3.574 \(\int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=44 \[ \frac{4 b \sqrt{a+b x}}{3 a^2 \sqrt{x}}-\frac{2 \sqrt{a+b x}}{3 a x^{3/2}} \]

[Out]

(-2*Sqrt[a + b*x])/(3*a*x^(3/2)) + (4*b*Sqrt[a + b*x])/(3*a^2*Sqrt[x])

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Rubi [A]  time = 0.0047552, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac{4 b \sqrt{a+b x}}{3 a^2 \sqrt{x}}-\frac{2 \sqrt{a+b x}}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x])/(3*a*x^(3/2)) + (4*b*Sqrt[a + b*x])/(3*a^2*Sqrt[x])

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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{1}{x^{5/2} \sqrt{a+b x}} \, dx &=-\frac{2 \sqrt{a+b x}}{3 a x^{3/2}}-\frac{(2 b) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{3 a}\\ &=-\frac{2 \sqrt{a+b x}}{3 a x^{3/2}}+\frac{4 b \sqrt{a+b x}}{3 a^2 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0074476, size = 27, normalized size = 0.61 \[ -\frac{2 (a-2 b x) \sqrt{a+b x}}{3 a^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*Sqrt[a + b*x]),x]

[Out]

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

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Maple [A]  time = 0.003, size = 22, normalized size = 0.5 \begin{align*} -{\frac{-4\,bx+2\,a}{3\,{a}^{2}}\sqrt{bx+a}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16657, size = 42, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (\frac{3 \, \sqrt{b x + a} b}{\sqrt{x}} - \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}}\right )}}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.71568, size = 61, normalized size = 1.39 \begin{align*} \frac{2 \,{\left (2 \, b x - a\right )} \sqrt{b x + a}}{3 \, a^{2} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.21835, size = 42, normalized size = 0.95 \begin{align*} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 a x} + \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07579, size = 68, normalized size = 1.55 \begin{align*} -\frac{\sqrt{b x + a} b{\left (\frac{2 \,{\left (b x + a\right )}}{a^{2} b^{3}} - \frac{3}{a b^{3}}\right )}}{24 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{3}{2}}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*sqrt(b*x + a)*b*(2*(b*x + a)/(a^2*b^3) - 3/(a*b^3))/(((b*x + a)*b - a*b)^(3/2)*abs(b))