3.3.23 \(\int \frac {x^{11} (c+d x^3)^{3/2}}{8 c-d x^3} \, dx\)

Optimal. Leaf size=130 \[ \frac {9216 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^4}-\frac {3072 c^4 \sqrt {c+d x^3}}{d^4}-\frac {1024 c^3 \left (c+d x^3\right )^{3/2}}{9 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {4 c \left (c+d x^3\right )^{7/2}}{7 d^4}-\frac {2 \left (c+d x^3\right )^{9/2}}{27 d^4} \]

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Rubi [A]  time = 0.11, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {446, 88, 50, 63, 206} \begin {gather*} -\frac {3072 c^4 \sqrt {c+d x^3}}{d^4}-\frac {1024 c^3 \left (c+d x^3\right )^{3/2}}{9 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{5/2}}{5 d^4}+\frac {9216 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^4}-\frac {4 c \left (c+d x^3\right )^{7/2}}{7 d^4}-\frac {2 \left (c+d x^3\right )^{9/2}}{27 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^11*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]

[Out]

(-3072*c^4*Sqrt[c + d*x^3])/d^4 - (1024*c^3*(c + d*x^3)^(3/2))/(9*d^4) - (38*c^2*(c + d*x^3)^(5/2))/(5*d^4) -
(4*c*(c + d*x^3)^(7/2))/(7*d^4) - (2*(c + d*x^3)^(9/2))/(27*d^4) + (9216*c^(9/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sq
rt[c])])/d^4

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^{11} \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3 (c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {57 c^2 (c+d x)^{3/2}}{d^3}+\frac {512 c^3 (c+d x)^{3/2}}{d^3 (8 c-d x)}-\frac {6 c (c+d x)^{5/2}}{d^3}-\frac {(c+d x)^{7/2}}{d^3}\right ) \, dx,x,x^3\right )\\ &=-\frac {38 c^2 \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {4 c \left (c+d x^3\right )^{7/2}}{7 d^4}-\frac {2 \left (c+d x^3\right )^{9/2}}{27 d^4}+\frac {\left (512 c^3\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )}{3 d^3}\\ &=-\frac {1024 c^3 \left (c+d x^3\right )^{3/2}}{9 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {4 c \left (c+d x^3\right )^{7/2}}{7 d^4}-\frac {2 \left (c+d x^3\right )^{9/2}}{27 d^4}+\frac {\left (1536 c^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )}{d^3}\\ &=-\frac {3072 c^4 \sqrt {c+d x^3}}{d^4}-\frac {1024 c^3 \left (c+d x^3\right )^{3/2}}{9 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {4 c \left (c+d x^3\right )^{7/2}}{7 d^4}-\frac {2 \left (c+d x^3\right )^{9/2}}{27 d^4}+\frac {\left (13824 c^5\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{d^3}\\ &=-\frac {3072 c^4 \sqrt {c+d x^3}}{d^4}-\frac {1024 c^3 \left (c+d x^3\right )^{3/2}}{9 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {4 c \left (c+d x^3\right )^{7/2}}{7 d^4}-\frac {2 \left (c+d x^3\right )^{9/2}}{27 d^4}+\frac {\left (27648 c^5\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d^4}\\ &=-\frac {3072 c^4 \sqrt {c+d x^3}}{d^4}-\frac {1024 c^3 \left (c+d x^3\right )^{3/2}}{9 d^4}-\frac {38 c^2 \left (c+d x^3\right )^{5/2}}{5 d^4}-\frac {4 c \left (c+d x^3\right )^{7/2}}{7 d^4}-\frac {2 \left (c+d x^3\right )^{9/2}}{27 d^4}+\frac {9216 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^4}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 93, normalized size = 0.72 \begin {gather*} \frac {9216 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^4}-\frac {2 \sqrt {c+d x^3} \left (1509176 c^4+61892 c^3 d x^3+4611 c^2 d^2 x^6+410 c d^3 x^9+35 d^4 x^{12}\right )}{945 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]

[Out]

(-2*Sqrt[c + d*x^3]*(1509176*c^4 + 61892*c^3*d*x^3 + 4611*c^2*d^2*x^6 + 410*c*d^3*x^9 + 35*d^4*x^12))/(945*d^4
) + (9216*c^(9/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^4

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IntegrateAlgebraic [A]  time = 0.07, size = 93, normalized size = 0.72 \begin {gather*} \frac {9216 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^4}-\frac {2 \sqrt {c+d x^3} \left (1509176 c^4+61892 c^3 d x^3+4611 c^2 d^2 x^6+410 c d^3 x^9+35 d^4 x^{12}\right )}{945 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^11*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]

[Out]

(-2*Sqrt[c + d*x^3]*(1509176*c^4 + 61892*c^3*d*x^3 + 4611*c^2*d^2*x^6 + 410*c*d^3*x^9 + 35*d^4*x^12))/(945*d^4
) + (9216*c^(9/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^4

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fricas [A]  time = 0.55, size = 191, normalized size = 1.47 \begin {gather*} \left [\frac {2 \, {\left (2177280 \, c^{\frac {9}{2}} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - {\left (35 \, d^{4} x^{12} + 410 \, c d^{3} x^{9} + 4611 \, c^{2} d^{2} x^{6} + 61892 \, c^{3} d x^{3} + 1509176 \, c^{4}\right )} \sqrt {d x^{3} + c}\right )}}{945 \, d^{4}}, -\frac {2 \, {\left (4354560 \, \sqrt {-c} c^{4} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + {\left (35 \, d^{4} x^{12} + 410 \, c d^{3} x^{9} + 4611 \, c^{2} d^{2} x^{6} + 61892 \, c^{3} d x^{3} + 1509176 \, c^{4}\right )} \sqrt {d x^{3} + c}\right )}}{945 \, d^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="fricas")

[Out]

[2/945*(2177280*c^(9/2)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) - (35*d^4*x^12 + 410*c*d
^3*x^9 + 4611*c^2*d^2*x^6 + 61892*c^3*d*x^3 + 1509176*c^4)*sqrt(d*x^3 + c))/d^4, -2/945*(4354560*sqrt(-c)*c^4*
arctan(1/3*sqrt(d*x^3 + c)*sqrt(-c)/c) + (35*d^4*x^12 + 410*c*d^3*x^9 + 4611*c^2*d^2*x^6 + 61892*c^3*d*x^3 + 1
509176*c^4)*sqrt(d*x^3 + c))/d^4]

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giac [A]  time = 0.17, size = 117, normalized size = 0.90 \begin {gather*} -\frac {9216 \, c^{5} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d^{4}} - \frac {2 \, {\left (35 \, {\left (d x^{3} + c\right )}^{\frac {9}{2}} d^{32} + 270 \, {\left (d x^{3} + c\right )}^{\frac {7}{2}} c d^{32} + 3591 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} c^{2} d^{32} + 53760 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c^{3} d^{32} + 1451520 \, \sqrt {d x^{3} + c} c^{4} d^{32}\right )}}{945 \, d^{36}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="giac")

[Out]

-9216*c^5*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^4) - 2/945*(35*(d*x^3 + c)^(9/2)*d^32 + 270*(d*x^3
+ c)^(7/2)*c*d^32 + 3591*(d*x^3 + c)^(5/2)*c^2*d^32 + 53760*(d*x^3 + c)^(3/2)*c^3*d^32 + 1451520*sqrt(d*x^3 +
c)*c^4*d^32)/d^36

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maple [C]  time = 0.25, size = 634, normalized size = 4.88 \begin {gather*} -\frac {512 \left (\frac {2 \sqrt {d \,x^{3}+c}\, x^{3}}{9}+\frac {56 \sqrt {d \,x^{3}+c}\, c}{9 d}+\frac {3 i c \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (2 \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d^{2}+i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -\left (-c \,d^{2}\right )^{\frac {1}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}-\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d +i \sqrt {3}\, c d -3 c d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )-3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{18 c d}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) d}}\right )}{d^{3} \sqrt {d \,x^{3}+c}}\right ) c^{3}}{d^{3}}-\frac {8 \left (\frac {2 \sqrt {d \,x^{3}+c}\, d \,x^{9}}{21}+\frac {16 \sqrt {d \,x^{3}+c}\, c \,x^{6}}{105}+\frac {2 \sqrt {d \,x^{3}+c}\, c^{2} x^{3}}{105 d}-\frac {4 \sqrt {d \,x^{3}+c}\, c^{3}}{105 d^{2}}\right ) c}{d^{2}}-\frac {\frac {2 \sqrt {d \,x^{3}+c}\, d \,x^{12}}{27}+\frac {20 \sqrt {d \,x^{3}+c}\, c \,x^{9}}{189}+\frac {2 \sqrt {d \,x^{3}+c}\, c^{2} x^{6}}{315 d}-\frac {8 \sqrt {d \,x^{3}+c}\, c^{3} x^{3}}{945 d^{2}}+\frac {16 \sqrt {d \,x^{3}+c}\, c^{4}}{945 d^{3}}}{d}-\frac {128 \left (d \,x^{3}+c \right )^{\frac {5}{2}} c^{2}}{15 d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x)

[Out]

-1/d*(2/27*d*x^12*(d*x^3+c)^(1/2)+20/189*c*x^9*(d*x^3+c)^(1/2)+2/315*c^2/d*x^6*(d*x^3+c)^(1/2)-8/945*c^3/d^2*x
^3*(d*x^3+c)^(1/2)+16/945*c^4/d^3*(d*x^3+c)^(1/2))-8*c/d^2*(2/21*d*x^9*(d*x^3+c)^(1/2)+16/105*c*x^6*(d*x^3+c)^
(1/2)+2/105*c^2/d*x^3*(d*x^3+c)^(1/2)-4/105*c^3/d^2*(d*x^3+c)^(1/2))-128/15*c^2*(d*x^3+c)^(5/2)/d^4-512*c^3/d^
3*(2/9*x^3*(d*x^3+c)^(1/2)+56/9*(d*x^3+c)^(1/2)*c/d+3*I*c/d^3*2^(1/2)*sum((-c*d^2)^(1/3)*(1/2*I*(2*x+(-I*3^(1/
2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3))/d)/(-c*d^2)^(1/3)*d)^(1/2)*((x-(-c*d^2)^(1/3)/d)/(-3*(-c*d^2)^(1/3)+I*3^(1/2
)*(-c*d^2)^(1/3))*d)^(1/2)*(-1/2*I*(2*x+(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3))/d)/(-c*d^2)^(1/3)*d)^(1/2)/(
d*x^3+c)^(1/2)*(2*_alpha^2*d^2+I*(-c*d^2)^(1/3)*3^(1/2)*_alpha*d-(-c*d^2)^(1/3)*_alpha*d-I*3^(1/2)*(-c*d^2)^(2
/3)-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2*(-c*d^2)^(1/3)/d-1/2*I*3^(1/2)*(-c*d^2)^(1/3)/d)*3^(1/2)/
(-c*d^2)^(1/3)*d)^(1/2),-1/18*(2*I*(-c*d^2)^(1/3)*3^(1/2)*_alpha^2*d+I*3^(1/2)*c*d-3*c*d-I*(-c*d^2)^(2/3)*3^(1
/2)*_alpha-3*(-c*d^2)^(2/3)*_alpha)/c/d,(I*3^(1/2)*(-c*d^2)^(1/3)/(-3/2*(-c*d^2)^(1/3)/d+1/2*I*3^(1/2)*(-c*d^2
)^(1/3)/d)/d)^(1/2)),_alpha=RootOf(_Z^3*d-8*c)))

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maxima [A]  time = 1.29, size = 110, normalized size = 0.85 \begin {gather*} -\frac {2 \, {\left (2177280 \, c^{\frac {9}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 35 \, {\left (d x^{3} + c\right )}^{\frac {9}{2}} + 270 \, {\left (d x^{3} + c\right )}^{\frac {7}{2}} c + 3591 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} c^{2} + 53760 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c^{3} + 1451520 \, \sqrt {d x^{3} + c} c^{4}\right )}}{945 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="maxima")

[Out]

-2/945*(2177280*c^(9/2)*log((sqrt(d*x^3 + c) - 3*sqrt(c))/(sqrt(d*x^3 + c) + 3*sqrt(c))) + 35*(d*x^3 + c)^(9/2
) + 270*(d*x^3 + c)^(7/2)*c + 3591*(d*x^3 + c)^(5/2)*c^2 + 53760*(d*x^3 + c)^(3/2)*c^3 + 1451520*sqrt(d*x^3 +
c)*c^4)/d^4

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mupad [B]  time = 3.53, size = 135, normalized size = 1.04 \begin {gather*} \frac {4608\,c^{9/2}\,\ln \left (\frac {10\,c+d\,x^3+6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{d^4}-\frac {2\,x^{12}\,\sqrt {d\,x^3+c}}{27}-\frac {3018352\,c^4\,\sqrt {d\,x^3+c}}{945\,d^4}-\frac {164\,c\,x^9\,\sqrt {d\,x^3+c}}{189\,d}-\frac {123784\,c^3\,x^3\,\sqrt {d\,x^3+c}}{945\,d^3}-\frac {3074\,c^2\,x^6\,\sqrt {d\,x^3+c}}{315\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^11*(c + d*x^3)^(3/2))/(8*c - d*x^3),x)

[Out]

(4608*c^(9/2)*log((10*c + d*x^3 + 6*c^(1/2)*(c + d*x^3)^(1/2))/(8*c - d*x^3)))/d^4 - (2*x^12*(c + d*x^3)^(1/2)
)/27 - (3018352*c^4*(c + d*x^3)^(1/2))/(945*d^4) - (164*c*x^9*(c + d*x^3)^(1/2))/(189*d) - (123784*c^3*x^3*(c
+ d*x^3)^(1/2))/(945*d^3) - (3074*c^2*x^6*(c + d*x^3)^(1/2))/(315*d^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(d*x**3+c)**(3/2)/(-d*x**3+8*c),x)

[Out]

Timed out

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