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

Optimal. Leaf size=41 \[ -\frac {\left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 a x^8} \]

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Rubi [A]  time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1111, 646, 37} \begin {gather*} -\frac {\left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 a x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*a*x^8)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[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 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 \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^3}{x^5} \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=-\frac {\left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 a x^8}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 59, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (a^3+4 a^2 b x^2+6 a b^2 x^4+4 b^3 x^6\right )}{8 x^8 \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(Sqrt[(a + b*x^2)^2]*(a^3 + 4*a^2*b*x^2 + 6*a*b^2*x^4 + 4*b^3*x^6))/(x^8*(a + b*x^2))

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IntegrateAlgebraic [B]  time = 1.19, size = 306, normalized size = 7.46 \begin {gather*} \frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-a^6 b-7 a^5 b^2 x^2-21 a^4 b^3 x^4-35 a^3 b^4 x^6-34 a^2 b^5 x^8-18 a b^6 x^{10}-4 b^7 x^{12}\right )+\sqrt {b^2} b^3 \left (a^7+8 a^6 b x^2+28 a^5 b^2 x^4+56 a^4 b^3 x^6+69 a^3 b^4 x^8+52 a^2 b^5 x^{10}+22 a b^6 x^{12}+4 b^7 x^{14}\right )}{\sqrt {b^2} x^8 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-8 a^3 b^3-24 a^2 b^4 x^2-24 a b^5 x^4-8 b^6 x^6\right )+x^8 \left (8 a^4 b^4+32 a^3 b^5 x^2+48 a^2 b^6 x^4+32 a b^7 x^6+8 b^8 x^8\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-(a^6*b) - 7*a^5*b^2*x^2 - 21*a^4*b^3*x^4 - 35*a^3*b^4*x^6 - 34*a^2*b^5*
x^8 - 18*a*b^6*x^10 - 4*b^7*x^12) + b^3*Sqrt[b^2]*(a^7 + 8*a^6*b*x^2 + 28*a^5*b^2*x^4 + 56*a^4*b^3*x^6 + 69*a^
3*b^4*x^8 + 52*a^2*b^5*x^10 + 22*a*b^6*x^12 + 4*b^7*x^14))/(Sqrt[b^2]*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-8*
a^3*b^3 - 24*a^2*b^4*x^2 - 24*a*b^5*x^4 - 8*b^6*x^6) + x^8*(8*a^4*b^4 + 32*a^3*b^5*x^2 + 48*a^2*b^6*x^4 + 32*a
*b^7*x^6 + 8*b^8*x^8))

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fricas [A]  time = 0.76, size = 35, normalized size = 0.85 \begin {gather*} -\frac {4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*b^3*x^6 + 6*a*b^2*x^4 + 4*a^2*b*x^2 + a^3)/x^8

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giac [B]  time = 0.17, size = 68, normalized size = 1.66 \begin {gather*} -\frac {4 \, b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*b^3*x^6*sgn(b*x^2 + a) + 6*a*b^2*x^4*sgn(b*x^2 + a) + 4*a^2*b*x^2*sgn(b*x^2 + a) + a^3*sgn(b*x^2 + a))
/x^8

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maple [A]  time = 0.00, size = 56, normalized size = 1.37 \begin {gather*} -\frac {\left (4 b^{3} x^{6}+6 a \,b^{2} x^{4}+4 a^{2} b \,x^{2}+a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{8 \left (b \,x^{2}+a \right )^{3} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*b^3*x^6+6*a*b^2*x^4+4*a^2*b*x^2+a^3)*((b*x^2+a)^2)^(3/2)/x^8/(b*x^2+a)^3

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maxima [A]  time = 1.36, size = 35, normalized size = 0.85 \begin {gather*} -\frac {b^{3}}{2 \, x^{2}} - \frac {3 \, a b^{2}}{4 \, x^{4}} - \frac {a^{2} b}{2 \, x^{6}} - \frac {a^{3}}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^3/x^2 - 3/4*a*b^2/x^4 - 1/2*a^2*b/x^6 - 1/8*a^3/x^8

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mupad [B]  time = 4.24, size = 151, normalized size = 3.68 \begin {gather*} -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{8\,x^8\,\left (b\,x^2+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,x^2\,\left (b\,x^2+a\right )}-\frac {3\,a\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^4\,\left (b\,x^2+a\right )}-\frac {a^2\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,x^6\,\left (b\,x^2+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (a^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(8*x^8*(a + b*x^2)) - (b^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(2*x^2
*(a + b*x^2)) - (3*a*b^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(4*x^4*(a + b*x^2)) - (a^2*b*(a^2 + b^2*x^4 + 2*a*
b*x^2)^(1/2))/(2*x^6*(a + b*x^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x^{9}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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