\(\int \frac {(c+d x^3)^{3/2}}{x^4 (a+b x^3)} \, dx\) [545]

Optimal result

Integrand size = 24, antiderivative size = 116 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (a+b x^3\right )} \, dx=-\frac {c \sqrt {c+d x^3}}{3 a x^3}+\frac {\sqrt {c} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2}-\frac {2 (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 \sqrt {b}} \] Output:

-1/3*c*(d*x^3+c)^(1/2)/a/x^3+1/3*c^(1/2)*(-3*a*d+2*b*c)*arctanh((d*x^3+c)^ 
(1/2)/c^(1/2))/a^2-2/3*(-a*d+b*c)^(3/2)*arctanh(b^(1/2)*(d*x^3+c)^(1/2)/(- 
a*d+b*c)^(1/2))/a^2/b^(1/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (a+b x^3\right )} \, dx=\frac {-\frac {a c \sqrt {c+d x^3}}{x^3}+\frac {2 (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{\sqrt {b}}+\sqrt {c} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2} \] Input:

Integrate[(c + d*x^3)^(3/2)/(x^4*(a + b*x^3)),x]
 

Output:

(-((a*c*Sqrt[c + d*x^3])/x^3) + (2*(-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[b]*Sq 
rt[c + d*x^3])/Sqrt[-(b*c) + a*d]])/Sqrt[b] + Sqrt[c]*(2*b*c - 3*a*d)*ArcT 
anh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^2)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {948, 109, 27, 174, 73, 221}

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.

\(\displaystyle \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (a+b x^3\right )} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{3} \int \frac {\left (d x^3+c\right )^{3/2}}{x^6 \left (b x^3+a\right )}dx^3\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {d (b c-2 a d) x^3+c (2 b c-3 a d)}{2 x^3 \left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{a}-\frac {c \sqrt {c+d x^3}}{a x^3}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {d (b c-2 a d) x^3+c (2 b c-3 a d)}{x^3 \left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{2 a}-\frac {c \sqrt {c+d x^3}}{a x^3}\right )\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {c (2 b c-3 a d) \int \frac {1}{x^3 \sqrt {d x^3+c}}dx^3}{a}-\frac {2 (b c-a d)^2 \int \frac {1}{\left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{a}}{2 a}-\frac {c \sqrt {c+d x^3}}{a x^3}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 c (2 b c-3 a d) \int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{a d}-\frac {4 (b c-a d)^2 \int \frac {1}{\frac {b x^6}{d}+a-\frac {b c}{d}}d\sqrt {d x^3+c}}{a d}}{2 a}-\frac {c \sqrt {c+d x^3}}{a x^3}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {4 (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{a \sqrt {b}}-\frac {2 \sqrt {c} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{a}}{2 a}-\frac {c \sqrt {c+d x^3}}{a x^3}\right )\)

Input:

Int[(c + d*x^3)^(3/2)/(x^4*(a + b*x^3)),x]
 

Output:

(-((c*Sqrt[c + d*x^3])/(a*x^3)) - ((-2*Sqrt[c]*(2*b*c - 3*a*d)*ArcTanh[Sqr 
t[c + d*x^3]/Sqrt[c]])/a + (4*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + 
d*x^3])/Sqrt[b*c - a*d]])/(a*Sqrt[b]))/(2*a))/3
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] + Simp[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 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(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 221

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

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[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]]
 

Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {c \left (-\frac {a \sqrt {d \,x^{3}+c}}{x^{3}}-\frac {\left (3 a d -2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )+\frac {2 \left (a d -b c \right )^{2} \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{3 a^{2}}\)	\(101\)
risch	\(-\frac {c \sqrt {d \,x^{3}+c}}{3 a \,x^{3}}+\frac {-\frac {2 \sqrt {c}\, \left (3 a d -2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 a}+\frac {4 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 a \sqrt {\left (a d -b c \right ) b}}}{2 a}\)	\(119\)
default	\(\frac {-\frac {c \sqrt {d \,x^{3}+c}}{3 x^{3}}+\frac {2 d \sqrt {d \,x^{3}+c}}{3}-\sqrt {c}\, d \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{a}-\frac {2 \left (-\left (a d -b c \right )^{2} \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\left (\frac {\left (-d \,x^{3}-4 c \right ) b}{3}+a d \right ) \sqrt {d \,x^{3}+c}\, \sqrt {\left (a d -b c \right ) b}\right )}{3 a^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {b \left (\frac {2 d \,x^{3} \sqrt {d \,x^{3}+c}}{9}+\frac {8 c \sqrt {d \,x^{3}+c}}{9}-\frac {2 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3}\right )}{a^{2}}\)	\(200\)
elliptic	\(\text {Expression too large to display}\)	\(1625\)

Input:

int((d*x^3+c)^(3/2)/x^4/(b*x^3+a),x,method=_RETURNVERBOSE)
 

Output:

1/3/a^2*(c*(-a*(d*x^3+c)^(1/2)/x^3-(3*a*d-2*b*c)/c^(1/2)*arctanh((d*x^3+c) 
^(1/2)/c^(1/2)))+2*(a*d-b*c)^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x^3+c)^(1/2 
)/((a*d-b*c)*b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 532, normalized size of antiderivative = 4.59 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (a+b x^3\right )} \, dx=\left [-\frac {2 \, {\left (b c - a d\right )} x^{3} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{3} + a}\right ) + {\left (2 \, b c - 3 \, a d\right )} \sqrt {c} x^{3} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt {d x^{3} + c} a c}{6 \, a^{2} x^{3}}, -\frac {4 \, {\left (b c - a d\right )} x^{3} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{3} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (2 \, b c - 3 \, a d\right )} \sqrt {c} x^{3} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt {d x^{3} + c} a c}{6 \, a^{2} x^{3}}, -\frac {{\left (2 \, b c - 3 \, a d\right )} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{3} + c}}\right ) + {\left (b c - a d\right )} x^{3} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{3} + a}\right ) + \sqrt {d x^{3} + c} a c}{3 \, a^{2} x^{3}}, -\frac {2 \, {\left (b c - a d\right )} x^{3} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{3} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (2 \, b c - 3 \, a d\right )} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{3} + c}}\right ) + \sqrt {d x^{3} + c} a c}{3 \, a^{2} x^{3}}\right ] \] Input:

integrate((d*x^3+c)^(3/2)/x^4/(b*x^3+a),x, algorithm="fricas")
 

Output:

[-1/6*(2*(b*c - a*d)*x^3*sqrt((b*c - a*d)/b)*log((b*d*x^3 + 2*b*c - a*d + 
2*sqrt(d*x^3 + c)*b*sqrt((b*c - a*d)/b))/(b*x^3 + a)) + (2*b*c - 3*a*d)*sq 
rt(c)*x^3*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) + 2*sqrt(d*x^ 
3 + c)*a*c)/(a^2*x^3), -1/6*(4*(b*c - a*d)*x^3*sqrt(-(b*c - a*d)/b)*arctan 
(-sqrt(d*x^3 + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + (2*b*c - 3*a*d)*sq 
rt(c)*x^3*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) + 2*sqrt(d*x^ 
3 + c)*a*c)/(a^2*x^3), -1/3*((2*b*c - 3*a*d)*sqrt(-c)*x^3*arctan(sqrt(-c)/ 
sqrt(d*x^3 + c)) + (b*c - a*d)*x^3*sqrt((b*c - a*d)/b)*log((b*d*x^3 + 2*b* 
c - a*d + 2*sqrt(d*x^3 + c)*b*sqrt((b*c - a*d)/b))/(b*x^3 + a)) + sqrt(d*x 
^3 + c)*a*c)/(a^2*x^3), -1/3*(2*(b*c - a*d)*x^3*sqrt(-(b*c - a*d)/b)*arcta 
n(-sqrt(d*x^3 + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + (2*b*c - 3*a*d)*s 
qrt(-c)*x^3*arctan(sqrt(-c)/sqrt(d*x^3 + c)) + sqrt(d*x^3 + c)*a*c)/(a^2*x 
^3)]
 

Sympy [F]

\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (a+b x^3\right )} \, dx=\int \frac {\left (c + d x^{3}\right )^{\frac {3}{2}}}{x^{4} \left (a + b x^{3}\right )}\, dx \] Input:

integrate((d*x**3+c)**(3/2)/x**4/(b*x**3+a),x)
 

Output:

Integral((c + d*x**3)**(3/2)/(x**4*(a + b*x**3)), x)
 

Maxima [F]

\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (a+b x^3\right )} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )} x^{4}} \,d x } \] Input:

integrate((d*x^3+c)^(3/2)/x^4/(b*x^3+a),x, algorithm="maxima")
 

Output:

integrate((d*x^3 + c)^(3/2)/((b*x^3 + a)*x^4), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.04 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (a+b x^3\right )} \, dx=\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, \sqrt {-b^{2} c + a b d} a^{2}} - \frac {{\left (2 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a^{2} \sqrt {-c}} - \frac {\sqrt {d x^{3} + c} c}{3 \, a x^{3}} \] Input:

integrate((d*x^3+c)^(3/2)/x^4/(b*x^3+a),x, algorithm="giac")
 

Output:

2/3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + 
 a*b*d))/(sqrt(-b^2*c + a*b*d)*a^2) - 1/3*(2*b*c^2 - 3*a*c*d)*arctan(sqrt( 
d*x^3 + c)/sqrt(-c))/(a^2*sqrt(-c)) - 1/3*sqrt(d*x^3 + c)*c/(a*x^3)
 

Mupad [B] (verification not implemented)

Time = 7.98 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.44 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (a+b x^3\right )} \, dx=\frac {\sqrt {c}\,\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}{x^6}\right )\,\left (3\,a\,d-2\,b\,c\right )}{6\,a^2}-\frac {c\,\sqrt {d\,x^3+c}}{3\,a\,x^3}+\frac {\ln \left (\frac {a^2\,d^2+2\,b^2\,c^2-a\,b\,d^2\,x^3+b^2\,c\,d\,x^3-3\,a\,b\,c\,d-\sqrt {b}\,\sqrt {d\,x^3+c}\,{\left (a\,d-b\,c\right )}^{3/2}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,{\left (a\,d-b\,c\right )}^{3/2}\,1{}\mathrm {i}}{3\,a^2\,\sqrt {b}} \] Input:

int((c + d*x^3)^(3/2)/(x^4*(a + b*x^3)),x)
 

Output:

(c^(1/2)*log((((c + d*x^3)^(1/2) - c^(1/2))^3*((c + d*x^3)^(1/2) + c^(1/2) 
))/x^6)*(3*a*d - 2*b*c))/(6*a^2) - (c*(c + d*x^3)^(1/2))/(3*a*x^3) + (log( 
(a^2*d^2 + 2*b^2*c^2 - b^(1/2)*(c + d*x^3)^(1/2)*(a*d - b*c)^(3/2)*2i - a* 
b*d^2*x^3 + b^2*c*d*x^3 - 3*a*b*c*d)/(a + b*x^3))*(a*d - b*c)^(3/2)*1i)/(3 
*a^2*b^(1/2))
 

Reduce [F]

\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (a+b x^3\right )} \, dx=\frac {-2 \sqrt {d \,x^{3}+c}\, a c d +4 \sqrt {d \,x^{3}+c}\, a \,d^{2} x^{3}-4 \sqrt {d \,x^{3}+c}\, b \,c^{2}+9 \left (\int \frac {\sqrt {d \,x^{3}+c}}{a b \,d^{2} x^{7}+2 b^{2} c d \,x^{7}+a^{2} d^{2} x^{4}+3 a b c d \,x^{4}+2 b^{2} c^{2} x^{4}+a^{2} c d x +2 a b \,c^{2} x}d x \right ) a^{3} c \,d^{3} x^{3}+30 \left (\int \frac {\sqrt {d \,x^{3}+c}}{a b \,d^{2} x^{7}+2 b^{2} c d \,x^{7}+a^{2} d^{2} x^{4}+3 a b c d \,x^{4}+2 b^{2} c^{2} x^{4}+a^{2} c d x +2 a b \,c^{2} x}d x \right ) a^{2} b \,c^{2} d^{2} x^{3}+12 \left (\int \frac {\sqrt {d \,x^{3}+c}}{a b \,d^{2} x^{7}+2 b^{2} c d \,x^{7}+a^{2} d^{2} x^{4}+3 a b c d \,x^{4}+2 b^{2} c^{2} x^{4}+a^{2} c d x +2 a b \,c^{2} x}d x \right ) a \,b^{2} c^{3} d \,x^{3}-24 \left (\int \frac {\sqrt {d \,x^{3}+c}}{a b \,d^{2} x^{7}+2 b^{2} c d \,x^{7}+a^{2} d^{2} x^{4}+3 a b c d \,x^{4}+2 b^{2} c^{2} x^{4}+a^{2} c d x +2 a b \,c^{2} x}d x \right ) b^{3} c^{4} x^{3}-6 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{a b \,d^{2} x^{6}+2 b^{2} c d \,x^{6}+a^{2} d^{2} x^{3}+3 a b c d \,x^{3}+2 b^{2} c^{2} x^{3}+a^{2} c d +2 a b \,c^{2}}d x \right ) a^{2} b \,d^{4} x^{3}-12 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{a b \,d^{2} x^{6}+2 b^{2} c d \,x^{6}+a^{2} d^{2} x^{3}+3 a b c d \,x^{3}+2 b^{2} c^{2} x^{3}+a^{2} c d +2 a b \,c^{2}}d x \right ) a \,b^{2} c \,d^{3} x^{3}+9 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{2}}{a b \,d^{2} x^{6}+2 b^{2} c d \,x^{6}+a^{2} d^{2} x^{3}+3 a b c d \,x^{3}+2 b^{2} c^{2} x^{3}+a^{2} c d +2 a b \,c^{2}}d x \right ) a^{2} b c \,d^{3} x^{3}+12 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{2}}{a b \,d^{2} x^{6}+2 b^{2} c d \,x^{6}+a^{2} d^{2} x^{3}+3 a b c d \,x^{3}+2 b^{2} c^{2} x^{3}+a^{2} c d +2 a b \,c^{2}}d x \right ) a \,b^{2} c^{2} d^{2} x^{3}-12 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{2}}{a b \,d^{2} x^{6}+2 b^{2} c d \,x^{6}+a^{2} d^{2} x^{3}+3 a b c d \,x^{3}+2 b^{2} c^{2} x^{3}+a^{2} c d +2 a b \,c^{2}}d x \right ) b^{3} c^{3} d \,x^{3}}{6 a \,x^{3} \left (a d +2 b c \right )} \] Input:

int((d*x^3+c)^(3/2)/x^4/(b*x^3+a),x)
 

Output:

( - 2*sqrt(c + d*x**3)*a*c*d + 4*sqrt(c + d*x**3)*a*d**2*x**3 - 4*sqrt(c + 
 d*x**3)*b*c**2 + 9*int(sqrt(c + d*x**3)/(a**2*c*d*x + a**2*d**2*x**4 + 2* 
a*b*c**2*x + 3*a*b*c*d*x**4 + a*b*d**2*x**7 + 2*b**2*c**2*x**4 + 2*b**2*c* 
d*x**7),x)*a**3*c*d**3*x**3 + 30*int(sqrt(c + d*x**3)/(a**2*c*d*x + a**2*d 
**2*x**4 + 2*a*b*c**2*x + 3*a*b*c*d*x**4 + a*b*d**2*x**7 + 2*b**2*c**2*x** 
4 + 2*b**2*c*d*x**7),x)*a**2*b*c**2*d**2*x**3 + 12*int(sqrt(c + d*x**3)/(a 
**2*c*d*x + a**2*d**2*x**4 + 2*a*b*c**2*x + 3*a*b*c*d*x**4 + a*b*d**2*x**7 
 + 2*b**2*c**2*x**4 + 2*b**2*c*d*x**7),x)*a*b**2*c**3*d*x**3 - 24*int(sqrt 
(c + d*x**3)/(a**2*c*d*x + a**2*d**2*x**4 + 2*a*b*c**2*x + 3*a*b*c*d*x**4 
+ a*b*d**2*x**7 + 2*b**2*c**2*x**4 + 2*b**2*c*d*x**7),x)*b**3*c**4*x**3 - 
6*int((sqrt(c + d*x**3)*x**5)/(a**2*c*d + a**2*d**2*x**3 + 2*a*b*c**2 + 3* 
a*b*c*d*x**3 + a*b*d**2*x**6 + 2*b**2*c**2*x**3 + 2*b**2*c*d*x**6),x)*a**2 
*b*d**4*x**3 - 12*int((sqrt(c + d*x**3)*x**5)/(a**2*c*d + a**2*d**2*x**3 + 
 2*a*b*c**2 + 3*a*b*c*d*x**3 + a*b*d**2*x**6 + 2*b**2*c**2*x**3 + 2*b**2*c 
*d*x**6),x)*a*b**2*c*d**3*x**3 + 9*int((sqrt(c + d*x**3)*x**2)/(a**2*c*d + 
 a**2*d**2*x**3 + 2*a*b*c**2 + 3*a*b*c*d*x**3 + a*b*d**2*x**6 + 2*b**2*c** 
2*x**3 + 2*b**2*c*d*x**6),x)*a**2*b*c*d**3*x**3 + 12*int((sqrt(c + d*x**3) 
*x**2)/(a**2*c*d + a**2*d**2*x**3 + 2*a*b*c**2 + 3*a*b*c*d*x**3 + a*b*d**2 
*x**6 + 2*b**2*c**2*x**3 + 2*b**2*c*d*x**6),x)*a*b**2*c**2*d**2*x**3 - 12* 
int((sqrt(c + d*x**3)*x**2)/(a**2*c*d + a**2*d**2*x**3 + 2*a*b*c**2 + 3...