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

Optimal. Leaf size=95 \[ \frac {2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt {c+d x^3}}-\frac {640 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{243 d^4}+\frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \]

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Rubi [A]  time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {446, 98, 146, 63, 206} \begin {gather*} \frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt {c+d x^3}}-\frac {640 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{243 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*x^6)/(27*d^2*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (2*(38*c + 39*d*x^3))/(81*d^4*Sqrt[c + d*x^3]) - (640*Sqrt[c]
*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(243*d^4)

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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 146

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)), x] - Dist[
(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m +
1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 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])

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 (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3}{(8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\operatorname {Subst}\left (\int \frac {x \left (16 c^2+13 c d x\right )}{(8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{27 c d^2}\\ &=\frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt {c+d x^3}}-\frac {(320 c) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{81 d^3}\\ &=\frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt {c+d x^3}}-\frac {(640 c) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{81 d^4}\\ &=\frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt {c+d x^3}}-\frac {640 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{243 d^4}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 88, normalized size = 0.93 \begin {gather*} \frac {6 \left (752 c^2-198 c d x^3+9 d^2 x^6\right )-640 c \left (8 c-d x^3\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3+c}{9 c}\right )}{81 d^4 \left (d x^3-8 c\right ) \sqrt {c+d x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(752*c^2 - 198*c*d*x^3 + 9*d^2*x^6) - 640*c*(8*c - d*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)
])/(81*d^4*(-8*c + d*x^3)*Sqrt[c + d*x^3])

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IntegrateAlgebraic [A]  time = 0.09, size = 127, normalized size = 1.34 \begin {gather*} \frac {\left (\frac {5120 c^{3/2}}{243 d^4}-\frac {640 \sqrt {c} x^3}{243 d^3}\right ) \sqrt {c+d x^3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-\frac {608 c^2}{81 d^4}-\frac {548 c x^3}{81 d^3}+\frac {2 x^6}{3 d^2}}{d x^3 \sqrt {c+d x^3}-8 c \sqrt {c+d x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-608*c^2)/(81*d^4) - (548*c*x^3)/(81*d^3) + (2*x^6)/(3*d^2) + ((5120*c^(3/2))/(243*d^4) - (640*Sqrt[c]*x^3)/
(243*d^3))*Sqrt[c + d*x^3]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(-8*c*Sqrt[c + d*x^3] + d*x^3*Sqrt[c + d*x^3]
)

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fricas [A]  time = 0.76, size = 233, normalized size = 2.45 \begin {gather*} \left [\frac {2 \, {\left (160 \, {\left (d^{2} x^{6} - 7 \, c d x^{3} - 8 \, c^{2}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 3 \, {\left (27 \, d^{2} x^{6} - 274 \, c d x^{3} - 304 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{243 \, {\left (d^{6} x^{6} - 7 \, c d^{5} x^{3} - 8 \, c^{2} d^{4}\right )}}, \frac {2 \, {\left (320 \, {\left (d^{2} x^{6} - 7 \, c d x^{3} - 8 \, c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 3 \, {\left (27 \, d^{2} x^{6} - 274 \, c d x^{3} - 304 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{243 \, {\left (d^{6} x^{6} - 7 \, c d^{5} x^{3} - 8 \, c^{2} d^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/243*(160*(d^2*x^6 - 7*c*d*x^3 - 8*c^2)*sqrt(c)*log((d*x^3 - 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)
) + 3*(27*d^2*x^6 - 274*c*d*x^3 - 304*c^2)*sqrt(d*x^3 + c))/(d^6*x^6 - 7*c*d^5*x^3 - 8*c^2*d^4), 2/243*(320*(d
^2*x^6 - 7*c*d*x^3 - 8*c^2)*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)*sqrt(-c)/c) + 3*(27*d^2*x^6 - 274*c*d*x^3 - 30
4*c^2)*sqrt(d*x^3 + c))/(d^6*x^6 - 7*c*d^5*x^3 - 8*c^2*d^4)]

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giac [A]  time = 0.19, size = 88, normalized size = 0.93 \begin {gather*} \frac {640 \, c \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{243 \, \sqrt {-c} d^{4}} + \frac {2 \, \sqrt {d x^{3} + c}}{3 \, d^{4}} - \frac {2 \, {\left (85 \, {\left (d x^{3} + c\right )} c + 3 \, c^{2}\right )}}{81 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {d x^{3} + c} c\right )} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

640/243*c*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^4) + 2/3*sqrt(d*x^3 + c)/d^4 - 2/81*(85*(d*x^3 + c)
*c + 3*c^2)/(((d*x^3 + c)^(3/2) - 9*sqrt(d*x^3 + c)*c)*d^4)

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maple [C]  time = 0.29, size = 970, normalized size = 10.21

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^3*((2/3/((x^3+c/d)*d)^(1/2)*c/d^2+2/3*(d*x^3+c)^(1/2)/d^2)*d-32/3/(d*x^3+c)^(1/2)*c/d)+512*c^3/d^3*(-1/243
/d/c^2*(d*x^3+c)^(1/2)/(d*x^3-8*c)-2/243/d/c^2/((x^3+c/d)*d)^(1/2)-1/1458*I/d^3/c^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)))+192*c^2/d^3*(2/27/((x^3+c/d)*d)^(1/2)/c
/d+1/243*I/c^2/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 = 1.21, size = 98, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (160 \, \sqrt {c} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 81 \, \sqrt {d x^{3} + c} - \frac {3 \, {\left (85 \, {\left (d x^{3} + c\right )} c + 3 \, c^{2}\right )}}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {d x^{3} + c} c}\right )}}{243 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/243*(160*sqrt(c)*log((sqrt(d*x^3 + c) - 3*sqrt(c))/(sqrt(d*x^3 + c) + 3*sqrt(c))) + 81*sqrt(d*x^3 + c) - 3*(
85*(d*x^3 + c)*c + 3*c^2)/((d*x^3 + c)^(3/2) - 9*sqrt(d*x^3 + c)*c))/d^4

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mupad [B]  time = 4.38, size = 111, normalized size = 1.17 \begin {gather*} \frac {2\,\sqrt {d\,x^3+c}}{3\,d^4}+\frac {320\,\sqrt {c}\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{243\,d^4}+\frac {\sqrt {d\,x^3+c}\,\left (\frac {176\,c^2}{81\,d^4}+\frac {170\,c\,x^3}{81\,d^3}\right )}{8\,c^2+7\,c\,d\,x^3-d^2\,x^6} \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)^2),x)

[Out]

(2*(c + d*x^3)^(1/2))/(3*d^4) + (320*c^(1/2)*log((10*c + d*x^3 - 6*c^(1/2)*(c + d*x^3)^(1/2))/(8*c - d*x^3)))/
(243*d^4) + ((c + d*x^3)^(1/2)*((176*c^2)/(81*d^4) + (170*c*x^3)/(81*d^3)))/(8*c^2 - d^2*x^6 + 7*c*d*x^3)

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

[Out]

Timed out

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