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

Optimal. Leaf size=49 \[ 6 a^2 b \sqrt{x}-\frac{2 a^3}{3 x^{3/2}}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{9} b^3 x^{9/2} \]

[Out]

(-2*a^3)/(3*x^(3/2)) + 6*a^2*b*Sqrt[x] + (6*a*b^2*x^(5/2))/5 + (2*b^3*x^(9/2))/9

________________________________________________________________________________________

Rubi [A]  time = 0.011726, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ 6 a^2 b \sqrt{x}-\frac{2 a^3}{3 x^{3/2}}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{9} b^3 x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^(5/2),x]

[Out]

(-2*a^3)/(3*x^(3/2)) + 6*a^2*b*Sqrt[x] + (6*a*b^2*x^(5/2))/5 + (2*b^3*x^(9/2))/9

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^3}{x^{5/2}} \, dx &=\int \left (\frac{a^3}{x^{5/2}}+\frac{3 a^2 b}{\sqrt{x}}+3 a b^2 x^{3/2}+b^3 x^{7/2}\right ) \, dx\\ &=-\frac{2 a^3}{3 x^{3/2}}+6 a^2 b \sqrt{x}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{9} b^3 x^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0112587, size = 41, normalized size = 0.84 \[ \frac{2 \left (135 a^2 b x^2-15 a^3+27 a b^2 x^4+5 b^3 x^6\right )}{45 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^(5/2),x]

[Out]

(2*(-15*a^3 + 135*a^2*b*x^2 + 27*a*b^2*x^4 + 5*b^3*x^6))/(45*x^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \begin{align*} -{\frac{-10\,{b}^{3}{x}^{6}-54\,a{b}^{2}{x}^{4}-270\,{a}^{2}b{x}^{2}+30\,{a}^{3}}{45}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*(-5*b^3*x^6-27*a*b^2*x^4-135*a^2*b*x^2+15*a^3)/x^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 1.88921, size = 47, normalized size = 0.96 \begin{align*} \frac{2}{9} \, b^{3} x^{\frac{9}{2}} + \frac{6}{5} \, a b^{2} x^{\frac{5}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b^3*x^(9/2) + 6/5*a*b^2*x^(5/2) + 6*a^2*b*sqrt(x) - 2/3*a^3/x^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.21198, size = 89, normalized size = 1.82 \begin{align*} \frac{2 \,{\left (5 \, b^{3} x^{6} + 27 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} - 15 \, a^{3}\right )}}{45 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(5*b^3*x^6 + 27*a*b^2*x^4 + 135*a^2*b*x^2 - 15*a^3)/x^(3/2)

________________________________________________________________________________________

Sympy [A]  time = 3.03934, size = 48, normalized size = 0.98 \begin{align*} - \frac{2 a^{3}}{3 x^{\frac{3}{2}}} + 6 a^{2} b \sqrt{x} + \frac{6 a b^{2} x^{\frac{5}{2}}}{5} + \frac{2 b^{3} x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**3/(3*x**(3/2)) + 6*a**2*b*sqrt(x) + 6*a*b**2*x**(5/2)/5 + 2*b**3*x**(9/2)/9

________________________________________________________________________________________

Giac [A]  time = 2.89049, size = 47, normalized size = 0.96 \begin{align*} \frac{2}{9} \, b^{3} x^{\frac{9}{2}} + \frac{6}{5} \, a b^{2} x^{\frac{5}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b^3*x^(9/2) + 6/5*a*b^2*x^(5/2) + 6*a^2*b*sqrt(x) - 2/3*a^3/x^(3/2)