3.84 \(\int \frac{x^{11/2}}{(a x+b x^3)^{9/2}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{x^{7/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]

[Out]

-x^(7/2)/(7*b*(a*x + b*x^3)^(7/2))

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Rubi [A]  time = 0.0370535, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2014} \[ -\frac{x^{7/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^(11/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-x^(7/2)/(7*b*(a*x + b*x^3)^(7/2))

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^{11/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=-\frac{x^{7/2}}{7 b \left (a x+b x^3\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0135167, size = 25, normalized size = 1. \[ -\frac{x^{7/2}}{7 b \left (x \left (a+b x^2\right )\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(11/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-x^(7/2)/(7*b*(x*(a + b*x^2))^(7/2))

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Maple [A]  time = 0.003, size = 27, normalized size = 1.1 \begin{align*} -{\frac{b{x}^{2}+a}{7\,b}{x}^{{\frac{9}{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^(11/2)/(b*x^3+a*x)^(9/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{11}{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^(11/2)/(b*x^3+a*x)^(9/2),x, algorithm="maxima")

[Out]

integrate(x^(11/2)/(b*x^3 + a*x)^(9/2), x)

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Fricas [B]  time = 1.70886, size = 132, normalized size = 5.28 \begin{align*} -\frac{\sqrt{b x^{3} + a x} \sqrt{x}}{7 \,{\left (b^{5} x^{9} + 4 \, a b^{4} x^{7} + 6 \, a^{2} b^{3} x^{5} + 4 \, a^{3} b^{2} x^{3} + a^{4} b x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(11/2)/(b*x^3+a*x)^(9/2),x, algorithm="fricas")

[Out]

-1/7*sqrt(b*x^3 + a*x)*sqrt(x)/(b^5*x^9 + 4*a*b^4*x^7 + 6*a^2*b^3*x^5 + 4*a^3*b^2*x^3 + a^4*b*x)

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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**(11/2)/(b*x**3+a*x)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.27413, size = 31, normalized size = 1.24 \begin{align*} -\frac{1}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{1}{7 \, a^{\frac{7}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(11/2)/(b*x^3+a*x)^(9/2),x, algorithm="giac")

[Out]

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