dx}{4 b^2 c^2 \log ^2(F)}\\ &=-\frac {3 e F^{c (a+b x)} \sqrt {d+e x}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}+\frac {(3 e) \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 )}{2 b^2 c^2 \log ^2(F)}\\ &=\frac {3 e^{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 )}{4 b^{5/2} c^{5/2} \log ^{\frac {5}{2}}(F)}-\frac {3 e F^{c (a+b x)} \sqrt {d+e x}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 63, normalized size = 0.46 \[ -\frac {(d+e x)^{5/2} F^{c \left (a-\frac {b d}{e}\right )} \Gamma \left (\frac {5}{2},-\frac {b c (d+e x) \log (F)}{e}\right )}{e \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 121, normalized size = 0.88 \[ -\frac {\frac {3 \, \sqrt {\pi } \sqrt {-\frac {b c \log \relax (F)}{e}} e^{2} \operatorname {erf}\left (\sqrt {e x + d} \sqrt {-\frac {b c \log \relax (F)}{e}}\right )}{F^{\frac {b c d - a c e}{e}}} + 2 \, {\left (3 \, b c e \log \relax (F) - 2 \, {\left (b^{2} c^{2} e x + b^{2} c^{2} d\right )} \log \relax (F)^{2}\right )} \sqrt {e x + d} F^{b c x + a c}}{4 \, b^{3} c^{3} \log \relax (F)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.91, size = 401, normalized size = 2.91 \[ -\frac {1}{4} \, {\left (\frac {4 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (-\sqrt {-b c e \log \relax (F)} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \relax (F) - a c e \log \relax (F)\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \relax (F)}} - 4 \, d {\left (\frac {\sqrt {\pi } {\left (2 \, b c d \log \relax (F) + e\right )} \operatorname {erf}\left (-\sqrt {-b c e \log \relax (F)} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \relax (F) - a c e \log \relax (F)\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \relax (F)} b c \log \relax (F)} + \frac {2 \, \sqrt {x e + d} e^{\left ({\left ({\left (x e + d\right )} b c \log \relax (F) - b c d \log \relax (F) + a c e \log \relax (F)\right )} e^{\left (-1\right )} + 1\right )}}{b c \log \relax (F)}\right )} + {\left (\frac {\sqrt {\pi } {\left (4 \, b^{2} c^{2} d^{2} \log \relax (F)^{2} + 4 \, b c d e \log \relax (F) + 3 \, e^{2}\right )} \operatorname {erf}\left (-\sqrt {-b c e \log \relax (F)} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \relax (F) - a c e \log \relax (F) + 2 \, e\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \relax (F)} b^{2} c^{2} \log \relax (F)^{2}} - \frac {2 \, {\left (2 \, {\left (x e + d\right )}^{\frac {3}{2}} b c e \log \relax (F) - 4 \, \sqrt {x e + d} b c d e \log \relax (F) - 3 \, \sqrt {x e + d} e^{2}\right )} e^{\left ({\left ({\left (x e + d\right )} b c \log \relax (F) - b c d \log \relax (F) + a c e \log \relax (F) - 2 \, e\right )} e^{\left (-1\right )}\right )}}{b^{2} c^{2} \log \relax (F)^{2}}\right )} e^{2}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{\frac {3}{2}} F^{\left (b x +a \right ) c}\, dx \]
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 \[ \int {\left (e x + d\right )}^{\frac {3}{2}} F^{{\left (b x + a\right )} c}\,{d x} \]
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 \[ \int F^{c\,\left (a+b\,x\right )}\,{\left (d+e\,x\right )}^{3/2} \,d x \]
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 \[ \int F^{c \left (a + b x\right )} \left (d + e x\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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