3.263 \(\int \frac{1}{(a f h+b g h x^2+h (b f x+a g x)) (A+B \log (e (a+b x)^n (c+d x)^{-n}))^2} \, dx\)

Optimal. Leaf size=82 \[ \frac{\operatorname{Subst}\left (\text{Unintegrable}\left (\frac{1}{(a+b x) (f+g x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2},x\right ),e \left (\frac{a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right )}{h} \]

[Out]

Defer[Subst][Unintegrable[1/((a + b*x)*(f + g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x], e*((a + b*x)/(
c + d*x))^n, (e*(a + b*x)^n)/(c + d*x)^n]/h

________________________________________________________________________________________

Rubi [A]  time = 0.459884, 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 = {} \[ \int \frac{1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((a*f*h + b*g*h*x^2 + h*(b*f*x + a*g*x))*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2),x]

[Out]

(b*Defer[Int][1/((a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2), x])/((b*f - a*g)*h) - (g*Defer[Int][1/
((f + g*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2), x])/((b*f - a*g)*h)

Rubi steps

\begin{align*} \int \frac{1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx &=\int \frac{1}{h (a+b x) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx\\ &=\frac{\int \frac{1}{(a+b x) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx}{h}\\ &=\frac{\int \left (\frac{b}{(b f-a g) (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}-\frac{g}{(b f-a g) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}\right ) \, dx}{h}\\ &=\frac{b \int \frac{1}{(a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx}{(b f-a g) h}-\frac{g \int \frac{1}{(f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx}{(b f-a g) h}\\ \end{align*}

Mathematica [A]  time = 0.268337, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((a*f*h + b*g*h*x^2 + h*(b*f*x + a*g*x))*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2),x]

[Out]

Integrate[1/((a*f*h + b*g*h*x^2 + h*(b*f*x + a*g*x))*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2), x]

________________________________________________________________________________________

Maple [A]  time = 4.622, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{afh+bgh{x}^{2}+h \left ( agx+bxf \right ) } \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x))/(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2,x)

[Out]

int(1/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x))/(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2,x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (d f - c g\right )} \int \frac{1}{{\left (b c f^{2} h n - a d f^{2} h n\right )} A B +{\left (b c f^{2} h n \log \left (e\right ) - a d f^{2} h n \log \left (e\right )\right )} B^{2} +{\left ({\left (b c g^{2} h n - a d g^{2} h n\right )} A B +{\left (b c g^{2} h n \log \left (e\right ) - a d g^{2} h n \log \left (e\right )\right )} B^{2}\right )} x^{2} + 2 \,{\left ({\left (b c f g h n - a d f g h n\right )} A B +{\left (b c f g h n \log \left (e\right ) - a d f g h n \log \left (e\right )\right )} B^{2}\right )} x +{\left ({\left (b c g^{2} h n - a d g^{2} h n\right )} B^{2} x^{2} + 2 \,{\left (b c f g h n - a d f g h n\right )} B^{2} x +{\left (b c f^{2} h n - a d f^{2} h n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left ({\left (b c g^{2} h n - a d g^{2} h n\right )} B^{2} x^{2} + 2 \,{\left (b c f g h n - a d f g h n\right )} B^{2} x +{\left (b c f^{2} h n - a d f^{2} h n\right )} B^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )}\,{d x} - \frac{d x + c}{{\left (b c f h n - a d f h n\right )} A B +{\left (b c f h n \log \left (e\right ) - a d f h n \log \left (e\right )\right )} B^{2} +{\left ({\left (b c g h n - a d g h n\right )} A B +{\left (b c g h n \log \left (e\right ) - a d g h n \log \left (e\right )\right )} B^{2}\right )} x +{\left ({\left (b c g h n - a d g h n\right )} B^{2} x +{\left (b c f h n - a d f h n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left ({\left (b c g h n - a d g h n\right )} B^{2} x +{\left (b c f h n - a d f h n\right )} B^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x))/(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2,x, algorithm="maxima")

[Out]

(d*f - c*g)*integrate(1/((b*c*f^2*h*n - a*d*f^2*h*n)*A*B + (b*c*f^2*h*n*log(e) - a*d*f^2*h*n*log(e))*B^2 + ((b
*c*g^2*h*n - a*d*g^2*h*n)*A*B + (b*c*g^2*h*n*log(e) - a*d*g^2*h*n*log(e))*B^2)*x^2 + 2*((b*c*f*g*h*n - a*d*f*g
*h*n)*A*B + (b*c*f*g*h*n*log(e) - a*d*f*g*h*n*log(e))*B^2)*x + ((b*c*g^2*h*n - a*d*g^2*h*n)*B^2*x^2 + 2*(b*c*f
*g*h*n - a*d*f*g*h*n)*B^2*x + (b*c*f^2*h*n - a*d*f^2*h*n)*B^2)*log((b*x + a)^n) - ((b*c*g^2*h*n - a*d*g^2*h*n)
*B^2*x^2 + 2*(b*c*f*g*h*n - a*d*f*g*h*n)*B^2*x + (b*c*f^2*h*n - a*d*f^2*h*n)*B^2)*log((d*x + c)^n)), x) - (d*x
+ c)/((b*c*f*h*n - a*d*f*h*n)*A*B + (b*c*f*h*n*log(e) - a*d*f*h*n*log(e))*B^2 + ((b*c*g*h*n - a*d*g*h*n)*A*B
+ (b*c*g*h*n*log(e) - a*d*g*h*n*log(e))*B^2)*x + ((b*c*g*h*n - a*d*g*h*n)*B^2*x + (b*c*f*h*n - a*d*f*h*n)*B^2)
*log((b*x + a)^n) - ((b*c*g*h*n - a*d*g*h*n)*B^2*x + (b*c*f*h*n - a*d*f*h*n)*B^2)*log((d*x + c)^n))

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{A^{2} b g h x^{2} + A^{2} a f h +{\left (A^{2} b f + A^{2} a g\right )} h x +{\left (B^{2} b g h x^{2} + B^{2} a f h +{\left (B^{2} b f + B^{2} a g\right )} h x\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \,{\left (A B b g h x^{2} + A B a f h +{\left (A B b f + A B a g\right )} h x\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x))/(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2,x, algorithm="fricas")

[Out]

integral(1/(A^2*b*g*h*x^2 + A^2*a*f*h + (A^2*b*f + A^2*a*g)*h*x + (B^2*b*g*h*x^2 + B^2*a*f*h + (B^2*b*f + B^2*
a*g)*h*x)*log((b*x + a)^n*e/(d*x + c)^n)^2 + 2*(A*B*b*g*h*x^2 + A*B*a*f*h + (A*B*b*f + A*B*a*g)*h*x)*log((b*x
+ a)^n*e/(d*x + c)^n)), x)

________________________________________________________________________________________

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(1/(a*f*h+b*g*h*x**2+h*(a*g*x+b*f*x))/(A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b g h x^{2} + a f h +{\left (b f x + a g x\right )} h\right )}{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x))/(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2,x, algorithm="giac")

[Out]

integrate(1/((b*g*h*x^2 + a*f*h + (b*f*x + a*g*x)*h)*(B*log((b*x + a)^n*e/(d*x + c)^n) + A)^2), x)