Integrand size = 42, antiderivative size = 103 \[ \int \frac {1}{\sqrt {a c^2+c (b c+2 a d) x+d (2 b c+a d) x^2+b d^2 x^3}} \, dx=\frac {2 \sqrt {a+b x} (c+d x) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {b c-a d} \sqrt {a c^2+c (b c+2 a d) x+d (2 b c+a d) x^2+b d^2 x^3}} \] Output:
2*(b*x+a)^(1/2)*(d*x+c)*arctan(d^(1/2)*(b*x+a)^(1/2)/(-a*d+b*c)^(1/2))/d^( 1/2)/(-a*d+b*c)^(1/2)/(a*c^2+c*(2*a*d+b*c)*x+d*(a*d+2*b*c)*x^2+b*d^2*x^3)^ (1/2)
Time = 0.00 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {a c^2+c (b c+2 a d) x+d (2 b c+a d) x^2+b d^2 x^3}} \, dx=\frac {2 \sqrt {a+b x} (c+d x) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {b c-a d} \sqrt {(a+b x) (c+d x)^2}} \] Input:
Integrate[1/Sqrt[a*c^2 + c*(b*c + 2*a*d)*x + d*(2*b*c + a*d)*x^2 + b*d^2*x ^3],x]
Output:
(2*Sqrt[a + b*x]*(c + d*x)*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]] )/(Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[(a + b*x)*(c + d*x)^2])
Time = 0.37 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.56, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2480, 27, 73, 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.
\(\displaystyle \int \frac {1}{\sqrt {d x^2 (a d+2 b c)+c x (2 a d+b c)+a c^2+b d^2 x^3}} \, dx\) |
\(\Big \downarrow \) 2480 |
\(\displaystyle \frac {2 d (c+d x) (b c-a d)^2 \sqrt {b d^4 x (b c-a d)^2+a d^4 (b c-a d)^2} \int \frac {1}{2 d (b c-a d)^2 (c+d x) \sqrt {a (b c-a d)^2 d^4+b (b c-a d)^2 x d^4}}dx}{\sqrt {d x^2 (a d+2 b c)+c x (2 a d+b c)+a c^2+b d^2 x^3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(c+d x) \sqrt {b d^4 x (b c-a d)^2+a d^4 (b c-a d)^2} \int \frac {1}{(c+d x) \sqrt {a (b c-a d)^2 d^4+b (b c-a d)^2 x d^4}}dx}{\sqrt {d x^2 (a d+2 b c)+c x (2 a d+b c)+a c^2+b d^2 x^3}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 (c+d x) \sqrt {b d^4 x (b c-a d)^2+a d^4 (b c-a d)^2} \int \frac {1}{c-\frac {a d}{b}+\frac {a (b c-a d)^2 d^4+b (b c-a d)^2 x d^4}{b d^3 (b c-a d)^2}}d\sqrt {a (b c-a d)^2 d^4+b (b c-a d)^2 x d^4}}{b d^4 (b c-a d)^2 \sqrt {d x^2 (a d+2 b c)+c x (2 a d+b c)+a c^2+b d^2 x^3}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 (c+d x) \sqrt {b d^4 x (b c-a d)^2+a d^4 (b c-a d)^2} \arctan \left (\frac {\sqrt {b d^4 x (b c-a d)^2+a d^4 (b c-a d)^2}}{d^{3/2} \sqrt {(b c-a d)^3}}\right )}{d^{5/2} \sqrt {(b c-a d)^3} \sqrt {d x^2 (a d+2 b c)+c x (2 a d+b c)+a c^2+b d^2 x^3}}\) |
Input:
Int[1/Sqrt[a*c^2 + c*(b*c + 2*a*d)*x + d*(2*b*c + a*d)*x^2 + b*d^2*x^3],x]
Output:
(2*(c + d*x)*Sqrt[a*d^4*(b*c - a*d)^2 + b*d^4*(b*c - a*d)^2*x]*ArcTan[Sqrt [a*d^4*(b*c - a*d)^2 + b*d^4*(b*c - a*d)^2*x]/(d^(3/2)*Sqrt[(b*c - a*d)^3] )])/(d^(5/2)*Sqrt[(b*c - a*d)^3]*Sqrt[a*c^2 + c*(b*c + 2*a*d)*x + d*(2*b*c + a*d)*x^2 + b*d^2*x^3])
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_)^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[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] , c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Simp[Px^p/((c^3 - 4*b*c*d + 9* a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3*b*d)*x)^(2*p)) Int [(c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3* b*d)*x)^(2*p), x], x] /; EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 27* a^2*d^2, 0] && NeQ[c^2 - 3*b*d, 0]] /; FreeQ[p, x] && PolyQ[Px, x, 3] && ! IntegerQ[p]
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {2 \left (d x +c \right ) \sqrt {b x +a}\, \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{\sqrt {b \,d^{2} x^{3}+a \,d^{2} x^{2}+2 b c d \,x^{2}+2 a c d x +b \,c^{2} x +a \,c^{2}}\, \sqrt {\left (a d -b c \right ) d}}\) | \(93\) |
Input:
int(1/(a*c^2+c*(2*a*d+b*c)*x+d*(a*d+2*b*c)*x^2+b*d^2*x^3)^(1/2),x,method=_ RETURNVERBOSE)
Output:
-2/(b*d^2*x^3+a*d^2*x^2+2*b*c*d*x^2+2*a*c*d*x+b*c^2*x+a*c^2)^(1/2)*(d*x+c) *(b*x+a)^(1/2)/((a*d-b*c)*d)^(1/2)*arctanh(d*(b*x+a)^(1/2)/((a*d-b*c)*d)^( 1/2))
Time = 0.10 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.29 \[ \int \frac {1}{\sqrt {a c^2+c (b c+2 a d) x+d (2 b c+a d) x^2+b d^2 x^3}} \, dx=\left [-\frac {\sqrt {-b c d + a d^{2}} \log \left (\frac {b d^{2} x^{2} + 2 \, a d^{2} x - b c^{2} + 2 \, a c d - 2 \, \sqrt {b d^{2} x^{3} + a c^{2} + {\left (2 \, b c d + a d^{2}\right )} x^{2} + {\left (b c^{2} + 2 \, a c d\right )} x} \sqrt {-b c d + a d^{2}}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b c d - a d^{2}}, -\frac {2 \, \arctan \left (\frac {\sqrt {b d^{2} x^{3} + a c^{2} + {\left (2 \, b c d + a d^{2}\right )} x^{2} + {\left (b c^{2} + 2 \, a c d\right )} x} \sqrt {b c d - a d^{2}}}{b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x}\right )}{\sqrt {b c d - a d^{2}}}\right ] \] Input:
integrate(1/(a*c^2+c*(2*a*d+b*c)*x+d*(a*d+2*b*c)*x^2+b*d^2*x^3)^(1/2),x, a lgorithm="fricas")
Output:
[-sqrt(-b*c*d + a*d^2)*log((b*d^2*x^2 + 2*a*d^2*x - b*c^2 + 2*a*c*d - 2*sq rt(b*d^2*x^3 + a*c^2 + (2*b*c*d + a*d^2)*x^2 + (b*c^2 + 2*a*c*d)*x)*sqrt(- b*c*d + a*d^2))/(d^2*x^2 + 2*c*d*x + c^2))/(b*c*d - a*d^2), -2*arctan(sqrt (b*d^2*x^3 + a*c^2 + (2*b*c*d + a*d^2)*x^2 + (b*c^2 + 2*a*c*d)*x)*sqrt(b*c *d - a*d^2)/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x))/sqrt(b*c*d - a*d^2)]
\[ \int \frac {1}{\sqrt {a c^2+c (b c+2 a d) x+d (2 b c+a d) x^2+b d^2 x^3}} \, dx=\int \frac {1}{\sqrt {a c^{2} + b d^{2} x^{3} + c x \left (2 a d + b c\right ) + d x^{2} \left (a d + 2 b c\right )}}\, dx \] Input:
integrate(1/(a*c**2+c*(2*a*d+b*c)*x+d*(a*d+2*b*c)*x**2+b*d**2*x**3)**(1/2) ,x)
Output:
Integral(1/sqrt(a*c**2 + b*d**2*x**3 + c*x*(2*a*d + b*c) + d*x**2*(a*d + 2 *b*c)), x)
\[ \int \frac {1}{\sqrt {a c^2+c (b c+2 a d) x+d (2 b c+a d) x^2+b d^2 x^3}} \, dx=\int { \frac {1}{\sqrt {b d^{2} x^{3} + {\left (2 \, b c + a d\right )} d x^{2} + a c^{2} + {\left (b c + 2 \, a d\right )} c x}} \,d x } \] Input:
integrate(1/(a*c^2+c*(2*a*d+b*c)*x+d*(a*d+2*b*c)*x^2+b*d^2*x^3)^(1/2),x, a lgorithm="maxima")
Output:
integrate(1/sqrt(b*d^2*x^3 + (2*b*c + a*d)*d*x^2 + a*c^2 + (b*c + 2*a*d)*c *x), x)
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\sqrt {a c^2+c (b c+2 a d) x+d (2 b c+a d) x^2+b d^2 x^3}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b x + a} d}{\sqrt {b c d - a d^{2}}}\right )}{\sqrt {b c d - a d^{2}} \mathrm {sgn}\left (d x + c\right )} \] Input:
integrate(1/(a*c^2+c*(2*a*d+b*c)*x+d*(a*d+2*b*c)*x^2+b*d^2*x^3)^(1/2),x, a lgorithm="giac")
Output:
2*arctan(sqrt(b*x + a)*d/sqrt(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^2)*sgn(d*x + c))
Timed out. \[ \int \frac {1}{\sqrt {a c^2+c (b c+2 a d) x+d (2 b c+a d) x^2+b d^2 x^3}} \, dx=\int \frac {1}{\sqrt {a\,c^2+d\,x^2\,\left (a\,d+2\,b\,c\right )+b\,d^2\,x^3+c\,x\,\left (2\,a\,d+b\,c\right )}} \,d x \] Input:
int(1/(a*c^2 + d*x^2*(a*d + 2*b*c) + b*d^2*x^3 + c*x*(2*a*d + b*c))^(1/2), x)
Output:
int(1/(a*c^2 + d*x^2*(a*d + 2*b*c) + b*d^2*x^3 + c*x*(2*a*d + b*c))^(1/2), x)
Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt {a c^2+c (b c+2 a d) x+d (2 b c+a d) x^2+b d^2 x^3}} \, dx=-\frac {2 \sqrt {d}\, \sqrt {-a d +b c}\, \mathit {atan} \left (\frac {\sqrt {b x +a}\, d}{\sqrt {d}\, \sqrt {-a d +b c}}\right )}{d \left (a d -b c \right )} \] Input:
int(1/(a*c^2+c*(2*a*d+b*c)*x+d*(a*d+2*b*c)*x^2+b*d^2*x^3)^(1/2),x)
Output:
( - 2*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a *d + b*c))))/(d*(a*d - b*c))