Integrand size = 31, antiderivative size = 82 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{\sqrt {b d-a e} \sqrt {b e-4 a f}} \] Output:
-2*e^(1/2)*arctanh((-a*e+b*d)^(1/2)*(2*f*x+e)/e^(1/2)/(-4*a*f+b*e)^(1/2)/( f*x^2+e*x+d)^(1/2))/(-a*e+b*d)^(1/2)/(-4*a*f+b*e)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.47 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.49 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=e \text {RootSum}\left [a e f^2-2 b \sqrt {d} e f \text {$\#$1}+b e^2 \text {$\#$1}^2+4 b d f \text {$\#$1}^2-2 a e f \text {$\#$1}^2-2 b \sqrt {d} e \text {$\#$1}^3+a e \text {$\#$1}^4\&,\frac {-f \log (x)+f \log \left (-\sqrt {d}+\sqrt {d+e x+f x^2}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^2-\log \left (-\sqrt {d}+\sqrt {d+e x+f x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-b \sqrt {d} e f+b e^2 \text {$\#$1}+4 b d f \text {$\#$1}-2 a e f \text {$\#$1}-3 b \sqrt {d} e \text {$\#$1}^2+2 a e \text {$\#$1}^3}\&\right ] \] Input:
Integrate[1/(Sqrt[d + e*x + f*x^2]*(a + b*x + (b*f*x^2)/e)),x]
Output:
e*RootSum[a*e*f^2 - 2*b*Sqrt[d]*e*f*#1 + b*e^2*#1^2 + 4*b*d*f*#1^2 - 2*a*e *f*#1^2 - 2*b*Sqrt[d]*e*#1^3 + a*e*#1^4 & , (-(f*Log[x]) + f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] + Log[x]*#1^2 - Log[-Sqrt[d] + Sqrt[d + e* x + f*x^2] - x*#1]*#1^2)/(-(b*Sqrt[d]*e*f) + b*e^2*#1 + 4*b*d*f*#1 - 2*a*e *f*#1 - 3*b*Sqrt[d]*e*#1^2 + 2*a*e*#1^3) & ]
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, 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.
\(\displaystyle \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+\frac {b f x^2}{e}+b x\right )} \, dx\) |
\(\Big \downarrow \) 1313 |
\(\displaystyle -2 e \int \frac {1}{e (b e-4 a f)-\frac {(b d-a e) (e+2 f x)^2}{f x^2+e x+d}}d\frac {e+2 f x}{\sqrt {f x^2+e x+d}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 \sqrt {e} \text {arctanh}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{\sqrt {b d-a e} \sqrt {b e-4 a f}}\) |
Input:
Int[1/(Sqrt[d + e*x + f*x^2]*(a + b*x + (b*f*x^2)/e)),x]
Output:
(-2*Sqrt[e]*ArcTanh[(Sqrt[b*d - a*e]*(e + 2*f*x))/(Sqrt[e]*Sqrt[b*e - 4*a* f]*Sqrt[d + e*x + f*x^2])])/(Sqrt[b*d - a*e]*Sqrt[b*e - 4*a*f])
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. \(490\) vs. \(2(68)=136\).
Time = 4.47 (sec) , antiderivative size = 491, normalized size of antiderivative = 5.99
method | result | size |
default | \(e \left (-\frac {\ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}+\frac {\sqrt {-b e \left (4 a f -b e \right )}\, \left (x -\frac {-b e +\sqrt {-b e \left (4 a f -b e \right )}}{2 f b}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x -\frac {-b e +\sqrt {-b e \left (4 a f -b e \right )}}{2 f b}\right )^{2} f +\frac {\sqrt {-b e \left (4 a f -b e \right )}\, \left (x -\frac {-b e +\sqrt {-b e \left (4 a f -b e \right )}}{2 f b}\right )}{b}-\frac {a e -b d}{b}}}{x -\frac {-b e +\sqrt {-b e \left (4 a f -b e \right )}}{2 f b}}\right )}{\sqrt {-b e \left (4 a f -b e \right )}\, \sqrt {-\frac {a e -b d}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}-\frac {\sqrt {-b e \left (4 a f -b e \right )}\, \left (x +\frac {b e +\sqrt {-b e \left (4 a f -b e \right )}}{2 f b}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x +\frac {b e +\sqrt {-b e \left (4 a f -b e \right )}}{2 f b}\right )^{2} f -\frac {\sqrt {-b e \left (4 a f -b e \right )}\, \left (x +\frac {b e +\sqrt {-b e \left (4 a f -b e \right )}}{2 f b}\right )}{b}-\frac {a e -b d}{b}}}{x +\frac {b e +\sqrt {-b e \left (4 a f -b e \right )}}{2 f b}}\right )}{\sqrt {-b e \left (4 a f -b e \right )}\, \sqrt {-\frac {a e -b d}{b}}}\right )\) | \(491\) |
Input:
int(1/(f*x^2+e*x+d)^(1/2)/(a+b*x+b*f*x^2/e),x,method=_RETURNVERBOSE)
Output:
e*(-1/(-b*e*(4*a*f-b*e))^(1/2)/(-(a*e-b*d)/b)^(1/2)*ln((-2*(a*e-b*d)/b+(-b *e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/f/b)+2*(-(a *e-b*d)/b)^(1/2)*((x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/f/b)^2*f+(-b*e*(4 *a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/f/b)-(a*e-b*d)/b )^(1/2))/(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/f/b))+1/(-b*e*(4*a*f-b*e)) ^(1/2)/(-(a*e-b*d)/b)^(1/2)*ln((-2*(a*e-b*d)/b-(-b*e*(4*a*f-b*e))^(1/2)/b* (x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/f/b)+2*(-(a*e-b*d)/b)^(1/2)*((x+1/2* (b*e+(-b*e*(4*a*f-b*e))^(1/2))/f/b)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2* (b*e+(-b*e*(4*a*f-b*e))^(1/2))/f/b)-(a*e-b*d)/b)^(1/2))/(x+1/2*(b*e+(-b*e* (4*a*f-b*e))^(1/2))/f/b)))
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (68) = 136\).
Time = 0.23 (sec) , antiderivative size = 1079, normalized size of antiderivative = 13.16 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/(f*x^2+e*x+d)^(1/2)/(a+b*x+b*f*x^2/e),x, algorithm="fricas")
Output:
[1/2*sqrt(e/(b^2*d*e - a*b*e^2 - 4*(a*b*d - a^2*e)*f))*log((8*b^2*d^2*e^4 - 8*a*b*d*e^5 + a^2*e^6 + 16*a^2*d^2*e^2*f^2 + (b^2*e^4*f^2 + 16*(b^2*d^2 - 8*a*b*d*e + 8*a^2*e^2)*f^4 + 8*(3*b^2*d*e^2 - 4*a*b*e^3)*f^3)*x^4 + 2*(b ^2*e^5*f + 16*(b^2*d^2*e - 8*a*b*d*e^2 + 8*a^2*e^3)*f^3 + 8*(3*b^2*d*e^3 - 4*a*b*e^4)*f^2)*x^3 + (b^2*e^6 - 32*(3*a*b*d^2*e - 4*a^2*d*e^2)*f^3 + 16* (3*b^2*d^2*e^2 - 13*a*b*d*e^3 + 10*a^2*e^4)*f^2 + 2*(16*b^2*d*e^4 - 19*a*b *e^5)*f)*x^2 - 8*(4*a*b*d^2*e^3 - 3*a^2*d*e^4)*f + 2*(4*b^2*d*e^5 - 3*a*b* e^6 - 16*(3*a*b*d^2*e^2 - 4*a^2*d*e^3)*f^2 + 8*(2*b^2*d^2*e^3 - 5*a*b*d*e^ 4 + 2*a^2*e^5)*f)*x - 4*(2*b^3*d^2*e^4 - 3*a*b^2*d*e^5 + a^2*b*e^6 - 2*(16 *(a*b^2*d^2 - 3*a^2*b*d*e + 2*a^3*e^2)*f^4 - 4*(b^3*d^2*e - 4*a*b^2*d*e^2 + 3*a^2*b*e^3)*f^3 - (b^3*d*e^3 - a*b^2*e^4)*f^2)*x^3 + 16*(a^2*b*d^2*e^2 - a^3*d*e^3)*f^2 - 3*(16*(a*b^2*d^2*e - 3*a^2*b*d*e^2 + 2*a^3*e^3)*f^3 - 4 *(b^3*d^2*e^2 - 4*a*b^2*d*e^3 + 3*a^2*b*e^4)*f^2 - (b^3*d*e^4 - a*b^2*e^5) *f)*x^2 - 4*(3*a*b^2*d^2*e^3 - 4*a^2*b*d*e^4 + a^3*e^5)*f + (b^3*d*e^5 - a *b^2*e^6 + 32*(a^2*b*d^2*e - a^3*d*e^2)*f^3 - 40*(a*b^2*d^2*e^2 - 2*a^2*b* d*e^3 + a^3*e^4)*f^2 + 2*(4*b^3*d^2*e^3 - 11*a*b^2*d*e^4 + 7*a^2*b*e^5)*f) *x)*sqrt(f*x^2 + e*x + d)*sqrt(e/(b^2*d*e - a*b*e^2 - 4*(a*b*d - a^2*e)*f) ))/(b^2*f^2*x^4 + 2*b^2*e*f*x^3 + 2*a*b*e^2*x + a^2*e^2 + (b^2*e^2 + 2*a*b *e*f)*x^2)), -sqrt(-e/(b^2*d*e - a*b*e^2 - 4*(a*b*d - a^2*e)*f))*arctan(-1 /2*(2*b*d*e^2 - a*e^3 - 4*a*d*e*f + (b*e^2*f + 4*(b*d - 2*a*e)*f^2)*x^2...
\[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=e \int \frac {1}{a e \sqrt {d + e x + f x^{2}} + b e x \sqrt {d + e x + f x^{2}} + b f x^{2} \sqrt {d + e x + f x^{2}}}\, dx \] Input:
integrate(1/(f*x**2+e*x+d)**(1/2)/(a+b*x+b*f*x**2/e),x)
Output:
e*Integral(1/(a*e*sqrt(d + e*x + f*x**2) + b*e*x*sqrt(d + e*x + f*x**2) + b*f*x**2*sqrt(d + e*x + f*x**2)), x)
Exception generated. \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(f*x^2+e*x+d)^(1/2)/(a+b*x+b*f*x^2/e),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(e*(4*a*f-b*e)>0)', see `assume?` for more
Leaf count of result is larger than twice the leaf count of optimal. 851 vs. \(2 (68) = 136\).
Time = 0.26 (sec) , antiderivative size = 851, normalized size of antiderivative = 10.38 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/(f*x^2+e*x+d)^(1/2)/(a+b*x+b*f*x^2/e),x, algorithm="giac")
Output:
-sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*log(abs(-(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*b*e^2*f - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d) )^2*b*d*f^2 + 8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*a*e*f^2 - (sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*e^3*sqrt(f) - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*d*e*f^(3/2) + 8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*a*e^2*f^(3/2) - 3*b*d*e^2*f + 2*a*e^3*f + 4*b*d^2*f^2 + 4*sqrt(b^2*d*e^2 - a*b*e^3 - 4*a *b*d*e*f + 4*a^2*e^2*f)*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*f^(3/2) + 4* sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*(sqrt(f)*x - sqrt(f* x^2 + e*x + d))*e*f + sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f )*e^2*sqrt(f)))/(b^2*d*e - a*b*e^2 - 4*a*b*d*f + 4*a^2*e*f) + sqrt(b^2*d*e ^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*log(abs(-(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*b*e^2*f - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*b*d*f^2 + 8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*a*e*f^2 - (sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*e^3*sqrt(f) - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*d*e*f ^(3/2) + 8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*a*e^2*f^(3/2) - 3*b*d*e^2*f + 2*a*e^3*f + 4*b*d^2*f^2 - 4*sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4* a^2*e^2*f)*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*f^(3/2) - 4*sqrt(b^2*d*e^ 2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*(sqrt(f)*x - sqrt(f*x^2 + e*x + d ))*e*f - sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*e^2*sqrt(f) ))/(b^2*d*e - a*b*e^2 - 4*a*b*d*f + 4*a^2*e*f)
Timed out. \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=\int \frac {1}{\sqrt {f\,x^2+e\,x+d}\,\left (a+b\,x+\frac {b\,f\,x^2}{e}\right )} \,d x \] Input:
int(1/((d + e*x + f*x^2)^(1/2)*(a + b*x + (b*f*x^2)/e)),x)
Output:
int(1/((d + e*x + f*x^2)^(1/2)*(a + b*x + (b*f*x^2)/e)), x)
Time = 0.20 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.85 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a+b x+\frac {b f x^2}{e}\right )} \, dx=\frac {\sqrt {e}\, \sqrt {4 a^{2} e f -4 a b d f -a b \,e^{2}+b^{2} d e}\, \left (\mathrm {log}\left (-\sqrt {4 \sqrt {f}\, \sqrt {e}\, \sqrt {4 a^{2} e f -4 a b d f -a b \,e^{2}+b^{2} d e}-8 a e f +4 b d f +b \,e^{2}}+2 \sqrt {f}\, \sqrt {b}\, \sqrt {f \,x^{2}+e x +d}+\sqrt {b}\, e +2 \sqrt {b}\, f x \right )+\mathrm {log}\left (\sqrt {4 \sqrt {f}\, \sqrt {e}\, \sqrt {4 a^{2} e f -4 a b d f -a b \,e^{2}+b^{2} d e}-8 a e f +4 b d f +b \,e^{2}}+2 \sqrt {f}\, \sqrt {b}\, \sqrt {f \,x^{2}+e x +d}+\sqrt {b}\, e +2 \sqrt {b}\, f x \right )-\mathrm {log}\left (4 \sqrt {f}\, \sqrt {e}\, \sqrt {4 a^{2} e f -4 a b d f -a b \,e^{2}+b^{2} d e}+4 \sqrt {f}\, \sqrt {f \,x^{2}+e x +d}\, b e +8 \sqrt {f}\, \sqrt {f \,x^{2}+e x +d}\, b f x +8 a e f +8 b e f x +8 b \,f^{2} x^{2}\right )\right )}{4 a^{2} e f -4 a b d f -a b \,e^{2}+b^{2} d e} \] Input:
int(1/(f*x^2+e*x+d)^(1/2)/(a+b*x+b*f*x^2/e),x)
Output:
(sqrt(e)*sqrt(4*a**2*e*f - 4*a*b*d*f - a*b*e**2 + b**2*d*e)*(log( - sqrt(4 *sqrt(f)*sqrt(e)*sqrt(4*a**2*e*f - 4*a*b*d*f - a*b*e**2 + b**2*d*e) - 8*a* e*f + 4*b*d*f + b*e**2) + 2*sqrt(f)*sqrt(b)*sqrt(d + e*x + f*x**2) + sqrt( b)*e + 2*sqrt(b)*f*x) + log(sqrt(4*sqrt(f)*sqrt(e)*sqrt(4*a**2*e*f - 4*a*b *d*f - a*b*e**2 + b**2*d*e) - 8*a*e*f + 4*b*d*f + b*e**2) + 2*sqrt(f)*sqrt (b)*sqrt(d + e*x + f*x**2) + sqrt(b)*e + 2*sqrt(b)*f*x) - log(4*sqrt(f)*sq rt(e)*sqrt(4*a**2*e*f - 4*a*b*d*f - a*b*e**2 + b**2*d*e) + 4*sqrt(f)*sqrt( d + e*x + f*x**2)*b*e + 8*sqrt(f)*sqrt(d + e*x + f*x**2)*b*f*x + 8*a*e*f + 8*b*e*f*x + 8*b*f**2*x**2)))/(4*a**2*e*f - 4*a*b*d*f - a*b*e**2 + b**2*d* e)