3.110 \(\int \frac {\sqrt {a+b \log (c (d+e x)^n)}}{(f+g x)^3} \, dx\)

Optimal. Leaf size=79 \[ \frac {b e n \text {Int}\left (\frac {1}{(d+e x) (f+g x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}},x\right )}{4 g}-\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 g (f+g x)^2} \]

[Out]

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

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Rubi [A]  time = 0.22, 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 = {} \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^3,x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^3} \, dx &=-\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 g (f+g x)^2}+\frac {(b e n) \int \frac {1}{(d+e x) (f+g x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{4 g}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^3,x]

[Out]

Integrate[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^3, x]

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^(1/2)/(g*x+f)^3,x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^(1/2)/(g*x+f)^3,x, algorithm="giac")

[Out]

integrate(sqrt(b*log((e*x + d)^n*c) + a)/(g*x + f)^3, x)

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maple [A]  time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \ln \left (c \left (e x +d \right )^{n}\right )+a}}{\left (g x +f \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*(e*x+d)^n)+a)^(1/2)/(g*x+f)^3,x)

[Out]

int((b*ln(c*(e*x+d)^n)+a)^(1/2)/(g*x+f)^3,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^(1/2)/(g*x+f)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(b*log((e*x + d)^n*c) + a)/(g*x + f)^3, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}}{{\left (f+g\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x)^n))^(1/2)/(f + g*x)^3,x)

[Out]

int((a + b*log(c*(d + e*x)^n))^(1/2)/(f + g*x)^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}{\left (f + g x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))**(1/2)/(g*x+f)**3,x)

[Out]

Integral(sqrt(a + b*log(c*(d + e*x)**n))/(f + g*x)**3, x)

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