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 = 0.19832, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{3 \sqrt{\pi } e^{3/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{4 b^{5/2} c^{5/2} \log ^{\frac{5}{2}}(F)}-\frac{3 e \sqrt{d+e x} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac{(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)} \]
Antiderivative was successfully verified.
[In] Int[F^(c*(a + b*x))*(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 31.8156, size = 131, normalized size = 0.95 \[ \frac{F^{c \left (a + b x\right )} \left (d + e x\right )^{\frac{3}{2}}}{b c \log{\left (F \right )}} - \frac{3 F^{c \left (a + b x\right )} e \sqrt{d + e x}}{2 b^{2} c^{2} \log{\left (F \right )}^{2}} + \frac{3 \sqrt{\pi } F^{\frac{c \left (a e - b d\right )}{e}} e^{\frac{3}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{c} \sqrt{d + e x} \sqrt{\log{\left (F \right )}}}{\sqrt{e}} \right )}}{4 b^{\frac{5}{2}} c^{\frac{5}{2}} \log{\left (F \right )}^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(c*(b*x+a))*(e*x+d)**(3/2),x)
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Mathematica [A] time = 0.332962, size = 169, normalized size = 1.22 \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \left (4 b^2 c^2 \log ^2(F) (d+e x)^2 F^{\frac{b c (d+e x)}{e}}-3 \sqrt{\pi } e^2 \sqrt{-\frac{b c \log (F) (d+e x)}{e}} \text{Erf}\left (\sqrt{-\frac{b c \log (F) (d+e x)}{e}}\right )+3 \sqrt{\pi } e^2 \sqrt{-\frac{b c \log (F) (d+e x)}{e}}-6 b c e \log (F) (d+e x) F^{\frac{b c (d+e x)}{e}}\right )}{4 b^3 c^3 \log ^3(F) \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[F^(c*(a + b*x))*(d + e*x)^(3/2),x]
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Maple [F] time = 0.026, size = 0, normalized size = 0. \[ \int{F}^{c \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(c*(b*x+a))*(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 0.816172, size = 181, normalized size = 1.31 \[ \frac{F^{a c}{\left (\frac{3 \, \sqrt{\pi } e^{2} \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{\sqrt{-\frac{b c \log \left (F\right )}{e}} F^{\frac{b c d}{e}} b^{2} c^{2} \log \left (F\right )^{2}} + \frac{2 \,{\left (2 \,{\left (e x + d\right )}^{\frac{3}{2}} b c e \log \left (F\right ) - 3 \, \sqrt{e x + d} e^{2}\right )} F^{\frac{{\left (e x + d\right )} b c}{e}}}{F^{\frac{b c d}{e}} b^{2} c^{2} \log \left (F\right )^{2}}\right )}}{4 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)*F^((b*x + a)*c),x, algorithm="maxima")
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Fricas [A] time = 0.269744, size = 157, normalized size = 1.14 \[ \frac{2 \, \sqrt{e x + d}{\left (2 \,{\left (b c e x + b c d\right )} \log \left (F\right ) - 3 \, e\right )} \sqrt{-\frac{b c \log \left (F\right )}{e}} F^{b c x + a c} + \frac{3 \, \sqrt{\pi } e \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{F^{\frac{b c d - a c e}{e}}}}{4 \, \sqrt{-\frac{b c \log \left (F\right )}{e}} b^{2} c^{2} \log \left (F\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)*F^((b*x + a)*c),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(c*(b*x+a))*(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.259327, size = 408, normalized size = 2.96 \[ \frac{1}{4} \,{\left (2 \, d{\left (\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b c e{\rm ln}\left (F\right )} \sqrt{x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d{\rm ln}\left (F\right ) - a c e{\rm ln}\left (F\right )\right )} e^{\left (-1\right )} + 2\right )}}{\sqrt{-b c e{\rm ln}\left (F\right )} b c{\rm ln}\left (F\right )} + \frac{2 \, \sqrt{x e + d} e^{\left ({\left ({\left (x e + d\right )} b c{\rm ln}\left (F\right ) - b c d{\rm ln}\left (F\right ) + a c e{\rm ln}\left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{b c{\rm ln}\left (F\right )}\right )} - \frac{\sqrt{\pi }{\left (2 \, b c d e{\rm ln}\left (F\right ) + 3 \, e^{2}\right )} \operatorname{erf}\left (-\sqrt{-b c e{\rm ln}\left (F\right )} \sqrt{x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d{\rm ln}\left (F\right ) - a c e{\rm ln}\left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt{-b c e{\rm ln}\left (F\right )} b^{2} c^{2}{\rm ln}\left (F\right )^{2}} + \frac{2 \,{\left (2 \,{\left (x e + d\right )}^{\frac{3}{2}} b c e{\rm ln}\left (F\right ) - 2 \, \sqrt{x e + d} b c d e{\rm ln}\left (F\right ) - 3 \, \sqrt{x e + d} e^{2}\right )} e^{\left ({\left ({\left (x e + d\right )} b c{\rm ln}\left (F\right ) - b c d{\rm ln}\left (F\right ) + a c e{\rm ln}\left (F\right )\right )} e^{\left (-1\right )}\right )}}{b^{2} c^{2}{\rm ln}\left (F\right )^{2}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)*F^((b*x + a)*c),x, algorithm="giac")
[Out]