3.726 \(\int \frac{x^7}{(a+b x^2)^{4/3}} \, dx\)

Optimal. Leaf size=80 \[ \frac{3 a^3}{2 b^4 \sqrt [3]{a+b x^2}}+\frac{9 a^2 \left (a+b x^2\right )^{2/3}}{4 b^4}-\frac{9 a \left (a+b x^2\right )^{5/3}}{10 b^4}+\frac{3 \left (a+b x^2\right )^{8/3}}{16 b^4} \]

[Out]

(3*a^3)/(2*b^4*(a + b*x^2)^(1/3)) + (9*a^2*(a + b*x^2)^(2/3))/(4*b^4) - (9*a*(a + b*x^2)^(5/3))/(10*b^4) + (3*
(a + b*x^2)^(8/3))/(16*b^4)

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Rubi [A]  time = 0.0438257, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{3 a^3}{2 b^4 \sqrt [3]{a+b x^2}}+\frac{9 a^2 \left (a+b x^2\right )^{2/3}}{4 b^4}-\frac{9 a \left (a+b x^2\right )^{5/3}}{10 b^4}+\frac{3 \left (a+b x^2\right )^{8/3}}{16 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^2)^(4/3),x]

[Out]

(3*a^3)/(2*b^4*(a + b*x^2)^(1/3)) + (9*a^2*(a + b*x^2)^(2/3))/(4*b^4) - (9*a*(a + b*x^2)^(5/3))/(10*b^4) + (3*
(a + b*x^2)^(8/3))/(16*b^4)

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 \frac{x^7}{\left (a+b x^2\right )^{4/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{4/3}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{4/3}}+\frac{3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac{3 a (a+b x)^{2/3}}{b^3}+\frac{(a+b x)^{5/3}}{b^3}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a^3}{2 b^4 \sqrt [3]{a+b x^2}}+\frac{9 a^2 \left (a+b x^2\right )^{2/3}}{4 b^4}-\frac{9 a \left (a+b x^2\right )^{5/3}}{10 b^4}+\frac{3 \left (a+b x^2\right )^{8/3}}{16 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0226396, size = 50, normalized size = 0.62 \[ \frac{3 \left (27 a^2 b x^2+81 a^3-9 a b^2 x^4+5 b^3 x^6\right )}{80 b^4 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^2)^(4/3),x]

[Out]

(3*(81*a^3 + 27*a^2*b*x^2 - 9*a*b^2*x^4 + 5*b^3*x^6))/(80*b^4*(a + b*x^2)^(1/3))

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Maple [A]  time = 0.006, size = 47, normalized size = 0.6 \begin{align*}{\frac{15\,{b}^{3}{x}^{6}-27\,a{b}^{2}{x}^{4}+81\,{a}^{2}b{x}^{2}+243\,{a}^{3}}{80\,{b}^{4}}{\frac{1}{\sqrt [3]{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(b*x^2+a)^(4/3),x)

[Out]

3/80/(b*x^2+a)^(1/3)*(5*b^3*x^6-9*a*b^2*x^4+27*a^2*b*x^2+81*a^3)/b^4

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Maxima [A]  time = 1.22713, size = 86, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}}}{16 \, b^{4}} - \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a}{10 \, b^{4}} + \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{2}}{4 \, b^{4}} + \frac{3 \, a^{3}}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^2+a)^(4/3),x, algorithm="maxima")

[Out]

3/16*(b*x^2 + a)^(8/3)/b^4 - 9/10*(b*x^2 + a)^(5/3)*a/b^4 + 9/4*(b*x^2 + a)^(2/3)*a^2/b^4 + 3/2*a^3/((b*x^2 +
a)^(1/3)*b^4)

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Fricas [A]  time = 1.68212, size = 124, normalized size = 1.55 \begin{align*} \frac{3 \,{\left (5 \, b^{3} x^{6} - 9 \, a b^{2} x^{4} + 27 \, a^{2} b x^{2} + 81 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{80 \,{\left (b^{5} x^{2} + a b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^2+a)^(4/3),x, algorithm="fricas")

[Out]

3/80*(5*b^3*x^6 - 9*a*b^2*x^4 + 27*a^2*b*x^2 + 81*a^3)*(b*x^2 + a)^(2/3)/(b^5*x^2 + a*b^4)

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Sympy [B]  time = 2.64196, size = 1584, normalized size = 19.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(b*x**2+a)**(4/3),x)

[Out]

243*a**(68/3)*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b
**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) - 243*a**(68/3)/(80*a**20*b**4
+ 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x*
*10 + 80*a**14*b**10*x**12) + 1296*a**(65/3)*b*x**2*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2
+ 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*
x**12) - 1458*a**(65/3)*b*x**2/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x
**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) + 2808*a**(62/3)*b**2*x**4*(1 + b*x*
*2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b
**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) - 3645*a**(62/3)*b**2*x**4/(80*a**20*b**4 + 480*a**19*
b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**
14*b**10*x**12) + 3120*a**(59/3)*b**3*x**6*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a
**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) -
4860*a**(59/3)*b**3*x**6/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 +
1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) + 1830*a**(56/3)*b**4*x**8*(1 + b*x**2/a)*
*(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x*
*8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) - 3645*a**(56/3)*b**4*x**8/(80*a**20*b**4 + 480*a**19*b**5*x
**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**
10*x**12) + 528*a**(53/3)*b**5*x**10*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b
**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) - 1458*a
**(53/3)*b**5*x**10/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*
a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) + 96*a**(50/3)*b**6*x**12*(1 + b*x**2/a)**(2/3)
/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 4
80*a**15*b**9*x**10 + 80*a**14*b**10*x**12) - 243*a**(50/3)*b**6*x**12/(80*a**20*b**4 + 480*a**19*b**5*x**2 +
1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**
12) + 48*a**(47/3)*b**7*x**14*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**
4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) + 15*a**(44/3)*
b**8*x**16*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7
*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12)

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Giac [A]  time = 2.00577, size = 77, normalized size = 0.96 \begin{align*} \frac{3 \,{\left (5 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} - 24 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a + 60 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{2} + \frac{40 \, a^{3}}{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}\right )}}{80 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^2+a)^(4/3),x, algorithm="giac")

[Out]

3/80*(5*(b*x^2 + a)^(8/3) - 24*(b*x^2 + a)^(5/3)*a + 60*(b*x^2 + a)^(2/3)*a^2 + 40*a^3/(b*x^2 + a)^(1/3))/b^4