Integrand size = 26, antiderivative size = 91 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx=-\frac {\sqrt {a+b x+c x^2}}{4 c d^3 (b+2 c x)^2}+\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{8 c^{3/2} \sqrt {b^2-4 a c} d^3} \] Output:
-1/4*(c*x^2+b*x+a)^(1/2)/c/d^3/(2*c*x+b)^2+1/8*arctan(2*c^(1/2)*(c*x^2+b*x +a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(3/2)/(-4*a*c+b^2)^(1/2)/d^3
Time = 10.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx=\frac {-\frac {2 c (a+x (b+c x))}{(b+2 c x)^2}-\sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \text {arctanh}\left (2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right )}{8 c^2 d^3 \sqrt {a+x (b+c x)}} \] Input:
Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^3,x]
Output:
((-2*c*(a + x*(b + c*x)))/(b + 2*c*x)^2 - Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*ArcTanh[2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]])/(8*c^2*d ^3*Sqrt[a + x*(b + c*x)])
Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1108, 27, 1112, 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.
\(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx\) |
\(\Big \downarrow \) 1108 |
\(\displaystyle \frac {\int \frac {1}{d (b+2 c x) \sqrt {c x^2+b x+a}}dx}{8 c d^2}-\frac {\sqrt {a+b x+c x^2}}{4 c d^3 (b+2 c x)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {1}{(b+2 c x) \sqrt {c x^2+b x+a}}dx}{8 c d^3}-\frac {\sqrt {a+b x+c x^2}}{4 c d^3 (b+2 c x)^2}\) |
\(\Big \downarrow \) 1112 |
\(\displaystyle \frac {\int \frac {1}{8 \left (c x^2+b x+a\right ) c^2+2 \left (b^2-4 a c\right ) c}d\sqrt {c x^2+b x+a}}{2 d^3}-\frac {\sqrt {a+b x+c x^2}}{4 c d^3 (b+2 c x)^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{8 c^{3/2} d^3 \sqrt {b^2-4 a c}}-\frac {\sqrt {a+b x+c x^2}}{4 c d^3 (b+2 c x)^2}\) |
Input:
Int[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^3,x]
Output:
-1/4*Sqrt[a + b*x + c*x^2]/(c*d^3*(b + 2*c*x)^2) + ArcTan[(2*Sqrt[c]*Sqrt[ a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]]/(8*c^(3/2)*Sqrt[b^2 - 4*a*c]*d^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_)^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[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[b*(p/(d*e*(m + 1))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x ], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] && LtQ[m, -1] && !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0] ) && IntegerQ[2*p]
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symb ol] :> Simp[4*c Subst[Int[1/(b^2*e - 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Time = 1.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {c \,x^{2}+b x +a}}{4 c^{2} x^{2}+4 c b x +b^{2}}-\frac {\operatorname {arctanh}\left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, c}{\sqrt {\left (4 a c -b^{2}\right ) c}}\right )}{2 \sqrt {\left (4 a c -b^{2}\right ) c}}}{4 c \,d^{3}}\) | \(89\) |
default | \(\frac {-\frac {2 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {2 c^{2} \left (\frac {\sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 a c -b^{2}}}{8 d^{3} c^{3}}\) | \(222\) |
Input:
int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^3,x,method=_RETURNVERBOSE)
Output:
1/4/c*(-(c*x^2+b*x+a)^(1/2)/(4*c^2*x^2+4*b*c*x+b^2)-1/2/((4*a*c-b^2)*c)^(1 /2)*arctanh(2*(c*x^2+b*x+a)^(1/2)*c/((4*a*c-b^2)*c)^(1/2)))/d^3
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (75) = 150\).
Time = 0.14 (sec) , antiderivative size = 371, normalized size of antiderivative = 4.08 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx=\left [-\frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt {-b^{2} c + 4 \, a c^{2}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {-b^{2} c + 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, {\left (4 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{3} x^{2} + 4 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{3} x + {\left (b^{4} c^{2} - 4 \, a b^{2} c^{3}\right )} d^{3}\right )}}, -\frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt {b^{2} c - 4 \, a c^{2}} \arctan \left (-\frac {2 \, \sqrt {b^{2} c - 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{8 \, {\left (4 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{3} x^{2} + 4 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{3} x + {\left (b^{4} c^{2} - 4 \, a b^{2} c^{3}\right )} d^{3}\right )}}\right ] \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^3,x, algorithm="fricas")
Output:
[-1/16*((4*c^2*x^2 + 4*b*c*x + b^2)*sqrt(-b^2*c + 4*a*c^2)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a)) /(4*c^2*x^2 + 4*b*c*x + b^2)) + 4*(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a)) /(4*(b^2*c^4 - 4*a*c^5)*d^3*x^2 + 4*(b^3*c^3 - 4*a*b*c^4)*d^3*x + (b^4*c^2 - 4*a*b^2*c^3)*d^3), -1/8*((4*c^2*x^2 + 4*b*c*x + b^2)*sqrt(b^2*c - 4*a*c ^2)*arctan(-2*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a)/(b^2 - 4*a*c)) + 2*(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a))/(4*(b^2*c^4 - 4*a*c^5)*d^3*x^2 + 4*(b^3*c^3 - 4*a*b*c^4)*d^3*x + (b^4*c^2 - 4*a*b^2*c^3)*d^3)]
\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx=\frac {\int \frac {\sqrt {a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx}{d^{3}} \] Input:
integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**3,x)
Output:
Integral(sqrt(a + b*x + c*x**2)/(b**3 + 6*b**2*c*x + 12*b*c**2*x**2 + 8*c* *3*x**3), x)/d**3
Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (75) = 150\).
Time = 0.19 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.70 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx=\frac {\arctan \left (-\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{4 \, \sqrt {b^{2} c - 4 \, a c^{2}} c d^{3}} + \frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} c + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b \sqrt {c} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{2} + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a c + a b \sqrt {c}}{4 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{2} c d^{3}} \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^3,x, algorithm="giac")
Output:
1/4*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt(b^2 *c - 4*a*c^2))/(sqrt(b^2*c - 4*a*c^2)*c*d^3) + 1/4*(2*(sqrt(c)*x - sqrt(c* x^2 + b*x + a))^3*c + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*sqrt(c) + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2 + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*c + a*b*sqrt(c))/((2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c + 2*( sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^2*c*d^3)
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (b\,d+2\,c\,d\,x\right )}^3} \,d x \] Input:
int((a + b*x + c*x^2)^(1/2)/(b*d + 2*c*d*x)^3,x)
Output:
int((a + b*x + c*x^2)^(1/2)/(b*d + 2*c*d*x)^3, x)
Time = 0.25 (sec) , antiderivative size = 494, normalized size of antiderivative = 5.43 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx=\frac {\sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2}+4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b c x +4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) c^{2} x^{2}-\sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2}-4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b c x -4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) c^{2} x^{2}-8 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2}+2 \sqrt {c \,x^{2}+b x +a}\, b^{2} c}{8 c^{2} d^{3} \left (16 a \,c^{3} x^{2}-4 b^{2} c^{2} x^{2}+16 a b \,c^{2} x -4 b^{3} c x +4 a \,b^{2} c -b^{4}\right )} \] Input:
int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^3,x)
Output:
(sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**2 + 4*sqrt(c)*sqrt(4*a *c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b*c*x + 4*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt (4*a*c - b**2))*c**2*x**2 - sqrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b **2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b **2 - 4*sqrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqr t(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b*c*x - 4*sqrt(c)*sqr t(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*c**2*x**2 - 8*sqrt(a + b*x + c*x**2)*a*c **2 + 2*sqrt(a + b*x + c*x**2)*b**2*c)/(8*c**2*d**3*(4*a*b**2*c + 16*a*b*c **2*x + 16*a*c**3*x**2 - b**4 - 4*b**3*c*x - 4*b**2*c**2*x**2))