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

Optimal. Leaf size=78 \[ \frac{g \text{Unintegrable}\left (\frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i}-\frac{i \text{Unintegrable}\left (\frac{1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i} \]

[Out]

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

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Rubi [A]  time = 0.182707, 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 = {} \[ \int \frac{1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

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 = 12.6754, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

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]

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Maple [A]  time = 15.503, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ix+h \right ) \left ( gx+f \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}}\, dx \end{align*}

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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{e x + d}{b^{2} e f h n \log \left (c\right ) + a b e f h n +{\left (b^{2} e g i n \log \left (c\right ) + a b e g i n\right )} x^{2} +{\left ({\left (g h n + f i n\right )} b^{2} e \log \left (c\right ) +{\left (g h n + f i n\right )} a b e\right )} x +{\left (b^{2} e g i n x^{2} + b^{2} e f h n +{\left (g h n + f i n\right )} b^{2} e x\right )} \log \left ({\left (e x + d\right )}^{n}\right )} - \int \frac{e g i x^{2} + 2 \, d g i x - e f h +{\left (g h + f i\right )} d}{b^{2} e f^{2} h^{2} n \log \left (c\right ) + a b e f^{2} h^{2} n +{\left (b^{2} e g^{2} i^{2} n \log \left (c\right ) + a b e g^{2} i^{2} n\right )} x^{4} + 2 \,{\left ({\left (g^{2} h i n + f g i^{2} n\right )} b^{2} e \log \left (c\right ) +{\left (g^{2} h i n + f g i^{2} n\right )} a b e\right )} x^{3} +{\left ({\left (g^{2} h^{2} n + 4 \, f g h i n + f^{2} i^{2} n\right )} b^{2} e \log \left (c\right ) +{\left (g^{2} h^{2} n + 4 \, f g h i n + f^{2} i^{2} n\right )} a b e\right )} x^{2} + 2 \,{\left ({\left (f g h^{2} n + f^{2} h i n\right )} b^{2} e \log \left (c\right ) +{\left (f g h^{2} n + f^{2} h i n\right )} a b e\right )} x +{\left (b^{2} e g^{2} i^{2} n x^{4} + b^{2} e f^{2} h^{2} n + 2 \,{\left (g^{2} h i n + f g i^{2} n\right )} b^{2} e x^{3} +{\left (g^{2} h^{2} n + 4 \, f g h i n + f^{2} i^{2} n\right )} b^{2} e x^{2} + 2 \,{\left (f g h^{2} n + f^{2} h i n\right )} b^{2} e x\right )} \log \left ({\left (e x + d\right )}^{n}\right )}\,{d x} \end{align*}

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]

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

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

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]

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

[Out]

Timed out

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

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)*(i*x + h)*(b*log((e*x + d)^n*c) + a)^2), x)