3.1.51 \(\int \sqrt [4]{x} \log (x) \, dx\) [51]

Optimal. Leaf size=21 \[ -\frac {16 x^{5/4}}{25}+\frac {4}{5} x^{5/4} \log (x) \]

[Out]

-16/25*x^(5/4)+4/5*x^(5/4)*ln(x)

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2341} \begin {gather*} \frac {4}{5} x^{5/4} \log (x)-\frac {16 x^{5/4}}{25} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(1/4)*Log[x],x]

[Out]

(-16*x^(5/4))/25 + (4*x^(5/4)*Log[x])/5

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=-\frac {16 x^{5/4}}{25}+\frac {4}{5} x^{5/4} \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 0.71 \begin {gather*} \frac {4}{25} x^{5/4} (-4+5 \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(1/4)*Log[x],x]

[Out]

(4*x^(5/4)*(-4 + 5*Log[x]))/25

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Maple [A]
time = 0.04, size = 14, normalized size = 0.67

method	result	size
derivativedivides	\(-\frac {16 x^{\frac {5}{4}}}{25}+\frac {4 x^{\frac {5}{4}} \ln \left (x \right )}{5}\)	\(14\)
default	\(-\frac {16 x^{\frac {5}{4}}}{25}+\frac {4 x^{\frac {5}{4}} \ln \left (x \right )}{5}\)	\(14\)
risch	\(-\frac {16 x^{\frac {5}{4}}}{25}+\frac {4 x^{\frac {5}{4}} \ln \left (x \right )}{5}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/4)*ln(x),x,method=_RETURNVERBOSE)

[Out]

-16/25*x^(5/4)+4/5*x^(5/4)*ln(x)

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Maxima [A]
time = 0.41, size = 13, normalized size = 0.62 \begin {gather*} \frac {4}{5} \, x^{\frac {5}{4}} \log \left (x\right ) - \frac {16}{25} \, x^{\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/4)*log(x),x, algorithm="maxima")

[Out]

4/5*x^(5/4)*log(x) - 16/25*x^(5/4)

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Fricas [A]
time = 0.59, size = 14, normalized size = 0.67 \begin {gather*} \frac {4}{25} \, {\left (5 \, x \log \left (x\right ) - 4 \, x\right )} x^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/4)*log(x),x, algorithm="fricas")

[Out]

4/25*(5*x*log(x) - 4*x)*x^(1/4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (19) = 38\).
time = 2.27, size = 105, normalized size = 5.00 \begin {gather*} \begin {cases} - \frac {4 x^{\frac {5}{4}} \log {\left (\frac {1}{x} \right )}}{5} + \frac {4 x^{\frac {5}{4}} \log {\left (x \right )}}{5} - \frac {32 x^{\frac {5}{4}}}{25} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\frac {4 x^{\frac {5}{4}} \log {\left (x \right )}}{5} - \frac {16 x^{\frac {5}{4}}}{25} & \text {for}\: \left |{x}\right | < 1 \\- \frac {4 x^{\frac {5}{4}} \log {\left (\frac {1}{x} \right )}}{5} - \frac {16 x^{\frac {5}{4}}}{25} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{3, 3}^{2, 1}\left (\begin {matrix} 1 & \frac {9}{4}, \frac {9}{4} \\\frac {5}{4}, \frac {5}{4} & 0 \end {matrix} \middle | {x} \right )} + {G_{3, 3}^{0, 3}\left (\begin {matrix} \frac {9}{4}, \frac {9}{4}, 1 & \\ & \frac {5}{4}, \frac {5}{4}, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/4)*ln(x),x)

[Out]

Piecewise((-4*x**(5/4)*log(1/x)/5 + 4*x**(5/4)*log(x)/5 - 32*x**(5/4)/25, (Abs(x) < 1) & (1/Abs(x) < 1)), (4*x
**(5/4)*log(x)/5 - 16*x**(5/4)/25, Abs(x) < 1), (-4*x**(5/4)*log(1/x)/5 - 16*x**(5/4)/25, 1/Abs(x) < 1), (-mei
jerg(((1,), (9/4, 9/4)), ((5/4, 5/4), (0,)), x) + meijerg(((9/4, 9/4, 1), ()), ((), (5/4, 5/4, 0)), x), True))

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Giac [A]
time = 0.42, size = 13, normalized size = 0.62 \begin {gather*} \frac {4}{5} \, x^{\frac {5}{4}} \log \left (x\right ) - \frac {16}{25} \, x^{\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/4)*log(x),x, algorithm="giac")

[Out]

4/5*x^(5/4)*log(x) - 16/25*x^(5/4)

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Mupad [B]
time = 0.03, size = 9, normalized size = 0.43 \begin {gather*} \frac {4\,x^{5/4}\,\left (\ln \left (x\right )-\frac {4}{5}\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/4)*log(x),x)

[Out]

(4*x^(5/4)*(log(x) - 4/5))/5

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Chatgpt [F] Failed to verify
time = 1.00, size = 13, normalized size = 0.62 \begin {gather*} \frac {4 \ln \left (x \right ) x^{\frac {5}{4}}}{5}-\frac {8 x^{\frac {5}{4}}}{15} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(x^(1/4)*ln(x),x)

[Out]

4/5*ln(x)*x^(5/4)-8/15*x^(5/4)

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