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

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

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Rubi [A]  time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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])

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]]

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*}

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Mathematica [A]  time = 0.03, size = 50, normalized size = 0.62 \begin {gather*} \frac {\left (a+b x^2\right )^{3/2} \left (-16 a^3+24 a^2 b x^2-30 a b^2 x^4+35 b^3 x^6\right )}{315 b^4} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.03, size = 61, normalized size = 0.76 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-16 a^4+8 a^3 b x^2-6 a^2 b^2 x^4+5 a b^3 x^6+35 b^4 x^8\right )}{315 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^7*Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(-16*a^4 + 8*a^3*b*x^2 - 6*a^2*b^2*x^4 + 5*a*b^3*x^6 + 35*b^4*x^8))/(315*b^4)

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fricas [A]  time = 1.01, size = 57, normalized size = 0.71 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.60, size = 57, normalized size = 0.71 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.01, size = 47, normalized size = 0.59 \begin {gather*} -\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (-35 b^{3} x^{6}+30 a \,b^{2} x^{4}-24 a^{2} b \,x^{2}+16 a^{3}\right )}{315 b^{4}} \end {gather*}

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 [A]  time = 1.36, size = 73, normalized size = 0.91 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{6}}{9 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{4}}{21 \, b^{2}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x^{2}}{105 \, b^{3}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}}{315 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(b*x^2 + a)^(3/2)*x^6/b - 2/21*(b*x^2 + a)^(3/2)*a*x^4/b^2 + 8/105*(b*x^2 + a)^(3/2)*a^2*x^2/b^3 - 16/315*
(b*x^2 + a)^(3/2)*a^3/b^4

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mupad [B]  time = 4.59, size = 55, normalized size = 0.69 \begin {gather*} \sqrt {b\,x^2+a}\,\left (\frac {x^8}{9}-\frac {16\,a^4}{315\,b^4}+\frac {a\,x^6}{63\,b}-\frac {2\,a^2\,x^4}{105\,b^2}+\frac {8\,a^3\,x^2}{315\,b^3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + b*x^2)^(1/2),x)

[Out]

(a + b*x^2)^(1/2)*(x^8/9 - (16*a^4)/(315*b^4) + (a*x^6)/(63*b) - (2*a^2*x^4)/(105*b^2) + (8*a^3*x^2)/(315*b^3)
)

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sympy [A]  time = 1.42, size = 110, normalized size = 1.38 \begin {gather*} \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 {gather*}

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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