3.558 \(\int x^9 (a^2+2 a b x^2+b^2 x^4)^{3/2} \, dx\)

Optimal. Leaf size=167 \[ \frac{b^3 x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 \left (a+b x^2\right )}+\frac{3 a b^2 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{a^2 b x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{a^3 x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )} \]

[Out]

(a^3*x^10*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(10*(a + b*x^2)) + (a^2*b*x^12*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*
(a + b*x^2)) + (3*a*b^2*x^14*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*(a + b*x^2)) + (b^3*x^16*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4])/(16*(a + b*x^2))

________________________________________________________________________________________

Rubi [A]  time = 0.113616, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1111, 646, 43} \[ \frac{b^3 x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 \left (a+b x^2\right )}+\frac{3 a b^2 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{a^2 b x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{a^3 x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(a^3*x^10*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(10*(a + b*x^2)) + (a^2*b*x^12*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*
(a + b*x^2)) + (3*a*b^2*x^14*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*(a + b*x^2)) + (b^3*x^16*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4])/(16*(a + b*x^2))

Rule 1111

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && Integ
erQ[(m - 1)/2] && (GtQ[m, 0] || LtQ[0, 4*p, -m - 1])

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

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^9 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \operatorname{Subst}\left (\int x^4 \left (a b+b^2 x\right )^3 \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \operatorname{Subst}\left (\int \left (a^3 b^3 x^4+3 a^2 b^4 x^5+3 a b^5 x^6+b^6 x^7\right ) \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{a^3 x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )}+\frac{a^2 b x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{3 a b^2 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{b^3 x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 \left (a+b x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0189648, size = 61, normalized size = 0.37 \[ \frac{x^{10} \sqrt{\left (a+b x^2\right )^2} \left (140 a^2 b x^2+56 a^3+120 a b^2 x^4+35 b^3 x^6\right )}{560 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(x^10*Sqrt[(a + b*x^2)^2]*(56*a^3 + 140*a^2*b*x^2 + 120*a*b^2*x^4 + 35*b^3*x^6))/(560*(a + b*x^2))

________________________________________________________________________________________

Maple [A]  time = 0.169, size = 58, normalized size = 0.4 \begin{align*}{\frac{{x}^{10} \left ( 35\,{b}^{3}{x}^{6}+120\,{b}^{2}a{x}^{4}+140\,{a}^{2}b{x}^{2}+56\,{a}^{3} \right ) }{560\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

1/560*x^10*(35*b^3*x^6+120*a*b^2*x^4+140*a^2*b*x^2+56*a^3)*((b*x^2+a)^2)^(3/2)/(b*x^2+a)^3

________________________________________________________________________________________

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^9*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.51797, size = 89, normalized size = 0.53 \begin{align*} \frac{1}{16} \, b^{3} x^{16} + \frac{3}{14} \, a b^{2} x^{14} + \frac{1}{4} \, a^{2} b x^{12} + \frac{1}{10} \, a^{3} x^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/16*b^3*x^16 + 3/14*a*b^2*x^14 + 1/4*a^2*b*x^12 + 1/10*a^3*x^10

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{9} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Integral(x**9*((a + b*x**2)**2)**(3/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.10875, size = 90, normalized size = 0.54 \begin{align*} \frac{1}{16} \, b^{3} x^{16} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{3}{14} \, a b^{2} x^{14} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{4} \, a^{2} b x^{12} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{10} \, a^{3} x^{10} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")

[Out]

1/16*b^3*x^16*sgn(b*x^2 + a) + 3/14*a*b^2*x^14*sgn(b*x^2 + a) + 1/4*a^2*b*x^12*sgn(b*x^2 + a) + 1/10*a^3*x^10*
sgn(b*x^2 + a)