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 = 0.201524, antiderivative size = 130, 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 \[ \frac{4 \sqrt{\pi } b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F) F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{3 e^{5/2}}-\frac{4 b c \log (F) F^{c (a+b x)}}{3 e^2 \sqrt{d+e x}}-\frac{2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[F^(c*(a + b*x))/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 28.4258, size = 126, normalized size = 0.97 \[ - \frac{4 F^{c \left (a + b x\right )} b c \log{\left (F \right )}}{3 e^{2} \sqrt{d + e x}} - \frac{2 F^{c \left (a + b x\right )}}{3 e \left (d + e x\right )^{\frac{3}{2}}} + \frac{4 \sqrt{\pi } F^{\frac{c \left (a e - b d\right )}{e}} b^{\frac{3}{2}} c^{\frac{3}{2}} \log{\left (F \right )}^{\frac{3}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{c} \sqrt{d + e x} \sqrt{\log{\left (F \right )}}}{\sqrt{e}} \right )}}{3 e^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(c*(b*x+a))/(e*x+d)**(5/2),x)
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Mathematica [A] time = 0.428486, size = 140, normalized size = 1.08 \[ -\frac{2 F^{c \left (a-\frac{b d}{e}\right )} \left (-2 \sqrt{\pi } e \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3/2} \text{Erf}\left (\sqrt{-\frac{b c \log (F) (d+e x)}{e}}\right )+e F^{\frac{b c (d+e x)}{e}}+2 b c \log (F) (d+e x) \left (F^{\frac{b c (d+e x)}{e}}-\sqrt{\pi } \sqrt{-\frac{b c \log (F) (d+e x)}{e}}\right )\right )}{3 e^2 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[F^(c*(a + b*x))/(d + e*x)^(5/2),x]
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Maple [F] time = 0.026, size = 0, normalized size = 0. \[ \int{{F}^{c \left ( bx+a \right ) } \left ( ex+d \right ) ^{-{\frac{5}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(c*(b*x+a))/(e*x+d)^(5/2),x)
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Maxima [A] time = 0.883238, size = 81, normalized size = 0.62 \[ -\frac{\left (-\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )^{\frac{3}{2}} F^{a c} \Gamma \left (-\frac{3}{2}, -\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )}{{\left (e x + d\right )}^{\frac{3}{2}} F^{\frac{b c d}{e}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*c)/(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254141, size = 203, normalized size = 1.56 \[ \frac{2 \,{\left (\frac{2 \, \sqrt{\pi }{\left (b^{2} c^{2} e x + b^{2} c^{2} d\right )} \sqrt{e x + d} \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right ) \log \left (F\right )^{2}}{F^{\frac{b c d - a c e}{e}}} -{\left (e^{2} + 2 \,{\left (b c e^{2} x + b c d e\right )} \log \left (F\right )\right )} \sqrt{-\frac{b c \log \left (F\right )}{e}} F^{b c x + a c}\right )}}{3 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*c)/(e*x + d)^(5/2),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)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*c)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]