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

Optimal. Leaf size=252 \[ \frac{b^5 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac{5 a b^4 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{a^3 b^2 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^5 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

[Out]

(a^5*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (5*a^4*b*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(
a + b*x^3)) + (a^3*b^2*x^10*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^3) + (10*a^2*b^3*x^13*Sqrt[a^2 + 2*a*b*x
^3 + b^2*x^6])/(13*(a + b*x^3)) + (5*a*b^4*x^16*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(16*(a + b*x^3)) + (b^5*x^19*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(19*(a + b*x^3))

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Rubi [A]  time = 0.0573269, antiderivative size = 252, normalized size of antiderivative = 1., 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} \[ \frac{b^5 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac{5 a b^4 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{a^3 b^2 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^5 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (5*a^4*b*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(
a + b*x^3)) + (a^3*b^2*x^10*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^3) + (10*a^2*b^3*x^13*Sqrt[a^2 + 2*a*b*x
^3 + b^2*x^6])/(13*(a + b*x^3)) + (5*a*b^4*x^16*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(16*(a + b*x^3)) + (b^5*x^19*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(19*(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^3 \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^3 \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^3+5 a^4 b^6 x^6+10 a^3 b^7 x^9+10 a^2 b^8 x^{12}+5 a b^9 x^{15}+b^{10} x^{18}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{a^5 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^3 b^2 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{10 a^2 b^3 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{5 a b^4 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac{b^5 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.02019, size = 83, normalized size = 0.33 \[ \frac{x^4 \sqrt{\left (a+b x^3\right )^2} \left (21280 a^2 b^3 x^9+27664 a^3 b^2 x^6+19760 a^4 b x^3+6916 a^5+8645 a b^4 x^{12}+1456 b^5 x^{15}\right )}{27664 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*Sqrt[(a + b*x^3)^2]*(6916*a^5 + 19760*a^4*b*x^3 + 27664*a^3*b^2*x^6 + 21280*a^2*b^3*x^9 + 8645*a*b^4*x^12
+ 1456*b^5*x^15))/(27664*(a + b*x^3))

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Maple [A]  time = 0.007, size = 80, normalized size = 0.3 \begin{align*}{\frac{{x}^{4} \left ( 1456\,{b}^{5}{x}^{15}+8645\,a{b}^{4}{x}^{12}+21280\,{a}^{2}{b}^{3}{x}^{9}+27664\,{a}^{3}{b}^{2}{x}^{6}+19760\,{a}^{4}b{x}^{3}+6916\,{a}^{5} \right ) }{27664\, \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^3*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/27664*x^4*(1456*b^5*x^15+8645*a*b^4*x^12+21280*a^2*b^3*x^9+27664*a^3*b^2*x^6+19760*a^4*b*x^3+6916*a^5)*((b*x
^3+a)^2)^(5/2)/(b*x^3+a)^5

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Maxima [A]  time = 1.02717, size = 76, normalized size = 0.3 \begin{align*} \frac{1}{19} \, b^{5} x^{19} + \frac{5}{16} \, a b^{4} x^{16} + \frac{10}{13} \, a^{2} b^{3} x^{13} + a^{3} b^{2} x^{10} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{4} \, a^{5} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*b^5*x^19 + 5/16*a*b^4*x^16 + 10/13*a^2*b^3*x^13 + a^3*b^2*x^10 + 5/7*a^4*b*x^7 + 1/4*a^5*x^4

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Fricas [A]  time = 1.72516, size = 134, normalized size = 0.53 \begin{align*} \frac{1}{19} \, b^{5} x^{19} + \frac{5}{16} \, a b^{4} x^{16} + \frac{10}{13} \, a^{2} b^{3} x^{13} + a^{3} b^{2} x^{10} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{4} \, a^{5} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*b^5*x^19 + 5/16*a*b^4*x^16 + 10/13*a^2*b^3*x^13 + a^3*b^2*x^10 + 5/7*a^4*b*x^7 + 1/4*a^5*x^4

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

[Out]

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

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Giac [A]  time = 1.11393, size = 140, normalized size = 0.56 \begin{align*} \frac{1}{19} \, b^{5} x^{19} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{16} \, a b^{4} x^{16} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{13} \, a^{2} b^{3} x^{13} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{3} b^{2} x^{10} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{7} \, a^{4} b x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{4} \, a^{5} x^{4} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*b^5*x^19*sgn(b*x^3 + a) + 5/16*a*b^4*x^16*sgn(b*x^3 + a) + 10/13*a^2*b^3*x^13*sgn(b*x^3 + a) + a^3*b^2*x^
10*sgn(b*x^3 + a) + 5/7*a^4*b*x^7*sgn(b*x^3 + a) + 1/4*a^5*x^4*sgn(b*x^3 + a)