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

Optimal. Leaf size=198 \[ \frac{2 b^2 d^2 \log ^2(F) (d e-c f)^2 \text{Unintegrable}\left (\frac{F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^4}+\frac{b d^2 \log (F) \text{Unintegrable}\left (\frac{F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^2}-\frac{\sqrt{\pi } b^{3/2} d^2 F^a \log ^{\frac{3}{2}}(F) (d e-c f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{f^4}+\frac{b d \log (F) (d e-c f) F^{a+b (c+d x)^2}}{f^3 (e+f x)}-\frac{F^{a+b (c+d x)^2}}{2 f (e+f x)^2} \]

[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]*Uni
ntegrable[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 = 0.313875, 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., Rules used = {} \[ \int \frac{F^{a+b (c+d x)^2}}{(e+f x)^3} \, dx \]

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]*Def
er[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

Rubi steps

\begin{align*} \int \frac{F^{a+b (c+d x)^2}}{(e+f x)^3} \, dx &=-\frac{F^{a+b (c+d x)^2}}{2 f (e+f x)^2}+\frac{\left (b d^2 \log (F)\right ) \int \frac{F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^2}-\frac{(b d (d e-c f) \log (F)) \int \frac{F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx}{f^2}\\ &=-\frac{F^{a+b (c+d x)^2}}{2 f (e+f x)^2}+\frac{b d (d e-c f) F^{a+b (c+d x)^2} \log (F)}{f^3 (e+f x)}+\frac{\left (b d^2 \log (F)\right ) \int \frac{F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^2}-\frac{\left (2 b^2 d^3 (d e-c f) \log ^2(F)\right ) \int F^{a+b (c+d x)^2} \, dx}{f^4}+\frac{\left (2 b^2 d^2 (d e-c f)^2 \log ^2(F)\right ) \int \frac{F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^4}\\ &=-\frac{F^{a+b (c+d x)^2}}{2 f (e+f x)^2}+\frac{b d (d e-c f) F^{a+b (c+d x)^2} \log (F)}{f^3 (e+f x)}-\frac{b^{3/2} d^2 (d e-c f) F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right ) \log ^{\frac{3}{2}}(F)}{f^4}+\frac{\left (b d^2 \log (F)\right ) \int \frac{F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^2}+\frac{\left (2 b^2 d^2 (d e-c f)^2 \log ^2(F)\right ) \int \frac{F^{a+b (c+d x)^2}}{e+f x} \, dx}{f^4}\\ \end{align*}

Mathematica [A]  time = 1.12085, 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 = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{2}}}{ \left ( fx+e \right ) ^{3}}}\, dx \end{align*}

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. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (f x + e\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)/(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. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)/(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. \begin{align*} \text{Timed out} \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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