3.2.97 \(\int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx\) [197]

Optimal. Leaf size=115 \[ \frac {f \log (a+b \log (c (e+f x)))}{b d (f h-e i)^2}-\frac {i \text {Int}\left (\frac {1}{(h+i x)^2 (a+b \log (c (e+f x)))},x\right )}{d (f h-e i)}-\frac {f i \text {Int}\left (\frac {1}{(h+i x) (a+b \log (c (e+f x)))},x\right )}{d (f h-e i)^2} \]

[Out]

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

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Rubi [A]
time = 0.20, 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}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{(h+197 x)^2 (d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\int \left (\frac {197}{d (197 e-f h) (h+197 x)^2 (a+b \log (c (e+f x)))}-\frac {197 f}{d (197 e-f h)^2 (h+197 x) (a+b \log (c (e+f x)))}+\frac {f^2}{d (197 e-f h)^2 (e+f x) (a+b \log (c (e+f x)))}\right ) \, dx\\ &=-\frac {(197 f) \int \frac {1}{(h+197 x) (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)^2}+\frac {f^2 \int \frac {1}{(e+f x) (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)^2}+\frac {197 \int \frac {1}{(h+197 x)^2 (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)}\\ &=\frac {f \text {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d (197 e-f h)^2}-\frac {(197 f) \int \frac {1}{(h+197 x) (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)^2}+\frac {197 \int \frac {1}{(h+197 x)^2 (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)}\\ &=-\frac {(197 f) \int \frac {1}{(h+197 x) (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)^2}+\frac {f \text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d (197 e-f h)^2}+\frac {197 \int \frac {1}{(h+197 x)^2 (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)}\\ &=\frac {f \log (a+b \log (c (e+f x)))}{b d (197 e-f h)^2}-\frac {(197 f) \int \frac {1}{(h+197 x) (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)^2}+\frac {197 \int \frac {1}{(h+197 x)^2 (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)}\\ \end {align*}

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Mathematica [A]
time = 6.84, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.95, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d f x +e d \right ) \left (i x +h \right )^{2} \left (a +b \ln \left (c \left (f x +e \right )\right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a e h^{2} + 2 a e h i x + a e i^{2} x^{2} + a f h^{2} x + 2 a f h i x^{2} + a f i^{2} x^{3} + b e h^{2} \log {\left (c e + c f x \right )} + 2 b e h i x \log {\left (c e + c f x \right )} + b e i^{2} x^{2} \log {\left (c e + c f x \right )} + b f h^{2} x \log {\left (c e + c f x \right )} + 2 b f h i x^{2} \log {\left (c e + c f x \right )} + b f i^{2} x^{3} \log {\left (c e + c f x \right )}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a*e*h**2 + 2*a*e*h*i*x + a*e*i**2*x**2 + a*f*h**2*x + 2*a*f*h*i*x**2 + a*f*i**2*x**3 + b*e*h**2*lo
g(c*e + c*f*x) + 2*b*e*h*i*x*log(c*e + c*f*x) + b*e*i**2*x**2*log(c*e + c*f*x) + b*f*h**2*x*log(c*e + c*f*x) +
2*b*f*h*i*x**2*log(c*e + c*f*x) + b*f*i**2*x**3*log(c*e + c*f*x)), x)/d

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

[Out]

integrate(1/((d*f*x + d*e)*(b*log((f*x + e)*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 (h+i\,x\right )}^2\,\left (d\,e+d\,f\,x\right )\,\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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