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

Optimal. Leaf size=61 \[ \frac {x \sqrt {a+b \log \left (c x^n\right )}}{d (d+e x)}-\frac {b n \text {Int}\left (\frac {1}{(d+e x) \sqrt {a+b \log \left (c x^n\right )}},x\right )}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.10, 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 = {} \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.80, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[Sqrt[a + b*Log[c*x^n]]/(d + e*x)^2, 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*x^n))^(1/2)/(e*x+d)^2,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 \[ \int \frac {\sqrt {b \log \left (c x^{n}\right ) + a}}{{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \ln \left (c \,x^{n}\right )+a}}{\left (e x +d \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+b\,\ln \left (c\,x^n\right )}}{{\left (d+e\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*log(c*x**n))/(d + e*x)**2, x)

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