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

Optimal. Leaf size=88 \[ -\frac{16 \sqrt{a-b x}}{3 a^3 x^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a-b x}}-\frac{32 b \sqrt{a-b x}}{3 a^4 \sqrt{x}}+\frac{2}{3 a x^{3/2} (a-b x)^{3/2}} \]

[Out]

2/(3*a*x^(3/2)*(a - b*x)^(3/2)) + 4/(a^2*x^(3/2)*Sqrt[a - b*x]) - (16*Sqrt[a - b*x])/(3*a^3*x^(3/2)) - (32*b*S
qrt[a - b*x])/(3*a^4*Sqrt[x])

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Rubi [A]  time = 0.0163789, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {45, 37} \[ -\frac{16 \sqrt{a-b x}}{3 a^3 x^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a-b x}}-\frac{32 b \sqrt{a-b x}}{3 a^4 \sqrt{x}}+\frac{2}{3 a x^{3/2} (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(a - b*x)^(5/2)),x]

[Out]

2/(3*a*x^(3/2)*(a - b*x)^(3/2)) + 4/(a^2*x^(3/2)*Sqrt[a - b*x]) - (16*Sqrt[a - b*x])/(3*a^3*x^(3/2)) - (32*b*S
qrt[a - b*x])/(3*a^4*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)^{5/2}} \, dx &=\frac{2}{3 a x^{3/2} (a-b x)^{3/2}}+\frac{2 \int \frac{1}{x^{5/2} (a-b x)^{3/2}} \, dx}{a}\\ &=\frac{2}{3 a x^{3/2} (a-b x)^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a-b x}}+\frac{8 \int \frac{1}{x^{5/2} \sqrt{a-b x}} \, dx}{a^2}\\ &=\frac{2}{3 a x^{3/2} (a-b x)^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a-b x}}-\frac{16 \sqrt{a-b x}}{3 a^3 x^{3/2}}+\frac{(16 b) \int \frac{1}{x^{3/2} \sqrt{a-b x}} \, dx}{3 a^3}\\ &=\frac{2}{3 a x^{3/2} (a-b x)^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a-b x}}-\frac{16 \sqrt{a-b x}}{3 a^3 x^{3/2}}-\frac{32 b \sqrt{a-b x}}{3 a^4 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0135244, size = 50, normalized size = 0.57 \[ -\frac{2 \left (6 a^2 b x+a^3-24 a b^2 x^2+16 b^3 x^3\right )}{3 a^4 x^{3/2} (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a^3 + 6*a^2*b*x - 24*a*b^2*x^2 + 16*b^3*x^3))/(3*a^4*x^(3/2)*(a - b*x)^(3/2))

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Maple [A]  time = 0.003, size = 45, normalized size = 0.5 \begin{align*} -{\frac{32\,{b}^{3}{x}^{3}-48\,a{b}^{2}{x}^{2}+12\,{a}^{2}bx+2\,{a}^{3}}{3\,{a}^{4}}{x}^{-{\frac{3}{2}}} \left ( -bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(16*b^3*x^3-24*a*b^2*x^2+6*a^2*b*x+a^3)/x^(3/2)/(-b*x+a)^(3/2)/a^4

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Maxima [A]  time = 1.02535, size = 92, normalized size = 1.05 \begin{align*} -\frac{2 \,{\left (\frac{9 \, \sqrt{-b x + a} b}{\sqrt{x}} + \frac{{\left (-b x + a\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}}\right )}}{3 \, a^{4}} + \frac{2 \,{\left (b^{3} - \frac{9 \,{\left (b x - a\right )} b^{2}}{x}\right )} x^{\frac{3}{2}}}{3 \,{\left (-b x + a\right )}^{\frac{3}{2}} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(9*sqrt(-b*x + a)*b/sqrt(x) + (-b*x + a)^(3/2)/x^(3/2))/a^4 + 2/3*(b^3 - 9*(b*x - a)*b^2/x)*x^(3/2)/((-b*
x + a)^(3/2)*a^4)

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Fricas [A]  time = 1.8497, size = 153, normalized size = 1.74 \begin{align*} -\frac{2 \,{\left (16 \, b^{3} x^{3} - 24 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}\right )} \sqrt{-b x + a} \sqrt{x}}{3 \,{\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(16*b^3*x^3 - 24*a*b^2*x^2 + 6*a^2*b*x + a^3)*sqrt(-b*x + a)*sqrt(x)/(a^4*b^2*x^4 - 2*a^5*b*x^3 + a^6*x^2
)

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Sympy [B]  time = 36.5423, size = 692, normalized size = 7.86 \begin{align*} \begin{cases} \frac{2 a^{4} b^{\frac{19}{2}} \sqrt{\frac{a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac{10 a^{3} b^{\frac{21}{2}} x \sqrt{\frac{a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac{60 a^{2} b^{\frac{23}{2}} x^{2} \sqrt{\frac{a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac{80 a b^{\frac{25}{2}} x^{3} \sqrt{\frac{a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac{32 b^{\frac{27}{2}} x^{4} \sqrt{\frac{a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\\frac{2 i a^{4} b^{\frac{19}{2}} \sqrt{- \frac{a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac{10 i a^{3} b^{\frac{21}{2}} x \sqrt{- \frac{a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac{60 i a^{2} b^{\frac{23}{2}} x^{2} \sqrt{- \frac{a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac{80 i a b^{\frac{25}{2}} x^{3} \sqrt{- \frac{a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac{32 i b^{\frac{27}{2}} x^{4} \sqrt{- \frac{a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a**4*b**(19/2)*sqrt(a/(b*x) - 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4
*b**12*x**4) + 10*a**3*b**(21/2)*x*sqrt(a/(b*x) - 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 +
3*a**4*b**12*x**4) - 60*a**2*b**(23/2)*x**2*sqrt(a/(b*x) - 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b*
*11*x**3 + 3*a**4*b**12*x**4) + 80*a*b**(25/2)*x**3*sqrt(a/(b*x) - 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*
a**5*b**11*x**3 + 3*a**4*b**12*x**4) - 32*b**(27/2)*x**4*sqrt(a/(b*x) - 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2
- 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4), Abs(a)/(Abs(b)*Abs(x)) > 1), (2*I*a**4*b**(19/2)*sqrt(-a/(b*x) + 1)
/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4) + 10*I*a**3*b**(21/2)*x*sqrt(-a/
(b*x) + 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4) - 60*I*a**2*b**(23/2)*
x**2*sqrt(-a/(b*x) + 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4) + 80*I*a*
b**(25/2)*x**3*sqrt(-a/(b*x) + 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4)
- 32*I*b**(27/2)*x**4*sqrt(-a/(b*x) + 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b**
12*x**4), True))

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Giac [B]  time = 1.15586, size = 277, normalized size = 3.15 \begin{align*} -\frac{\sqrt{-b x + a}{\left (\frac{8 \,{\left (b x - a\right )} a{\left | b \right |}}{b^{2}} + \frac{9 \, a^{2}{\left | b \right |}}{b^{2}}\right )}}{24 \,{\left ({\left (b x - a\right )} b + a b\right )}^{\frac{3}{2}}} - \frac{8 \,{\left (3 \,{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt{-b} b^{3} - 9 \, a{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} \sqrt{-b} b^{4} + 4 \, a^{2} \sqrt{-b} b^{5}\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^{3}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*sqrt(-b*x + a)*(8*(b*x - a)*a*abs(b)/b^2 + 9*a^2*abs(b)/b^2)/((b*x - a)*b + a*b)^(3/2) - 8/3*(3*(sqrt(-b
*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^4*sqrt(-b)*b^3 - 9*a*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b +
a*b))^2*sqrt(-b)*b^4 + 4*a^2*sqrt(-b)*b^5)/(((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)^3*a
^3*abs(b))