3.188 \(\int x^5 \sqrt{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=108 \[ \frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{5/2}}-\frac{b \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{24 c^2}+\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 c} \]

[Out]

-(b*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(24*c^2) + (a + b*x^3 + c*x^6)^(3/2)/(9*c) + (b*(b^2 - 4*a*c)*ArcTa
nh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(48*c^(5/2))

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Rubi [A]  time = 0.0853352, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1357, 640, 612, 621, 206} \[ \frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{5/2}}-\frac{b \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{24 c^2}+\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 c} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[a + b*x^3 + c*x^6],x]

[Out]

-(b*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(24*c^2) + (a + b*x^3 + c*x^6)^(3/2)/(9*c) + (b*(b^2 - 4*a*c)*ArcTa
nh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(48*c^(5/2))

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^5 \sqrt{a+b x^3+c x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x \sqrt{a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 c}-\frac{b \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^3\right )}{6 c}\\ &=-\frac{b \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{24 c^2}+\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 c}+\frac{\left (b \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{48 c^2}\\ &=-\frac{b \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{24 c^2}+\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 c}+\frac{\left (b \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^3}{\sqrt{a+b x^3+c x^6}}\right )}{24 c^2}\\ &=-\frac{b \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{24 c^2}+\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 c}+\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0760385, size = 99, normalized size = 0.92 \[ \frac{\sqrt{a+b x^3+c x^6} \left (8 c \left (a+c x^6\right )-3 b^2+2 b c x^3\right )}{72 c^2}+\frac{\left (b^3-4 a b c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[a + b*x^3 + c*x^6],x]

[Out]

(Sqrt[a + b*x^3 + c*x^6]*(-3*b^2 + 2*b*c*x^3 + 8*c*(a + c*x^6)))/(72*c^2) + ((b^3 - 4*a*b*c)*ArcTanh[(b + 2*c*
x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(48*c^(5/2))

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{x}^{5}\sqrt{c{x}^{6}+b{x}^{3}+a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(x^5*(c*x^6+b*x^3+a)^(1/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^6+b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.64762, size = 554, normalized size = 5.13 \begin{align*} \left [-\frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} + 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{c} - 4 \, a c\right ) - 4 \,{\left (8 \, c^{3} x^{6} + 2 \, b c^{2} x^{3} - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt{c x^{6} + b x^{3} + a}}{288 \, c^{3}}, -\frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \,{\left (8 \, c^{3} x^{6} + 2 \, b c^{2} x^{3} - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt{c x^{6} + b x^{3} + a}}{144 \, c^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^6+b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/288*(3*(b^3 - 4*a*b*c)*sqrt(c)*log(-8*c^2*x^6 - 8*b*c*x^3 - b^2 + 4*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*
sqrt(c) - 4*a*c) - 4*(8*c^3*x^6 + 2*b*c^2*x^3 - 3*b^2*c + 8*a*c^2)*sqrt(c*x^6 + b*x^3 + a))/c^3, -1/144*(3*(b^
3 - 4*a*b*c)*sqrt(-c)*arctan(1/2*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(-c)/(c^2*x^6 + b*c*x^3 + a*c)) - 2
*(8*c^3*x^6 + 2*b*c^2*x^3 - 3*b^2*c + 8*a*c^2)*sqrt(c*x^6 + b*x^3 + a))/c^3]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \sqrt{a + b x^{3} + c x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(x**5*sqrt(a + b*x**3 + c*x**6), x)

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Giac [A]  time = 1.16761, size = 132, normalized size = 1.22 \begin{align*} \frac{1}{72} \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \,{\left (4 \, x^{3} + \frac{b}{c}\right )} x^{3} - \frac{3 \, b^{2} - 8 \, a c}{c^{2}}\right )} - \frac{{\left (b^{3} - 4 \, a b c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{3} - \sqrt{c x^{6} + b x^{3} + a}\right )} \sqrt{c} - b \right |}\right )}{48 \, c^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^6+b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

1/72*sqrt(c*x^6 + b*x^3 + a)*(2*(4*x^3 + b/c)*x^3 - (3*b^2 - 8*a*c)/c^2) - 1/48*(b^3 - 4*a*b*c)*log(abs(-2*(sq
rt(c)*x^3 - sqrt(c*x^6 + b*x^3 + a))*sqrt(c) - b))/c^(5/2)