Integrand size = 27, antiderivative size = 66 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a-d} \sqrt {b^2-4 c d}} \] Output:
-2*arctanh((a-d)^(1/2)*(2*c*x+b)/(b^2-4*c*d)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a -d)^(1/2)/(b^2-4*c*d)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.45 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.86 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx=\text {RootSum}\left [c^2 d-2 \sqrt {a} b c \text {$\#$1}+b^2 \text {$\#$1}^2+4 a c \text {$\#$1}^2-2 c d \text {$\#$1}^2-2 \sqrt {a} b \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {-c \log (x)+c \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^2-\log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} b c+b^2 \text {$\#$1}+4 a c \text {$\#$1}-2 c d \text {$\#$1}-3 \sqrt {a} b \text {$\#$1}^2+2 d \text {$\#$1}^3}\&\right ] \] Input:
Integrate[1/(Sqrt[a + b*x + c*x^2]*(d + b*x + c*x^2)),x]
Output:
RootSum[c^2*d - 2*Sqrt[a]*b*c*#1 + b^2*#1^2 + 4*a*c*#1^2 - 2*c*d*#1^2 - 2* Sqrt[a]*b*#1^3 + d*#1^4 & , (-(c*Log[x]) + c*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + Log[x]*#1^2 - Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*# 1]*#1^2)/(-(Sqrt[a]*b*c) + b^2*#1 + 4*a*c*#1 - 2*c*d*#1 - 3*Sqrt[a]*b*#1^2 + 2*d*#1^3) & ]
Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1313, 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.
\(\displaystyle \int \frac {1}{\sqrt {a+b x+c x^2} \left (b x+c x^2+d\right )} \, dx\) |
\(\Big \downarrow \) 1313 |
\(\displaystyle -2 b \int \frac {1}{b \left (b^2-4 c d\right )-\frac {b (a-d) (b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\frac {\sqrt {a-d} (b+2 c x)}{\sqrt {b^2-4 c d} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a-d} \sqrt {b^2-4 c d}}\) |
Input:
Int[1/(Sqrt[a + b*x + c*x^2]*(d + b*x + c*x^2)),x]
Output:
(-2*ArcTanh[(Sqrt[a - d]*(b + 2*c*x))/(Sqrt[b^2 - 4*c*d]*Sqrt[a + b*x + c* x^2])])/(Sqrt[a - d]*Sqrt[b^2 - 4*c*d])
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[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*( x_)^2]), x_Symbol] :> Simp[-2*e Subst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e )*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(306\) vs. \(2(56)=112\).
Time = 2.58 (sec) , antiderivative size = 307, normalized size of antiderivative = 4.65
method | result | size |
default | \(-\frac {\ln \left (\frac {2 a -2 d +\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {c {\left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )}^{2}+\sqrt {b^{2}-4 c d}\, \left (x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{x -\frac {-b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{\sqrt {b^{2}-4 c d}\, \sqrt {a -d}}+\frac {\ln \left (\frac {2 a -2 d -\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+2 \sqrt {a -d}\, \sqrt {c {\left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )}^{2}-\sqrt {b^{2}-4 c d}\, \left (x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}\right )+a -d}}{x +\frac {b +\sqrt {b^{2}-4 c d}}{2 c}}\right )}{\sqrt {b^{2}-4 c d}\, \sqrt {a -d}}\) | \(307\) |
Input:
int(1/(c*x^2+b*x+a)^(1/2)/(c*x^2+b*x+d),x,method=_RETURNVERBOSE)
Output:
-1/(b^2-4*c*d)^(1/2)/(a-d)^(1/2)*ln((2*a-2*d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+ (b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*(c*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2+ (b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(x-1/2*(-b+ (b^2-4*c*d)^(1/2))/c))+1/(b^2-4*c*d)^(1/2)/(a-d)^(1/2)*ln((2*a-2*d-(b^2-4* c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*(c*(x+1/2*(b+(b^2 -4*c*d)^(1/2))/c)^2-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)+a-d) ^(1/2))/(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (56) = 112\).
Time = 0.12 (sec) , antiderivative size = 813, normalized size of antiderivative = 12.32 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/(c*x^2+b*x+a)^(1/2)/(c*x^2+b*x+d),x, algorithm="fricas")
Output:
[1/2*log((8*a^2*b^4 + (b^4*c^2 + 24*a*b^2*c^3 + 16*a^2*c^4 + 128*c^4*d^2 - 32*(b^2*c^3 + 4*a*c^4)*d)*x^4 + 2*(b^5*c + 24*a*b^3*c^2 + 16*a^2*b*c^3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)*x^3 + (b^4 + 24*a*b^2*c + 16*a ^2*c^2)*d^2 + (b^6 + 32*a*b^4*c + 48*a^2*b^2*c^2 + 32*(5*b^2*c^2 + 4*a*c^3 )*d^2 - 2*(19*b^4*c + 104*a*b^2*c^2 + 48*a^2*c^3)*d)*x^2 - 4*(2*a*b^3 + 2* (b^2*c^2 + 4*a*c^3 - 8*c^3*d)*x^3 + 3*(b^3*c + 4*a*b*c^2 - 8*b*c^2*d)*x^2 - (b^3 + 4*a*b*c)*d + (b^4 + 8*a*b^2*c - 2*(5*b^2*c + 4*a*c^2)*d)*x)*sqrt( a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d)*sqrt(c*x^2 + b*x + a) - 8*(a*b^4 + 4*a^ 2*b^2*c)*d + 2*(4*a*b^5 + 16*a^2*b^3*c + 16*(b^3*c + 4*a*b*c^2)*d^2 - (3*b ^5 + 40*a*b^3*c + 48*a^2*b*c^2)*d)*x)/(c^2*x^4 + 2*b*c*x^3 + 2*b*d*x + (b^ 2 + 2*c*d)*x^2 + d^2))/sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d), -sqrt(-a*b ^2 - 4*c*d^2 + (b^2 + 4*a*c)*d)*arctan(-1/2*(2*a*b^2 + (b^2*c + 4*a*c^2 - 8*c^2*d)*x^2 - (b^2 + 4*a*c)*d + (b^3 + 4*a*b*c - 8*b*c*d)*x)*sqrt(-a*b^2 - 4*c*d^2 + (b^2 + 4*a*c)*d)*sqrt(c*x^2 + b*x + a)/(a^2*b^3 + 4*a*b*c*d^2 + 2*(a*b^2*c^2 + 4*c^3*d^2 - (b^2*c^2 + 4*a*c^3)*d)*x^3 + 3*(a*b^3*c + 4*b *c^2*d^2 - (b^3*c + 4*a*b*c^2)*d)*x^2 - (a*b^3 + 4*a^2*b*c)*d + (a*b^4 + 2 *a^2*b^2*c + 4*(b^2*c + 2*a*c^2)*d^2 - (b^4 + 6*a*b^2*c + 8*a^2*c^2)*d)*x) )/(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d)]
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx=\int \frac {1}{\sqrt {a + b x + c x^{2}} \left (b x + c x^{2} + d\right )}\, dx \] Input:
integrate(1/(c*x**2+b*x+a)**(1/2)/(c*x**2+b*x+d),x)
Output:
Integral(1/(sqrt(a + b*x + c*x**2)*(b*x + c*x**2 + d)), x)
Exception generated. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(c*x^2+b*x+a)^(1/2)/(c*x^2+b*x+d),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*c*d-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (56) = 112\).
Time = 0.22 (sec) , antiderivative size = 703, normalized size of antiderivative = 10.65 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx=-\frac {\log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} c - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{2} d - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c^{\frac {3}{2}} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{\frac {3}{2}} d - 3 \, a b^{2} c + 4 \, \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {3}{2}} + 4 \, a^{2} c^{2} + 2 \, b^{2} c d + 4 \, \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c + \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b^{2} \sqrt {c} \right |}\right )}{\sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}}} + \frac {\log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} c - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{2} d - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c^{\frac {3}{2}} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{\frac {3}{2}} d - 3 \, a b^{2} c - 4 \, \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {3}{2}} + 4 \, a^{2} c^{2} + 2 \, b^{2} c d - 4 \, \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c - \sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}} b^{2} \sqrt {c} \right |}\right )}{\sqrt {a b^{2} - b^{2} d - 4 \, a c d + 4 \, c d^{2}}} \] Input:
integrate(1/(c*x^2+b*x+a)^(1/2)/(c*x^2+b*x+d),x, algorithm="giac")
Output:
-log(abs(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c - 4*(sqrt(c)*x - sqr t(c*x^2 + b*x + a))^2*a*c^2 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^2* d - (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*sqrt(c) - 4*(sqrt(c)*x - sqrt( c*x^2 + b*x + a))*a*b*c^(3/2) + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^ (3/2)*d - 3*a*b^2*c + 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(3/2) + 4*a^2*c^2 + 2*b^2*c*d + 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c + sq rt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*b^2*sqrt(c)))/sqrt(a*b^2 - b^2*d - 4 *a*c*d + 4*c*d^2) + log(abs(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^2 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^2*d - (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*sqrt(c) - 4 *(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^(3/2) + 8*(sqrt(c)*x - sqrt(c*x ^2 + b*x + a))*b*c^(3/2)*d - 3*a*b^2*c - 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(3/2) + 4*a^2*c^2 + 2*b^2 *c*d - 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c - sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*b^2*sqrt(c)))/sqr t(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)
Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx=\int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,\left (c\,x^2+b\,x+d\right )} \,d x \] Input:
int(1/((a + b*x + c*x^2)^(1/2)*(d + b*x + c*x^2)),x)
Output:
int(1/((a + b*x + c*x^2)^(1/2)*(d + b*x + c*x^2)), x)
Time = 0.22 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.41 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx=\frac {\sqrt {a -d}\, \sqrt {b^{2}-4 c d}\, \left (\mathrm {log}\left (-\sqrt {4 \sqrt {c}\, \sqrt {a -d}\, \sqrt {b^{2}-4 c d}+4 a c +b^{2}-8 c d}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x \right )+\mathrm {log}\left (\sqrt {4 \sqrt {c}\, \sqrt {a -d}\, \sqrt {b^{2}-4 c d}+4 a c +b^{2}-8 c d}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x \right )-\mathrm {log}\left (4 \sqrt {c}\, \sqrt {a -d}\, \sqrt {b^{2}-4 c d}+4 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}\, b +8 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}\, c x +8 b c x +8 c^{2} x^{2}+8 c d \right )\right )}{a \,b^{2}-4 a c d -b^{2} d +4 c \,d^{2}} \] Input:
int(1/(c*x^2+b*x+a)^(1/2)/(c*x^2+b*x+d),x)
Output:
(sqrt(a - d)*sqrt(b**2 - 4*c*d)*(log( - sqrt(4*sqrt(c)*sqrt(a - d)*sqrt(b* *2 - 4*c*d) + 4*a*c + b**2 - 8*c*d) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x) + log(sqrt(4*sqrt(c)*sqrt(a - d)*sqrt(b**2 - 4*c*d) + 4*a*c + b* *2 - 8*c*d) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x) - log(4*sqrt(c )*sqrt(a - d)*sqrt(b**2 - 4*c*d) + 4*sqrt(c)*sqrt(a + b*x + c*x**2)*b + 8* sqrt(c)*sqrt(a + b*x + c*x**2)*c*x + 8*b*c*x + 8*c**2*x**2 + 8*c*d)))/(a*b **2 - 4*a*c*d - b**2*d + 4*c*d**2)