3.353 \(\int x^7 \sqrt{a+b x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0452027, 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} \[ \frac{3 a^2 \left (a+b x^2\right )^{5/2}}{5 b^4}-\frac{a^3 \left (a+b x^2\right )^{3/2}}{3 b^4}+\frac{\left (a+b x^2\right )^{9/2}}{9 b^4}-\frac{3 a \left (a+b x^2\right )^{7/2}}{7 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^7*Sqrt[a + b*x^2],x]

[Out]

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

Mathematica [A]  time = 0.0258305, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^2\right )^{3/2} \left (24 a^2 b x^2-16 a^3-30 a b^2 x^4+35 b^3 x^6\right )}{315 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*Sqrt[a + b*x^2],x]

[Out]

((a + b*x^2)^(3/2)*(-16*a^3 + 24*a^2*b*x^2 - 30*a*b^2*x^4 + 35*b^3*x^6))/(315*b^4)

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Maple [A]  time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-35\,{b}^{3}{x}^{6}+30\,a{b}^{2}{x}^{4}-24\,{a}^{2}b{x}^{2}+16\,{a}^{3}}{315\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b*x^2+a)^(1/2),x)

[Out]

-1/315*(b*x^2+a)^(3/2)*(-35*b^3*x^6+30*a*b^2*x^4-24*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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61312, size = 126, normalized size = 1.58 \begin{align*} \frac{{\left (35 \, b^{4} x^{8} + 5 \, a b^{3} x^{6} - 6 \, a^{2} b^{2} x^{4} + 8 \, a^{3} b x^{2} - 16 \, a^{4}\right )} \sqrt{b x^{2} + a}}{315 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

1/315*(35*b^4*x^8 + 5*a*b^3*x^6 - 6*a^2*b^2*x^4 + 8*a^3*b*x^2 - 16*a^4)*sqrt(b*x^2 + a)/b^4

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Sympy [A]  time = 1.29625, size = 110, normalized size = 1.38 \begin{align*} \begin{cases} - \frac{16 a^{4} \sqrt{a + b x^{2}}}{315 b^{4}} + \frac{8 a^{3} x^{2} \sqrt{a + b x^{2}}}{315 b^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{6} \sqrt{a + b x^{2}}}{63 b} + \frac{x^{8} \sqrt{a + b x^{2}}}{9} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} 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)**(1/2),x)

[Out]

Piecewise((-16*a**4*sqrt(a + b*x**2)/(315*b**4) + 8*a**3*x**2*sqrt(a + b*x**2)/(315*b**3) - 2*a**2*x**4*sqrt(a
+ b*x**2)/(105*b**2) + a*x**6*sqrt(a + b*x**2)/(63*b) + x**8*sqrt(a + b*x**2)/9, Ne(b, 0)), (sqrt(a)*x**8/8,
True))

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Giac [A]  time = 1.86378, size = 77, normalized size = 0.96 \begin{align*} \frac{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}}{315 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/315*(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)
/b^4