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

Optimal. Leaf size=80 \[ \frac{3 a^2 \left (a+b x^2\right )^{7/2}}{7 b^4}-\frac{a^3 \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac{\left (a+b x^2\right )^{11/2}}{11 b^4}-\frac{a \left (a+b x^2\right )^{9/2}}{3 b^4} \]

[Out]

-(a^3*(a + b*x^2)^(5/2))/(5*b^4) + (3*a^2*(a + b*x^2)^(7/2))/(7*b^4) - (a*(a + b*x^2)^(9/2))/(3*b^4) + (a + b*
x^2)^(11/2)/(11*b^4)

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Rubi [A]  time = 0.0476189, 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} \[ \frac{3 a^2 \left (a+b x^2\right )^{7/2}}{7 b^4}-\frac{a^3 \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac{\left (a+b x^2\right )^{11/2}}{11 b^4}-\frac{a \left (a+b x^2\right )^{9/2}}{3 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^2)^(3/2),x]

[Out]

-(a^3*(a + b*x^2)^(5/2))/(5*b^4) + (3*a^2*(a + b*x^2)^(7/2))/(7*b^4) - (a*(a + b*x^2)^(9/2))/(3*b^4) + (a + b*
x^2)^(11/2)/(11*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 )^{3/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^{3/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{3/2}}{b^3}+\frac{3 a^2 (a+b x)^{5/2}}{b^3}-\frac{3 a (a+b x)^{7/2}}{b^3}+\frac{(a+b x)^{9/2}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3 \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac{3 a^2 \left (a+b x^2\right )^{7/2}}{7 b^4}-\frac{a \left (a+b x^2\right )^{9/2}}{3 b^4}+\frac{\left (a+b x^2\right )^{11/2}}{11 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0253097, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^2\right )^{5/2} \left (40 a^2 b x^2-16 a^3-70 a b^2 x^4+105 b^3 x^6\right )}{1155 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^2)^(3/2),x]

[Out]

((a + b*x^2)^(5/2)*(-16*a^3 + 40*a^2*b*x^2 - 70*a*b^2*x^4 + 105*b^3*x^6))/(1155*b^4)

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Maple [A]  time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-105\,{b}^{3}{x}^{6}+70\,a{b}^{2}{x}^{4}-40\,{a}^{2}b{x}^{2}+16\,{a}^{3}}{1155\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1155*(b*x^2+a)^(5/2)*(-105*b^3*x^6+70*a*b^2*x^4-40*a^2*b*x^2+16*a^3)/b^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55586, size = 154, normalized size = 1.92 \begin{align*} \frac{{\left (105 \, b^{5} x^{10} + 140 \, a b^{4} x^{8} + 5 \, a^{2} b^{3} x^{6} - 6 \, a^{3} b^{2} x^{4} + 8 \, a^{4} b x^{2} - 16 \, a^{5}\right )} \sqrt{b x^{2} + a}}{1155 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1155*(105*b^5*x^10 + 140*a*b^4*x^8 + 5*a^2*b^3*x^6 - 6*a^3*b^2*x^4 + 8*a^4*b*x^2 - 16*a^5)*sqrt(b*x^2 + a)/b
^4

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Sympy [A]  time = 3.69204, size = 133, normalized size = 1.66 \begin{align*} \begin{cases} - \frac{16 a^{5} \sqrt{a + b x^{2}}}{1155 b^{4}} + \frac{8 a^{4} x^{2} \sqrt{a + b x^{2}}}{1155 b^{3}} - \frac{2 a^{3} x^{4} \sqrt{a + b x^{2}}}{385 b^{2}} + \frac{a^{2} x^{6} \sqrt{a + b x^{2}}}{231 b} + \frac{4 a x^{8} \sqrt{a + b x^{2}}}{33} + \frac{b x^{10} \sqrt{a + b x^{2}}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} 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)**(3/2),x)

[Out]

Piecewise((-16*a**5*sqrt(a + b*x**2)/(1155*b**4) + 8*a**4*x**2*sqrt(a + b*x**2)/(1155*b**3) - 2*a**3*x**4*sqrt
(a + b*x**2)/(385*b**2) + a**2*x**6*sqrt(a + b*x**2)/(231*b) + 4*a*x**8*sqrt(a + b*x**2)/33 + b*x**10*sqrt(a +
b*x**2)/11, Ne(b, 0)), (a**(3/2)*x**8/8, True))

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Giac [B]  time = 2.03377, size = 181, normalized size = 2.26 \begin{align*} \frac{\frac{11 \,{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} a}{b^{3}} + \frac{315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4}}{b^{3}}}{3465 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3465*(11*(35*(b*x^2 + a)^(9/2) - 135*(b*x^2 + a)^(7/2)*a + 189*(b*x^2 + a)^(5/2)*a^2 - 105*(b*x^2 + a)^(3/2)
*a^3)*a/b^3 + (315*(b*x^2 + a)^(11/2) - 1540*(b*x^2 + a)^(9/2)*a + 2970*(b*x^2 + a)^(7/2)*a^2 - 2772*(b*x^2 +
a)^(5/2)*a^3 + 1155*(b*x^2 + a)^(3/2)*a^4)/b^3)/b