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

Optimal. Leaf size=252 \[ \frac{b^5 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{5 a b^4 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{a^4 b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]

[Out]

(a^5*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (a^4*b*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b
*x^3) + (5*a^3*b^2*x^8*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (10*a^2*b^3*x^11*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(11*(a + b*x^3)) + (5*a*b^4*x^14*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*(a + b*x^3)) + (b^5*x^17*S
qrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(17*(a + b*x^3))

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Rubi [A]  time = 0.0544888, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1355, 270} \[ \frac{b^5 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{5 a b^4 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{a^4 b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (a^4*b*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b
*x^3) + (5*a^3*b^2*x^8*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (10*a^2*b^3*x^11*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(11*(a + b*x^3)) + (5*a*b^4*x^14*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*(a + b*x^3)) + (b^5*x^17*S
qrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(17*(a + b*x^3))

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 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]

Rubi steps

\begin{align*} \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int x \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (a^5 b^5 x+5 a^4 b^6 x^4+10 a^3 b^7 x^7+10 a^2 b^8 x^{10}+5 a b^9 x^{13}+b^{10} x^{16}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{a^4 b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a b^4 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{b^5 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.0204332, size = 83, normalized size = 0.33 \[ \frac{x^2 \sqrt{\left (a+b x^3\right )^2} \left (4760 a^2 b^3 x^9+6545 a^3 b^2 x^6+5236 a^4 b x^3+2618 a^5+1870 a b^4 x^{12}+308 b^5 x^{15}\right )}{5236 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*Sqrt[(a + b*x^3)^2]*(2618*a^5 + 5236*a^4*b*x^3 + 6545*a^3*b^2*x^6 + 4760*a^2*b^3*x^9 + 1870*a*b^4*x^12 +
308*b^5*x^15))/(5236*(a + b*x^3))

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Maple [A]  time = 0.004, size = 80, normalized size = 0.3 \begin{align*}{\frac{{x}^{2} \left ( 308\,{b}^{5}{x}^{15}+1870\,a{b}^{4}{x}^{12}+4760\,{a}^{2}{b}^{3}{x}^{9}+6545\,{a}^{3}{b}^{2}{x}^{6}+5236\,{a}^{4}b{x}^{3}+2618\,{a}^{5} \right ) }{5236\, \left ( b{x}^{3}+a \right ) ^{5}} \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*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/5236*x^2*(308*b^5*x^15+1870*a*b^4*x^12+4760*a^2*b^3*x^9+6545*a^3*b^2*x^6+5236*a^4*b*x^3+2618*a^5)*((b*x^3+a)
^2)^(5/2)/(b*x^3+a)^5

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Maxima [A]  time = 1.0165, size = 76, normalized size = 0.3 \begin{align*} \frac{1}{17} \, b^{5} x^{17} + \frac{5}{14} \, a b^{4} x^{14} + \frac{10}{11} \, a^{2} b^{3} x^{11} + \frac{5}{4} \, a^{3} b^{2} x^{8} + a^{4} b x^{5} + \frac{1}{2} \, a^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*b^5*x^17 + 5/14*a*b^4*x^14 + 10/11*a^2*b^3*x^11 + 5/4*a^3*b^2*x^8 + a^4*b*x^5 + 1/2*a^5*x^2

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Fricas [A]  time = 1.7524, size = 132, normalized size = 0.52 \begin{align*} \frac{1}{17} \, b^{5} x^{17} + \frac{5}{14} \, a b^{4} x^{14} + \frac{10}{11} \, a^{2} b^{3} x^{11} + \frac{5}{4} \, a^{3} b^{2} x^{8} + a^{4} b x^{5} + \frac{1}{2} \, a^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*b^5*x^17 + 5/14*a*b^4*x^14 + 10/11*a^2*b^3*x^11 + 5/4*a^3*b^2*x^8 + a^4*b*x^5 + 1/2*a^5*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \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*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

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

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Giac [A]  time = 1.13603, size = 140, normalized size = 0.56 \begin{align*} \frac{1}{17} \, b^{5} x^{17} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{14} \, a b^{4} x^{14} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{11} \, a^{2} b^{3} x^{11} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{4} \, a^{3} b^{2} x^{8} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{4} b x^{5} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{2} \, a^{5} x^{2} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*b^5*x^17*sgn(b*x^3 + a) + 5/14*a*b^4*x^14*sgn(b*x^3 + a) + 10/11*a^2*b^3*x^11*sgn(b*x^3 + a) + 5/4*a^3*b^
2*x^8*sgn(b*x^3 + a) + a^4*b*x^5*sgn(b*x^3 + a) + 1/2*a^5*x^2*sgn(b*x^3 + a)