Integrand size = 22, antiderivative size = 77 \[ \int \frac {A+B x^3}{x \left (a+b x^3\right )^{5/2}} \, dx=\frac {2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}}+\frac {2 A}{3 a^2 \sqrt {a+b x^3}}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}} \]
2/9*(A*b-B*a)/a/b/(b*x^3+a)^(3/2)-2/3*A*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a ^(5/2)+2/3*A/a^2/(b*x^3+a)^(1/2)
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x^3}{x \left (a+b x^3\right )^{5/2}} \, dx=-\frac {2 \left (-4 a A b+a^2 B-3 A b^2 x^3\right )}{9 a^2 b \left (a+b x^3\right )^{3/2}}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}} \]
(-2*(-4*a*A*b + a^2*B - 3*A*b^2*x^3))/(9*a^2*b*(a + b*x^3)^(3/2)) - (2*A*A rcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*a^(5/2))
Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {948, 87, 61, 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.
\(\displaystyle \int \frac {A+B x^3}{x \left (a+b x^3\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {B x^3+A}{x^3 \left (b x^3+a\right )^{5/2}}dx^3\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{3} \left (\frac {A \int \frac {1}{x^3 \left (b x^3+a\right )^{3/2}}dx^3}{a}+\frac {2 (A b-a B)}{3 a b \left (a+b x^3\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{3} \left (\frac {A \left (\frac {\int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3}{a}+\frac {2}{a \sqrt {a+b x^3}}\right )}{a}+\frac {2 (A b-a B)}{3 a b \left (a+b x^3\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {A \left (\frac {2 \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{a b}+\frac {2}{a \sqrt {a+b x^3}}\right )}{a}+\frac {2 (A b-a B)}{3 a b \left (a+b x^3\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (\frac {A \left (\frac {2}{a \sqrt {a+b x^3}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{a}+\frac {2 (A b-a B)}{3 a b \left (a+b x^3\right )^{3/2}}\right )\) |
((2*(A*b - a*B))/(3*a*b*(a + b*x^3)^(3/2)) + (A*(2/(a*Sqrt[a + b*x^3]) - ( 2*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/a^(3/2)))/a)/3
3.3.47.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 4.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(-\frac {2 \left (-3 A \sqrt {a}\, b^{2} x^{3}+3 A \left (b \,x^{3}+a \right )^{\frac {3}{2}} b \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )-4 A \,a^{\frac {3}{2}} b +B \,a^{\frac {5}{2}}\right )}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}} a^{\frac {5}{2}} b}\) | \(70\) |
elliptic | \(\frac {2 \left (A b -B a \right ) \sqrt {b \,x^{3}+a}}{9 b^{3} a \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {2 A}{3 a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {5}{2}}}\) | \(77\) |
default | \(-\frac {2 B}{9 b \left (b \,x^{3}+a \right )^{\frac {3}{2}}}+A \left (\frac {2 \sqrt {b \,x^{3}+a}}{9 a \,b^{2} \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {2}{3 a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {5}{2}}}\right )\) | \(85\) |
-2/9*(-3*A*a^(1/2)*b^2*x^3+3*A*(b*x^3+a)^(3/2)*b*arctanh((b*x^3+a)^(1/2)/a ^(1/2))-4*A*a^(3/2)*b+B*a^(5/2))/(b*x^3+a)^(3/2)/a^(5/2)/b
Time = 0.29 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.16 \[ \int \frac {A+B x^3}{x \left (a+b x^3\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{3} + A a^{2} b\right )} \sqrt {a} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (3 \, A a b^{2} x^{3} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x^{3} + a}}{9 \, {\left (a^{3} b^{3} x^{6} + 2 \, a^{4} b^{2} x^{3} + a^{5} b\right )}}, \frac {2 \, {\left (3 \, {\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{3} + A a^{2} b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, A a b^{2} x^{3} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x^{3} + a}\right )}}{9 \, {\left (a^{3} b^{3} x^{6} + 2 \, a^{4} b^{2} x^{3} + a^{5} b\right )}}\right ] \]
[1/9*(3*(A*b^3*x^6 + 2*A*a*b^2*x^3 + A*a^2*b)*sqrt(a)*log((b*x^3 - 2*sqrt( b*x^3 + a)*sqrt(a) + 2*a)/x^3) + 2*(3*A*a*b^2*x^3 - B*a^3 + 4*A*a^2*b)*sqr t(b*x^3 + a))/(a^3*b^3*x^6 + 2*a^4*b^2*x^3 + a^5*b), 2/9*(3*(A*b^3*x^6 + 2 *A*a*b^2*x^3 + A*a^2*b)*sqrt(-a)*arctan(sqrt(b*x^3 + a)*sqrt(-a)/a) + (3*A *a*b^2*x^3 - B*a^3 + 4*A*a^2*b)*sqrt(b*x^3 + a))/(a^3*b^3*x^6 + 2*a^4*b^2* x^3 + a^5*b)]
Time = 6.74 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.30 \[ \int \frac {A+B x^3}{x \left (a+b x^3\right )^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {A b}{3 a^{2} \sqrt {a + b x^{3}}} + \frac {A b \operatorname {atan}{\left (\frac {\sqrt {a + b x^{3}}}{\sqrt {- a}} \right )}}{3 a^{2} \sqrt {- a}} - \frac {- A b + B a}{9 a \left (a + b x^{3}\right )^{\frac {3}{2}}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {A \log {\left (B x^{3} \right )} + B x^{3}}{3 a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(A*b/(3*a**2*sqrt(a + b*x**3)) + A*b*atan(sqrt(a + b*x**3)/sq rt(-a))/(3*a**2*sqrt(-a)) - (-A*b + B*a)/(9*a*(a + b*x**3)**(3/2)))/b, Ne( b, 0)), ((A*log(B*x**3) + B*x**3)/(3*a**(5/2)), True))
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x^3}{x \left (a+b x^3\right )^{5/2}} \, dx=\frac {1}{9} \, A {\left (\frac {3 \, \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x^{3} + 4 \, a\right )}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}\right )} - \frac {2 \, B}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b} \]
1/9*A*(3*log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a)))/a^(5 /2) + 2*(3*b*x^3 + 4*a)/((b*x^3 + a)^(3/2)*a^2)) - 2/9*B/((b*x^3 + a)^(3/2 )*b)
Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^3}{x \left (a+b x^3\right )^{5/2}} \, dx=\frac {2 \, A \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a} a^{2}} - \frac {2 \, {\left (B a^{2} - 3 \, {\left (b x^{3} + a\right )} A b - A a b\right )}}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2} b} \]
2/3*A*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*a^2) - 2/9*(B*a^2 - 3*(b* x^3 + a)*A*b - A*a*b)/((b*x^3 + a)^(3/2)*a^2*b)
Time = 7.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x^3}{x \left (a+b x^3\right )^{5/2}} \, dx=\frac {\frac {2\,A}{9\,a}-\frac {2\,B}{9\,b}}{{\left (b\,x^3+a\right )}^{3/2}}+\frac {2\,A}{3\,a^2\,\sqrt {b\,x^3+a}}+\frac {A\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{3\,a^{5/2}} \]