3.2.83 \(\int \frac {x}{\sqrt {a+b x}} \, dx\) [183]

Optimal. Leaf size=32 \[ -\frac {2 a \sqrt {a+b x}}{b^2}+\frac {2 (a+b x)^{3/2}}{3 b^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \begin {gather*} \frac {2 (a+b x)^{3/2}}{3 b^2}-\frac {2 a \sqrt {a+b x}}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + b*x],x]

[Out]

(-2*a*Sqrt[a + b*x])/b^2 + (2*(a + b*x)^(3/2))/(3*b^2)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.72 \begin {gather*} \frac {2 (-2 a+b x) \sqrt {a+b x}}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a + b*x],x]

[Out]

(2*(-2*a + b*x)*Sqrt[a + b*x])/(3*b^2)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(32)=64\).
time = 3.34, size = 73, normalized size = 2.28 \begin {gather*} \frac {2 \sqrt {a} \left (2 a^2 \left (1-\sqrt {\frac {a+b x}{a}}\right )+a b x \left (2-\sqrt {\frac {a+b x}{a}}\right )+b^2 x^2 \sqrt {\frac {a+b x}{a}}\right )}{3 b^2 \left (a+b x\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x/Sqrt[a+b*x],x]')

[Out]

2 Sqrt[a] (2 a ^ 2 (1 - Sqrt[(a + b x) / a]) + a b x (2 - Sqrt[(a + b x) / a]) + b ^ 2 x ^ 2 Sqrt[(a + b x) /
a]) / (3 b ^ 2 (a + b x))

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Maple [A]
time = 0.03, size = 26, normalized size = 0.81

method	result	size
gosper	\(-\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}}\)	\(21\)
trager	\(-\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}}\)	\(21\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}}\)	\(21\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \sqrt {b x +a}}{b^{2}}\)	\(26\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \sqrt {b x +a}}{b^{2}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, size = 26, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, b^{2}} - \frac {2 \, \sqrt {b x + a} a}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.30, size = 19, normalized size = 0.59 \begin {gather*} \frac {2 \, \sqrt {b x + a} {\left (b x - 2 \, a\right )}}{3 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(b*x + a)*(b*x - 2*a)/b^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (29) = 58\)
time = 0.55, size = 162, normalized size = 5.06 \begin {gather*} - \frac {4 a^{\frac {7}{2}} \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {4 a^{\frac {7}{2}}}{3 a^{2} b^{2} + 3 a b^{3} x} - \frac {2 a^{\frac {5}{2}} b x \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {4 a^{\frac {5}{2}} b x}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {2 a^{\frac {3}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*a**(7/2)*sqrt(1 + b*x/a)/(3*a**2*b**2 + 3*a*b**3*x) + 4*a**(7/2)/(3*a**2*b**2 + 3*a*b**3*x) - 2*a**(5/2)*b*
x*sqrt(1 + b*x/a)/(3*a**2*b**2 + 3*a*b**3*x) + 4*a**(5/2)*b*x/(3*a**2*b**2 + 3*a*b**3*x) + 2*a**(3/2)*b**2*x**
2*sqrt(1 + b*x/a)/(3*a**2*b**2 + 3*a*b**3*x)

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Giac [A]
time = 0.00, size = 36, normalized size = 1.12 \begin {gather*} \frac {2 \left (\frac {1}{3} \sqrt {a+b x} \left (a+b x\right )-a \sqrt {a+b x}\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.03, size = 25, normalized size = 0.78 \begin {gather*} -\frac {6\,a\,\sqrt {a+b\,x}-2\,{\left (a+b\,x\right )}^{3/2}}{3\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x)^(1/2),x)

[Out]

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

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