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

Optimal. Leaf size=80 \[ \frac{9 a^2 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac{3 a^3 \left (a+b x^2\right )^{7/3}}{14 b^4}+\frac{3 \left (a+b x^2\right )^{16/3}}{32 b^4}-\frac{9 a \left (a+b x^2\right )^{13/3}}{26 b^4} \]

[Out]

(-3*a^3*(a + b*x^2)^(7/3))/(14*b^4) + (9*a^2*(a + b*x^2)^(10/3))/(20*b^4) - (9*a*(a + b*x^2)^(13/3))/(26*b^4)
+ (3*(a + b*x^2)^(16/3))/(32*b^4)

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Rubi [A]  time = 0.048907, 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} \[ \frac{9 a^2 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac{3 a^3 \left (a+b x^2\right )^{7/3}}{14 b^4}+\frac{3 \left (a+b x^2\right )^{16/3}}{32 b^4}-\frac{9 a \left (a+b x^2\right )^{13/3}}{26 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^2)^(4/3),x]

[Out]

(-3*a^3*(a + b*x^2)^(7/3))/(14*b^4) + (9*a^2*(a + b*x^2)^(10/3))/(20*b^4) - (9*a*(a + b*x^2)^(13/3))/(26*b^4)
+ (3*(a + b*x^2)^(16/3))/(32*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 x^7 \left (a+b x^2\right )^{4/3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^{4/3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{4/3}}{b^3}+\frac{3 a^2 (a+b x)^{7/3}}{b^3}-\frac{3 a (a+b x)^{10/3}}{b^3}+\frac{(a+b x)^{13/3}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a^3 \left (a+b x^2\right )^{7/3}}{14 b^4}+\frac{9 a^2 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac{9 a \left (a+b x^2\right )^{13/3}}{26 b^4}+\frac{3 \left (a+b x^2\right )^{16/3}}{32 b^4}\\ \end{align*}

Mathematica [A]  time = 0.026091, size = 50, normalized size = 0.62 \[ \frac{3 \left (a+b x^2\right )^{7/3} \left (189 a^2 b x^2-81 a^3-315 a b^2 x^4+455 b^3 x^6\right )}{14560 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^2)^(4/3),x]

[Out]

(3*(a + b*x^2)^(7/3)*(-81*a^3 + 189*a^2*b*x^2 - 315*a*b^2*x^4 + 455*b^3*x^6))/(14560*b^4)

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Maple [A]  time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-1365\,{b}^{3}{x}^{6}+945\,a{b}^{2}{x}^{4}-567\,{a}^{2}b{x}^{2}+243\,{a}^{3}}{14560\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/14560*(b*x^2+a)^(7/3)*(-455*b^3*x^6+315*a*b^2*x^4-189*a^2*b*x^2+81*a^3)/b^4

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Maxima [A]  time = 1.85292, size = 86, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{16}{3}}}{32 \, b^{4}} - \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} a}{26 \, b^{4}} + \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a^{2}}{20 \, b^{4}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{3}}{14 \, 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]

3/32*(b*x^2 + a)^(16/3)/b^4 - 9/26*(b*x^2 + a)^(13/3)*a/b^4 + 9/20*(b*x^2 + a)^(10/3)*a^2/b^4 - 3/14*(b*x^2 +
a)^(7/3)*a^3/b^4

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Fricas [A]  time = 1.74078, size = 162, normalized size = 2.02 \begin{align*} \frac{3 \,{\left (455 \, b^{5} x^{10} + 595 \, a b^{4} x^{8} + 14 \, a^{2} b^{3} x^{6} - 18 \, a^{3} b^{2} x^{4} + 27 \, a^{4} b x^{2} - 81 \, a^{5}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{14560 \, 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="fricas")

[Out]

3/14560*(455*b^5*x^10 + 595*a*b^4*x^8 + 14*a^2*b^3*x^6 - 18*a^3*b^2*x^4 + 27*a^4*b*x^2 - 81*a^5)*(b*x^2 + a)^(
1/3)/b^4

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Sympy [A]  time = 5.78559, size = 136, normalized size = 1.7 \begin{align*} \begin{cases} - \frac{243 a^{5} \sqrt [3]{a + b x^{2}}}{14560 b^{4}} + \frac{81 a^{4} x^{2} \sqrt [3]{a + b x^{2}}}{14560 b^{3}} - \frac{27 a^{3} x^{4} \sqrt [3]{a + b x^{2}}}{7280 b^{2}} + \frac{3 a^{2} x^{6} \sqrt [3]{a + b x^{2}}}{1040 b} + \frac{51 a x^{8} \sqrt [3]{a + b x^{2}}}{416} + \frac{3 b x^{10} \sqrt [3]{a + b x^{2}}}{32} & \text{for}\: b \neq 0 \\\frac{a^{\frac{4}{3}} x^{8}}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-243*a**5*(a + b*x**2)**(1/3)/(14560*b**4) + 81*a**4*x**2*(a + b*x**2)**(1/3)/(14560*b**3) - 27*a**
3*x**4*(a + b*x**2)**(1/3)/(7280*b**2) + 3*a**2*x**6*(a + b*x**2)**(1/3)/(1040*b) + 51*a*x**8*(a + b*x**2)**(1
/3)/416 + 3*b*x**10*(a + b*x**2)**(1/3)/32, Ne(b, 0)), (a**(4/3)*x**8/8, True))

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Giac [B]  time = 2.53501, size = 181, normalized size = 2.26 \begin{align*} \frac{3 \,{\left (\frac{4 \,{\left (140 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} - 546 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a + 780 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{2} - 455 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{3}\right )} a}{b^{3}} + \frac{455 \,{\left (b x^{2} + a\right )}^{\frac{16}{3}} - 2240 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} a + 4368 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a^{2} - 4160 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{3} + 1820 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{4}}{b^{3}}\right )}}{14560 \, b} \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]

3/14560*(4*(140*(b*x^2 + a)^(13/3) - 546*(b*x^2 + a)^(10/3)*a + 780*(b*x^2 + a)^(7/3)*a^2 - 455*(b*x^2 + a)^(4
/3)*a^3)*a/b^3 + (455*(b*x^2 + a)^(16/3) - 2240*(b*x^2 + a)^(13/3)*a + 4368*(b*x^2 + a)^(10/3)*a^2 - 4160*(b*x
^2 + a)^(7/3)*a^3 + 1820*(b*x^2 + a)^(4/3)*a^4)/b^3)/b