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

Optimal. Leaf size=120 \[ \frac{g^2 \text{Unintegrable}\left (\frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{(g h-f i)^2}-\frac{g i \text{Unintegrable}\left (\frac{1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{(g h-f i)^2}-\frac{i \text{Unintegrable}\left (\frac{1}{(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i} \]

[Out]

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

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Rubi [A]  time = 0.224048, 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 = {} \[ \int \frac{1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{1}{(h+241 x)^2 (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=\int \left (\frac{241}{(241 f-g h) (h+241 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac{241 g}{(241 f-g h)^2 (h+241 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac{g^2}{(241 f-g h)^2 (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\right ) \, dx\\ &=-\frac{(241 g) \int \frac{1}{(h+241 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{(241 f-g h)^2}+\frac{g^2 \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{(241 f-g h)^2}+\frac{241 \int \frac{1}{(h+241 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{241 f-g h}\\ \end{align*}

Mathematica [A]  time = 26.6407, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 21.281, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) \left ( ix+h \right ) ^{2} \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)^2/(a+b*ln(c*(e*x+d)^n))^2,x)

[Out]

int(1/(g*x+f)/(i*x+h)^2/(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*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(e*x + d)/(b^2*e*f*h^2*n*log(c) + a*b*e*f*h^2*n + (b^2*e*g*i^2*n*log(c) + a*b*e*g*i^2*n)*x^3 + ((2*g*h*i*n +
f*i^2*n)*b^2*e*log(c) + (2*g*h*i*n + f*i^2*n)*a*b*e)*x^2 + ((g*h^2*n + 2*f*h*i*n)*b^2*e*log(c) + (g*h^2*n + 2*
f*h*i*n)*a*b*e)*x + (b^2*e*g*i^2*n*x^3 + b^2*e*f*h^2*n + (2*g*h*i*n + f*i^2*n)*b^2*e*x^2 + (g*h^2*n + 2*f*h*i*
n)*b^2*e*x)*log((e*x + d)^n)) - integrate((2*e*g*i*x^2 - e*f*h + (g*h + 2*f*i)*d + (e*f*i + 3*d*g*i)*x)/(b^2*e
*f^2*h^3*n*log(c) + a*b*e*f^2*h^3*n + (b^2*e*g^2*i^3*n*log(c) + a*b*e*g^2*i^3*n)*x^5 + ((3*g^2*h*i^2*n + 2*f*g
*i^3*n)*b^2*e*log(c) + (3*g^2*h*i^2*n + 2*f*g*i^3*n)*a*b*e)*x^4 + ((3*g^2*h^2*i*n + 6*f*g*h*i^2*n + f^2*i^3*n)
*b^2*e*log(c) + (3*g^2*h^2*i*n + 6*f*g*h*i^2*n + f^2*i^3*n)*a*b*e)*x^3 + ((g^2*h^3*n + 6*f*g*h^2*i*n + 3*f^2*h
*i^2*n)*b^2*e*log(c) + (g^2*h^3*n + 6*f*g*h^2*i*n + 3*f^2*h*i^2*n)*a*b*e)*x^2 + ((2*f*g*h^3*n + 3*f^2*h^2*i*n)
*b^2*e*log(c) + (2*f*g*h^3*n + 3*f^2*h^2*i*n)*a*b*e)*x + (b^2*e*g^2*i^3*n*x^5 + b^2*e*f^2*h^3*n + (3*g^2*h*i^2
*n + 2*f*g*i^3*n)*b^2*e*x^4 + (3*g^2*h^2*i*n + 6*f*g*h*i^2*n + f^2*i^3*n)*b^2*e*x^3 + (g^2*h^3*n + 6*f*g*h^2*i
*n + 3*f^2*h*i^2*n)*b^2*e*x^2 + (2*f*g*h^3*n + 3*f^2*h^2*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^{2} x^{3} + a^{2} f h^{2} +{\left (2 \, a^{2} g h i + a^{2} f i^{2}\right )} x^{2} +{\left (b^{2} g i^{2} x^{3} + b^{2} f h^{2} +{\left (2 \, b^{2} g h i + b^{2} f i^{2}\right )} x^{2} +{\left (b^{2} g h^{2} + 2 \, b^{2} f h i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} +{\left (a^{2} g h^{2} + 2 \, a^{2} f h i\right )} x + 2 \,{\left (a b g i^{2} x^{3} + a b f h^{2} +{\left (2 \, a b g h i + a b f i^{2}\right )} x^{2} +{\left (a b g h^{2} + 2 \, a b f h 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)^2/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="fricas")

[Out]

integral(1/(a^2*g*i^2*x^3 + a^2*f*h^2 + (2*a^2*g*h*i + a^2*f*i^2)*x^2 + (b^2*g*i^2*x^3 + b^2*f*h^2 + (2*b^2*g*
h*i + b^2*f*i^2)*x^2 + (b^2*g*h^2 + 2*b^2*f*h*i)*x)*log((e*x + d)^n*c)^2 + (a^2*g*h^2 + 2*a^2*f*h*i)*x + 2*(a*
b*g*i^2*x^3 + a*b*f*h^2 + (2*a*b*g*h*i + a*b*f*i^2)*x^2 + (a*b*g*h^2 + 2*a*b*f*h*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)**2/(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 )}^{2}{\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)^2/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="giac")

[Out]

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