\(\int \frac {(a+b x^3)^{4/3}}{x^{11} (c+d x^3)} \, dx\) [722]

Optimal result

Integrand size = 24, antiderivative size = 318 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx=-\frac {a \sqrt [3]{a+b x^3}}{10 c x^{10}}-\frac {(11 b c-10 a d) \sqrt [3]{a+b x^3}}{70 c^2 x^7}-\frac {\left (2 b^2 c^2-40 a b c d+35 a^2 d^2\right ) \sqrt [3]{a+b x^3}}{140 a c^3 x^4}+\frac {\left (6 b^3 c^3+20 a b^2 c^2 d-175 a^2 b c d^2+140 a^3 d^3\right ) \sqrt [3]{a+b x^3}}{140 a^2 c^4 x}-\frac {d^2 (b c-a d)^{4/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{13/3}}+\frac {d^2 (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^{13/3}}-\frac {d^2 (b c-a d)^{4/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{13/3}} \] Output:

-1/10*a*(b*x^3+a)^(1/3)/c/x^10-1/70*(-10*a*d+11*b*c)*(b*x^3+a)^(1/3)/c^2/x 
^7-1/140*(35*a^2*d^2-40*a*b*c*d+2*b^2*c^2)*(b*x^3+a)^(1/3)/a/c^3/x^4+1/140 
*(140*a^3*d^3-175*a^2*b*c*d^2+20*a*b^2*c^2*d+6*b^3*c^3)*(b*x^3+a)^(1/3)/a^ 
2/c^4/x-1/3*d^2*(-a*d+b*c)^(4/3)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3 
)/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/c^(13/3)+1/6*d^2*(-a*d+b*c)^(4/3)*ln(d 
*x^3+c)/c^(13/3)-1/2*d^2*(-a*d+b*c)^(4/3)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b 
*x^3+a)^(1/3))/c^(13/3)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.54 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx=\frac {\frac {3 \sqrt [3]{c} \sqrt [3]{a+b x^3} \left (6 b^3 c^3 x^9-2 a b^2 c^2 x^6 \left (c-10 d x^3\right )+a^2 b c x^3 \left (-22 c^2+40 c d x^3-175 d^2 x^6\right )+a^3 \left (-14 c^3+20 c^2 d x^3-35 c d^2 x^6+140 d^3 x^9\right )\right )}{a^2 x^{10}}+70 \sqrt {-6-6 i \sqrt {3}} d^2 (b c-a d)^{4/3} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )+70 \left (1-i \sqrt {3}\right ) d^2 (b c-a d)^{4/3} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )+35 i \left (i+\sqrt {3}\right ) d^2 (b c-a d)^{4/3} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{420 c^{13/3}} \] Input:

Integrate[(a + b*x^3)^(4/3)/(x^11*(c + d*x^3)),x]
 

Output:

((3*c^(1/3)*(a + b*x^3)^(1/3)*(6*b^3*c^3*x^9 - 2*a*b^2*c^2*x^6*(c - 10*d*x 
^3) + a^2*b*c*x^3*(-22*c^2 + 40*c*d*x^3 - 175*d^2*x^6) + a^3*(-14*c^3 + 20 
*c^2*d*x^3 - 35*c*d^2*x^6 + 140*d^3*x^9)))/(a^2*x^10) + 70*Sqrt[-6 - (6*I) 
*Sqrt[3]]*d^2*(b*c - a*d)^(4/3)*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*(b 
*c - a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))] + 70*(1 - 
I*Sqrt[3])*d^2*(b*c - a*d)^(4/3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3 
])*c^(1/3)*(a + b*x^3)^(1/3)] + (35*I)*(I + Sqrt[3])*d^2*(b*c - a*d)^(4/3) 
*Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[3])*c^(1/3)*(b*c - a*d)^(1/3)* 
x*(a + b*x^3)^(1/3) + I*(I + Sqrt[3])*c^(2/3)*(a + b*x^3)^(2/3)])/(420*c^( 
13/3))
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {974, 1053, 27, 1053, 1053, 27, 992}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x^3\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx\)

\(\Big \downarrow \) 974

\(\displaystyle \frac {\int \frac {b (10 b c-9 a d) x^3+a (11 b c-10 a d)}{x^8 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{10 c}-\frac {a \sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {-\frac {\int -\frac {2 a \left (-3 b d (11 b c-10 a d) x^3+2 b^2 c^2+35 a^2 d^2-40 a b c d\right )}{x^5 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{7 a c}-\frac {\sqrt [3]{a+b x^3} (11 b c-10 a d)}{7 c x^7}}{10 c}-\frac {a \sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \int \frac {-3 b d (11 b c-10 a d) x^3+2 b^2 c^2+35 a^2 d^2-40 a b c d}{x^5 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{7 c}-\frac {\sqrt [3]{a+b x^3} (11 b c-10 a d)}{7 c x^7}}{10 c}-\frac {a \sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {\frac {2 \left (-\frac {\int \frac {6 b^3 c^3+20 a b^2 d c^2-175 a^2 b d^2 c+140 a^3 d^3+3 b d \left (2 b^2 c^2-40 a b d c+35 a^2 d^2\right ) x^3}{x^2 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {2 b^2 c}{a}+\frac {35 a d^2}{c}-40 b d\right )}{4 x^4}\right )}{7 c}-\frac {\sqrt [3]{a+b x^3} (11 b c-10 a d)}{7 c x^7}}{10 c}-\frac {a \sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {\frac {2 \left (-\frac {-\frac {\int \frac {140 a^2 d^2 (b c-a d)^2 x}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{a c}-\frac {\sqrt [3]{a+b x^3} \left (140 a^3 d^3-175 a^2 b c d^2+20 a b^2 c^2 d+6 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {2 b^2 c}{a}+\frac {35 a d^2}{c}-40 b d\right )}{4 x^4}\right )}{7 c}-\frac {\sqrt [3]{a+b x^3} (11 b c-10 a d)}{7 c x^7}}{10 c}-\frac {a \sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (-\frac {-\frac {140 a d^2 (b c-a d)^2 \int \frac {x}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{c}-\frac {\sqrt [3]{a+b x^3} \left (140 a^3 d^3-175 a^2 b c d^2+20 a b^2 c^2 d+6 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {2 b^2 c}{a}+\frac {35 a d^2}{c}-40 b d\right )}{4 x^4}\right )}{7 c}-\frac {\sqrt [3]{a+b x^3} (11 b c-10 a d)}{7 c x^7}}{10 c}-\frac {a \sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 992

\(\displaystyle \frac {\frac {2 \left (-\frac {-\frac {\sqrt [3]{a+b x^3} \left (140 a^3 d^3-175 a^2 b c d^2+20 a b^2 c^2 d+6 b^3 c^3\right )}{a c x}-\frac {140 a d^2 (b c-a d)^2 \left (-\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)^{2/3}}+\frac {\log \left (c+d x^3\right )}{6 \sqrt [3]{c} (b c-a d)^{2/3}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} (b c-a d)^{2/3}}\right )}{c}}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {2 b^2 c}{a}+\frac {35 a d^2}{c}-40 b d\right )}{4 x^4}\right )}{7 c}-\frac {\sqrt [3]{a+b x^3} (11 b c-10 a d)}{7 c x^7}}{10 c}-\frac {a \sqrt [3]{a+b x^3}}{10 c x^{10}}\)

Input:

Int[(a + b*x^3)^(4/3)/(x^11*(c + d*x^3)),x]
 

Output:

-1/10*(a*(a + b*x^3)^(1/3))/(c*x^10) + (-1/7*((11*b*c - 10*a*d)*(a + b*x^3 
)^(1/3))/(c*x^7) + (2*(-1/4*(((2*b^2*c)/a - 40*b*d + (35*a*d^2)/c)*(a + b* 
x^3)^(1/3))/x^4 - (-(((6*b^3*c^3 + 20*a*b^2*c^2*d - 175*a^2*b*c*d^2 + 140* 
a^3*d^3)*(a + b*x^3)^(1/3))/(a*c*x)) - (140*a*d^2*(b*c - a*d)^2*(-(ArcTan[ 
(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3 
]*c^(1/3)*(b*c - a*d)^(2/3))) + Log[c + d*x^3]/(6*c^(1/3)*(b*c - a*d)^(2/3 
)) - Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)]/(2*c^(1/3)*(b* 
c - a*d)^(2/3))))/c)/(4*a*c)))/(7*c))/(10*c)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 974

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^ 
(q - 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1))   Int[(e*x)^(m + n)*(a 
+ b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1 
) + a*d*(q - 1)) + d*((c*b - a*d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] 
/; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q 
, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
 

rule 992

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> 
With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3 
))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c* 
q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && 
 NeQ[b*c - a*d, 0]
 

rule 1053

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

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(-\frac {c \left (\left (b \,x^{3}+a \right )^{2} \left (-\frac {3 b \,x^{3}}{7}+a \right ) c^{3}-\frac {10 a d \,x^{3} \left (b \,x^{3}+a \right )^{2} c^{2}}{7}+\frac {5 a^{2} d^{2} x^{6} \left (5 b \,x^{3}+a \right ) c}{2}-10 a^{3} d^{3} x^{9}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}-\frac {5 a^{2} d^{2} x^{10} \left (a d -b c \right )^{2} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right )+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right )}{3}}{10 \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{10} c^{5} a^{2}}\)	\(294\)

Input:

int((b*x^3+a)^(4/3)/x^11/(d*x^3+c),x,method=_RETURNVERBOSE)
 

Output:

-1/10/((a*d-b*c)/c)^(2/3)*(c*((b*x^3+a)^2*(-3/7*b*x^3+a)*c^3-10/7*a*d*x^3* 
(b*x^3+a)^2*c^2+5/2*a^2*d^2*x^6*(5*b*x^3+a)*c-10*a^3*d^3*x^9)*((a*d-b*c)/c 
)^(2/3)*(b*x^3+a)^(1/3)-5/3*a^2*d^2*x^10*(a*d-b*c)^2*(2*3^(1/2)*arctan(1/3 
*3^(1/2)*(((a*d-b*c)/c)^(1/3)*x-2*(b*x^3+a)^(1/3))/((a*d-b*c)/c)^(1/3)/x)+ 
ln((((a*d-b*c)/c)^(2/3)*x^2-((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a 
)^(2/3))/x^2)-2*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)))/x^10/c^5/a 
^2
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^3+a)^(4/3)/x^11/(d*x^3+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x**3+a)**(4/3)/x**11/(d*x**3+c),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{11}} \,d x } \] Input:

integrate((b*x^3+a)^(4/3)/x^11/(d*x^3+c),x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^11), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{11}} \,d x } \] Input:

integrate((b*x^3+a)^(4/3)/x^11/(d*x^3+c),x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^11), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{4/3}}{x^{11}\,\left (d\,x^3+c\right )} \,d x \] Input:

int((a + b*x^3)^(4/3)/(x^11*(c + d*x^3)),x)
 

Output:

int((a + b*x^3)^(4/3)/(x^11*(c + d*x^3)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx=\frac {-14 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} c^{2}+20 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} c d \,x^{3}-35 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} d^{2} x^{6}-22 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b \,c^{2} x^{3}+40 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b c d \,x^{6}+105 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b \,d^{2} x^{9}-2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{2} c^{2} x^{6}-120 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{2} c d \,x^{9}+6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{3} c^{2} x^{9}-140 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b d \,x^{8}+a d \,x^{5}+b c \,x^{5}+a c \,x^{2}}d x \right ) a^{4} d^{3} x^{10}+280 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b d \,x^{8}+a d \,x^{5}+b c \,x^{5}+a c \,x^{2}}d x \right ) a^{3} b c \,d^{2} x^{10}-140 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b d \,x^{8}+a d \,x^{5}+b c \,x^{5}+a c \,x^{2}}d x \right ) a^{2} b^{2} c^{2} d \,x^{10}}{140 a^{2} c^{3} x^{10}} \] Input:

int((b*x^3+a)^(4/3)/x^11/(d*x^3+c),x)
 

Output:

( - 14*(a + b*x**3)**(1/3)*a**3*c**2 + 20*(a + b*x**3)**(1/3)*a**3*c*d*x** 
3 - 35*(a + b*x**3)**(1/3)*a**3*d**2*x**6 - 22*(a + b*x**3)**(1/3)*a**2*b* 
c**2*x**3 + 40*(a + b*x**3)**(1/3)*a**2*b*c*d*x**6 + 105*(a + b*x**3)**(1/ 
3)*a**2*b*d**2*x**9 - 2*(a + b*x**3)**(1/3)*a*b**2*c**2*x**6 - 120*(a + b* 
x**3)**(1/3)*a*b**2*c*d*x**9 + 6*(a + b*x**3)**(1/3)*b**3*c**2*x**9 - 140* 
int((a + b*x**3)**(1/3)/(a*c*x**2 + a*d*x**5 + b*c*x**5 + b*d*x**8),x)*a** 
4*d**3*x**10 + 280*int((a + b*x**3)**(1/3)/(a*c*x**2 + a*d*x**5 + b*c*x**5 
 + b*d*x**8),x)*a**3*b*c*d**2*x**10 - 140*int((a + b*x**3)**(1/3)/(a*c*x** 
2 + a*d*x**5 + b*c*x**5 + b*d*x**8),x)*a**2*b**2*c**2*d*x**10)/(140*a**2*c 
**3*x**10)