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

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

[Out]

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

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Rubi [A]  time = 0.233583, 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 = {} \[ \int \frac{1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

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])),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{1}{(h+237 x)^2 (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx &=\int \left (\frac{237}{(237 f-g h) (h+237 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{237 g}{(237 f-g h)^2 (h+237 x) \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{g^2}{(237 f-g h)^2 (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx\\ &=-\frac{(237 g) \int \frac{1}{(h+237 x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{(237 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 )} \, dx}{(237 f-g h)^2}+\frac{237 \int \frac{1}{(h+237 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{237 f-g h}\\ \end{align*}

Mathematica [A]  time = 3.06919, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

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])),x]

[Out]

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

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Maple [A]  time = 1.711, 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 ) }}\, 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)),x)

[Out]

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

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Maxima [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 )}}\,{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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a g i^{2} x^{3} + a f h^{2} +{\left (2 \, a g h i + a f i^{2}\right )} x^{2} +{\left (a g h^{2} + 2 \, a f h i\right )} x +{\left (b g i^{2} x^{3} + b f h^{2} +{\left (2 \, b g h i + b f i^{2}\right )} x^{2} +{\left (b g h^{2} + 2 \, 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)),x, algorithm="fricas")

[Out]

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

[Out]

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