Integrand size = 22, antiderivative size = 64 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x} \, dx=\frac {2}{3} A \sqrt {a+b x^3}+\frac {2 B \left (a+b x^3\right )^{3/2}}{9 b}-\frac {2}{3} \sqrt {a} A \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) \] Output:
2/3*A*(b*x^3+a)^(1/2)+2/9*B*(b*x^3+a)^(3/2)/b-2/3*a^(1/2)*A*arctanh((b*x^3 +a)^(1/2)/a^(1/2))
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x} \, dx=\frac {2 \sqrt {a+b x^3} \left (3 A b+a B+b B x^3\right )}{9 b}-\frac {2}{3} \sqrt {a} A \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) \] Input:
Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x,x]
Output:
(2*Sqrt[a + b*x^3]*(3*A*b + a*B + b*B*x^3))/(9*b) - (2*Sqrt[a]*A*ArcTanh[S qrt[a + b*x^3]/Sqrt[a]])/3
Time = 0.32 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {948, 90, 60, 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.
\(\displaystyle \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt {b x^3+a} \left (B x^3+A\right )}{x^3}dx^3\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{3} \left (A \int \frac {\sqrt {b x^3+a}}{x^3}dx^3+\frac {2 B \left (a+b x^3\right )^{3/2}}{3 b}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} \left (A \left (a \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3+2 \sqrt {a+b x^3}\right )+\frac {2 B \left (a+b x^3\right )^{3/2}}{3 b}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (A \left (\frac {2 a \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{b}+2 \sqrt {a+b x^3}\right )+\frac {2 B \left (a+b x^3\right )^{3/2}}{3 b}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (A \left (2 \sqrt {a+b x^3}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )+\frac {2 B \left (a+b x^3\right )^{3/2}}{3 b}\right )\) |
Input:
Int[(Sqrt[a + b*x^3]*(A + B*x^3))/x,x]
Output:
((2*B*(a + b*x^3)^(3/2))/(3*b) + A*(2*Sqrt[a + b*x^3] - 2*Sqrt[a]*ArcTanh[ Sqrt[a + b*x^3]/Sqrt[a]]))/3
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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 = 0.80 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.78
method | result | size |
default | \(A \left (\frac {2 \sqrt {b \,x^{3}+a}}{3}-\frac {2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3}\right )+\frac {2 B \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{9 b}\) | \(50\) |
elliptic | \(\frac {2 B \,x^{3} \sqrt {b \,x^{3}+a}}{9}+\frac {2 \left (A b +\frac {B a}{3}\right ) \sqrt {b \,x^{3}+a}}{3 b}-\frac {2 \sqrt {a}\, A \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3}\) | \(59\) |
pseudoelliptic | \(\frac {2 B b \,x^{3} \sqrt {b \,x^{3}+a}-6 \sqrt {a}\, b A \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )+6 A b \sqrt {b \,x^{3}+a}+2 B a \sqrt {b \,x^{3}+a}}{9 b}\) | \(70\) |
Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x,x,method=_RETURNVERBOSE)
Output:
A*(2/3*(b*x^3+a)^(1/2)-2/3*a^(1/2)*arctanh((b*x^3+a)^(1/2)/a^(1/2)))+2/9*B *(b*x^3+a)^(3/2)/b
Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x} \, dx=\left [\frac {3 \, A \sqrt {a} b \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (B b x^{3} + B a + 3 \, A b\right )} \sqrt {b x^{3} + a}}{9 \, b}, \frac {2 \, {\left (3 \, A \sqrt {-a} b \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{3} + a}}\right ) + {\left (B b x^{3} + B a + 3 \, A b\right )} \sqrt {b x^{3} + a}\right )}}{9 \, b}\right ] \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x,x, algorithm="fricas")
Output:
[1/9*(3*A*sqrt(a)*b*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*a)/x^3) + 2 *(B*b*x^3 + B*a + 3*A*b)*sqrt(b*x^3 + a))/b, 2/9*(3*A*sqrt(-a)*b*arctan(sq rt(-a)/sqrt(b*x^3 + a)) + (B*b*x^3 + B*a + 3*A*b)*sqrt(b*x^3 + a))/b]
Time = 5.61 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x} \, dx=\frac {A \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + b x^{3}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + b x^{3}} & \text {for}\: b \neq 0 \\- \sqrt {a} \log {\left (\frac {1}{x^{3}} \right )} & \text {otherwise} \end {cases}\right )}{3} - \frac {B \left (\begin {cases} - \sqrt {a} x^{3} & \text {for}\: b = 0 \\- \frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right )}{3} \] Input:
integrate((b*x**3+a)**(1/2)*(B*x**3+A)/x,x)
Output:
A*Piecewise((2*a*atan(sqrt(a + b*x**3)/sqrt(-a))/sqrt(-a) + 2*sqrt(a + b*x **3), Ne(b, 0)), (-sqrt(a)*log(x**(-3)), True))/3 - B*Piecewise((-sqrt(a)* x**3, Eq(b, 0)), (-2*(a + b*x**3)**(3/2)/(3*b), True))/3
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x} \, dx=\frac {1}{3} \, {\left (\sqrt {a} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right ) + 2 \, \sqrt {b x^{3} + a}\right )} A + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} B}{9 \, b} \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x,x, algorithm="maxima")
Output:
1/3*(sqrt(a)*log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a))) + 2*sqrt(b*x^3 + a))*A + 2/9*(b*x^3 + a)^(3/2)*B/b
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x} \, dx=\frac {2 \, A a \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} + \frac {2 \, {\left ({\left (b x^{3} + a\right )}^{\frac {3}{2}} B b^{2} + 3 \, \sqrt {b x^{3} + a} A b^{3}\right )}}{9 \, b^{3}} \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x,x, algorithm="giac")
Output:
2/3*A*a*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) + 2/9*((b*x^3 + a)^(3/2) *B*b^2 + 3*sqrt(b*x^3 + a)*A*b^3)/b^3
Time = 1.01 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x} \, dx=\frac {2\,B\,x^3\,\sqrt {b\,x^3+a}}{9}+\frac {\sqrt {b\,x^3+a}\,\left (2\,A\,b+\frac {2\,B\,a}{3}\right )}{3\,b}+\frac {A\,\sqrt {a}\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{3} \] Input:
int(((A + B*x^3)*(a + b*x^3)^(1/2))/x,x)
Output:
(2*B*x^3*(a + b*x^3)^(1/2))/9 + ((a + b*x^3)^(1/2)*(2*A*b + (2*B*a)/3))/(3 *b) + (A*a^(1/2)*log((((a + b*x^3)^(1/2) - a^(1/2))^3*((a + b*x^3)^(1/2) + a^(1/2)))/x^6))/3
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x} \, dx=\frac {8 \sqrt {b \,x^{3}+a}\, a}{9}+\frac {2 \sqrt {b \,x^{3}+a}\, b \,x^{3}}{9}+\frac {\sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {a}\right ) a}{3}-\frac {\sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {a}\right ) a}{3} \] Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x,x)
Output:
(8*sqrt(a + b*x**3)*a + 2*sqrt(a + b*x**3)*b*x**3 + 3*sqrt(a)*log(sqrt(a + b*x**3) - sqrt(a))*a - 3*sqrt(a)*log(sqrt(a + b*x**3) + sqrt(a))*a)/9