Integrand size = 28, antiderivative size = 454 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d x \sqrt {a+b x+c x^2}}-\frac {2 f \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) d \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) d x}+\frac {3 b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{5/2} d}+\frac {f^2 \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\frac {f^2 \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}} \]
3/2*b*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(5/2)/d+1/2*f^2 *arctanh(1/2*(b*d^(1/2)-2*a*f^(1/2)+x*(2*c*d^(1/2)-b*f^(1/2)))/(c*x^2+b*x+ a)^(1/2)/(c*d+a*f-b*d^(1/2)*f^(1/2))^(1/2))/d^(3/2)/(c*d+a*f-b*d^(1/2)*f^( 1/2))^(3/2)+1/2*f^2*arctanh(1/2*(b*d^(1/2)+2*a*f^(1/2)+x*(2*c*d^(1/2)+b*f^ (1/2)))/(c*x^2+b*x+a)^(1/2)/(c*d+a*f+b*d^(1/2)*f^(1/2))^(1/2))/d^(3/2)/(c* d+a*f+b*d^(1/2)*f^(1/2))^(3/2)+2*(b*c*x-2*a*c+b^2)/a/(-4*a*c+b^2)/d/x/(c*x ^2+b*x+a)^(1/2)-2*f*(b*(b^2*f-c*(3*a*f+c*d))-c*(2*a*c*f-b^2*f+2*c^2*d)*x)/ (-4*a*c+b^2)/d/(b^2*d*f-(a*f+c*d)^2)/(c*x^2+b*x+a)^(1/2)-(-8*a*c+3*b^2)*(c *x^2+b*x+a)^(1/2)/a^2/(-4*a*c+b^2)/d/x
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.10 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\frac {-2 \sqrt {a} \left (4 a^4 c f^2+3 b^2 d \left (-c^2 d+b^2 f\right ) x (b+c x)+a^3 f \left (-b^2 f+4 b c f x+4 c^2 \left (2 d+f x^2\right )\right )+a^2 \left (18 b c^2 d f x-b^3 f^2 x-b^2 c f \left (6 d+f x^2\right )+4 c^3 d \left (d+3 f x^2\right )\right )+a d \left (b^4 f+10 b c^3 d x-16 b^3 c f x+8 c^4 d x^2-b^2 c^2 \left (d+14 f x^2\right )\right )\right )-6 b \left (b^2-4 a c\right ) \left (-c^2 d^2-2 a c d f+f \left (b^2 d-a^2 f\right )\right ) x \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+a^{5/2} \left (b^2-4 a c\right ) f^2 x \sqrt {a+x (b+c x)} \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+2 a b f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{3/2} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a \sqrt {c} f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-b f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 a^{5/2} \left (-b^2+4 a c\right ) d \left (c^2 d^2+2 a c d f+f \left (-b^2 d+a^2 f\right )\right ) x \sqrt {a+x (b+c x)}} \]
(-2*Sqrt[a]*(4*a^4*c*f^2 + 3*b^2*d*(-(c^2*d) + b^2*f)*x*(b + c*x) + a^3*f* (-(b^2*f) + 4*b*c*f*x + 4*c^2*(2*d + f*x^2)) + a^2*(18*b*c^2*d*f*x - b^3*f ^2*x - b^2*c*f*(6*d + f*x^2) + 4*c^3*d*(d + 3*f*x^2)) + a*d*(b^4*f + 10*b* c^3*d*x - 16*b^3*c*f*x + 8*c^4*d*x^2 - b^2*c^2*(d + 14*f*x^2))) - 6*b*(b^2 - 4*a*c)*(-(c^2*d^2) - 2*a*c*d*f + f*(b^2*d - a^2*f))*x*Sqrt[a + x*(b + c *x)]*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] + a^(5/2)*(b^2 - 4*a*c)*f^2*x*Sqrt[a + x*(b + c*x)]*RootSum[b^2*d - a^2*f - 4*b*Sqrt[c]*d* #1 + 4*c*d*#1^2 + 2*a*f*#1^2 - f*#1^4 & , (b*c*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + 2*a*b*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*c^(3/2)*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*a *Sqrt[c]*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - b*f*Log[-(S qrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2)/(b*Sqrt[c]*d - 2*c*d*#1 - a* f*#1 + f*#1^3) & ])/(2*a^(5/2)*(-b^2 + 4*a*c)*d*(c^2*d^2 + 2*a*c*d*f + f*( -(b^2*d) + a^2*f))*x*Sqrt[a + x*(b + c*x)])
Time = 1.25 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7276, 2009}
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.
| \(\displaystyle \int \frac {1}{x^2 \left (d-f x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {f}{d \left (d-f x^2\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{d x^2 \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{5/2} d}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a+b x+c x^2}}{a^2 d x \left (b^2-4 a c\right )}+\frac {f^2 \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^{3/2} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2}}+\frac {f^2 \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^{3/2} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2}}-\frac {2 f \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a d x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*x*Sqrt[a + b*x + c*x^2]) - (2 *f*(b*(b^2*f - c*(c*d + 3*a*f)) - c*(2*c^2*d - b^2*f + 2*a*c*f)*x))/((b^2 - 4*a*c)*d*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) - ((3*b^2 - 8* a*c)*Sqrt[a + b*x + c*x^2])/(a^2*(b^2 - 4*a*c)*d*x) + (3*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(5/2)*d) + (f^2*ArcTanh[(b*S qrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d ]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2)*(c*d - b*Sqrt[d]*Sqrt [f] + a*f)^(3/2)) + (f^2*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2 ])])/(2*d^(3/2)*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))
3.2.8.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1064\) vs. \(2(382)=764\).
Time = 0.95 (sec) , antiderivative size = 1065, normalized size of antiderivative = 2.35
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1065\) |
| risch | \(\text {Expression too large to display}\) | \(3015\) |
1/d*(-1/a/x/(c*x^2+b*x+a)^(1/2)-3/2*b/a*(1/a/(c*x^2+b*x+a)^(1/2)-b/a*(2*c* x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^ 2+b*x+a)^(1/2))/x))-4*c/a*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))-1/2*f /d/(d*f)^(1/2)*(1/(b*(d*f)^(1/2)+f*a+c*d)*f/((x-(d*f)^(1/2)/f)^2*c+(2*c*(d *f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)-(2*c*( d*f)^(1/2)+b*f)/(b*(d*f)^(1/2)+f*a+c*d)*(2*c*(x-(d*f)^(1/2)/f)+(2*c*(d*f)^ (1/2)+b*f)/f)/(4*c*(b*(d*f)^(1/2)+f*a+c*d)/f-(2*c*(d*f)^(1/2)+b*f)^2/f^2)/ ((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^ (1/2)+f*a+c*d)/f)^(1/2)-1/(b*(d*f)^(1/2)+f*a+c*d)*f/((b*(d*f)^(1/2)+f*a+c* d)/f)^(1/2)*ln((2*(b*(d*f)^(1/2)+f*a+c*d)/f+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d* f)^(1/2)/f)+2*((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*((x-(d*f)^(1/2)/f)^2*c+(2* c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2))/( x-(d*f)^(1/2)/f)))+1/2*f/d/(d*f)^(1/2)*(f/(-b*(d*f)^(1/2)+f*a+c*d)/((x+(d* f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f) ^(1/2)+f*a+c*d))^(1/2)-(-2*c*(d*f)^(1/2)+b*f)/(-b*(d*f)^(1/2)+f*a+c*d)*(2* c*(x+(d*f)^(1/2)/f)+1/f*(-2*c*(d*f)^(1/2)+b*f))/(4*c/f*(-b*(d*f)^(1/2)+f*a +c*d)-1/f^2*(-2*c*(d*f)^(1/2)+b*f)^2)/((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d* f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)-f/(-b* (d*f)^(1/2)+f*a+c*d)/(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*ln((2/f*(-b*(d*f )^(1/2)+f*a+c*d)+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+2*(1/f*(-...
Timed out. \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=- \int \frac {1}{- a d x^{2} \sqrt {a + b x + c x^{2}} + a f x^{4} \sqrt {a + b x + c x^{2}} - b d x^{3} \sqrt {a + b x + c x^{2}} + b f x^{5} \sqrt {a + b x + c x^{2}} - c d x^{4} \sqrt {a + b x + c x^{2}} + c f x^{6} \sqrt {a + b x + c x^{2}}}\, dx \]
-Integral(1/(-a*d*x**2*sqrt(a + b*x + c*x**2) + a*f*x**4*sqrt(a + b*x + c* x**2) - b*d*x**3*sqrt(a + b*x + c*x**2) + b*f*x**5*sqrt(a + b*x + c*x**2) - c*d*x**4*sqrt(a + b*x + c*x**2) + c*f*x**6*sqrt(a + b*x + c*x**2)), x)
\[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\int { -\frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (f x^{2} - d\right )} x^{2}} \,d x } \]
Exception generated. \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueDone
Timed out. \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\int \frac {1}{x^2\,\left (d-f\,x^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]