3.582 \(\int \frac{1}{x^{5/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=63 \[ -\frac{8 \sqrt{a+b x}}{3 a^2 x^{3/2}}+\frac{16 b \sqrt{a+b x}}{3 a^3 \sqrt{x}}+\frac{2}{a x^{3/2} \sqrt{a+b x}} \]

[Out]

2/(a*x^(3/2)*Sqrt[a + b*x]) - (8*Sqrt[a + b*x])/(3*a^2*x^(3/2)) + (16*b*Sqrt[a + b*x])/(3*a^3*Sqrt[x])

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Rubi [A]  time = 0.0098531, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{8 \sqrt{a+b x}}{3 a^2 x^{3/2}}+\frac{16 b \sqrt{a+b x}}{3 a^3 \sqrt{x}}+\frac{2}{a x^{3/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(a + b*x)^(3/2)),x]

[Out]

2/(a*x^(3/2)*Sqrt[a + b*x]) - (8*Sqrt[a + b*x])/(3*a^2*x^(3/2)) + (16*b*Sqrt[a + b*x])/(3*a^3*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} (a+b x)^{3/2}} \, dx &=\frac{2}{a x^{3/2} \sqrt{a+b x}}+\frac{4 \int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx}{a}\\ &=\frac{2}{a x^{3/2} \sqrt{a+b x}}-\frac{8 \sqrt{a+b x}}{3 a^2 x^{3/2}}-\frac{(8 b) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{3 a^2}\\ &=\frac{2}{a x^{3/2} \sqrt{a+b x}}-\frac{8 \sqrt{a+b x}}{3 a^2 x^{3/2}}+\frac{16 b \sqrt{a+b x}}{3 a^3 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0094522, size = 38, normalized size = 0.6 \[ -\frac{2 \left (a^2-4 a b x-8 b^2 x^2\right )}{3 a^3 x^{3/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(a^2 - 4*a*b*x - 8*b^2*x^2))/(3*a^3*x^(3/2)*Sqrt[a + b*x])

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Maple [A]  time = 0.005, size = 33, normalized size = 0.5 \begin{align*} -{\frac{-16\,{b}^{2}{x}^{2}-8\,abx+2\,{a}^{2}}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(b*x+a)^(3/2),x)

[Out]

-2/3*(-8*b^2*x^2-4*a*b*x+a^2)/x^(3/2)/(b*x+a)^(1/2)/a^3

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Maxima [A]  time = 1.16904, size = 68, normalized size = 1.08 \begin{align*} \frac{2 \, b^{2} \sqrt{x}}{\sqrt{b x + a} a^{3}} + \frac{2 \,{\left (\frac{6 \, \sqrt{b x + a} b}{\sqrt{x}} - \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}}\right )}}{3 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.11559, size = 104, normalized size = 1.65 \begin{align*} \frac{2 \,{\left (8 \, b^{2} x^{2} + 4 \, a b x - a^{2}\right )} \sqrt{b x + a} \sqrt{x}}{3 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

2/3*(8*b^2*x^2 + 4*a*b*x - a^2)*sqrt(b*x + a)*sqrt(x)/(a^3*b*x^3 + a^4*x^2)

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Sympy [B]  time = 16.9072, size = 219, normalized size = 3.48 \begin{align*} - \frac{2 a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{6 a^{2} b^{\frac{11}{2}} x \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{24 a b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{16 b^{\frac{15}{2}} x^{3} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**3*b**(9/2)*sqrt(a/(b*x) + 1)/(3*a**5*b**4*x + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) + 6*a**2*b**(11/2)*x*
sqrt(a/(b*x) + 1)/(3*a**5*b**4*x + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) + 24*a*b**(13/2)*x**2*sqrt(a/(b*x) + 1
)/(3*a**5*b**4*x + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) + 16*b**(15/2)*x**3*sqrt(a/(b*x) + 1)/(3*a**5*b**4*x +
6*a**4*b**5*x**2 + 3*a**3*b**6*x**3)

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Giac [A]  time = 1.09024, size = 126, normalized size = 2. \begin{align*} -\frac{\sqrt{b x + a}{\left (\frac{5 \,{\left (b x + a\right )}{\left | b \right |}}{b^{2}} - \frac{6 \, a{\left | b \right |}}{b^{2}}\right )}}{24 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{3}{2}}} + \frac{4 \, b^{\frac{7}{2}}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{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)^(3/2),x, algorithm="giac")

[Out]

-1/24*sqrt(b*x + a)*(5*(b*x + a)*abs(b)/b^2 - 6*a*abs(b)/b^2)/((b*x + a)*b - a*b)^(3/2) + 4*b^(7/2)/(((sqrt(b*
x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*a^2*abs(b))