3.79 \(\int \frac{x^{21/2}}{(a x+b x^3)^{9/2}} \, dx\)

Optimal. Leaf size=25 \[ \frac{x^{21/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]

[Out]

x^(21/2)/(7*a*(a*x + b*x^3)^(7/2))

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Rubi [A]  time = 0.0369776, 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} \[ \frac{x^{21/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^(21/2)/(a*x + b*x^3)^(9/2),x]

[Out]

x^(21/2)/(7*a*(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^{21/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac{x^{21/2}}{7 a \left (a x+b x^3\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0168311, size = 25, normalized size = 1. \[ \frac{x^{21/2}}{7 a \left (x \left (a+b x^2\right )\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(21/2)/(a*x + b*x^3)^(9/2),x]

[Out]

x^(21/2)/(7*a*(x*(a + b*x^2))^(7/2))

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Maple [A]  time = 0.004, size = 27, normalized size = 1.1 \begin{align*}{\frac{b{x}^{2}+a}{7\,a}{x}^{{\frac{23}{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^(21/2)/(b*x^3+a*x)^(9/2),x)

[Out]

1/7*(b*x^2+a)/a*x^(23/2)/(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{21}{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^(21/2)/(b*x^3+a*x)^(9/2),x, algorithm="maxima")

[Out]

integrate(x^(21/2)/(b*x^3 + a*x)^(9/2), x)

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Fricas [B]  time = 1.42907, size = 130, normalized size = 5.2 \begin{align*} \frac{\sqrt{b x^{3} + a x} x^{\frac{13}{2}}}{7 \,{\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(21/2)/(b*x^3+a*x)^(9/2),x, algorithm="fricas")

[Out]

1/7*sqrt(b*x^3 + a*x)*x^(13/2)/(a*b^4*x^8 + 4*a^2*b^3*x^6 + 6*a^3*b^2*x^4 + 4*a^4*b*x^2 + a^5)

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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**(21/2)/(b*x**3+a*x)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.2361, size = 23, normalized size = 0.92 \begin{align*} \frac{x^{7}}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(21/2)/(b*x^3+a*x)^(9/2),x, algorithm="giac")

[Out]

1/7*x^7/((b*x^2 + a)^(7/2)*a)