3.1.41 \(\int \frac {(a^2+2 a b x^3+b^2 x^6)^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=162 \[ -\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 270} \begin {gather*} -\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^9,x]

[Out]

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

Rule 270

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

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^9} \, 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^9} \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (b^6+\frac {a^3 b^3}{x^9}+\frac {3 a^2 b^4}{x^6}+\frac {3 a b^5}{x^3}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 61, normalized size = 0.38 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (5 a^3+24 a^2 b x^3+60 a b^2 x^6-40 b^3 x^9\right )}{40 x^8 \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^9,x]

[Out]

-1/40*(Sqrt[(a + b*x^3)^2]*(5*a^3 + 24*a^2*b*x^3 + 60*a*b^2*x^6 - 40*b^3*x^9))/(x^8*(a + b*x^3))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 20.49, size = 61, normalized size = 0.38 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (-5 a^3-24 a^2 b x^3-60 a b^2 x^6+40 b^3 x^9\right )}{40 x^8 \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^9,x]

[Out]

(Sqrt[(a + b*x^3)^2]*(-5*a^3 - 24*a^2*b*x^3 - 60*a*b^2*x^6 + 40*b^3*x^9))/(40*x^8*(a + b*x^3))

________________________________________________________________________________________

fricas [A]  time = 1.26, size = 37, normalized size = 0.23 \begin {gather*} \frac {40 \, b^{3} x^{9} - 60 \, a b^{2} x^{6} - 24 \, a^{2} b x^{3} - 5 \, a^{3}}{40 \, x^{8}} \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^9,x, algorithm="fricas")

[Out]

1/40*(40*b^3*x^9 - 60*a*b^2*x^6 - 24*a^2*b*x^3 - 5*a^3)/x^8

________________________________________________________________________________________

giac [A]  time = 0.39, size = 67, normalized size = 0.41 \begin {gather*} b^{3} x \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {60 \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 24 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 5 \, a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{40 \, x^{8}} \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^9,x, algorithm="giac")

[Out]

b^3*x*sgn(b*x^3 + a) - 1/40*(60*a*b^2*x^6*sgn(b*x^3 + a) + 24*a^2*b*x^3*sgn(b*x^3 + a) + 5*a^3*sgn(b*x^3 + a))
/x^8

________________________________________________________________________________________

maple [A]  time = 0.01, size = 58, normalized size = 0.36 \begin {gather*} -\frac {\left (-40 b^{3} x^{9}+60 a \,b^{2} x^{6}+24 a^{2} b \,x^{3}+5 a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}}}{40 \left (b \,x^{3}+a \right )^{3} x^{8}} \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^9,x)

[Out]

-1/40*(-40*b^3*x^9+60*a*b^2*x^6+24*a^2*b*x^3+5*a^3)*((b*x^3+a)^2)^(3/2)/x^8/(b*x^3+a)^3

________________________________________________________________________________________

maxima [A]  time = 0.68, size = 37, normalized size = 0.23 \begin {gather*} \frac {40 \, b^{3} x^{9} - 60 \, a b^{2} x^{6} - 24 \, a^{2} b x^{3} - 5 \, a^{3}}{40 \, x^{8}} \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^9,x, algorithm="maxima")

[Out]

1/40*(40*b^3*x^9 - 60*a*b^2*x^6 - 24*a^2*b*x^3 - 5*a^3)/x^8

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}}{x^9} \,d x \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^9,x)

[Out]

int((a^2 + b^2*x^6 + 2*a*b*x^3)^(3/2)/x^9, x)

________________________________________________________________________________________

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^{9}}\, 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**9,x)

[Out]

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

________________________________________________________________________________________