3.3.40 \(\int \frac {1}{(f+g x) (h+i x) (a+b \log (c (d+e x)^n))^2} \, dx\) [240]

Optimal. Leaf size=80 \[ \frac {g \text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i}-\frac {i \text {Int}\left (\frac {1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i} \]

[Out]

g*Unintegrable(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^2,x)/(-f*i+g*h)-i*Unintegrable(1/(i*x+h)/(a+b*ln(c*(e*x+d)^n))^
2,x)/(-f*i+g*h)

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{(h+240 x) (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=\int \left (\frac {240}{(240 f-g h) (h+240 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {g}{(240 f-g h) (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\right ) \, dx\\ &=\frac {240 \int \frac {1}{(h+240 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{240 f-g h}-\frac {g \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{240 f-g h}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 6.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 88.96, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (g x +f \right ) \left (i x +h \right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*e + d)/((b^2*g*n*log(c) + a*b*g*n)*x^2*e + ((-I*g*h*n + f*n)*b^2*log(c) + (-I*g*h*n + f*n)*a*b)*x*e + (-I*
b^2*f*h*n*log(c) - I*a*b*f*h*n)*e + (b^2*g*n*x^2*e - I*b^2*f*h*n*e + (-I*g*h*n + f*n)*b^2*x*e)*log((x*e + d)^n
)) - integrate((g*x^2*e + 2*d*g*x + I*f*h*e + (-I*g*h + f)*d)/((b^2*g^2*n*log(c) + a*b*g^2*n)*x^4*e - 2*((I*g^
2*h*n - f*g*n)*b^2*log(c) + (I*g^2*h*n - f*g*n)*a*b)*x^3*e - ((g^2*h^2*n + 4*I*f*g*h*n - f^2*n)*b^2*log(c) + (
g^2*h^2*n + 4*I*f*g*h*n - f^2*n)*a*b)*x^2*e - 2*((f*g*h^2*n + I*f^2*h*n)*b^2*log(c) + (f*g*h^2*n + I*f^2*h*n)*
a*b)*x*e - (b^2*f^2*h^2*n*log(c) + a*b*f^2*h^2*n)*e + (b^2*g^2*n*x^4*e - b^2*f^2*h^2*n*e - 2*(I*g^2*h*n - f*g*
n)*b^2*x^3*e - (g^2*h^2*n + 4*I*f*g*h*n - f^2*n)*b^2*x^2*e - 2*(f*g*h^2*n + I*f^2*h*n)*b^2*x*e)*log((x*e + d)^
n)), x)

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(I*x*e + ((b^2*g*n^2*x^2 - I*b^2*f*h*n^2 + (-I*b^2*g*h + b^2*f)*n^2*x)*e*log(x*e + d) + (b^2*g*n*x^2 - I*b^2*f
*h*n + (-I*b^2*g*h + b^2*f)*n*x)*e*log(c) + (a*b*g*n*x^2 - I*a*b*f*h*n + (-I*a*b*g*h + a*b*f)*n*x)*e)*integral
((d*g*h + 2*I*d*g*x + I*d*f + (I*g*x^2 - f*h)*e)/((b^2*g^2*n^2*x^4 - b^2*f^2*h^2*n^2 - 2*(I*b^2*g^2*h - b^2*f*
g)*n^2*x^3 - (b^2*g^2*h^2 + 4*I*b^2*f*g*h - b^2*f^2)*n^2*x^2 - 2*(b^2*f*g*h^2 + I*b^2*f^2*h)*n^2*x)*e*log(x*e
+ d) + (b^2*g^2*n*x^4 - b^2*f^2*h^2*n - 2*(I*b^2*g^2*h - b^2*f*g)*n*x^3 - (b^2*g^2*h^2 + 4*I*b^2*f*g*h - b^2*f
^2)*n*x^2 - 2*(b^2*f*g*h^2 + I*b^2*f^2*h)*n*x)*e*log(c) + (a*b*g^2*n*x^4 - a*b*f^2*h^2*n - 2*(I*a*b*g^2*h - a*
b*f*g)*n*x^3 - (a*b*g^2*h^2 + 4*I*a*b*f*g*h - a*b*f^2)*n*x^2 - 2*(a*b*f*g*h^2 + I*a*b*f^2*h)*n*x)*e), x) + I*d
)/((b^2*g*n^2*x^2 - I*b^2*f*h*n^2 + (-I*b^2*g*h + b^2*f)*n^2*x)*e*log(x*e + d) + (b^2*g*n*x^2 - I*b^2*f*h*n +
(-I*b^2*g*h + b^2*f)*n*x)*e*log(c) + (a*b*g*n*x^2 - I*a*b*f*h*n + (-I*a*b*g*h + a*b*f)*n*x)*e)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (f + g x\right ) \left (h + i x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + b*log(c*(d + e*x)**n))**2*(f + g*x)*(h + i*x)), x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (f+g\,x\right )\,\left (h+i\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________