Integrand size = 24, antiderivative size = 114 \[ \int \frac {1}{x \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=-\frac {2 d}{3 c (b c-a d) \sqrt {c+d x^3}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a c^{3/2}}+\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a (b c-a d)^{3/2}} \]
-2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))/a/c^(3/2)+2/3*b^(3/2)*arctanh(b^(1/2 )*(d*x^3+c)^(1/2)/(-a*d+b*c)^(1/2))/a/(-a*d+b*c)^(3/2)-2/3*d/c/(-a*d+b*c)/ (d*x^3+c)^(1/2)
Time = 0.60 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\frac {2}{3} \left (\frac {d}{c (-b c+a d) \sqrt {c+d x^3}}+\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{a (-b c+a d)^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{a c^{3/2}}\right ) \]
(2*(d/(c*(-(b*c) + a*d)*Sqrt[c + d*x^3]) + (b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[-(b*c) + a*d]])/(a*(-(b*c) + a*d)^(3/2)) - ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]]/(a*c^(3/2))))/3
Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {948, 96, 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.
| \(\displaystyle \int \frac {1}{x \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{x^3 \left (b x^3+a\right ) \left (d x^3+c\right )^{3/2}}dx^3\) |
| \(\Big \downarrow \) 96 |
| \(\displaystyle \frac {1}{3} \left (\frac {\int \frac {-b d x^3+b c-a d}{x^3 \left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{c (b c-a d)}-\frac {2 d}{c \sqrt {c+d x^3} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{3} \left (\frac {\frac {(b c-a d) \int \frac {1}{x^3 \sqrt {d x^3+c}}dx^3}{a}-\frac {b^2 c \int \frac {1}{\left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{a}}{c (b c-a d)}-\frac {2 d}{c \sqrt {c+d x^3} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (\frac {\frac {2 (b c-a d) \int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{a d}-\frac {2 b^2 c \int \frac {1}{\frac {b x^6}{d}+a-\frac {b c}{d}}d\sqrt {d x^3+c}}{a d}}{c (b c-a d)}-\frac {2 d}{c \sqrt {c+d x^3} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{3} \left (\frac {\frac {2 b^{3/2} c \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{a \sqrt {c}}}{c (b c-a d)}-\frac {2 d}{c \sqrt {c+d x^3} (b c-a d)}\right )\) |
((-2*d)/(c*(b*c - a*d)*Sqrt[c + d*x^3]) + ((-2*(b*c - a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(a*Sqrt[c]) + (2*b^(3/2)*c*ArcTanh[(Sqrt[b]*Sqrt[c + d* x^3])/Sqrt[b*c - a*d]])/(a*Sqrt[b*c - a*d]))/(c*(b*c - a*d)))/3
3.4.91.3.1 Defintions of rubi rules used
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[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[f*((e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + S imp[1/((b*e - a*f)*(d*e - c*f)) Int[(b*d*e - b*c*f - a*d*f - b*d*f*x)*((e + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -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 = 4.91 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(\frac {2 b^{2} \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 \left (a d -b c \right ) a \sqrt {\left (a d -b c \right ) b}}+\frac {2 d}{3 \left (a d -b c \right ) c \sqrt {d \,x^{3}+c}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 a \,c^{\frac {3}{2}}}\) | \(103\) |
| default | \(\frac {\frac {2}{3 c \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}}{a}+\frac {2 b \left (b \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) \sqrt {d \,x^{3}+c}+\sqrt {\left (a d -b c \right ) b}\right )}{a \sqrt {\left (a d -b c \right ) b}\, \sqrt {d \,x^{3}+c}\, \left (3 a d -3 b c \right )}\) | \(130\) |
| elliptic | \(\text {Expression too large to display}\) | \(1637\) |
2/3/(a*d-b*c)*b^2/a/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x^3+c)^(1/2)/((a*d-b*c )*b)^(1/2))+2/3*d/(a*d-b*c)/c/(d*x^3+c)^(1/2)-2/3*arctanh((d*x^3+c)^(1/2)/ c^(1/2))/a/c^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (90) = 180\).
Time = 0.38 (sec) , antiderivative size = 790, normalized size of antiderivative = 6.93 \[ \int \frac {1}{x \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\left [-\frac {2 \, \sqrt {d x^{3} + c} a c d + {\left (b c^{2} d x^{3} + b c^{3}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{3} + a}\right ) - {\left ({\left (b c d - a d^{2}\right )} x^{3} + b c^{2} - a c d\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right )}{3 \, {\left (a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{3}\right )}}, -\frac {2 \, \sqrt {d x^{3} + c} a c d - 2 \, {\left (b c^{2} d x^{3} + b c^{3}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{3} + b c}\right ) - {\left ({\left (b c d - a d^{2}\right )} x^{3} + b c^{2} - a c d\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right )}{3 \, {\left (a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{3}\right )}}, -\frac {2 \, \sqrt {d x^{3} + c} a c d - 2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + b c^{2} - a c d\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left (b c^{2} d x^{3} + b c^{3}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{3} + a}\right )}{3 \, {\left (a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{3}\right )}}, -\frac {2 \, {\left (\sqrt {d x^{3} + c} a c d - {\left (b c^{2} d x^{3} + b c^{3}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{3} + b c}\right ) - {\left ({\left (b c d - a d^{2}\right )} x^{3} + b c^{2} - a c d\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right )\right )}}{3 \, {\left (a b c^{4} - a^{2} c^{3} d + {\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{3}\right )}}\right ] \]
[-1/3*(2*sqrt(d*x^3 + c)*a*c*d + (b*c^2*d*x^3 + b*c^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)) - ((b*c*d - a*d^2)*x^3 + b*c^2 - a*c*d)*sqrt(c)*log((d *x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3))/(a*b*c^4 - a^2*c^3*d + (a*b* c^3*d - a^2*c^2*d^2)*x^3), -1/3*(2*sqrt(d*x^3 + c)*a*c*d - 2*(b*c^2*d*x^3 + b*c^3)*sqrt(-b/(b*c - a*d))*arctan(-sqrt(d*x^3 + c)*(b*c - a*d)*sqrt(-b/ (b*c - a*d))/(b*d*x^3 + b*c)) - ((b*c*d - a*d^2)*x^3 + b*c^2 - a*c*d)*sqrt (c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3))/(a*b*c^4 - a^2*c^3 *d + (a*b*c^3*d - a^2*c^2*d^2)*x^3), -1/3*(2*sqrt(d*x^3 + c)*a*c*d - 2*((b *c*d - a*d^2)*x^3 + b*c^2 - a*c*d)*sqrt(-c)*arctan(sqrt(d*x^3 + c)*sqrt(-c )/c) + (b*c^2*d*x^3 + b*c^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)))/(a*b* c^4 - a^2*c^3*d + (a*b*c^3*d - a^2*c^2*d^2)*x^3), -2/3*(sqrt(d*x^3 + c)*a* c*d - (b*c^2*d*x^3 + b*c^3)*sqrt(-b/(b*c - a*d))*arctan(-sqrt(d*x^3 + c)*( b*c - a*d)*sqrt(-b/(b*c - a*d))/(b*d*x^3 + b*c)) - ((b*c*d - a*d^2)*x^3 + b*c^2 - a*c*d)*sqrt(-c)*arctan(sqrt(d*x^3 + c)*sqrt(-c)/c))/(a*b*c^4 - a^2 *c^3*d + (a*b*c^3*d - a^2*c^2*d^2)*x^3)]
Time = 6.87 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {d^{2}}{3 c \sqrt {c + d x^{3}} \left (a d - b c\right )} + \frac {b d \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{3 a \sqrt {\frac {a d - b c}{b}} \left (a d - b c\right )} + \frac {d \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{\sqrt {- c}} \right )}}{3 a c \sqrt {- c}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {2 \operatorname {atan}{\left (\frac {2 \left (\frac {a}{2 b} + x^{3}\right )}{\sqrt {- \frac {a^{2}}{b^{2}}}} \right )}}{3 b c^{\frac {3}{2}} \sqrt {- \frac {a^{2}}{b^{2}}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(d**2/(3*c*sqrt(c + d*x**3)*(a*d - b*c)) + b*d*atan(sqrt(c + d*x**3)/sqrt((a*d - b*c)/b))/(3*a*sqrt((a*d - b*c)/b)*(a*d - b*c)) + d*ata n(sqrt(c + d*x**3)/sqrt(-c))/(3*a*c*sqrt(-c)))/d, Ne(d, 0)), (2*atan(2*(a/ (2*b) + x**3)/sqrt(-a**2/b**2))/(3*b*c**(3/2)*sqrt(-a**2/b**2)), True))
\[ \int \frac {1}{x \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )} {\left (d x^{3} + c\right )}^{\frac {3}{2}} x} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=-\frac {2 \, b^{2} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, {\left (a b c - a^{2} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, d}{3 \, \sqrt {d x^{3} + c} {\left (b c^{2} - a c d\right )}} + \frac {2 \, \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a \sqrt {-c} c} \]
-2/3*b^2*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/((a*b*c - a^2*d)*s qrt(-b^2*c + a*b*d)) - 2/3*d/(sqrt(d*x^3 + c)*(b*c^2 - a*c*d)) + 2/3*arcta n(sqrt(d*x^3 + c)/sqrt(-c))/(a*sqrt(-c)*c)
Time = 12.98 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\frac {\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}{x^6}\right )}{3\,a\,c^{3/2}}+\frac {2\,d}{3\,c\,\sqrt {d\,x^3+c}\,\left (a\,d-b\,c\right )}+\frac {b^{3/2}\,\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,1{}\mathrm {i}}{3\,a\,{\left (a\,d-b\,c\right )}^{3/2}} \]