3.21.15 \(\int \frac {-12+e^{\frac {1}{4} (40+\log (2 x))} x}{4 x^2} \, dx\)

Optimal. Leaf size=25 \[ -e^8+\frac {3}{x}+\sqrt [4]{2} e^{10} \sqrt [4]{x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 14} \begin {gather*} \sqrt [4]{2} e^{10} \sqrt [4]{x}+\frac {3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12 + E^((40 + Log[2*x])/4)*x)/(4*x^2),x]

[Out]

3/x + 2^(1/4)*E^10*x^(1/4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {-12+e^{\frac {1}{4} (40+\log (2 x))} x}{x^2} \, dx\\ &=\frac {1}{4} \int \left (-\frac {12}{x^2}+\frac {\sqrt [4]{2} e^{10}}{x^{3/4}}\right ) \, dx\\ &=\frac {3}{x}+\sqrt [4]{2} e^{10} \sqrt [4]{x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{4} \left (\frac {12}{x}+4 \sqrt [4]{2} e^{10} \sqrt [4]{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 + E^((40 + Log[2*x])/4)*x)/(4*x^2),x]

[Out]

(12/x + 4*2^(1/4)*E^10*x^(1/4))/4

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 15, normalized size = 0.60 \begin {gather*} \frac {2^{\frac {1}{4}} x^{\frac {5}{4}} e^{10} + 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(x*exp(1/4*log(2*x)+10)-12)/x^2,x, algorithm="fricas")

[Out]

(2^(1/4)*x^(5/4)*e^10 + 3)/x

________________________________________________________________________________________

giac [A]  time = 0.26, size = 15, normalized size = 0.60 \begin {gather*} \frac {3}{x} + e^{\left (\frac {1}{4} \, \log \left (2 \, x\right ) + 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(x*exp(1/4*log(2*x)+10)-12)/x^2,x, algorithm="giac")

[Out]

3/x + e^(1/4*log(2*x) + 10)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 16, normalized size = 0.64

method	result	size
default	\({\mathrm e}^{\frac {\ln \left (2 x \right )}{4}+10}+\frac {3}{x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(x*exp(1/4*ln(2*x)+10)-12)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(1/4*ln(2*x)+10)+3/x

________________________________________________________________________________________

maxima [A]  time = 0.48, size = 15, normalized size = 0.60 \begin {gather*} 2^{\frac {1}{4}} x^{\frac {1}{4}} e^{10} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(x*exp(1/4*log(2*x)+10)-12)/x^2,x, algorithm="maxima")

[Out]

2^(1/4)*x^(1/4)*e^10 + 3/x

________________________________________________________________________________________

mupad [B]  time = 1.17, size = 15, normalized size = 0.60 \begin {gather*} \frac {3}{x}+2^{1/4}\,x^{1/4}\,{\mathrm {e}}^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x*exp(log(2*x)/4 + 10))/4 - 3)/x^2,x)

[Out]

3/x + 2^(1/4)*x^(1/4)*exp(10)

________________________________________________________________________________________

sympy [A]  time = 2.30, size = 15, normalized size = 0.60 \begin {gather*} \sqrt [4]{2} \sqrt [4]{x} e^{10} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(x*exp(1/4*ln(2*x)+10)-12)/x**2,x)

[Out]

2**(1/4)*x**(1/4)*exp(10) + 3/x

________________________________________________________________________________________