3.346 \(\int \frac{1}{(d x+e x^2) \log (c (a+b x)^n)} \, dx\)

Optimal. Leaf size=51 \[ \frac{\text{Unintegrable}\left (\frac{1}{x \log \left (c (a+b x)^n\right )},x\right )}{d}-\frac{e \text{Unintegrable}\left (\frac{1}{(d+e x) \log \left (c (a+b x)^n\right )},x\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.10989, 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 = {} \[ \int \frac{1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.625283, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.825, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( e{x}^{2}+dx \right ) \ln \left ( c \left ( bx+a \right ) ^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((e*x^2 + d*x)*log((b*x + a)^n*c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*(d + e*x)*log(c*(a + b*x)**n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d*x)/log(c*(b*x+a)^n),x, algorithm="giac")

[Out]

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