3.2.71 \(\int \frac {\sqrt {b x^{2/3}+a x}}{x} \, dx\) [171]

Optimal. Leaf size=23 \[ \frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{a x} \]

[Out]

2*(b*x^(2/3)+a*x)^(3/2)/a/x

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Rubi [A]
time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2039} \begin {gather*} \frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^(2/3) + a*x]/x,x]

[Out]

(2*(b*x^(2/3) + a*x)^(3/2))/(a*x)

Rule 2039

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-c^(j - 1))*(c*x)^(m - j
+ 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {\sqrt {b x^{2/3}+a x}}{x} \, dx &=\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{a x}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^(2/3) + a*x]/x,x]

[Out]

(2*(b*x^(2/3) + a*x)^(3/2))/(a*x)

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Maple [A]
time = 0.36, size = 27, normalized size = 1.17

method	result	size
derivativedivides	\(\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (b +a \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{3}} a}\)	\(27\)
default	\(\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (b +a \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{3}} a}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^(2/3)+a*x)^(1/2)/x,x,method=_RETURNVERBOSE)

[Out]

2*(b*x^(2/3)+a*x)^(1/2)/x^(1/3)*(b+a*x^(1/3))/a

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Maxima [F]
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((b*x^(2/3)+a*x)^(1/2)/x,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + b*x^(2/3))/x, x)

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Fricas [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((b*x^(2/3)+a*x)^(1/2)/x,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**(2/3)+a*x)**(1/2)/x,x)

[Out]

Integral(sqrt(a*x + b*x**(2/3))/x, x)

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Giac [A]
time = 1.51, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}}}{a} - \frac {2 \, b^{\frac {3}{2}}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(2/3)+a*x)^(1/2)/x,x, algorithm="giac")

[Out]

2*(a*x^(1/3) + b)^(3/2)/a - 2*b^(3/2)/a

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {a\,x+b\,x^{2/3}}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^(2/3))^(1/2)/x,x)

[Out]

int((a*x + b*x^(2/3))^(1/2)/x, x)

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