3.272 \(\int F^{a+b (c+d x)^2} (c+d x)^2 \, dx\)

Optimal. Leaf size=77 \[ \frac{(c+d x) F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d \log ^{\frac{3}{2}}(F)} \]

[Out]

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

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Rubi [A]  time = 0.137965, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{(c+d x) F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d \log ^{\frac{3}{2}}(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.23363, size = 66, normalized size = 0.86 \[ - \frac{\sqrt{\pi } F^{a} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{4 b^{\frac{3}{2}} d \log{\left (F \right )}^{\frac{3}{2}}} + \frac{F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )}{2 b d \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0694123, size = 77, normalized size = 1. \[ \frac{(c+d x) F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d \log ^{\frac{3}{2}}(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.042, size = 121, normalized size = 1.6 \[{\frac{x{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{2\,b\ln \left ( F \right ) }}+{\frac{c{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{2\,d\ln \left ( F \right ) b}}+{\frac{\sqrt{\pi }{F}^{a}}{4\,d\ln \left ( F \right ) b}{\it Erf} \left ( -d\sqrt{-b\ln \left ( F \right ) }x+{cb\ln \left ( F \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/ln(F)/b*x*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+1/2/d*c/ln(F)/b*F^(b*d^2*x^2+2*b*c
*d*x+b*c^2+a)+1/4/d/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))

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Maxima [A]  time = 1.03472, size = 699, normalized size = 9.08 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(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)*c*d/(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^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*l
og(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*l
og(F))^2/(b*d^2*log(F)))^(3/2)))*F^(b*c^2 + a)*d^2/(sqrt(b*d^2*log(F))*F^(b*c^2)
) + 1/2*sqrt(pi)*F^(b*c^2 + a)*c^2*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 = 0.282548, size = 123, normalized size = 1.6 \[ -\frac{\sqrt{\pi } F^{a} d \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) - 2 \, \sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{4 \, \sqrt{-b d^{2} \log \left (F\right )} b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23159, size = 126, normalized size = 1.64 \[ \frac{{\left (x + \frac{c}{d}\right )} e^{\left (b d^{2} x^{2}{\rm ln}\left (F\right ) + 2 \, b c d x{\rm ln}\left (F\right ) + b c^{2}{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right )\right )}}{2 \, b{\rm ln}\left (F\right )} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a{\rm ln}\left (F\right )\right )}}{4 \, \sqrt{-b{\rm ln}\left (F\right )} b d{\rm ln}\left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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