Optimal. Leaf size=97 \[ \frac {2 \sqrt {\pi } \sqrt {b} \sqrt {c} \sqrt {\log (F)} F^{c \left (a-\frac {b d}{e}\right )} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {2 F^{c (a+b x)}}{e \sqrt {d+e x}} \]
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Rubi [A] time = 0.09, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2177, 2180, 2204} \[ \frac {2 \sqrt {\pi } \sqrt {b} \sqrt {c} \sqrt {\log (F)} F^{c \left (a-\frac {b d}{e}\right )} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {2 F^{c (a+b x)}}{e \sqrt {d+e x}} \]
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)^{3/2}} \, dx &=-\frac {2 F^{c (a+b x)}}{e \sqrt {d+e x}}+\frac {(2 b c \log (F)) \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}} \, dx}{e}\\ &=-\frac {2 F^{c (a+b x)}}{e \sqrt {d+e x}}+\frac {(4 b c \log (F)) \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 )}{e^2}\\ &=-\frac {2 F^{c (a+b x)}}{e \sqrt {d+e x}}+\frac {2 \sqrt {b} \sqrt {c} 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 ) \sqrt {\log (F)}}{e^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 75, normalized size = 0.77 \[ -\frac {2 \left (F^{c (a+b x)}-F^{c \left (a-\frac {b d}{e}\right )} \sqrt {-\frac {b c \log (F) (d+e x)}{e}} \Gamma \left (\frac {1}{2},-\frac {b c (d+e x) \log (F)}{e}\right )\right )}{e \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 90, normalized size = 0.93 \[ -\frac {2 \, {\left (\frac {\sqrt {\pi } {\left (e x + d\right )} \sqrt {-\frac {b c \log \relax (F)}{e}} \operatorname {erf}\left (\sqrt {e x + d} \sqrt {-\frac {b c \log \relax (F)}{e}}\right )}{F^{\frac {b c d - a c e}{e}}} + \sqrt {e x + d} F^{b c x + a c}\right )}}{e^{2} x + d e} \]
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 \[ \int \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {F^{\left (b x +a \right ) c}}{\left (e x +d \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {F^{c \left (a + b x\right )}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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