3.5.16 \(\int x^7 (a^2+2 a b x^2+b^2 x^4)^{5/2} \, dx\)

Optimal. Leaf size=160 \[ \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{18 b^4}-\frac {3 a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^4}+\frac {3 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{14 b^4}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^4} \]

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Rubi [A]  time = 0.12, antiderivative size = 160, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{18 b^4}-\frac {3 a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^4}+\frac {3 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{14 b^4}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

-(a^3*(a + b*x^2)^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*b^4) + (3*a^2*(a + b*x^2)^6*Sqrt[a^2 + 2*a*b*x^2 + b^
2*x^4])/(14*b^4) - (3*a*(a + b*x^2)^7*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*b^4) + ((a + b*x^2)^8*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4])/(18*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 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 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])

Rubi steps

\begin {align*} \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int x^3 \left (a b+b^2 x\right )^5 \, dx,x,x^2\right )}{2 b^4 \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 (-\frac {a^3 \left (a b+b^2 x\right )^5}{b^3}+\frac {3 a^2 \left (a b+b^2 x\right )^6}{b^4}-\frac {3 a \left (a b+b^2 x\right )^7}{b^5}+\frac {\left (a b+b^2 x\right )^8}{b^6}\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=-\frac {a^3 \left (a+b x^2\right )^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 b^4}+\frac {3 a^2 \left (a+b x^2\right )^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 b^4}-\frac {3 a \left (a+b x^2\right )^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 b^4}+\frac {\left (a+b x^2\right )^8 \sqrt {a^2+2 a b x^2+b^2 x^4}}{18 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 83, normalized size = 0.52 \begin {gather*} \frac {x^8 \sqrt {\left (a+b x^2\right )^2} \left (126 a^5+504 a^4 b x^2+840 a^3 b^2 x^4+720 a^2 b^3 x^6+315 a b^4 x^8+56 b^5 x^{10}\right )}{1008 \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x^8*Sqrt[(a + b*x^2)^2]*(126*a^5 + 504*a^4*b*x^2 + 840*a^3*b^2*x^4 + 720*a^2*b^3*x^6 + 315*a*b^4*x^8 + 56*b^5
*x^10))/(1008*(a + b*x^2))

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IntegrateAlgebraic [A]  time = 14.64, size = 83, normalized size = 0.52 \begin {gather*} \frac {x^8 \sqrt {\left (a+b x^2\right )^2} \left (126 a^5+504 a^4 b x^2+840 a^3 b^2 x^4+720 a^2 b^3 x^6+315 a b^4 x^8+56 b^5 x^{10}\right )}{1008 \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x^8*Sqrt[(a + b*x^2)^2]*(126*a^5 + 504*a^4*b*x^2 + 840*a^3*b^2*x^4 + 720*a^2*b^3*x^6 + 315*a*b^4*x^8 + 56*b^5
*x^10))/(1008*(a + b*x^2))

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fricas [A]  time = 1.11, size = 57, normalized size = 0.36 \begin {gather*} \frac {1}{18} \, b^{5} x^{18} + \frac {5}{16} \, a b^{4} x^{16} + \frac {5}{7} \, a^{2} b^{3} x^{14} + \frac {5}{6} \, a^{3} b^{2} x^{12} + \frac {1}{2} \, a^{4} b x^{10} + \frac {1}{8} \, a^{5} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/18*b^5*x^18 + 5/16*a*b^4*x^16 + 5/7*a^2*b^3*x^14 + 5/6*a^3*b^2*x^12 + 1/2*a^4*b*x^10 + 1/8*a^5*x^8

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giac [A]  time = 0.16, size = 105, normalized size = 0.66 \begin {gather*} \frac {1}{18} \, b^{5} x^{18} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{16} \, a b^{4} x^{16} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{7} \, a^{2} b^{3} x^{14} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{6} \, a^{3} b^{2} x^{12} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{2} \, a^{4} b x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{8} \, a^{5} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

1/18*b^5*x^18*sgn(b*x^2 + a) + 5/16*a*b^4*x^16*sgn(b*x^2 + a) + 5/7*a^2*b^3*x^14*sgn(b*x^2 + a) + 5/6*a^3*b^2*
x^12*sgn(b*x^2 + a) + 1/2*a^4*b*x^10*sgn(b*x^2 + a) + 1/8*a^5*x^8*sgn(b*x^2 + a)

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maple [A]  time = 0.01, size = 80, normalized size = 0.50 \begin {gather*} \frac {\left (56 b^{5} x^{10}+315 a \,b^{4} x^{8}+720 a^{2} b^{3} x^{6}+840 a^{3} b^{2} x^{4}+504 a^{4} b \,x^{2}+126 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x^{8}}{1008 \left (b \,x^{2}+a \right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/1008*x^8*(56*b^5*x^10+315*a*b^4*x^8+720*a^2*b^3*x^6+840*a^3*b^2*x^4+504*a^4*b*x^2+126*a^5)*((b*x^2+a)^2)^(5/
2)/(b*x^2+a)^5

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maxima [A]  time = 1.40, size = 57, normalized size = 0.36 \begin {gather*} \frac {1}{18} \, b^{5} x^{18} + \frac {5}{16} \, a b^{4} x^{16} + \frac {5}{7} \, a^{2} b^{3} x^{14} + \frac {5}{6} \, a^{3} b^{2} x^{12} + \frac {1}{2} \, a^{4} b x^{10} + \frac {1}{8} \, a^{5} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/18*b^5*x^18 + 5/16*a*b^4*x^16 + 5/7*a^2*b^3*x^14 + 5/6*a^3*b^2*x^12 + 1/2*a^4*b*x^10 + 1/8*a^5*x^8

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^7\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)

[Out]

int(x^7*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{7} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(x**7*((a + b*x**2)**2)**(5/2), x)

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