Integrand size = 31, antiderivative size = 80 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=a \sqrt {-c+d x} \sqrt {c+d x}+\frac {b (-c+d x)^{3/2} (c+d x)^{3/2}}{3 d^2}-a c \arctan \left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right ) \] Output:
a*(d*x-c)^(1/2)*(d*x+c)^(1/2)+1/3*b*(d*x-c)^(3/2)*(d*x+c)^(3/2)/d^2-a*c*ar ctan((d*x-c)^(1/2)*(d*x+c)^(1/2)/c)
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=\frac {\sqrt {-c+d x} \sqrt {c+d x} \left (-b c^2+3 a d^2+b d^2 x^2\right )}{3 d^2}-2 a c \arctan \left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right ) \] Input:
Integrate[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x,x]
Output:
(Sqrt[-c + d*x]*Sqrt[c + d*x]*(-(b*c^2) + 3*a*d^2 + b*d^2*x^2))/(3*d^2) - 2*a*c*ArcTan[Sqrt[-c + d*x]/Sqrt[c + d*x]]
Time = 0.37 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {960, 112, 27, 103, 218}
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.
\(\displaystyle \int \frac {\left (a+b x^2\right ) \sqrt {d x-c} \sqrt {c+d x}}{x} \, dx\) |
\(\Big \downarrow \) 960 |
\(\displaystyle a \int \frac {\sqrt {d x-c} \sqrt {c+d x}}{x}dx+\frac {b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2}\) |
\(\Big \downarrow \) 112 |
\(\displaystyle a \left (\sqrt {d x-c} \sqrt {c+d x}-\int \frac {c^2}{x \sqrt {d x-c} \sqrt {c+d x}}dx\right )+\frac {b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a \left (\sqrt {d x-c} \sqrt {c+d x}-c^2 \int \frac {1}{x \sqrt {d x-c} \sqrt {c+d x}}dx\right )+\frac {b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle a \left (\sqrt {d x-c} \sqrt {c+d x}-c^2 d \int \frac {1}{d c^2+d (d x-c) (c+d x)}d\left (\sqrt {d x-c} \sqrt {c+d x}\right )\right )+\frac {b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle a \left (\sqrt {d x-c} \sqrt {c+d x}-c \arctan \left (\frac {\sqrt {d x-c} \sqrt {c+d x}}{c}\right )\right )+\frac {b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2}\) |
Input:
Int[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x,x]
Output:
(b*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(3*d^2) + a*(Sqrt[-c + d*x]*Sqrt[c + d*x] - c*ArcTan[(Sqrt[-c + d*x]*Sqrt[c + d*x])/c])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) *(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n *(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ (b1*b2*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n /2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(66)=132\).
Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.18
method | result | size |
default | \(\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (b \,d^{2} x^{2} \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}+3 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) a \,c^{2} d^{2}+3 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\, a \,d^{2}-b \,c^{2} \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{3 \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}\, d^{2}}\) | \(174\) |
Input:
int((d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a)/x,x,method=_RETURNVERBOSE)
Output:
1/3*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*d^2*x^2*(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2 )+3*ln(-2*(c^2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)*a*c^2*d^2+3*(-c^2)^(1/ 2)*(d^2*x^2-c^2)^(1/2)*a*d^2-b*c^2*(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/(d^2* x^2-c^2)^(1/2)/(-c^2)^(1/2)/d^2
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=-\frac {6 \, a c d^{2} \arctan \left (-\frac {d x - \sqrt {d x + c} \sqrt {d x - c}}{c}\right ) - {\left (b d^{2} x^{2} - b c^{2} + 3 \, a d^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{3 \, d^{2}} \] Input:
integrate((d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a)/x,x, algorithm="fricas")
Output:
-1/3*(6*a*c*d^2*arctan(-(d*x - sqrt(d*x + c)*sqrt(d*x - c))/c) - (b*d^2*x^ 2 - b*c^2 + 3*a*d^2)*sqrt(d*x + c)*sqrt(d*x - c))/d^2
\[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=\int \frac {\left (a + b x^{2}\right ) \sqrt {- c + d x} \sqrt {c + d x}}{x}\, dx \] Input:
integrate((d*x-c)**(1/2)*(d*x+c)**(1/2)*(b*x**2+a)/x,x)
Output:
Integral((a + b*x**2)*sqrt(-c + d*x)*sqrt(c + d*x)/x, x)
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=a c \arcsin \left (\frac {c}{d {\left | x \right |}}\right ) + \sqrt {d^{2} x^{2} - c^{2}} a + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b}{3 \, d^{2}} \] Input:
integrate((d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a)/x,x, algorithm="maxima")
Output:
a*c*arcsin(c/(d*abs(x))) + sqrt(d^2*x^2 - c^2)*a + 1/3*(d^2*x^2 - c^2)^(3/ 2)*b/d^2
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=2 \, a c \arctan \left (\frac {{\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}}{2 \, c}\right ) + \frac {1}{3} \, \sqrt {d x + c} \sqrt {d x - c} {\left ({\left (d x + c\right )} {\left (\frac {{\left (d x + c\right )} b}{d^{2}} - \frac {2 \, b c}{d^{2}}\right )} + 3 \, a\right )} \] Input:
integrate((d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a)/x,x, algorithm="giac")
Output:
2*a*c*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c) + 1/3*sqrt(d*x + c)* sqrt(d*x - c)*((d*x + c)*((d*x + c)*b/d^2 - 2*b*c/d^2) + 3*a)
Time = 7.05 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.10 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=a\,\sqrt {-c}\,\sqrt {c}\,\ln \left (\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+1\right )-a\,\sqrt {-c}\,\sqrt {c}\,\ln \left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )-\frac {b\,\left (c^2-d^2\,x^2\right )\,\sqrt {c+d\,x}\,\sqrt {d\,x-c}}{3\,d^2}-\frac {8\,a\,\sqrt {-c}\,\sqrt {c}\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2\,\left (\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}-\frac {2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+1\right )} \] Input:
int(((a + b*x^2)*(c + d*x)^(1/2)*(d*x - c)^(1/2))/x,x)
Output:
a*(-c)^(1/2)*c^(1/2)*log(((c + d*x)^(1/2) - c^(1/2))^2/((-c)^(1/2) - (d*x - c)^(1/2))^2 + 1) - a*(-c)^(1/2)*c^(1/2)*log(((c + d*x)^(1/2) - c^(1/2))/ ((-c)^(1/2) - (d*x - c)^(1/2))) - (b*(c^2 - d^2*x^2)*(c + d*x)^(1/2)*(d*x - c)^(1/2))/(3*d^2) - (8*a*(-c)^(1/2)*c^(1/2)*((c + d*x)^(1/2) - c^(1/2))^ 2)/(((-c)^(1/2) - (d*x - c)^(1/2))^2*(((c + d*x)^(1/2) - c^(1/2))^4/((-c)^ (1/2) - (d*x - c)^(1/2))^4 - (2*((c + d*x)^(1/2) - c^(1/2))^2)/((-c)^(1/2) - (d*x - c)^(1/2))^2 + 1))
Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x} \, dx=\frac {-6 \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}-\sqrt {c}}{\sqrt {c}}\right ) a c \,d^{2}+6 \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}+\sqrt {c}}{\sqrt {c}}\right ) a c \,d^{2}+3 \sqrt {d x +c}\, \sqrt {d x -c}\, a \,d^{2}-\sqrt {d x +c}\, \sqrt {d x -c}\, b \,c^{2}+\sqrt {d x +c}\, \sqrt {d x -c}\, b \,d^{2} x^{2}}{3 d^{2}} \] Input:
int((d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a)/x,x)
Output:
( - 6*atan((sqrt( - c + d*x) + sqrt(c + d*x) - sqrt(c))/sqrt(c))*a*c*d**2 + 6*atan((sqrt( - c + d*x) + sqrt(c + d*x) + sqrt(c))/sqrt(c))*a*c*d**2 + 3*sqrt(c + d*x)*sqrt( - c + d*x)*a*d**2 - sqrt(c + d*x)*sqrt( - c + d*x)*b *c**2 + sqrt(c + d*x)*sqrt( - c + d*x)*b*d**2*x**2)/(3*d**2)