3.3.37 \(\int \frac {1}{(f+g x) (h+i x)^2 (a+b \log (c (d+e x)^n))} \, dx\) [237]

Optimal. Leaf size=123 \[ \frac {g^2 \text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{(g h-f i)^2}-\frac {i \text {Int}\left (\frac {1}{(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{g h-f i}-\frac {g i \text {Int}\left (\frac {1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{(g h-f i)^2} \]

[Out]

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

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Rubi [A]
time = 0.16, 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)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 1.75, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \end {gather*}

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 = 0.34, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (g x +f \right ) \left (i x +h \right )^{2} \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}\, dx\]

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 = 0.00, 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)^2/(a+b*log(c*(e*x+d)^n)),x, algorithm="maxima")

[Out]

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

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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)^2/(a+b*log(c*(e*x+d)^n)),x, algorithm="fricas")

[Out]

integral(-1/(a*g*x^3 - a*f*h^2 + (-2*I*a*g*h + a*f)*x^2 - (a*g*h^2 + 2*I*a*f*h)*x + (b*g*n*x^3 - b*f*h^2*n + (
-2*I*b*g*h + b*f)*n*x^2 - (b*g*h^2 + 2*I*b*f*h)*n*x)*log(x*e + d) + (b*g*x^3 - b*f*h^2 + (-2*I*b*g*h + b*f)*x^
2 - (b*g*h^2 + 2*I*b*f*h)*x)*log(c)), x)

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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 ) \left (f + g x\right ) \left (h + i x\right )^{2}}\, dx \end {gather*}

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]

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

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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)^2/(a+b*log(c*(e*x+d)^n)),x, algorithm="giac")

[Out]

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

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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 )}^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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