3.108 \(\int \frac{x^5}{(a^2+2 a b x^3+b^2 x^6)^{5/2}} \, dx\)

Optimal. Leaf size=78 \[ \frac{a}{12 b^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{1}{9 b^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

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

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Rubi [A]  time = 0.0526029, antiderivative size = 78, 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 = {1355, 266, 43} \[ \frac{a}{12 b^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{1}{9 b^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

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

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]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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

Mathematica [A]  time = 0.016172, size = 39, normalized size = 0.5 \[ \frac{-a-4 b x^3}{36 b^2 \left (a+b x^3\right )^3 \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(-a - 4*b*x^3)/(36*b^2*(a + b*x^3)^3*Sqrt[(a + b*x^3)^2])

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Maple [A]  time = 0.007, size = 32, normalized size = 0.4 \begin{align*} -{\frac{ \left ( b{x}^{3}+a \right ) \left ( 4\,b{x}^{3}+a \right ) }{36\,{b}^{2}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

-1/36*(b*x^3+a)*(4*b*x^3+a)/b^2/((b*x^3+a)^2)^(5/2)

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Maxima [A]  time = 1.02845, size = 65, normalized size = 0.83 \begin{align*} -\frac{1}{9 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac{3}{2}} b^{2}} + \frac{a}{12 \,{\left (x^{3} + \frac{a}{b}\right )}^{4}{\left (b^{2}\right )}^{\frac{5}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.68398, size = 119, normalized size = 1.53 \begin{align*} -\frac{4 \, b x^{3} + a}{36 \,{\left (b^{6} x^{12} + 4 \, a b^{5} x^{9} + 6 \, a^{2} b^{4} x^{6} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/36*(4*b*x^3 + a)/(b^6*x^12 + 4*a*b^5*x^9 + 6*a^2*b^4*x^6 + 4*a^3*b^3*x^3 + a^4*b^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**5/((a + b*x**3)**2)**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x