Optimal. Leaf size=105 \[ -\frac {\sqrt {e} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)} \]
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Rubi [A]
time = 0.07, antiderivative size = 105, 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 = {2207, 2211,
2235} \begin {gather*} \frac {\sqrt {d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac {\sqrt {\pi } \sqrt {e} 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 )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2211
Rule 2235
Rubi steps
\begin {align*} \int F^{c (a+b x)} \sqrt {d+e x} \, dx &=\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)}-\frac {e \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}} \, dx}{2 b c \log (F)}\\ &=\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)}-\frac {\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 )}{b c \log (F)}\\ &=-\frac {\sqrt {e} 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 )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 63, normalized size = 0.60 \begin {gather*} -\frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^{3/2} \Gamma \left (\frac {3}{2},-\frac {b c (d+e x) \log (F)}{e}\right )}{e \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \sqrt {e x +d}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 91, normalized size = 0.87 \begin {gather*} \frac {2 \, \sqrt {x e + d} F^{b c x + a c} b c \log \left (F\right ) + \frac {\sqrt {\pi } \sqrt {-b c e^{\left (-1\right )} \log \left (F\right )} \operatorname {erf}\left (\sqrt {-b c e^{\left (-1\right )} \log \left (F\right )} \sqrt {x e + d}\right ) e}{F^{{\left (b c d - a c e\right )} e^{\left (-1\right )}}}}{2 \, b^{2} c^{2} \log \left (F\right )^{2}} \end {gather*}
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 \begin {gather*} \int F^{c \left (a + b x\right )} \sqrt {d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 198 vs.
\(2 (83) = 166\).
time = 4.24, size = 198, normalized size = 1.89 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, \sqrt {\pi } d \operatorname {erf}\left (-\sqrt {-b c e \log \left (F\right )} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \left (F\right )}} - \frac {\sqrt {\pi } {\left (2 \, b c d \log \left (F\right ) + e\right )} \operatorname {erf}\left (-\sqrt {-b c e \log \left (F\right )} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \left (F\right )} b c \log \left (F\right )} - \frac {2 \, \sqrt {x e + d} e^{\left ({\left ({\left (x e + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{b c \log \left (F\right )}\right )} e^{\left (-1\right )} \end {gather*}
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 \begin {gather*} \int F^{c\,\left (a+b\,x\right )}\,\sqrt {d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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