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

Optimal. Leaf size=103 \[ \frac{2 f \text{Unintegrable}\left (\frac{1}{\left (-\sqrt{e^2-4 d f}+e+2 f x\right ) \log \left (c (a+b x)^n\right )},x\right )}{\sqrt{e^2-4 d f}}-\frac{2 f \text{Unintegrable}\left (\frac{1}{\left (\sqrt{e^2-4 d f}+e+2 f x\right ) \log \left (c (a+b x)^n\right )},x\right )}{\sqrt{e^2-4 d f}} \]

[Out]

(2*f*Unintegrable[1/((e - Sqrt[e^2 - 4*d*f] + 2*f*x)*Log[c*(a + b*x)^n]), x])/Sqrt[e^2 - 4*d*f] - (2*f*Uninteg
rable[1/((e + Sqrt[e^2 - 4*d*f] + 2*f*x)*Log[c*(a + b*x)^n]), x])/Sqrt[e^2 - 4*d*f]

________________________________________________________________________________________

Rubi [A]  time = 0.199032, 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 = {} \[ \int \frac{1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(2*f*Defer[Int][1/((e - Sqrt[e^2 - 4*d*f] + 2*f*x)*Log[c*(a + b*x)^n]), x])/Sqrt[e^2 - 4*d*f] - (2*f*Defer[Int
][1/((e + Sqrt[e^2 - 4*d*f] + 2*f*x)*Log[c*(a + b*x)^n]), x])/Sqrt[e^2 - 4*d*f]

Rubi steps

\begin{align*} \int \frac{1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx &=\int \left (\frac{2 f}{\sqrt{e^2-4 d f} \left (e-\sqrt{e^2-4 d f}+2 f x\right ) \log \left (c (a+b x)^n\right )}-\frac{2 f}{\sqrt{e^2-4 d f} \left (e+\sqrt{e^2-4 d f}+2 f x\right ) \log \left (c (a+b x)^n\right )}\right ) \, dx\\ &=\frac{(2 f) \int \frac{1}{\left (e-\sqrt{e^2-4 d f}+2 f x\right ) \log \left (c (a+b x)^n\right )} \, dx}{\sqrt{e^2-4 d f}}-\frac{(2 f) \int \frac{1}{\left (e+\sqrt{e^2-4 d f}+2 f x\right ) \log \left (c (a+b x)^n\right )} \, dx}{\sqrt{e^2-4 d f}}\\ \end{align*}

Mathematica [A]  time = 0.565202, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 1.796, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( f{x}^{2}+ex+d \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/(f*x^2+e*x+d)/ln(c*(b*x+a)^n),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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