3.41 \(\int F^{c (a+b x)} (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=138 \[ \frac{3 \sqrt{\pi } e^{3/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{4 b^{5/2} c^{5/2} \log ^{\frac{5}{2}}(F)}-\frac{3 e \sqrt{d+e x} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac{(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)} \]

[Out]

(3*e^(3/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt
[Log[F]])/Sqrt[e]])/(4*b^(5/2)*c^(5/2)*Log[F]^(5/2)) - (3*e*F^(c*(a + b*x))*Sqrt
[d + e*x])/(2*b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^(3/2))/(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]

(3*e^(3/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt
[Log[F]])/Sqrt[e]])/(4*b^(5/2)*c^(5/2)*Log[F]^(5/2)) - (3*e*F^(c*(a + b*x))*Sqrt
[d + e*x])/(2*b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^(3/2))/(b*c*Log[F])

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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)

[Out]

F**(c*(a + b*x))*(d + e*x)**(3/2)/(b*c*log(F)) - 3*F**(c*(a + b*x))*e*sqrt(d + e
*x)/(2*b**2*c**2*log(F)**2) + 3*sqrt(pi)*F**(c*(a*e - b*d)/e)*e**(3/2)*erfi(sqrt
(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/sqrt(e))/(4*b**(5/2)*c**(5/2)*log(F)**(5/
2))

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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]

[Out]

(F^(c*(a - (b*d)/e))*(-6*b*c*e*F^((b*c*(d + e*x))/e)*(d + e*x)*Log[F] + 4*b^2*c^
2*F^((b*c*(d + e*x))/e)*(d + e*x)^2*Log[F]^2 + 3*e^2*Sqrt[Pi]*Sqrt[-((b*c*(d + e
*x)*Log[F])/e)] - 3*e^2*Sqrt[Pi]*Erf[Sqrt[-((b*c*(d + e*x)*Log[F])/e)]]*Sqrt[-((
b*c*(d + e*x)*Log[F])/e)]))/(4*b^3*c^3*Sqrt[d + e*x]*Log[F]^3)

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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]

int(F^(c*(b*x+a))*(e*x+d)^(3/2),x)

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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")

[Out]

1/4*F^(a*c)*(3*sqrt(pi)*e^2*erf(sqrt(e*x + d)*sqrt(-b*c*log(F)/e))/(sqrt(-b*c*lo
g(F)/e)*F^(b*c*d/e)*b^2*c^2*log(F)^2) + 2*(2*(e*x + d)^(3/2)*b*c*e*log(F) - 3*sq
rt(e*x + d)*e^2)*F^((e*x + d)*b*c/e)/(F^(b*c*d/e)*b^2*c^2*log(F)^2))/e

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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]

1/4*(2*sqrt(e*x + d)*(2*(b*c*e*x + b*c*d)*log(F) - 3*e)*sqrt(-b*c*log(F)/e)*F^(b
*c*x + a*c) + 3*sqrt(pi)*e*erf(sqrt(e*x + d)*sqrt(-b*c*log(F)/e))/F^((b*c*d - a*
c*e)/e))/(sqrt(-b*c*log(F)/e)*b^2*c^2*log(F)^2)

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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]

Timed 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]

1/4*(2*d*(sqrt(pi)*erf(-sqrt(-b*c*e*ln(F))*sqrt(x*e + d)*e^(-1))*e^(-(b*c*d*ln(F
) - a*c*e*ln(F))*e^(-1) + 2)/(sqrt(-b*c*e*ln(F))*b*c*ln(F)) + 2*sqrt(x*e + d)*e^
(((x*e + d)*b*c*ln(F) - b*c*d*ln(F) + a*c*e*ln(F))*e^(-1) + 1)/(b*c*ln(F))) - sq
rt(pi)*(2*b*c*d*e*ln(F) + 3*e^2)*erf(-sqrt(-b*c*e*ln(F))*sqrt(x*e + d)*e^(-1))*e
^(-(b*c*d*ln(F) - a*c*e*ln(F))*e^(-1) + 1)/(sqrt(-b*c*e*ln(F))*b^2*c^2*ln(F)^2)
+ 2*(2*(x*e + d)^(3/2)*b*c*e*ln(F) - 2*sqrt(x*e + d)*b*c*d*e*ln(F) - 3*sqrt(x*e
+ d)*e^2)*e^(((x*e + d)*b*c*ln(F) - b*c*d*ln(F) + a*c*e*ln(F))*e^(-1))/(b^2*c^2*
ln(F)^2))*e^(-1)