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

Optimal. Leaf size=113 \[ -\frac{i \text{Unintegrable}\left (\frac{1}{(h+i x)^2 (a+b \log (c (e+f x)))},x\right )}{d (f h-e i)}-\frac{f i \text{Unintegrable}\left (\frac{1}{(h+i x) (a+b \log (c (e+f x)))},x\right )}{d (f h-e i)^2}+\frac{f \log (a+b \log (c (e+f x)))}{b d (f h-e i)^2} \]

[Out]

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

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Rubi [A]  time = 0.294842, 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 = {} \[ \int \frac{1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx \]

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 \operatorname{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 \operatorname{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*}

Mathematica [A]  time = 4.01106, size = 0, normalized size = 0. \[ \int \frac{1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx \]

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 = 0.918, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dfx+de \right ) \left ( ix+h \right ) ^{2} \left ( a+b\ln \left ( c \left ( fx+e \right ) \right ) \right ) }}\, dx \end{align*}

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

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a d f i^{2} x^{3} + a d e h^{2} +{\left (2 \, a d f h i + a d e i^{2}\right )} x^{2} +{\left (a d f h^{2} + 2 \, a d e h i\right )} x +{\left (b d f i^{2} x^{3} + b d e h^{2} +{\left (2 \, b d f h i + b d e i^{2}\right )} x^{2} +{\left (b d f h^{2} + 2 \, b d e h i\right )} x\right )} \log \left (c f x + c e\right )}, x\right ) \end{align*}

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*i^2*x^3 + a*d*e*h^2 + (2*a*d*f*h*i + a*d*e*i^2)*x^2 + (a*d*f*h^2 + 2*a*d*e*h*i)*x + (b*d*f*i
^2*x^3 + b*d*e*h^2 + (2*b*d*f*h*i + b*d*e*i^2)*x^2 + (b*d*f*h^2 + 2*b*d*e*h*i)*x)*log(c*f*x + c*e)), 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/(d*f*x+d*e)/(i*x+h)**2/(a+b*ln(c*(f*x+e))),x)

[Out]

Timed out

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

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