3.1.45 \(\int \frac {(a^2+2 a b x^3+b^2 x^6)^{3/2}}{x^{13}} \, dx\)

Optimal. Leaf size=41 \[ -\frac {\left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}} \]

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Rubi [A]  time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 264} \begin {gather*} -\frac {\left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^13,x]

[Out]

-((a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*a*x^12)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{13}} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^3}{x^{13}} \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac {\left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 59, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (a^3+4 a^2 b x^3+6 a b^2 x^6+4 b^3 x^9\right )}{12 x^{12} \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^13,x]

[Out]

-1/12*(Sqrt[(a + b*x^3)^2]*(a^3 + 4*a^2*b*x^3 + 6*a*b^2*x^6 + 4*b^3*x^9))/(x^12*(a + b*x^3))

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IntegrateAlgebraic [B]  time = 1.35, size = 310, normalized size = 7.56 \begin {gather*} \frac {2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-a^6 b-7 a^5 b^2 x^3-21 a^4 b^3 x^6-35 a^3 b^4 x^9-34 a^2 b^5 x^{12}-18 a b^6 x^{15}-4 b^7 x^{18}\right )+2 \sqrt {b^2} b^3 \left (a^7+8 a^6 b x^3+28 a^5 b^2 x^6+56 a^4 b^3 x^9+69 a^3 b^4 x^{12}+52 a^2 b^5 x^{15}+22 a b^6 x^{18}+4 b^7 x^{21}\right )}{3 \sqrt {b^2} x^{12} \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-8 a^3 b^3-24 a^2 b^4 x^3-24 a b^5 x^6-8 b^6 x^9\right )+3 x^{12} \left (8 a^4 b^4+32 a^3 b^5 x^3+48 a^2 b^6 x^6+32 a b^7 x^9+8 b^8 x^{12}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^13,x]

[Out]

(2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*(-(a^6*b) - 7*a^5*b^2*x^3 - 21*a^4*b^3*x^6 - 35*a^3*b^4*x^9 - 34*a^2*b^
5*x^12 - 18*a*b^6*x^15 - 4*b^7*x^18) + 2*b^3*Sqrt[b^2]*(a^7 + 8*a^6*b*x^3 + 28*a^5*b^2*x^6 + 56*a^4*b^3*x^9 +
69*a^3*b^4*x^12 + 52*a^2*b^5*x^15 + 22*a*b^6*x^18 + 4*b^7*x^21))/(3*Sqrt[b^2]*x^12*Sqrt[a^2 + 2*a*b*x^3 + b^2*
x^6]*(-8*a^3*b^3 - 24*a^2*b^4*x^3 - 24*a*b^5*x^6 - 8*b^6*x^9) + 3*x^12*(8*a^4*b^4 + 32*a^3*b^5*x^3 + 48*a^2*b^
6*x^6 + 32*a*b^7*x^9 + 8*b^8*x^12))

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fricas [A]  time = 1.42, size = 35, normalized size = 0.85 \begin {gather*} -\frac {4 \, b^{3} x^{9} + 6 \, a b^{2} x^{6} + 4 \, a^{2} b x^{3} + a^{3}}{12 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(3/2)/x^13,x, algorithm="fricas")

[Out]

-1/12*(4*b^3*x^9 + 6*a*b^2*x^6 + 4*a^2*b*x^3 + a^3)/x^12

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giac [B]  time = 0.36, size = 68, normalized size = 1.66 \begin {gather*} -\frac {4 \, b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 6 \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 4 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{12 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(3/2)/x^13,x, algorithm="giac")

[Out]

-1/12*(4*b^3*x^9*sgn(b*x^3 + a) + 6*a*b^2*x^6*sgn(b*x^3 + a) + 4*a^2*b*x^3*sgn(b*x^3 + a) + a^3*sgn(b*x^3 + a)
)/x^12

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maple [A]  time = 0.01, size = 56, normalized size = 1.37 \begin {gather*} -\frac {\left (4 b^{3} x^{9}+6 a \,b^{2} x^{6}+4 a^{2} b \,x^{3}+a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}}}{12 \left (b \,x^{3}+a \right )^{3} x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^6+2*a*b*x^3+a^2)^(3/2)/x^13,x)

[Out]

-1/12*(4*b^3*x^9+6*a*b^2*x^6+4*a^2*b*x^3+a^3)*((b*x^3+a)^2)^(3/2)/x^12/(b*x^3+a)^3

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maxima [B]  time = 0.55, size = 148, normalized size = 3.61 \begin {gather*} \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{4}}{12 \, a^{4}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{3}}{12 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{2}}{12 \, a^{4} x^{6}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b}{12 \, a^{3} x^{9}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}}}{12 \, a^{2} x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(3/2)/x^13,x, algorithm="maxima")

[Out]

1/12*(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*b^4/a^4 + 1/12*(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*b^3/(a^3*x^3) - 1/12*(
b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*b^2/(a^4*x^6) + 1/12*(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*b/(a^3*x^9) - 1/12*(b^
2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/(a^2*x^12)

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mupad [B]  time = 1.21, size = 151, normalized size = 3.68 \begin {gather*} -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{12\,x^{12}\,\left (b\,x^3+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^3\,\left (b\,x^3+a\right )}-\frac {a\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{2\,x^6\,\left (b\,x^3+a\right )}-\frac {a^2\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^9\,\left (b\,x^3+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^6 + 2*a*b*x^3)^(3/2)/x^13,x)

[Out]

- (a^3*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(12*x^12*(a + b*x^3)) - (b^3*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(3*x
^3*(a + b*x^3)) - (a*b^2*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/(2*x^6*(a + b*x^3)) - (a^2*b*(a^2 + b^2*x^6 + 2*a*
b*x^3)^(1/2))/(3*x^9*(a + b*x^3))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x^{13}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**6+2*a*b*x**3+a**2)**(3/2)/x**13,x)

[Out]

Integral(((a + b*x**3)**2)**(3/2)/x**13, x)

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