3.466 \(\int x^3 (1+x^2)^{9/14} \, dx\)

Optimal. Leaf size=27 \[ \frac{7}{37} \left (x^2+1\right )^{37/14}-\frac{7}{23} \left (x^2+1\right )^{23/14} \]

[Out]

(-7*(1 + x^2)^(23/14))/23 + (7*(1 + x^2)^(37/14))/37

________________________________________________________________________________________

Rubi [A]  time = 0.0114618, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{7}{37} \left (x^2+1\right )^{37/14}-\frac{7}{23} \left (x^2+1\right )^{23/14} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(1 + x^2)^(9/14),x]

[Out]

(-7*(1 + x^2)^(23/14))/23 + (7*(1 + x^2)^(37/14))/37

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^3 \left (1+x^2\right )^{9/14} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (1+x)^{9/14} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-(1+x)^{9/14}+(1+x)^{23/14}\right ) \, dx,x,x^2\right )\\ &=-\frac{7}{23} \left (1+x^2\right )^{23/14}+\frac{7}{37} \left (1+x^2\right )^{37/14}\\ \end{align*}

Mathematica [A]  time = 0.0063217, size = 20, normalized size = 0.74 \[ \frac{7}{851} \left (x^2+1\right )^{23/14} \left (23 x^2-14\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(1 + x^2)^(9/14),x]

[Out]

(7*(1 + x^2)^(23/14)*(-14 + 23*x^2))/851

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 17, normalized size = 0.6 \begin{align*}{\frac{161\,{x}^{2}-98}{851} \left ({x}^{2}+1 \right ) ^{{\frac{23}{14}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^2+1)^(9/14),x)

[Out]

7/851*(x^2+1)^(23/14)*(23*x^2-14)

________________________________________________________________________________________

Maxima [A]  time = 0.945455, size = 26, normalized size = 0.96 \begin{align*} \frac{7}{37} \,{\left (x^{2} + 1\right )}^{\frac{37}{14}} - \frac{7}{23} \,{\left (x^{2} + 1\right )}^{\frac{23}{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^2+1)^(9/14),x, algorithm="maxima")

[Out]

7/37*(x^2 + 1)^(37/14) - 7/23*(x^2 + 1)^(23/14)

________________________________________________________________________________________

Fricas [A]  time = 1.95499, size = 62, normalized size = 2.3 \begin{align*} \frac{7}{851} \,{\left (23 \, x^{4} + 9 \, x^{2} - 14\right )}{\left (x^{2} + 1\right )}^{\frac{9}{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^2+1)^(9/14),x, algorithm="fricas")

[Out]

7/851*(23*x^4 + 9*x^2 - 14)*(x^2 + 1)^(9/14)

________________________________________________________________________________________

Sympy [A]  time = 18.7864, size = 41, normalized size = 1.52 \begin{align*} \frac{7 x^{4} \left (x^{2} + 1\right )^{\frac{9}{14}}}{37} + \frac{63 x^{2} \left (x^{2} + 1\right )^{\frac{9}{14}}}{851} - \frac{98 \left (x^{2} + 1\right )^{\frac{9}{14}}}{851} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(x**2+1)**(9/14),x)

[Out]

7*x**4*(x**2 + 1)**(9/14)/37 + 63*x**2*(x**2 + 1)**(9/14)/851 - 98*(x**2 + 1)**(9/14)/851

________________________________________________________________________________________

Giac [A]  time = 1.07927, size = 26, normalized size = 0.96 \begin{align*} \frac{7}{37} \,{\left (x^{2} + 1\right )}^{\frac{37}{14}} - \frac{7}{23} \,{\left (x^{2} + 1\right )}^{\frac{23}{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^2+1)^(9/14),x, algorithm="giac")

[Out]

7/37*(x^2 + 1)^(37/14) - 7/23*(x^2 + 1)^(23/14)