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

Optimal. Leaf size=84 \[ -\frac{16 \sqrt{a+b x}}{3 a^3 x^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a+b x}}+\frac{32 b \sqrt{a+b x}}{3 a^4 \sqrt{x}}+\frac{2}{3 a x^{3/2} (a+b x)^{3/2}} \]

[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 = 0.0153343, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{16 \sqrt{a+b x}}{3 a^3 x^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a+b x}}+\frac{32 b \sqrt{a+b x}}{3 a^4 \sqrt{x}}+\frac{2}{3 a x^{3/2} (a+b x)^{3/2}} \]

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 = 0.0234266, size = 49, normalized size = 0.58 \[ -\frac{2 \left (-6 a^2 b x+a^3-24 a b^2 x^2-16 b^3 x^3\right )}{3 a^4 x^{3/2} (a+b x)^{3/2}} \]

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 = 44, 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 = 1.12725, size = 86, normalized size = 1.02 \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 = 2.05912, size = 150, normalized size = 1.79 \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 = 40.0572, size = 337, normalized size = 4.01 \begin{align*} - \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}} \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]

-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**1
2*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)

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Giac [B]  time = 1.15531, size = 234, normalized size = 2.79 \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} b^{\frac{7}{2}} + 9 \, a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{9}{2}} + 4 \, a^{2} b^{\frac{11}{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^{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*b^(7/2) + 9*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b
^(9/2) + 4*a^2*b^(11/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*a^3*abs(b))