3.203 \(\int x^8 (a+b x^3+c x^6)^{3/2} \, dx\)

Optimal. Leaf size=204 \[ \frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3072 c^{9/2}}-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c} \]

[Out]

-((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1536*c^4) + ((7*b^2 - 4*a*c)*(b + 2*c*
x^3)*(a + b*x^3 + c*x^6)^(3/2))/(576*c^3) - (7*b*(a + b*x^3 + c*x^6)^(5/2))/(180*c^2) + (x^3*(a + b*x^3 + c*x^
6)^(5/2))/(18*c) + ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]
)/(3072*c^(9/2))

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Rubi [A]  time = 0.187612, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1357, 742, 640, 612, 621, 206} \[ \frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3072 c^{9/2}}-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

-((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1536*c^4) + ((7*b^2 - 4*a*c)*(b + 2*c*
x^3)*(a + b*x^3 + c*x^6)^(3/2))/(576*c^3) - (7*b*(a + b*x^3 + c*x^6)^(5/2))/(180*c^2) + (x^3*(a + b*x^3 + c*x^
6)^(5/2))/(18*c) + ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]
)/(3072*c^(9/2))

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 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

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^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )\\ &=\frac{x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac{\operatorname{Subst}\left (\int \left (-a-\frac{7 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{18 c}\\ &=-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac{\left (7 b^2-4 a c\right ) \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{72 c^2}\\ &=\frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}-\frac{\left (\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^3\right )}{384 c^3}\\ &=-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^4}+\frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 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 )}{3072 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^4}+\frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 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 )}{1536 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^4}+\frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac{\left (b^2-4 a c\right )^2 \left (7 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 )}{3072 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.169545, size = 175, normalized size = 0.86 \[ \frac{\frac{\left (7 b^2-4 a c\right ) \left (2 \sqrt{c} \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6} \left (4 c \left (5 a+2 c x^6\right )-3 b^2+8 b c x^3\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )\right )}{512 c^{7/2}}+x^3 \left (a+b x^3+c x^6\right )^{5/2}-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{10 c}}{18 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

((-7*b*(a + b*x^3 + c*x^6)^(5/2))/(10*c) + x^3*(a + b*x^3 + c*x^6)^(5/2) + ((7*b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*
c*x^3)*Sqrt[a + b*x^3 + c*x^6]*(-3*b^2 + 8*b*c*x^3 + 4*c*(5*a + 2*c*x^6)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c
*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]))/(512*c^(7/2)))/(18*c)

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{x}^{8} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(c*x^6+b*x^3+a)^(3/2),x)

[Out]

int(x^8*(c*x^6+b*x^3+a)^(3/2),x)

________________________________________________________________________________________

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^8*(c*x^6+b*x^3+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.76723, size = 1064, normalized size = 5.22 \begin{align*} \left [-\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 (1280 \, c^{6} x^{15} + 1664 \, b c^{5} x^{12} + 16 \,{\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{9} - 8 \,{\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 2 \,{\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{92160 \, c^{5}}, -\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 (1280 \, c^{6} x^{15} + 1664 \, b c^{5} x^{12} + 16 \,{\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{9} - 8 \,{\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 2 \,{\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{46080 \, c^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(c*x^6+b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/92160*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*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*(1280*c^6*x^15 + 1664*b*c^5*x^12 + 16*(3*b^2*c^4 +
140*a*c^5)*x^9 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^6 - 105*b^5*c + 760*a*b^3*c^2 - 1296*a^2*b*c^3 + 2*(35*b^4*c^2
- 216*a*b^2*c^3 + 240*a^2*c^4)*x^3)*sqrt(c*x^6 + b*x^3 + a))/c^5, -1/46080*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b
^2*c^2 - 64*a^3*c^3)*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*(1280*c^6*x^15 + 1664*b*c^5*x^12 + 16*(3*b^2*c^4 + 140*a*c^5)*x^9 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^6 -
105*b^5*c + 760*a*b^3*c^2 - 1296*a^2*b*c^3 + 2*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*x^3)*sqrt(c*x^6 + b*
x^3 + a))/c^5]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{8} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**8*(a + b*x**3 + c*x**6)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{8}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(c*x^6+b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)*x^8, x)