3.4.32 \(\int \frac {1}{(d+e x^2) \log (c (a+b x)^n)} \, dx\) [332]

Optimal. Leaf size=90 \[ -\frac {\text {Int}\left (\frac {1}{\left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (c (a+b x)^n\right )},x\right )}{2 \sqrt {-d}}-\frac {\text {Int}\left (\frac {1}{\left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (c (a+b x)^n\right )},x\right )}{2 \sqrt {-d}} \]

[Out]

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

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Rubi [A]
time = 0.08, 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}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \end {gather*}

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 = 0.18, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e \,x^{2}+d \right ) \ln \left (c \left (b x +a \right )^{n}\right )}\, dx\]

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 = 0.00, 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/(e*x^2+d)/log(c*(b*x+a)^n),x, algorithm="maxima")

[Out]

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

[Out]

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

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

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 = 0.00, 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/(e*x^2+d)/log(c*(b*x+a)^n),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\ln \left (c\,{\left (a+b\,x\right )}^n\right )\,\left (e\,x^2+d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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