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

Optimal. Leaf size=88 \[ -\frac{\text{Unintegrable}\left (\frac{1}{\left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (c (a+b x)^n\right )},x\right )}{2 \sqrt{-d}}-\frac{\text{Unintegrable}\left (\frac{1}{\left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (c (a+b x)^n\right )},x\right )}{2 \sqrt{-d}} \]

[Out]

-Unintegrable[1/((Sqrt[-d] - Sqrt[e]*x)*Log[c*(a + b*x)^n]), x]/(2*Sqrt[-d]) - Unintegrable[1/((Sqrt[-d] + Sqr
t[e]*x)*Log[c*(a + b*x)^n]), x]/(2*Sqrt[-d])

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Rubi [A]  time = 0.114693, 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 = {} \[ \int \frac{1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.395655, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.838, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( e{x}^{2}+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/(e*x^2+d)/ln(c*(b*x+a)^n),x)

[Out]

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

[Out]

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

[Out]

integral(1/((e*x^2 + d)*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}{\left (d + e x^{2}\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)/ln(c*(b*x+a)**n),x)

[Out]

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

[Out]

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