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

Optimal. Leaf size=79 \[ \frac{b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.0235534, antiderivative size = 79, 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, 14} \[ \frac{b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

(a*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (b*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x^3 \sqrt{a^2+2 a b x^3+b^2 x^6} \, 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 ) \, dx}{a b+b^2 x^3}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (a b x^3+b^2 x^6\right ) \, dx}{a b+b^2 x^3}\\ &=\frac{a x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.0072048, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (7 a x^4+4 b x^7\right )}{28 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

(Sqrt[(a + b*x^3)^2]*(7*a*x^4 + 4*b*x^7))/(28*(a + b*x^3))

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Maple [A]  time = 0.003, size = 36, normalized size = 0.5 \begin{align*}{\frac{{x}^{4} \left ( 4\,b{x}^{3}+7\,a \right ) }{28\,b{x}^{3}+28\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*((b*x^3+a)^2)^(1/2),x)

[Out]

1/28*x^4*(4*b*x^3+7*a)*((b*x^3+a)^2)^(1/2)/(b*x^3+a)

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Maxima [A]  time = 1.08544, size = 18, normalized size = 0.23 \begin{align*} \frac{1}{7} \, b x^{7} + \frac{1}{4} \, a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*((b*x^3+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/7*b*x^7 + 1/4*a*x^4

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Fricas [A]  time = 1.71453, size = 31, normalized size = 0.39 \begin{align*} \frac{1}{7} \, b x^{7} + \frac{1}{4} \, a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*((b*x^3+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/7*b*x^7 + 1/4*a*x^4

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Sympy [A]  time = 0.098831, size = 12, normalized size = 0.15 \begin{align*} \frac{a x^{4}}{4} + \frac{b x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*((b*x**3+a)**2)**(1/2),x)

[Out]

a*x**4/4 + b*x**7/7

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Giac [A]  time = 1.10824, size = 39, normalized size = 0.49 \begin{align*} \frac{1}{7} \, b x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{4} \, a 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*x^3+a)^2)^(1/2),x, algorithm="giac")

[Out]

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