3.306 \(\int \frac{1}{x^{9/2} \sqrt{a x^2+b x^5}} \, dx\)

Optimal. Leaf size=56 \[ \frac{4 b \sqrt{a x^2+b x^5}}{9 a^2 x^{5/2}}-\frac{2 \sqrt{a x^2+b x^5}}{9 a x^{11/2}} \]

[Out]

(-2*Sqrt[a*x^2 + b*x^5])/(9*a*x^(11/2)) + (4*b*Sqrt[a*x^2 + b*x^5])/(9*a^2*x^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0830774, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2016, 2014} \[ \frac{4 b \sqrt{a x^2+b x^5}}{9 a^2 x^{5/2}}-\frac{2 \sqrt{a x^2+b x^5}}{9 a x^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(9/2)*Sqrt[a*x^2 + b*x^5]),x]

[Out]

(-2*Sqrt[a*x^2 + b*x^5])/(9*a*x^(11/2)) + (4*b*Sqrt[a*x^2 + b*x^5])/(9*a^2*x^(5/2))

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^{9/2} \sqrt{a x^2+b x^5}} \, dx &=-\frac{2 \sqrt{a x^2+b x^5}}{9 a x^{11/2}}-\frac{(2 b) \int \frac{1}{x^{3/2} \sqrt{a x^2+b x^5}} \, dx}{3 a}\\ &=-\frac{2 \sqrt{a x^2+b x^5}}{9 a x^{11/2}}+\frac{4 b \sqrt{a x^2+b x^5}}{9 a^2 x^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0173151, size = 35, normalized size = 0.62 \[ -\frac{2 \left (a-2 b x^3\right ) \sqrt{x^2 \left (a+b x^3\right )}}{9 a^2 x^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(9/2)*Sqrt[a*x^2 + b*x^5]),x]

[Out]

(-2*(a - 2*b*x^3)*Sqrt[x^2*(a + b*x^3)])/(9*a^2*x^(11/2))

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 37, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,b{x}^{3}+2\,a \right ) \left ( -2\,b{x}^{3}+a \right ) }{9\,{a}^{2}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(9/2)/(b*x^5+a*x^2)^(1/2),x)

[Out]

-2/9*(b*x^3+a)*(-2*b*x^3+a)/x^(7/2)/a^2/(b*x^5+a*x^2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.10407, size = 51, normalized size = 0.91 \begin{align*} \frac{2 \,{\left (2 \, b^{2} x^{7} + a b x^{4} - a^{2} x\right )}}{9 \, \sqrt{b x^{3} + a} a^{2} x^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="maxima")

[Out]

2/9*(2*b^2*x^7 + a*b*x^4 - a^2*x)/(sqrt(b*x^3 + a)*a^2*x^(11/2))

________________________________________________________________________________________

Fricas [A]  time = 0.822707, size = 73, normalized size = 1.3 \begin{align*} \frac{2 \, \sqrt{b x^{5} + a x^{2}}{\left (2 \, b x^{3} - a\right )}}{9 \, a^{2} x^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/9*sqrt(b*x^5 + a*x^2)*(2*b*x^3 - a)/(a^2*x^(11/2))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(9/2)/(b*x**5+a*x**2)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.20074, size = 49, normalized size = 0.88 \begin{align*} -\frac{4 \, b^{\frac{3}{2}}}{9 \, a^{2}} - \frac{2 \,{\left ({\left (b + \frac{a}{x^{3}}\right )}^{\frac{3}{2}} - 3 \, \sqrt{b + \frac{a}{x^{3}}} b\right )}}{9 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")

[Out]

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