3.83 \(\int \frac{x^{13/2}}{(a x+b x^3)^{9/2}} \, dx\)

Optimal. Leaf size=76 \[ \frac{4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 x^{9/2}}{105 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]

[Out]

x^(13/2)/(7*a*(a*x + b*x^3)^(7/2)) + (4*x^(11/2))/(35*a^2*(a*x + b*x^3)^(5/2)) + (8*x^(9/2))/(105*a^3*(a*x + b
*x^3)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.112928, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2015, 2014} \[ \frac{4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 x^{9/2}}{105 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^(13/2)/(a*x + b*x^3)^(9/2),x]

[Out]

x^(13/2)/(7*a*(a*x + b*x^3)^(7/2)) + (4*x^(11/2))/(35*a^2*(a*x + b*x^3)^(5/2)) + (8*x^(9/2))/(105*a^3*(a*x + b
*x^3)^(3/2))

Rule 2015

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] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)),
Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n, j
] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || 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{x^{13/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac{x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{4 \int \frac{x^{11/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac{x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 \int \frac{x^{9/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac{x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{8 x^{9/2}}{105 a^3 \left (a x+b x^3\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.020886, size = 55, normalized size = 0.72 \[ \frac{x^{5/2} \sqrt{x \left (a+b x^2\right )} \left (35 a^2+28 a b x^2+8 b^2 x^4\right )}{105 a^3 \left (a+b x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(13/2)/(a*x + b*x^3)^(9/2),x]

[Out]

(x^(5/2)*Sqrt[x*(a + b*x^2)]*(35*a^2 + 28*a*b*x^2 + 8*b^2*x^4))/(105*a^3*(a + b*x^2)^4)

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 48, normalized size = 0.6 \begin{align*}{\frac{ \left ( b{x}^{2}+a \right ) \left ( 8\,{b}^{2}{x}^{4}+28\,ab{x}^{2}+35\,{a}^{2} \right ) }{105\,{a}^{3}}{x}^{{\frac{15}{2}}} \left ( b{x}^{3}+ax \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(13/2)/(b*x^3+a*x)^(9/2),x)

[Out]

1/105*(b*x^2+a)*x^(15/2)*(8*b^2*x^4+28*a*b*x^2+35*a^2)/a^3/(b*x^3+a*x)^(9/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{13}{2}}}{{\left (b x^{3} + a x\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/(b*x^3+a*x)^(9/2),x, algorithm="maxima")

[Out]

integrate(x^(13/2)/(b*x^3 + a*x)^(9/2), x)

________________________________________________________________________________________

Fricas [A]  time = 1.65983, size = 185, normalized size = 2.43 \begin{align*} \frac{{\left (8 \, b^{2} x^{6} + 28 \, a b x^{4} + 35 \, a^{2} x^{2}\right )} \sqrt{b x^{3} + a x} \sqrt{x}}{105 \,{\left (a^{3} b^{4} x^{8} + 4 \, a^{4} b^{3} x^{6} + 6 \, a^{5} b^{2} x^{4} + 4 \, a^{6} b x^{2} + a^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/(b*x^3+a*x)^(9/2),x, algorithm="fricas")

[Out]

1/105*(8*b^2*x^6 + 28*a*b*x^4 + 35*a^2*x^2)*sqrt(b*x^3 + a*x)*sqrt(x)/(a^3*b^4*x^8 + 4*a^4*b^3*x^6 + 6*a^5*b^2
*x^4 + 4*a^6*b*x^2 + a^7)

________________________________________________________________________________________

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(x**(13/2)/(b*x**3+a*x)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.27394, size = 58, normalized size = 0.76 \begin{align*} \frac{{\left (4 \, x^{2}{\left (\frac{2 \, b^{2} x^{2}}{a^{3}} + \frac{7 \, b}{a^{2}}\right )} + \frac{35}{a}\right )} x^{3}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/(b*x^3+a*x)^(9/2),x, algorithm="giac")

[Out]

1/105*(4*x^2*(2*b^2*x^2/a^3 + 7*b/a^2) + 35/a)*x^3/(b*x^2 + a)^(7/2)