Integrand size = 24, antiderivative size = 161 \[ \int \frac {\sqrt {c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx=-\frac {2 b \sqrt {c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a x^3 \left (a+b x^3\right )}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 \sqrt {c}}-\frac {\sqrt {b} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 \sqrt {b c-a d}} \] Output:
-2/3*b*(d*x^3+c)^(1/2)/a^2/(b*x^3+a)-1/3*(d*x^3+c)^(1/2)/a/x^3/(b*x^3+a)+1 /3*(-a*d+4*b*c)*arctanh((d*x^3+c)^(1/2)/c^(1/2))/a^3/c^(1/2)-1/3*b^(1/2)*( -3*a*d+4*b*c)*arctanh(b^(1/2)*(d*x^3+c)^(1/2)/(-a*d+b*c)^(1/2))/a^3/(-a*d+ b*c)^(1/2)
Time = 0.98 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx=\frac {-\frac {a \left (a+2 b x^3\right ) \sqrt {c+d x^3}}{x^3 \left (a+b x^3\right )}+\frac {\sqrt {b} (4 b c-3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d}}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{\sqrt {c}}}{3 a^3} \] Input:
Integrate[Sqrt[c + d*x^3]/(x^4*(a + b*x^3)^2),x]
Output:
(-((a*(a + 2*b*x^3)*Sqrt[c + d*x^3])/(x^3*(a + b*x^3))) + (Sqrt[b]*(4*b*c - 3*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[-(b*c) + a*d]])/Sqrt[-(b*c) + a*d] + ((4*b*c - a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/Sqrt[c])/(3*a^3 )
Time = 0.56 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {948, 110, 27, 168, 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.
\(\displaystyle \int \frac {\sqrt {c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt {d x^3+c}}{x^6 \left (b x^3+a\right )^2}dx^3\) |
\(\Big \downarrow \) 110 |
\(\displaystyle \frac {1}{3} \left (\frac {\int -\frac {3 b d x^3+4 b c-a d}{2 x^3 \left (b x^3+a\right )^2 \sqrt {d x^3+c}}dx^3}{a}-\frac {\sqrt {c+d x^3}}{a x^3 \left (a+b x^3\right )}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {3 b d x^3+4 b c-a d}{x^3 \left (b x^3+a\right )^2 \sqrt {d x^3+c}}dx^3}{2 a}-\frac {\sqrt {c+d x^3}}{a x^3 \left (a+b x^3\right )}\right )\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {1}{3} \left (-\frac {\frac {\int \frac {(b c-a d) \left (2 b d x^3+4 b c-a d\right )}{x^3 \left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{a (b c-a d)}+\frac {4 b \sqrt {c+d x^3}}{a \left (a+b x^3\right )}}{2 a}-\frac {\sqrt {c+d x^3}}{a x^3 \left (a+b x^3\right )}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (-\frac {\frac {\int \frac {2 b d x^3+4 b c-a d}{x^3 \left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{a}+\frac {4 b \sqrt {c+d x^3}}{a \left (a+b x^3\right )}}{2 a}-\frac {\sqrt {c+d x^3}}{a x^3 \left (a+b x^3\right )}\right )\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{3} \left (-\frac {\frac {\frac {(4 b c-a d) \int \frac {1}{x^3 \sqrt {d x^3+c}}dx^3}{a}-\frac {b (4 b c-3 a d) \int \frac {1}{\left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{a}}{a}+\frac {4 b \sqrt {c+d x^3}}{a \left (a+b x^3\right )}}{2 a}-\frac {\sqrt {c+d x^3}}{a x^3 \left (a+b x^3\right )}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (-\frac {\frac {\frac {2 (4 b c-a d) \int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{a d}-\frac {2 b (4 b c-3 a d) \int \frac {1}{\frac {b x^6}{d}+a-\frac {b c}{d}}d\sqrt {d x^3+c}}{a d}}{a}+\frac {4 b \sqrt {c+d x^3}}{a \left (a+b x^3\right )}}{2 a}-\frac {\sqrt {c+d x^3}}{a x^3 \left (a+b x^3\right )}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (-\frac {\frac {\frac {2 \sqrt {b} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{a \sqrt {c}}}{a}+\frac {4 b \sqrt {c+d x^3}}{a \left (a+b x^3\right )}}{2 a}-\frac {\sqrt {c+d x^3}}{a x^3 \left (a+b x^3\right )}\right )\) |
Input:
Int[Sqrt[c + d*x^3]/(x^4*(a + b*x^3)^2),x]
Output:
(-(Sqrt[c + d*x^3]/(a*x^3*(a + b*x^3))) - ((4*b*Sqrt[c + d*x^3])/(a*(a + b *x^3)) + ((-2*(4*b*c - a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(a*Sqrt[c]) + (2*Sqrt[b]*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(a*Sqrt[b*c - a*d]))/a)/(2*a))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S imp[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] && ILtQ[m, -1]
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]
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]
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]]
Time = 1.43 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {\frac {4 \sqrt {c}\, \left (b \,x^{3}+a \right ) b \,x^{3} \left (b c -\frac {3 a d}{4}\right ) \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{3}-\frac {\sqrt {\left (a d -b c \right ) b}\, \left (x^{3} \left (b \,x^{3}+a \right ) \left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )+\sqrt {c}\, a \left (2 b \,x^{3}+a \right ) \sqrt {d \,x^{3}+c}\right )}{3}}{x^{3} \sqrt {c}\, a^{3} \left (b \,x^{3}+a \right ) \sqrt {\left (a d -b c \right ) b}}\) | \(151\) |
risch | \(-\frac {\sqrt {d \,x^{3}+c}}{3 a^{2} x^{3}}-\frac {\frac {2 \left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 a \sqrt {c}}+\frac {2 b \left (\frac {\sqrt {d \,x^{3}+c}}{b \,x^{3}+a}+\frac {d \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}\right )}{3}+\frac {4 b \left (a d -2 b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 a \sqrt {\left (a d -b c \right ) b}}}{2 a^{2}}\) | \(166\) |
default | \(\frac {-\frac {\sqrt {d \,x^{3}+c}}{3 x^{3}}-\frac {d \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 \sqrt {c}}}{a^{2}}+\frac {b \left (-\frac {\sqrt {d \,x^{3}+c}}{b \,x^{3}+a}+\frac {d \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 a^{2}}-\frac {2 b \left (\frac {2 \sqrt {d \,x^{3}+c}}{3}-\frac {2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3}\right )}{a^{3}}+\frac {4 b \left (\sqrt {d \,x^{3}+c}-\frac {\left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 a^{3}}\) | \(205\) |
elliptic | \(\text {Expression too large to display}\) | \(1708\) |
Input:
int((d*x^3+c)^(1/2)/x^4/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
4/3/((a*d-b*c)*b)^(1/2)/c^(1/2)*(c^(1/2)*(b*x^3+a)*b*x^3*(b*c-3/4*a*d)*arc tan(b*(d*x^3+c)^(1/2)/((a*d-b*c)*b)^(1/2))-1/4*((a*d-b*c)*b)^(1/2)*(x^3*(b *x^3+a)*(a*d-4*b*c)*arctanh((d*x^3+c)^(1/2)/c^(1/2))+c^(1/2)*a*(2*b*x^3+a) *(d*x^3+c)^(1/2)))/x^3/a^3/(b*x^3+a)
Time = 0.14 (sec) , antiderivative size = 825, normalized size of antiderivative = 5.12 \[ \int \frac {\sqrt {c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((d*x^3+c)^(1/2)/x^4/(b*x^3+a)^2,x, algorithm="fricas")
Output:
[-1/6*(((4*b^2*c^2 - 3*a*b*c*d)*x^6 + (4*a*b*c^2 - 3*a^2*c*d)*x^3)*sqrt(b/ (b*c - a*d))*log((b*d*x^3 + 2*b*c - a*d + 2*sqrt(d*x^3 + c)*(b*c - a*d)*sq rt(b/(b*c - a*d)))/(b*x^3 + a)) + ((4*b^2*c - a*b*d)*x^6 + (4*a*b*c - a^2* d)*x^3)*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) + 2*(2* a*b*c*x^3 + a^2*c)*sqrt(d*x^3 + c))/(a^3*b*c*x^6 + a^4*c*x^3), 1/6*(2*((4* b^2*c^2 - 3*a*b*c*d)*x^6 + (4*a*b*c^2 - 3*a^2*c*d)*x^3)*sqrt(-b/(b*c - a*d ))*arctan(sqrt(d*x^3 + c)*sqrt(-b/(b*c - a*d))) - ((4*b^2*c - a*b*d)*x^6 + (4*a*b*c - a^2*d)*x^3)*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2 *c)/x^3) - 2*(2*a*b*c*x^3 + a^2*c)*sqrt(d*x^3 + c))/(a^3*b*c*x^6 + a^4*c*x ^3), -1/6*(2*((4*b^2*c - a*b*d)*x^6 + (4*a*b*c - a^2*d)*x^3)*sqrt(-c)*arct an(sqrt(-c)/sqrt(d*x^3 + c)) + ((4*b^2*c^2 - 3*a*b*c*d)*x^6 + (4*a*b*c^2 - 3*a^2*c*d)*x^3)*sqrt(b/(b*c - a*d))*log((b*d*x^3 + 2*b*c - a*d + 2*sqrt(d *x^3 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^3 + a)) + 2*(2*a*b*c*x^3 + a^2*c)*sqrt(d*x^3 + c))/(a^3*b*c*x^6 + a^4*c*x^3), 1/3*(((4*b^2*c^2 - 3*a *b*c*d)*x^6 + (4*a*b*c^2 - 3*a^2*c*d)*x^3)*sqrt(-b/(b*c - a*d))*arctan(sqr t(d*x^3 + c)*sqrt(-b/(b*c - a*d))) - ((4*b^2*c - a*b*d)*x^6 + (4*a*b*c - a ^2*d)*x^3)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^3 + c)) - (2*a*b*c*x^3 + a^2* c)*sqrt(d*x^3 + c))/(a^3*b*c*x^6 + a^4*c*x^3)]
\[ \int \frac {\sqrt {c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx=\int \frac {\sqrt {c + d x^{3}}}{x^{4} \left (a + b x^{3}\right )^{2}}\, dx \] Input:
integrate((d*x**3+c)**(1/2)/x**4/(b*x**3+a)**2,x)
Output:
Integral(sqrt(c + d*x**3)/(x**4*(a + b*x**3)**2), x)
\[ \int \frac {\sqrt {c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{{\left (b x^{3} + a\right )}^{2} x^{4}} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/x^4/(b*x^3+a)^2,x, algorithm="maxima")
Output:
integrate(sqrt(d*x^3 + c)/((b*x^3 + a)^2*x^4), x)
Time = 0.13 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx=\frac {{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, \sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} b d - 2 \, \sqrt {d x^{3} + c} b c d + \sqrt {d x^{3} + c} a d^{2}}{3 \, {\left ({\left (d x^{3} + c\right )}^{2} b - 2 \, {\left (d x^{3} + c\right )} b c + b c^{2} + {\left (d x^{3} + c\right )} a d - a c d\right )} a^{2}} \] Input:
integrate((d*x^3+c)^(1/2)/x^4/(b*x^3+a)^2,x, algorithm="giac")
Output:
1/3*(4*b^2*c - 3*a*b*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sq rt(-b^2*c + a*b*d)*a^3) - 1/3*(4*b*c - a*d)*arctan(sqrt(d*x^3 + c)/sqrt(-c ))/(a^3*sqrt(-c)) - 1/3*(2*(d*x^3 + c)^(3/2)*b*d - 2*sqrt(d*x^3 + c)*b*c*d + sqrt(d*x^3 + c)*a*d^2)/(((d*x^3 + c)^2*b - 2*(d*x^3 + c)*b*c + b*c^2 + (d*x^3 + c)*a*d - a*c*d)*a^2)
Time = 9.20 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.72 \[ \int \frac {\sqrt {c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx=\frac {\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {b^2\,d^2}{2\,a^3\,c^2}-\frac {b^2\,d^2\,\left (3\,a\,d-4\,b\,c\right )}{6\,a^2\,c^2\,\left (a^2\,d-a\,b\,c\right )}+\frac {b^2\,d\,\left (2\,a\,d-b\,c\right )\,\left (3\,a\,d-4\,b\,c\right )}{6\,a^3\,c^2\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}-\frac {b\,d\,\left (2\,a\,d-b\,c\right )}{2\,a^3\,c^2}+\frac {b\,\left (3\,a\,d-4\,b\,c\right )\,\left (-a^2\,d^2+2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{6\,a^3\,c^2\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}-\frac {-a^2\,d^2+2\,a\,b\,c\,d+2\,b^2\,c^2}{2\,a^3\,c^2}+\frac {b\,\left (a\,d-4\,b\,c\right )\,\left (3\,a\,d-4\,b\,c\right )}{6\,a^2\,c\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}-\frac {a\,d-4\,b\,c}{2\,a^2\,c}\right )\,\sqrt {d\,x^3+c}}{b\,x^3+a}-\frac {\sqrt {d\,x^3+c}}{3\,a^2\,x^3}+\frac {\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}{x^6}\right )\,\left (a\,d-4\,b\,c\right )}{6\,a^3\,\sqrt {c}}+\frac {\sqrt {b}\,\ln \left (\frac {a\,d-2\,b\,c-b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\left (3\,a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{6\,a^3\,\sqrt {a\,d-b\,c}} \] Input:
int((c + d*x^3)^(1/2)/(x^4*(a + b*x^3)^2),x)
Output:
(((a*((a*((a*((b^2*d^2)/(2*a^3*c^2) - (b^2*d^2*(3*a*d - 4*b*c))/(6*a^2*c^2 *(a^2*d - a*b*c)) + (b^2*d*(2*a*d - b*c)*(3*a*d - 4*b*c))/(6*a^3*c^2*(a^2* d - a*b*c))))/b - (b*d*(2*a*d - b*c))/(2*a^3*c^2) + (b*(3*a*d - 4*b*c)*(2* b^2*c^2 - a^2*d^2 + 2*a*b*c*d))/(6*a^3*c^2*(a^2*d - a*b*c))))/b - (2*b^2*c ^2 - a^2*d^2 + 2*a*b*c*d)/(2*a^3*c^2) + (b*(a*d - 4*b*c)*(3*a*d - 4*b*c))/ (6*a^2*c*(a^2*d - a*b*c))))/b - (a*d - 4*b*c)/(2*a^2*c))*(c + d*x^3)^(1/2) )/(a + b*x^3) - (c + d*x^3)^(1/2)/(3*a^2*x^3) + (log((((c + d*x^3)^(1/2) - c^(1/2))^3*((c + d*x^3)^(1/2) + c^(1/2)))/x^6)*(a*d - 4*b*c))/(6*a^3*c^(1 /2)) + (b^(1/2)*log((a*d - 2*b*c + b^(1/2)*(c + d*x^3)^(1/2)*(a*d - b*c)^( 1/2)*2i - b*d*x^3)/(a + b*x^3))*(3*a*d - 4*b*c)*1i)/(6*a^3*(a*d - b*c)^(1/ 2))
\[ \int \frac {\sqrt {c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx=\int \frac {\sqrt {d \,x^{3}+c}}{b^{2} x^{10}+2 a b \,x^{7}+a^{2} x^{4}}d x \] Input:
int((d*x^3+c)^(1/2)/x^4/(b*x^3+a)^2,x)
Output:
int(sqrt(c + d*x**3)/(a**2*x**4 + 2*a*b*x**7 + b**2*x**10),x)