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

Optimal. Leaf size=64 \[ -\frac{16 \sqrt{a+b x}}{3 a^3 \sqrt{x}}+\frac{8}{3 a^2 \sqrt{x} \sqrt{a+b x}}+\frac{2}{3 a \sqrt{x} (a+b x)^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0091767, antiderivative size = 64, 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} \[ -\frac{16 \sqrt{a+b x}}{3 a^3 \sqrt{x}}+\frac{8}{3 a^2 \sqrt{x} \sqrt{a+b x}}+\frac{2}{3 a \sqrt{x} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(a + b*x)^(5/2)),x]

[Out]

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

Mathematica [A]  time = 0.014205, size = 40, normalized size = 0.62 \[ -\frac{2 \left (3 a^2+12 a b x+8 b^2 x^2\right )}{3 a^3 \sqrt{x} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3*a^2 + 12*a*b*x + 8*b^2*x^2))/(3*a^3*Sqrt[x]*(a + b*x)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 35, normalized size = 0.6 \begin{align*} -{\frac{16\,{b}^{2}{x}^{2}+24\,abx+6\,{a}^{2}}{3\,{a}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(8*b^2*x^2+12*a*b*x+3*a^2)/x^(1/2)/(b*x+a)^(3/2)/a^3

________________________________________________________________________________________

Maxima [A]  time = 1.1227, size = 62, normalized size = 0.97 \begin{align*} \frac{2 \,{\left (b^{2} - \frac{6 \,{\left (b x + a\right )} b}{x}\right )} x^{\frac{3}{2}}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}} - \frac{2 \, \sqrt{b x + a}}{a^{3} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.10661, size = 128, normalized size = 2. \begin{align*} -\frac{2 \,{\left (8 \, b^{2} x^{2} + 12 \, a b x + 3 \, a^{2}\right )} \sqrt{b x + a} \sqrt{x}}{3 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(8*b^2*x^2 + 12*a*b*x + 3*a^2)*sqrt(b*x + a)*sqrt(x)/(a^3*b^2*x^3 + 2*a^4*b*x^2 + a^5*x)

________________________________________________________________________________________

Sympy [B]  time = 17.4037, size = 153, normalized size = 2.39 \begin{align*} - \frac{6 a^{2} b^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac{24 a b^{\frac{11}{2}} x \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac{16 b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.11258, size = 215, normalized size = 3.36 \begin{align*} -\frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{{\left (b x + a\right )} b - a b} a^{3}{\left | b \right |}} - \frac{4 \,{\left (3 \,{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{5}{2}} + 12 \, a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{7}{2}} + 5 \, a^{2} b^{\frac{9}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{2}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b*x + a)*b^2/(sqrt((b*x + a)*b - a*b)*a^3*abs(b)) - 4/3*(3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b -
a*b))^4*b^(5/2) + 12*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(7/2) + 5*a^2*b^(9/2))/(((sqrt(b
*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*a^2*abs(b))