3.6.37 \(\int \frac {(a+b \log (c (d+e \sqrt {x})))^p}{x^2} \, dx\) [537]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{x^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, 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 {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

2*Defer[Subst][Defer[Int][(a + b*Log[c*(d + e*x)])^p/x^3, x], x, Sqrt[x]]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{x^2} \, dx &=2 \text {Subst}\left (\int \frac {(a+b \log (c (d+e x)))^p}{x^3} \, dx,x,\sqrt {x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,\left (d+e\,\sqrt {x}\right )\right )\right )}^p}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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