3.202 \(\int x^{11} (a+b x^3+c x^6)^{3/2} \, dx\)

Optimal. Leaf size=223 \[ \frac{\left (-16 a c+21 b^2-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac{b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac{b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1024 c^5}-\frac{b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{2048 c^{11/2}}+\frac{x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c} \]

[Out]

(b*(b^2 - 4*a*c)*(3*b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1024*c^5) - (b*(3*b^2 - 4*a*c)*(b + 2
*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(384*c^4) + (x^6*(a + b*x^3 + c*x^6)^(5/2))/(21*c) + ((21*b^2 - 16*a*c - 30
*b*c*x^3)*(a + b*x^3 + c*x^6)^(5/2))/(840*c^3) - (b*(b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*S
qrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(2048*c^(11/2))

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Rubi [A]  time = 0.205978, antiderivative size = 223, 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, 779, 612, 621, 206} \[ \frac{\left (-16 a c+21 b^2-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac{b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac{b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1024 c^5}-\frac{b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{2048 c^{11/2}}+\frac{x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c} \]

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(b*(b^2 - 4*a*c)*(3*b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1024*c^5) - (b*(3*b^2 - 4*a*c)*(b + 2
*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(384*c^4) + (x^6*(a + b*x^3 + c*x^6)^(5/2))/(21*c) + ((21*b^2 - 16*a*c - 30
*b*c*x^3)*(a + b*x^3 + c*x^6)^(5/2))/(840*c^3) - (b*(b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*S
qrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(2048*c^(11/2))

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 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[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^{11} \left (a+b x^3+c x^6\right )^{3/2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^3 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )\\ &=\frac{x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac{\operatorname{Subst}\left (\int x \left (-2 a-\frac{9 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{21 c}\\ &=\frac{x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac{\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac{\left (b \left (3 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{48 c^3}\\ &=-\frac{b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac{x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac{\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}+\frac{\left (b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^3\right )}{256 c^4}\\ &=\frac{b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1024 c^5}-\frac{b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac{x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac{\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac{\left (b \left (b^2-4 a c\right )^2 \left (3 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 )}{2048 c^5}\\ &=\frac{b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1024 c^5}-\frac{b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac{x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac{\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac{\left (b \left (b^2-4 a c\right )^2 \left (3 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 )}{1024 c^5}\\ &=\frac{b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1024 c^5}-\frac{b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac{x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac{\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac{b \left (b^2-4 a c\right )^2 \left (3 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 )}{2048 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.196137, size = 192, normalized size = 0.86 \[ \frac{-\frac{\left (16 a c-21 b^2+30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{40 c^2}+\frac{7 \left (4 a b c-3 b^3\right ) \left (2 \sqrt{c} \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6} \left (4 c \left (5 a+2 c x^6\right )-3 b^2+8 b c x^3\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )\right )}{2048 c^{9/2}}+x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(x^6*(a + b*x^3 + c*x^6)^(5/2) - ((-21*b^2 + 16*a*c + 30*b*c*x^3)*(a + b*x^3 + c*x^6)^(5/2))/(40*c^2) + (7*(-3
*b^3 + 4*a*b*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])]))/(2048*c^(9/2)))/(21*c)

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{x}^{11} \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^11*(c*x^6+b*x^3+a)^(3/2),x)

[Out]

int(x^11*(c*x^6+b*x^3+a)^(3/2),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^11*(c*x^6+b*x^3+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.07128, size = 1268, normalized size = 5.69 \begin{align*} \left [-\frac{105 \,{\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b 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 (5120 \, c^{7} x^{18} + 6400 \, b c^{6} x^{15} + 128 \,{\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{12} - 16 \,{\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{9} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} + 8 \,{\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{6} - 2 \,{\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{430080 \, c^{6}}, \frac{105 \,{\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b 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 (5120 \, c^{7} x^{18} + 6400 \, b c^{6} x^{15} + 128 \,{\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{12} - 16 \,{\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{9} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} + 8 \,{\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{6} - 2 \,{\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{215040 \, c^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(c*x^6+b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/430080*(105*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*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*(5120*c^7*x^18 + 6400*b*c^6*x^15 + 128*(b^2*c^5
+ 64*a*c^6)*x^12 - 16*(9*b^3*c^4 - 44*a*b*c^5)*x^9 + 315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3
*c^4 + 8*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*x^6 - 2*(105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*x^3
)*sqrt(c*x^6 + b*x^3 + a))/c^6, 1/215040*(105*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*sqrt(-c)*ar
ctan(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*(5120*c^7*x^18 + 6400*b
*c^6*x^15 + 128*(b^2*c^5 + 64*a*c^6)*x^12 - 16*(9*b^3*c^4 - 44*a*b*c^5)*x^9 + 315*b^6*c - 2520*a*b^4*c^2 + 548
8*a^2*b^2*c^3 - 2048*a^3*c^4 + 8*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*x^6 - 2*(105*b^5*c^2 - 728*a*b^3*c
^3 + 1168*a^2*b*c^4)*x^3)*sqrt(c*x^6 + b*x^3 + a))/c^6]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{11} \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**11*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**11*(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^{11}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(c*x^6+b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)*x^11, x)