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

Optimal. Leaf size=105 \[ \frac {2 f \text {Int}\left (\frac {1}{\left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (c (a+b x)^n\right )},x\right )}{\sqrt {e^2-4 d f}}-\frac {2 f \text {Int}\left (\frac {1}{\left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (c (a+b x)^n\right )},x\right )}{\sqrt {e^2-4 d f}} \]

[Out]

2*f*Unintegrable(1/ln(c*(b*x+a)^n)/(e+2*f*x-(-4*d*f+e^2)^(1/2)),x)/(-4*d*f+e^2)^(1/2)-2*f*Unintegrable(1/ln(c*
(b*x+a)^n)/(e+2*f*x+(-4*d*f+e^2)^(1/2)),x)/(-4*d*f+e^2)^(1/2)

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Rubi [A]  time = 0.20, 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 = {} \[ \int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

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*}

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Mathematica [A]  time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d+e x+f x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]

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]

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (f x^{2} + e x + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )}, x\right ) \]

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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (f x^{2} + e x + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )}\,{d x} \]

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)

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maple [A]  time = 1.46, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (f \,x^{2}+e x +d \right ) \ln \left (c \left (b x +a \right )^{n}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (f x^{2} + e x + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )}\,{d x} \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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