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

Optimal. Leaf size=258 \[ -\frac{3 \sqrt{\pi } f^2 F^a (d e-c f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d^4 \log ^{\frac{3}{2}}(F)}-\frac{f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^3 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^4 \sqrt{\log (F)}}+\frac{3 f^2 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac{3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac{f^3 (c+d x)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.712696, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{3 \sqrt{\pi } f^2 F^a (d e-c f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d^4 \log ^{\frac{3}{2}}(F)}-\frac{f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^3 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^4 \sqrt{\log (F)}}+\frac{3 f^2 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac{3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac{f^3 (c+d x)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^2)*(e + f*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 60.9533, size = 241, normalized size = 0.93 \[ - \frac{\sqrt{\pi } F^{a} \left (c f - d e\right )^{3} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{2 \sqrt{b} d^{4} \sqrt{\log{\left (F \right )}}} + \frac{3 \sqrt{\pi } F^{a} f^{2} \left (c f - d e\right ) \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{4 b^{\frac{3}{2}} d^{4} \log{\left (F \right )}^{\frac{3}{2}}} + \frac{F^{a + b \left (c + d x\right )^{2}} f^{3} \left (c + d x\right )^{2}}{2 b d^{4} \log{\left (F \right )}} - \frac{3 F^{a + b \left (c + d x\right )^{2}} f^{2} \left (c + d x\right ) \left (c f - d e\right )}{2 b d^{4} \log{\left (F \right )}} + \frac{3 F^{a + b \left (c + d x\right )^{2}} f \left (c f - d e\right )^{2}}{2 b d^{4} \log{\left (F \right )}} - \frac{F^{a + b \left (c + d x\right )^{2}} f^{3}}{2 b^{2} d^{4} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**2)*(f*x+e)**3,x)

[Out]

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

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Mathematica [A]  time = 0.303604, size = 148, normalized size = 0.57 \[ \frac{F^a \left (2 f F^{b (c+d x)^2} \left (b \log (F) \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )-f^2\right )+\sqrt{\pi } \sqrt{b} \sqrt{\log (F)} (d e-c f) \left (2 b \log (F) (d e-c f)^2-3 f^2\right ) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )\right )}{4 b^2 d^4 \log ^2(F)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^2)*(e + f*x)^3,x]

[Out]

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

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Maple [B]  time = 0.053, size = 582, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^2)*(f*x+e)^3,x)

[Out]

-1/2*e^3*Pi^(1/2)*F^a/d/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b
*ln(F))^(1/2))+1/2*f^3/ln(F)/b/d^2*x^2*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-1/2*f^3*c
/d^3/ln(F)/b*x*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+1/2*f^3*c^2/d^4/ln(F)/b*F^(b*d^2*
x^2+2*b*c*d*x+b*c^2+a)+1/2*f^3*c^3/d^4*Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf(-d*(-b*
ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))-3/4*f^3*c/d^4/ln(F)/b*Pi^(1/2)*F^a/(-
b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))-1/2*f^3/ln(
F)^2/b^2/d^4*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+3/2*e*f^2/ln(F)/b/d^2*x*F^(b*d^2*x^
2+2*b*c*d*x+b*c^2+a)-3/2*e*f^2*c/d^3/ln(F)/b*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-3/2
*e*f^2*c^2/d^3*Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)
/(-b*ln(F))^(1/2))+3/4*e*f^2/ln(F)/b/d^3*Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf(-d*(-
b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))+3/2*e^2*f/ln(F)/b/d^2*F^(b*d^2*x^2+
2*b*c*d*x+b*c^2+a)+3/2*e^2*f*c/d^2*Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F
))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))

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Maxima [A]  time = 0.965483, size = 1160, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*F^((d*x + c)^2*b + a),x, algorithm="maxima")

[Out]

-3/2*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b*c*d*(erf(sqrt(-(b*d^2*x*log(F)
+ b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)/((b*d^2*log(F))^(3/2)*sqrt(-(b*d^
2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - b*d^2*e^((b*d^2*x*log(F) + b*c*d
*log(F))^2/(b*d^2*log(F)))*log(F)/(b*d^2*log(F))^(3/2))*F^(b*c^2 + a)*e^2*f/(sqr
t(b*d^2*log(F))*F^(b*c^2)) + 3/2*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^2*c
^2*d^2*(erf(sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)
^2/((b*d^2*log(F))^(5/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))
) - 2*b^2*c*d^3*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^2/(b
*d^2*log(F))^(5/2) - (b*d^2*x*log(F) + b*c*d*log(F))^3*gamma(3/2, -(b*d^2*x*log(
F) + b*c*d*log(F))^2/(b*d^2*log(F)))/((b*d^2*log(F))^(5/2)*(-(b*d^2*x*log(F) + b
*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)))*F^(b*c^2 + a)*e*f^2/(sqrt(b*d^2*log(F))*F
^(b*c^2)) - 1/2*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^3*c^3*d^3*(erf(sqrt(
-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^3/((b*d^2*log(F)
)^(7/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 3*b^3*c^2*d^4
*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/(b*d^2*log(F))^(7
/2) + b^2*d^4*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)
^2/(b*d^2*log(F))^(7/2) - 3*(b*d^2*x*log(F) + b*c*d*log(F))^3*b*c*d*gamma(3/2, -
(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)/((b*d^2*log(F))^(7/2)*(
-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)))*F^(b*c^2 + a)*f^3/(sq
rt(b*d^2*log(F))*F^(b*c^2)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*e^3*erf(sqrt(-b*log(F))
*d*x - b*c*log(F)/sqrt(-b*log(F)))/(sqrt(-b*log(F))*F^(b*c^2)*d)

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Fricas [A]  time = 0.288504, size = 315, normalized size = 1.22 \[ \frac{\sqrt{\pi }{\left (2 \,{\left (b^{2} d^{4} e^{3} - 3 \, b^{2} c d^{3} e^{2} f + 3 \, b^{2} c^{2} d^{2} e f^{2} - b^{2} c^{3} d f^{3}\right )} \log \left (F\right )^{2} - 3 \,{\left (b d^{2} e f^{2} - b c d f^{3}\right )} \log \left (F\right )\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) - 2 \, \sqrt{-b d^{2} \log \left (F\right )}{\left (f^{3} -{\left (b d^{2} f^{3} x^{2} + 3 \, b d^{2} e^{2} f - 3 \, b c d e f^{2} + b c^{2} f^{3} +{\left (3 \, b d^{2} e f^{2} - b c d f^{3}\right )} x\right )} \log \left (F\right )\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{4 \, \sqrt{-b d^{2} \log \left (F\right )} b^{2} d^{4} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*F^((d*x + c)^2*b + a),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int F^{a + b \left (c + d x\right )^{2}} \left (e + f x\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b*(d*x+c)**2)*(f*x+e)**3,x)

[Out]

Integral(F**(a + b*(c + d*x)**2)*(e + f*x)**3, x)

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GIAC/XCAS [A]  time = 0.248633, size = 578, normalized size = 2.24 \[ -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a{\rm ln}\left (F\right ) + 3\right )}}{2 \, \sqrt{-b{\rm ln}\left (F\right )} d} + \frac{3 \,{\left (\frac{\sqrt{\pi } c f \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a{\rm ln}\left (F\right ) + 2\right )}}{\sqrt{-b{\rm ln}\left (F\right )} d} + \frac{f e^{\left (b d^{2} x^{2}{\rm ln}\left (F\right ) + 2 \, b c d x{\rm ln}\left (F\right ) + b c^{2}{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right ) + 2\right )}}{b d{\rm ln}\left (F\right )}\right )}}{2 \, d} - \frac{3 \,{\left (\frac{\sqrt{\pi }{\left (2 \, b c^{2} f^{2}{\rm ln}\left (F\right ) - f^{2}\right )} \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a{\rm ln}\left (F\right ) + 1\right )}}{\sqrt{-b{\rm ln}\left (F\right )} b d{\rm ln}\left (F\right )} - \frac{2 \,{\left (d f^{2}{\left (x + \frac{c}{d}\right )} - 2 \, c f^{2}\right )} e^{\left (b d^{2} x^{2}{\rm ln}\left (F\right ) + 2 \, b c d x{\rm ln}\left (F\right ) + b c^{2}{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right ) + 1\right )}}{b d{\rm ln}\left (F\right )}\right )}}{4 \, d^{2}} + \frac{\frac{\sqrt{\pi }{\left (2 \, b c^{3} f^{3}{\rm ln}\left (F\right ) - 3 \, c f^{3}\right )} \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a{\rm ln}\left (F\right )\right )}}{\sqrt{-b{\rm ln}\left (F\right )} b d{\rm ln}\left (F\right )} + \frac{2 \,{\left (b d^{2} f^{3}{\left (x + \frac{c}{d}\right )}^{2}{\rm ln}\left (F\right ) - 3 \, b c d f^{3}{\left (x + \frac{c}{d}\right )}{\rm ln}\left (F\right ) + 3 \, b c^{2} f^{3}{\rm ln}\left (F\right ) - f^{3}\right )} e^{\left (b d^{2} x^{2}{\rm ln}\left (F\right ) + 2 \, b c d x{\rm ln}\left (F\right ) + b c^{2}{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right )\right )}}{b^{2} d{\rm ln}\left (F\right )^{2}}}{4 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*F^((d*x + c)^2*b + a),x, algorithm="giac")

[Out]

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