Integrand size = 24, antiderivative size = 131 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x \left (a+b x^3\right )^2} \, dx=\frac {(b c-a d) \sqrt {c+d x^3}}{3 a b \left (a+b x^3\right )}-\frac {2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2}+\frac {\sqrt {b c-a d} (2 b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 b^{3/2}} \]
-2/3*c^(3/2)*arctanh((d*x^3+c)^(1/2)/c^(1/2))/a^2+1/3*(a*d+2*b*c)*arctanh( b^(1/2)*(d*x^3+c)^(1/2)/(-a*d+b*c)^(1/2))*(-a*d+b*c)^(1/2)/a^2/b^(3/2)+1/3 *(-a*d+b*c)*(d*x^3+c)^(1/2)/a/b/(b*x^3+a)
Time = 0.52 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x \left (a+b x^3\right )^2} \, dx=\frac {\frac {a (b c-a d) \sqrt {c+d x^3}}{b \left (a+b x^3\right )}+\frac {\sqrt {-b c+a d} (2 b c+a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}-2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2} \]
((a*(b*c - a*d)*Sqrt[c + d*x^3])/(b*(a + b*x^3)) + (Sqrt[-(b*c) + a*d]*(2* b*c + a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[-(b*c) + a*d]])/b^(3/2) - 2*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^2)
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.07, 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.
| \(\displaystyle \int \frac {\left (c+d x^3\right )^{3/2}}{x \left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (d x^3+c\right )^{3/2}}{x^3 \left (b x^3+a\right )^2}dx^3\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{3} \left (\frac {\int \frac {d (b c+a d) x^3+2 b c^2}{2 x^3 \left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{a b}+\frac {\sqrt {c+d x^3} (b c-a d)}{a b \left (a+b x^3\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {\int \frac {d (b c+a d) x^3+2 b c^2}{x^3 \left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{2 a b}+\frac {\sqrt {c+d x^3} (b c-a d)}{a b \left (a+b x^3\right )}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{3} \left (\frac {\frac {2 b c^2 \int \frac {1}{x^3 \sqrt {d x^3+c}}dx^3}{a}-\frac {(b c-a d) (a d+2 b c) \int \frac {1}{\left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{a}}{2 a b}+\frac {\sqrt {c+d x^3} (b c-a d)}{a b \left (a+b x^3\right )}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (\frac {\frac {4 b c^2 \int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{a d}-\frac {2 (b c-a d) (a d+2 b c) \int \frac {1}{\frac {b x^6}{d}+a-\frac {b c}{d}}d\sqrt {d x^3+c}}{a d}}{2 a b}+\frac {\sqrt {c+d x^3} (b c-a d)}{a b \left (a+b x^3\right )}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{3} \left (\frac {\frac {2 \sqrt {b c-a d} (a d+2 b c) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{a \sqrt {b}}-\frac {4 b c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{a}}{2 a b}+\frac {\sqrt {c+d x^3} (b c-a d)}{a b \left (a+b x^3\right )}\right )\) |
(((b*c - a*d)*Sqrt[c + d*x^3])/(a*b*(a + b*x^3)) + ((-4*b*c^(3/2)*ArcTanh[ Sqrt[c + d*x^3]/Sqrt[c]])/a + (2*Sqrt[b*c - a*d]*(2*b*c + a*d)*ArcTanh[(Sq rt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(a*Sqrt[b]))/(2*a*b))/3
3.5.73.3.1 Defintions of rubi rules used
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[(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])
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 = 4.66 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.07
| method | result | size |
| pseudoelliptic | \(-\frac {-\left (b \,x^{3}+a \right ) \left (a d +2 b c \right ) \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {\left (a d -b c \right ) b}\, \left (2 b \,c^{\frac {3}{2}} \left (b \,x^{3}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )+a \sqrt {d \,x^{3}+c}\, \left (a d -b c \right )\right )}{3 \sqrt {\left (a d -b c \right ) b}\, a^{2} b \left (b \,x^{3}+a \right )}\) | \(140\) |
| default | \(\frac {\frac {2 d \,x^{3} \sqrt {d \,x^{3}+c}}{9}+\frac {8 c \sqrt {d \,x^{3}+c}}{9}-\frac {2 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3}}{a^{2}}+\frac {-\frac {2 \left (a d -b c \right )^{2} \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{3}+\frac {2 \sqrt {d \,x^{3}+c}\, \left (\frac {\left (-d \,x^{3}-4 c \right ) b}{3}+a d \right ) \sqrt {\left (a d -b c \right ) b}}{3}}{b \,a^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {-d \left (b \,x^{3}+a \right ) \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\left (\frac {\left (2 d \,x^{3}-c \right ) b}{3}+a d \right ) \sqrt {d \,x^{3}+c}\, \sqrt {\left (a d -b c \right ) b}}{b a \sqrt {\left (a d -b c \right ) b}\, \left (b \,x^{3}+a \right )}\) | \(260\) |
| elliptic | \(\text {Expression too large to display}\) | \(1632\) |
-1/3*(-(b*x^3+a)*(a*d+2*b*c)*(a*d-b*c)*arctan(b*(d*x^3+c)^(1/2)/((a*d-b*c) *b)^(1/2))+((a*d-b*c)*b)^(1/2)*(2*b*c^(3/2)*(b*x^3+a)*arctanh((d*x^3+c)^(1 /2)/c^(1/2))+a*(d*x^3+c)^(1/2)*(a*d-b*c)))/((a*d-b*c)*b)^(1/2)/a^2/b/(b*x^ 3+a)
Time = 0.42 (sec) , antiderivative size = 686, normalized size of antiderivative = 5.24 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x \left (a+b x^3\right )^2} \, dx=\left [\frac {{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{3} + a}\right ) + 2 \, {\left (b^{2} c x^{3} + a b c\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt {d x^{3} + c} {\left (a b c - a^{2} d\right )}}{6 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac {{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{3} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (b^{2} c x^{3} + a b c\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + \sqrt {d x^{3} + c} {\left (a b c - a^{2} d\right )}}{3 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac {4 \, {\left (b^{2} c x^{3} + a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{3} + a}\right ) + 2 \, \sqrt {d x^{3} + c} {\left (a b c - a^{2} d\right )}}{6 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac {{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{3} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + 2 \, {\left (b^{2} c x^{3} + a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + \sqrt {d x^{3} + c} {\left (a b c - a^{2} d\right )}}{3 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}\right ] \]
[1/6*(((2*b^2*c + a*b*d)*x^3 + 2*a*b*c + a^2*d)*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^2*c*x^3 + a*b*c)*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) + 2*sqrt(d*x^3 + c)*(a*b*c - a^2*d))/(a^2*b^2*x^3 + a^3*b), 1/ 3*(((2*b^2*c + a*b*d)*x^3 + 2*a*b*c + a^2*d)*sqrt(-(b*c - a*d)/b)*arctan(- sqrt(d*x^3 + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + (b^2*c*x^3 + a*b*c)* sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) + sqrt(d*x^3 + c)*(a*b*c - a^2*d))/(a^2*b^2*x^3 + a^3*b), 1/6*(4*(b^2*c*x^3 + a*b*c)*sqrt (-c)*arctan(sqrt(d*x^3 + c)*sqrt(-c)/c) + ((2*b^2*c + a*b*d)*x^3 + 2*a*b*c + a^2*d)*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*sqrt(d*x^3 + c)*(a*b*c - a^2*d) )/(a^2*b^2*x^3 + a^3*b), 1/3*(((2*b^2*c + a*b*d)*x^3 + 2*a*b*c + a^2*d)*sq rt(-(b*c - a*d)/b)*arctan(-sqrt(d*x^3 + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a *d)) + 2*(b^2*c*x^3 + a*b*c)*sqrt(-c)*arctan(sqrt(d*x^3 + c)*sqrt(-c)/c) + sqrt(d*x^3 + c)*(a*b*c - a^2*d))/(a^2*b^2*x^3 + a^3*b)]
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x \left (a+b x^3\right )^2} \, dx=\int \frac {\left (c + d x^{3}\right )^{\frac {3}{2}}}{x \left (a + b x^{3}\right )^{2}}\, dx \]
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x \left (a+b x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{2} x} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.18 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x \left (a+b x^3\right )^2} \, dx=\frac {2 \, c^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a^{2} \sqrt {-c}} - \frac {{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, \sqrt {-b^{2} c + a b d} a^{2} b} + \frac {\sqrt {d x^{3} + c} b c d - \sqrt {d x^{3} + c} a d^{2}}{3 \, {\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} a b} \]
2/3*c^2*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(a^2*sqrt(-c)) - 1/3*(2*b^2*c^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*b) + 1/3*(sqrt(d*x^3 + c)*b*c*d - sqrt(d*x^3 + c)*a*d^2 )/(((d*x^3 + c)*b - b*c + a*d)*a*b)
Time = 13.94 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.63 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x \left (a+b x^3\right )^2} \, dx=\frac {c^{3/2}\,\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}{x^6}\right )}{3\,a^2}+\frac {\sqrt {d\,x^3+c}\,\left (\frac {a\,\left (\frac {b\,d^2}{3\,\left (b^2\,c-a\,b\,d\right )}-\frac {2\,b^2\,c\,d}{3\,a\,\left (b^2\,c-a\,b\,d\right )}\right )}{b}+\frac {b^2\,c^2}{3\,a\,\left (b^2\,c-a\,b\,d\right )}\right )}{b\,x^3+a}+\frac {\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\sqrt {a\,d-b\,c}\,\left (a\,d+2\,b\,c\right )\,1{}\mathrm {i}}{6\,a^2\,b^{3/2}} \]
(c^(3/2)*log((((c + d*x^3)^(1/2) - c^(1/2))^3*((c + d*x^3)^(1/2) + c^(1/2) ))/x^6))/(3*a^2) + ((c + d*x^3)^(1/2)*((a*((b*d^2)/(3*(b^2*c - a*b*d)) - ( 2*b^2*c*d)/(3*a*(b^2*c - a*b*d))))/b + (b^2*c^2)/(3*a*(b^2*c - a*b*d))))/( a + b*x^3) + (log((2*b*c - a*d + b^(1/2)*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/ 2)*2i + b*d*x^3)/(a + b*x^3))*(a*d - b*c)^(1/2)*(a*d + 2*b*c)*1i)/(6*a^2*b ^(3/2))