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

Optimal. Leaf size=200 \[ \frac{2 b^2 d^2 \log ^2(F) (d e-c f)^2 \text{Int}\left (\frac{F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^4}+\frac{b d^2 \log (F) \text{Int}\left (\frac{F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^2}-\frac{\sqrt{\pi } b^{3/2} d^2 F^a \log ^{\frac{3}{2}}(F) (d e-c f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{f^4}+\frac{b d \log (F) (d e-c f) F^{a+b (c+d x)^2}}{f^3 (e+f x)}-\frac{F^{a+b (c+d x)^2}}{2 f (e+f x)^2} \]

[Out]

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

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Rubi [A]  time = 0.5096, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{F^{a+b (c+d x)^2}}{(e+f x)^3},x\right ) \]

Verification is Not applicable to the result.

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

[Out]

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

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Rubi in Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{\pi } F^{a} b^{\frac{3}{2}} d^{2} \left (c f - d e\right ) \log{\left (F \right )}^{\frac{3}{2}} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{f^{4}} - \frac{F^{a + b \left (c + d x\right )^{2}} b d \left (c f - d e\right ) \log{\left (F \right )}}{f^{3} \left (e + f x\right )} - \frac{F^{a + b \left (c + d x\right )^{2}}}{2 f \left (e + f x\right )^{2}} + \frac{2 b^{2} d^{2} \left (c f - d e\right )^{2} \log{\left (F \right )}^{2} \int \frac{F^{a + b \left (c + d x\right )^{2}}}{e + f x}\, dx}{f^{4}} + \frac{b d^{2} \log{\left (F \right )} \int \frac{F^{a + b \left (c + d x\right )^{2}}}{e + f x}\, dx}{f^{2}} \]

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*b**(3/2)*d**2*(c*f - d*e)*log(F)**(3/2)*erfi(sqrt(b)*(c + d*x)*sqr
t(log(F)))/f**4 - F**(a + b*(c + d*x)**2)*b*d*(c*f - d*e)*log(F)/(f**3*(e + f*x)
) - F**(a + b*(c + d*x)**2)/(2*f*(e + f*x)**2) + 2*b**2*d**2*(c*f - d*e)**2*log(
F)**2*Integral(F**(a + b*(c + d*x)**2)/(e + f*x), x)/f**4 + b*d**2*log(F)*Integr
al(F**(a + b*(c + d*x)**2)/(e + f*x), x)/f**2

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Mathematica [A]  time = 1.28949, size = 0, normalized size = 0. \[ \int \frac{F^{a+b (c+d x)^2}}{(e+f x)^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{2}}}{ \left ( fx+e \right ) ^{3}}}\, dx \]

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)/(f*x+e)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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