3.271 \(\int F^{a+b (c+d x)^2} (c+d x)^4 \, dx\)

Optimal. Leaf size=111 \[ \frac{3 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{8 b^{5/2} d \log ^{\frac{5}{2}}(F)}-\frac{3 (c+d x) F^{a+b (c+d x)^2}}{4 b^2 d \log ^2(F)}+\frac{(c+d x)^3 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]

[Out]

(3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(8*b^(5/2)*d*Log[F]^(5/2))
- (3*F^(a + b*(c + d*x)^2)*(c + d*x))/(4*b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^
2)*(c + d*x)^3)/(2*b*d*Log[F])

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Rubi [A]  time = 0.248928, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{3 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{8 b^{5/2} d \log ^{\frac{5}{2}}(F)}-\frac{3 (c+d x) F^{a+b (c+d x)^2}}{4 b^2 d \log ^2(F)}+\frac{(c+d x)^3 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^2)*(c + d*x)^4,x]

[Out]

(3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(8*b^(5/2)*d*Log[F]^(5/2))
- (3*F^(a + b*(c + d*x)^2)*(c + d*x))/(4*b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^
2)*(c + d*x)^3)/(2*b*d*Log[F])

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Rubi in Sympy [A]  time = 17.3632, size = 100, normalized size = 0.9 \[ \frac{3 \sqrt{\pi } F^{a} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{8 b^{\frac{5}{2}} d \log{\left (F \right )}^{\frac{5}{2}}} + \frac{F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )^{3}}{2 b d \log{\left (F \right )}} - \frac{3 F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )}{4 b^{2} d \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**4,x)

[Out]

3*sqrt(pi)*F**a*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/(8*b**(5/2)*d*log(F)**(5/2)
) + F**(a + b*(c + d*x)**2)*(c + d*x)**3/(2*b*d*log(F)) - 3*F**(a + b*(c + d*x)*
*2)*(c + d*x)/(4*b**2*d*log(F)**2)

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Mathematica [A]  time = 0.137197, size = 88, normalized size = 0.79 \[ \frac{\frac{3 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{b^{5/2} \log ^{\frac{5}{2}}(F)}+\frac{2 (c+d x) F^{a+b (c+d x)^2} \left (2 b \log (F) (c+d x)^2-3\right )}{b^2 \log ^2(F)}}{8 d} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^4,x]

[Out]

((3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(b^(5/2)*Log[F]^(5/2)) +
(2*F^(a + b*(c + d*x)^2)*(c + d*x)*(-3 + 2*b*(c + d*x)^2*Log[F]))/(b^2*Log[F]^2)
)/(8*d)

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Maple [B]  time = 0.053, size = 270, normalized size = 2.4 \[{\frac{{d}^{2}{x}^{3}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{2\,b\ln \left ( F \right ) }}+{\frac{3\,cd{x}^{2}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{2\,b\ln \left ( F \right ) }}+{\frac{3\,{c}^{2}x{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{2\,b\ln \left ( F \right ) }}+{\frac{{c}^{3}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{2\,d\ln \left ( F \right ) b}}-{\frac{3\,c{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{4\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}d}}-{\frac{3\,x{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{4\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}}}-{\frac{3\,\sqrt{\pi }{F}^{a}}{8\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}d}{\it Erf} \left ( -d\sqrt{-b\ln \left ( F \right ) }x+{cb\ln \left ( F \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^2)*(d*x+c)^4,x)

[Out]

1/2*d^2/ln(F)/b*x^3*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+3/2*d*c/ln(F)/b*x^2*F^(b*d^2
*x^2+2*b*c*d*x+b*c^2+a)+3/2*c^2/ln(F)/b*x*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+1/2/d*
c^3/ln(F)/b*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-3/4/d*c/ln(F)^2/b^2*F^(b*d^2*x^2+2*b
*c*d*x+b*c^2+a)-3/4/ln(F)^2/b^2*x*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-3/8/d/ln(F)^2/
b^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 = 1.15663, size = 1755, normalized size = 15.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^4*F^((d*x + c)^2*b + a),x, algorithm="maxima")

[Out]

-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*l
og(F))^2/(b*d^2*log(F)))*log(F)/(b*d^2*log(F))^(3/2))*F^(b*c^2 + a)*c^3*d/(sqrt(
b*d^2*log(F))*F^(b*c^2)) + 3*(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)*c^2*d^2/(sqrt(b*d^2*log(F))*F^(
b*c^2)) - 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)*c*d^3/(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^4*c^
4*d^4*(erf(sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^
4/((b*d^2*log(F))^(9/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F))))
- 4*b^4*c^3*d^5*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^4/(
b*d^2*log(F))^(9/2) + 4*b^3*c*d^5*gamma(2, -(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))^(9/2) - 6*(b*d^2*x*log(F) + b*c*d*log(F))^
3*b^2*c^2*d^2*gamma(3/2, -(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))^(9/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(
3/2)) - (b*d^2*x*log(F) + b*c*d*log(F))^5*gamma(5/2, -(b*d^2*x*log(F) + b*c*d*lo
g(F))^2/(b*d^2*log(F)))/((b*d^2*log(F))^(9/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^
2/(b*d^2*log(F)))^(5/2)))*F^(b*c^2 + a)*d^4/(sqrt(b*d^2*log(F))*F^(b*c^2)) + 1/2
*sqrt(pi)*F^(b*c^2 + a)*c^4*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 = 0.26962, size = 177, normalized size = 1.59 \[ \frac{3 \, \sqrt{\pi } F^{a} d \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) - 2 \, \sqrt{-b d^{2} \log \left (F\right )}{\left (3 \, d x - 2 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) + 3 \, c\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{8 \, \sqrt{-b d^{2} \log \left (F\right )} b^{2} d \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^4*F^((d*x + c)^2*b + a),x, algorithm="fricas")

[Out]

1/8*(3*sqrt(pi)*F^a*d*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d) - 2*sqrt(-b*d^2*log(F
))*(3*d*x - 2*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(F) + 3*c)*F^
(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(sqrt(-b*d^2*log(F))*b^2*d*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**(a+b*(d*x+c)**2)*(d*x+c)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265232, size = 153, normalized size = 1.38 \[ \frac{{\left (2 \, b d^{2}{\left (x + \frac{c}{d}\right )}^{3}{\rm ln}\left (F\right ) - 3 \, x - \frac{3 \, c}{d}\right )} e^{\left (b d^{2} x^{2}{\rm ln}\left (F\right ) + 2 \, b c d x{\rm ln}\left (F\right ) + b c^{2}{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right )\right )}}{4 \, b^{2}{\rm ln}\left (F\right )^{2}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a{\rm ln}\left (F\right )\right )}}{8 \, \sqrt{-b{\rm ln}\left (F\right )} b^{2} d{\rm ln}\left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^4*F^((d*x + c)^2*b + a),x, algorithm="giac")

[Out]

1/4*(2*b*d^2*(x + c/d)^3*ln(F) - 3*x - 3*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^2*ln(F)^2) - 3/8*sqrt(pi)*erf(-sqrt(-b*ln(F))*d*(
x + c/d))*e^(a*ln(F))/(sqrt(-b*ln(F))*b^2*d*ln(F)^2)