\(\int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx\) [389]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx=-\frac {F^{a+b (c+d x)^2}}{f (e+f x)}+\frac {\sqrt {b} d F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \sqrt {\log (F)}}{f^2}-\frac {2 b d (d e-c f) \log (F) \text {Int}\left (\frac {F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^2} \]

[Out]

-F^(a+b*(d*x+c)^2)/f/(f*x+e)+d*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*b^(1/2)*Pi^(1/2)*ln(F)^(1/2)/f^2-2*b*d*(-
c*f+d*e)*ln(F)*Unintegrable(F^(a+b*(d*x+c)^2)/(f*x+e),x)/f^2

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx=\int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx=\int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

\[\int \frac {F^{a +b \left (d x +c \right )^{2}}}{\left (f x +e \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.10 \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (f x + e\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx=\int \frac {F^{a + b \left (c + d x\right )^{2}}}{\left (e + f x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (f x + e\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (f x + e\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.60 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx=\int \frac {F^{a+b\,{\left (c+d\,x\right )}^2}}{{\left (e+f\,x\right )}^2} \,d x \]

[In]

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

[Out]

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