3.6.86 \(\int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx\) [586]

Optimal. Leaf size=21 \[ \frac {2 x^{3/2}}{3 a (a+b x)^{3/2}} \]

[Out]

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

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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \begin {gather*} \frac {2 x^{3/2}}{3 a (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + b*x)^(5/2),x]

[Out]

(2*x^(3/2))/(3*a*(a + b*x)^(3/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin {align*} \int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx &=\frac {2 x^{3/2}}{3 a (a+b x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} \frac {2 x^{3/2}}{3 a (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + b*x)^(5/2),x]

[Out]

(2*x^(3/2))/(3*a*(a + b*x)^(3/2))

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Mathics [A]
time = 2.55, size = 26, normalized size = 1.24 \begin {gather*} \frac {2 x^{\frac {3}{2}}}{3 a^{\frac {3}{2}} \sqrt {\frac {a+b x}{a}} \left (a+b x\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[x]/(a + b*x)^(5/2),x]')

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(15)=30\).
time = 0.12, size = 54, normalized size = 2.57

method	result	size
gosper	\(\frac {2 x^{\frac {3}{2}}}{3 a \left (b x +a \right )^{\frac {3}{2}}}\)	\(16\)
default	\(-\frac {\sqrt {x}}{b \left (b x +a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {2 \sqrt {x}}{3 a \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {b x +a}}\right )}{2 b}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/b*x^(1/2)/(b*x+a)^(3/2)+1/2*a/b*(2/3*x^(1/2)/a/(b*x+a)^(3/2)+4/3*x^(1/2)/a^2/(b*x+a)^(1/2))

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Maxima [A]
time = 0.25, size = 15, normalized size = 0.71 \begin {gather*} \frac {2 \, x^{\frac {3}{2}}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

2/3*x^(3/2)/((b*x + a)^(3/2)*a)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).
time = 0.30, size = 33, normalized size = 1.57 \begin {gather*} \frac {2 \, \sqrt {b x + a} x^{\frac {3}{2}}}{3 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (17) = 34\).
time = 0.78, size = 42, normalized size = 2.00 \begin {gather*} \frac {2 x^{\frac {3}{2}}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {3}{2}} b x \sqrt {1 + \frac {b x}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)/(b*x+a)**(5/2),x)

[Out]

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

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Giac [A]
time = 0.01, size = 42, normalized size = 2.00 \begin {gather*} \frac {\frac {1}{18}\cdot 12 b \sqrt {x} \sqrt {x} \sqrt {x} \sqrt {a+b x}}{b a \left (a+b x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(b*x+a)^(5/2),x)

[Out]

2/3*x^(3/2)/((b*x + a)^(3/2)*a)

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Mupad [B]
time = 0.24, size = 36, normalized size = 1.71 \begin {gather*} \frac {2\,x^{3/2}\,\sqrt {a+b\,x}}{3\,\left (a^3+2\,a^2\,b\,x+a\,b^2\,x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a + b*x)^(5/2),x)

[Out]

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

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