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

Optimal. Leaf size=167 \[ \frac{b^3 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{3 a b^2 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac{3 a^2 b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.0409833, antiderivative size = 167, 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} \[ \frac{b^3 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{3 a b^2 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac{3 a^2 b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0165013, size = 61, normalized size = 0.37 \[ \frac{x^4 \sqrt{\left (a+b x^3\right )^2} \left (780 a^2 b x^3+455 a^3+546 a b^2 x^6+140 b^3 x^9\right )}{1820 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*Sqrt[(a + b*x^3)^2]*(455*a^3 + 780*a^2*b*x^3 + 546*a*b^2*x^6 + 140*b^3*x^9))/(1820*(a + b*x^3))

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Maple [A]  time = 0.004, size = 58, normalized size = 0.4 \begin{align*}{\frac{{x}^{4} \left ( 140\,{b}^{3}{x}^{9}+546\,a{b}^{2}{x}^{6}+780\,{a}^{2}b{x}^{3}+455\,{a}^{3} \right ) }{1820\, \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{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)^(3/2),x)

[Out]

1/1820*x^4*(140*b^3*x^9+546*a*b^2*x^6+780*a^2*b*x^3+455*a^3)*((b*x^3+a)^2)^(3/2)/(b*x^3+a)^3

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Maxima [A]  time = 1.02238, size = 47, normalized size = 0.28 \begin{align*} \frac{1}{13} \, b^{3} x^{13} + \frac{3}{10} \, a b^{2} x^{10} + \frac{3}{7} \, a^{2} b x^{7} + \frac{1}{4} \, a^{3} 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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.71503, size = 85, normalized size = 0.51 \begin{align*} \frac{1}{13} \, b^{3} x^{13} + \frac{3}{10} \, a b^{2} x^{10} + \frac{3}{7} \, a^{2} b x^{7} + \frac{1}{4} \, a^{3} 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)^(3/2),x, algorithm="fricas")

[Out]

1/13*b^3*x^13 + 3/10*a*b^2*x^10 + 3/7*a^2*b*x^7 + 1/4*a^3*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{3}{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)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.11049, size = 90, normalized size = 0.54 \begin{align*} \frac{1}{13} \, b^{3} x^{13} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{3}{10} \, a b^{2} x^{10} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{3}{7} \, a^{2} b x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{4} \, a^{3} 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)^(3/2),x, algorithm="giac")

[Out]

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