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

Optimal. Leaf size=67 \[ -\frac{16 \sqrt{a-b x}}{3 a^3 \sqrt{x}}+\frac{8}{3 a^2 \sqrt{x} \sqrt{a-b x}}+\frac{2}{3 a \sqrt{x} (a-b x)^{3/2}} \]

[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])

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Rubi [A]  time = 0.009829, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, 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 = 0.0116426, size = 41, normalized size = 0.61 \[ -\frac{2 \left (3 a^2-12 a b x+8 b^2 x^2\right )}{3 a^3 \sqrt{x} (a-b x)^{3/2}} \]

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))

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Maple [A]  time = 0.004, size = 36, normalized size = 0.5 \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

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Maxima [A]  time = 1.11762, size = 68, normalized size = 1.01 \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))

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Fricas [A]  time = 1.85453, size = 130, normalized size = 1.94 \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)

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Sympy [B]  time = 16.0072, size = 318, normalized size = 4.75 \begin{align*} \begin{cases} - \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}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{6 i 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 i 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 i 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}} & \text{otherwise} \end{cases} \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]

Piecewise((-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), Abs(a)/(Abs(b)*Abs(x)) > 1), (-6*I*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*I*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*I*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), True))

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Giac [B]  time = 1.15958, size = 255, normalized size = 3.81 \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} \sqrt{-b} b^{2} - 12 \, a{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} \sqrt{-b} b^{3} + 5 \, a^{2} \sqrt{-b} b^{4}\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*sqrt(-b)*b^2 - 12*a*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2*sqrt(-b)*b^3 + 5*a^2*sqr
t(-b)*b^4)/(((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)^3*a^2*abs(b))