3.544 \(\int x (a+b \log (c (d+\frac {e}{\sqrt {x}})))^p \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left (x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.16, size = 0, normalized size = 0.00 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (\frac {c d x + c e \sqrt {x}}{x}\right ) + a\right )}^{p} x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.12, size = 0, normalized size = 0.00 \[ \int x \left (b \ln \left (\left (d +\frac {e}{\sqrt {x}}\right ) c \right )+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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