3.3.2 \(\int x^{11} (a+b x^3+c x^6)^{3/2} \, dx\) [202]

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

[Out]

-1/384*b*(-4*a*c+3*b^2)*(2*c*x^3+b)*(c*x^6+b*x^3+a)^(3/2)/c^4+1/21*x^6*(c*x^6+b*x^3+a)^(5/2)/c+1/840*(-30*b*c*
x^3-16*a*c+21*b^2)*(c*x^6+b*x^3+a)^(5/2)/c^3-1/2048*b*(-4*a*c+b^2)^2*(-4*a*c+3*b^2)*arctanh(1/2*(2*c*x^3+b)/c^
(1/2)/(c*x^6+b*x^3+a)^(1/2))/c^(11/2)+1/1024*b*(-4*a*c+b^2)*(-4*a*c+3*b^2)*(2*c*x^3+b)*(c*x^6+b*x^3+a)^(1/2)/c
^5

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Rubi [A]
time = 0.14, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1371, 756, 793, 626, 635, 212} \begin {gather*} -\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 {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 {\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 {x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c} \end {gather*}

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 212

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

Rule 626

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 635

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 756

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 793

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 1371

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]]

Rubi steps

\begin {align*} \int x^{11} \left (a+b x^3+c x^6\right )^{3/2} \, dx &=\frac {1}{3} \text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.64, size = 220, normalized size = 0.99 \begin {gather*} \frac {\sqrt {a+b x^3+c x^6} \left (315 b^6-210 b^5 c x^3+16 b^3 c^2 x^3 \left (91 a-9 c x^6\right )+168 b^4 c \left (-15 a+c x^6\right )+1024 c^3 \left (a+c x^6\right )^2 \left (-2 a+5 c x^6\right )+16 b^2 c^2 \left (343 a^2-62 a c x^6+8 c^2 x^{12}\right )+32 b c^3 x^3 \left (-73 a^2+22 a c x^6+200 c^2 x^{12}\right )\right )}{107520 c^5}+\frac {b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \log \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right )}{2048 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^3 + c*x^6]*(315*b^6 - 210*b^5*c*x^3 + 16*b^3*c^2*x^3*(91*a - 9*c*x^6) + 168*b^4*c*(-15*a + c*x^6
) + 1024*c^3*(a + c*x^6)^2*(-2*a + 5*c*x^6) + 16*b^2*c^2*(343*a^2 - 62*a*c*x^6 + 8*c^2*x^12) + 32*b*c^3*x^3*(-
73*a^2 + 22*a*c*x^6 + 200*c^2*x^12)))/(107520*c^5) + (b*(b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*Log[b + 2*c*x^3 - 2*Sq
rt[c]*Sqrt[a + b*x^3 + c*x^6]])/(2048*c^(11/2))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{11} \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [A]
time = 0.43, size = 535, normalized size = 2.40 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{11} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^{11}\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(a + b*x^3 + c*x^6)^(3/2),x)

[Out]

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

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