3.3.93 \(\int \frac {1}{x^7 (8 c-d x^3)^2 \sqrt {c+d x^3}} \, dx\)

Optimal. Leaf size=164 \[ \frac {31 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{165888 c^{9/2}}-\frac {19 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6144 c^{9/2}}-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )} \]

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Rubi [A]  time = 0.13, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {446, 103, 151, 156, 63, 208, 206} \begin {gather*} -\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}+\frac {31 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{165888 c^{9/2}}-\frac {19 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6144 c^{9/2}}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]

[Out]

(-35*d^2*Sqrt[c + d*x^3])/(13824*c^4*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(48*c^2*x^6*(8*c - d*x^3)) + (3*d*Sqrt[c
+ d*x^3])/(128*c^3*x^3*(8*c - d*x^3)) + (31*d^2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(165888*c^(9/2)) - (19*
d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(6144*c^(9/2))

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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 208

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

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 {1}{x^7 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^3 (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {9 c d-\frac {5 d^2 x}{2}}{x^2 (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{48 c^2}\\ &=-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {38 c^2 d^2-\frac {27}{2} c d^3 x}{x (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{384 c^4}\\ &=-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {-342 c^3 d^3+35 c^2 d^4 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27648 c^6 d}\\ &=-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac {\left (19 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{12288 c^4}+\frac {\left (31 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{110592 c^4}\\ &=-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac {(19 d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{6144 c^4}+\frac {\left (31 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{55296 c^4}\\ &=-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac {31 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{165888 c^{9/2}}-\frac {19 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6144 c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 112, normalized size = 0.68 \begin {gather*} \frac {\frac {12 \sqrt {c} \sqrt {c+d x^3} \left (288 c^2-324 c d x^3+35 d^2 x^6\right )}{d x^9-8 c x^6}+31 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-513 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{165888 c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]

[Out]

((12*Sqrt[c]*Sqrt[c + d*x^3]*(288*c^2 - 324*c*d*x^3 + 35*d^2*x^6))/(-8*c*x^6 + d*x^9) + 31*d^2*ArcTanh[Sqrt[c
+ d*x^3]/(3*Sqrt[c])] - 513*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(165888*c^(9/2))

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IntegrateAlgebraic [A]  time = 0.15, size = 118, normalized size = 0.72 \begin {gather*} \frac {31 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{165888 c^{9/2}}-\frac {19 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6144 c^{9/2}}+\frac {\sqrt {c+d x^3} \left (-288 c^2+324 c d x^3-35 d^2 x^6\right )}{13824 c^4 x^6 \left (8 c-d x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^7*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]

[Out]

(Sqrt[c + d*x^3]*(-288*c^2 + 324*c*d*x^3 - 35*d^2*x^6))/(13824*c^4*x^6*(8*c - d*x^3)) + (31*d^2*ArcTanh[Sqrt[c
+ d*x^3]/(3*Sqrt[c])])/(165888*c^(9/2)) - (19*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(6144*c^(9/2))

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fricas [A]  time = 0.65, size = 310, normalized size = 1.89 \begin {gather*} \left [\frac {31 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\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 ) + 513 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 24 \, {\left (35 \, c d^{2} x^{6} - 324 \, c^{2} d x^{3} + 288 \, c^{3}\right )} \sqrt {d x^{3} + c}}{331776 \, {\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )}}, \frac {513 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - 31 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (35 \, c d^{2} x^{6} - 324 \, c^{2} d x^{3} + 288 \, c^{3}\right )} \sqrt {d x^{3} + c}}{165888 \, {\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="fricas")

[Out]

[1/331776*(31*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(c)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) +
513*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) + 24*(35*c*d^2*x^6 - 32
4*c^2*d*x^3 + 288*c^3)*sqrt(d*x^3 + c))/(c^5*d*x^9 - 8*c^6*x^6), 1/165888*(513*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(-c
)*arctan(sqrt(d*x^3 + c)*sqrt(-c)/c) - 31*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)*sqrt(-c)
/c) + 12*(35*c*d^2*x^6 - 324*c^2*d*x^3 + 288*c^3)*sqrt(d*x^3 + c))/(c^5*d*x^9 - 8*c^6*x^6)]

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giac [A]  time = 0.18, size = 128, normalized size = 0.78 \begin {gather*} \frac {19 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{6144 \, \sqrt {-c} c^{4}} - \frac {31 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{165888 \, \sqrt {-c} c^{4}} - \frac {\sqrt {d x^{3} + c} d^{2}}{13824 \, {\left (d x^{3} - 8 \, c\right )} c^{4}} + \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{2} - 2 \, \sqrt {d x^{3} + c} c d^{2}}{384 \, c^{4} d^{2} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="giac")

[Out]

19/6144*d^2*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^4) - 31/165888*d^2*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c
))/(sqrt(-c)*c^4) - 1/13824*sqrt(d*x^3 + c)*d^2/((d*x^3 - 8*c)*c^4) + 1/384*((d*x^3 + c)^(3/2)*d^2 - 2*sqrt(d*
x^3 + c)*c*d^2)/(c^4*d^2*x^6)

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maple [C]  time = 0.20, size = 989, normalized size = 6.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x)

[Out]

1/512/c^3*d^3*(-1/27*(d*x^3+c)^(1/2)/(d*x^3-8*c)/c/d-1/486*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)))+1/256/c^3*d*(-1/3*(d*x^3+c)^(1/2)/c/x^3+1/3*d*arctanh(
(d*x^3+c)^(1/2)/c^(1/2))/c^(3/2))+1/64/c^2*(-1/6*(d*x^3+c)^(1/2)/c/x^6+1/4*(d*x^3+c)^(1/2)/c^2*d/x^3-1/4*d^2*a
rctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(5/2))-1/36864*I/c^5*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)*_al
pha-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))-1/2048*d^2*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(9/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}^{2} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2*x^7), x)

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mupad [B]  time = 4.37, size = 155, normalized size = 0.95 \begin {gather*} -\frac {\frac {647\,d^2\,\sqrt {d\,x^3+c}}{4608\,c^2}-\frac {197\,d^2\,{\left (d\,x^3+c\right )}^{3/2}}{2304\,c^3}+\frac {35\,d^2\,{\left (d\,x^3+c\right )}^{5/2}}{4608\,c^4}}{33\,c\,{\left (d\,x^3+c\right )}^2-57\,c^2\,\left (d\,x^3+c\right )-3\,{\left (d\,x^3+c\right )}^3+27\,c^3}+\frac {d^2\,\left (\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{\sqrt {c^9}}\right )\,1{}\mathrm {i}-\frac {\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^9}}\right )\,31{}\mathrm {i}}{513}\right )\,19{}\mathrm {i}}{6144\,\sqrt {c^9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(c + d*x^3)^(1/2)*(8*c - d*x^3)^2),x)

[Out]

(d^2*(atanh((c^4*(c + d*x^3)^(1/2))/(c^9)^(1/2))*1i - (atanh((c^4*(c + d*x^3)^(1/2))/(3*(c^9)^(1/2)))*31i)/513
)*19i)/(6144*(c^9)^(1/2)) - ((647*d^2*(c + d*x^3)^(1/2))/(4608*c^2) - (197*d^2*(c + d*x^3)^(3/2))/(2304*c^3) +
(35*d^2*(c + d*x^3)^(5/2))/(4608*c^4))/(33*c*(c + d*x^3)^2 - 57*c^2*(c + d*x^3) - 3*(c + d*x^3)^3 + 27*c^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{7} \left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)

[Out]

Integral(1/(x**7*(-8*c + d*x**3)**2*sqrt(c + d*x**3)), x)

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