Integrand size = 28, antiderivative size = 394 \[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}-\frac {2 f \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) d \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}-\frac {\text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2} d}-\frac {f^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\frac {f^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}} \]
-arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(3/2)/d-1/2*f^(3/2)* 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/(c*d+a*f-b*d^(1/2)*f^(1/2))^( 3/2)+1/2*f^(3/2)*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/(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/(c*x^2+b*x+a)^( 1/2)-2*f*(a*(2*a*c*f-b^2*f+2*c^2*d)+b*c*(-a*f+c*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)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.23 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\frac {2 \left (b^4 f+2 a c^2 (c d+a f)-b^2 c (c d+4 a f)+b^3 c f x-b c^2 (c d+3 a f) x\right )}{a \left (-b^2+4 a c\right ) \left (c^2 d^2+2 a c d f+f \left (-b^2 d+a^2 f\right )\right ) \sqrt {a+x (b+c x)}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {f^2 \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b^2 d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 b \sqrt {c} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 c^2 d^3-2 b^2 d^2 f+4 a c d^2 f+2 a^2 d f^2} \]
(2*(b^4*f + 2*a*c^2*(c*d + a*f) - b^2*c*(c*d + 4*a*f) + b^3*c*f*x - b*c^2* (c*d + 3*a*f)*x))/(a*(-b^2 + 4*a*c)*(c^2*d^2 + 2*a*c*d*f + f*(-(b^2*d) + a ^2*f))*Sqrt[a + x*(b + c*x)]) + (2*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c* x)])/Sqrt[a]])/(a^(3/2)*d) + (f^2*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^2*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + a*c*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + a^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*b*Sqrt[c]*d*Lo g[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - c*d*Log[-(Sqrt[c]*x) + S qrt[a + b*x + c*x^2] - #1]*#1^2 - a*f*Log[-(Sqrt[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*c^2*d^ 3 - 2*b^2*d^2*f + 4*a*c*d^2*f + 2*a^2*d*f^2)
Time = 1.19 (sec) , antiderivative size = 394, 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.
| \(\displaystyle \int \frac {1}{x \left (d-f x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}-\frac {f x}{d \left (f x^2-d\right ) \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2} d}-\frac {f^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2}}+\frac {f^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2}}-\frac {2 f \left (a \left (2 a c f+b^2 (-f)+2 c^2 d\right )+b c x (c d-a f)\right )}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) - (2*f *(a*(2*c^2*d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*d*(b^ 2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) - ArcTanh[(2*a + b*x)/(2*Sqr t[a]*Sqrt[a + b*x + c*x^2])]/(a^(3/2)*d) - (f^(3/2)*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*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2) ) + (f^(3/2)*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*(c *d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))
3.2.7.3.1 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. \(989\) vs. \(2(326)=652\).
Time = 0.75 (sec) , antiderivative size = 990, normalized size of antiderivative = 2.51
| method | result | size |
| default | \(\frac {\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}}{d}-\frac {\frac {f}{\left (b \sqrt {d f}+f a +c d \right ) \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}-\frac {\left (2 c \sqrt {d f}+b f \right ) \left (2 c \left (x -\frac {\sqrt {d f}}{f}\right )+\frac {2 c \sqrt {d f}+b f}{f}\right )}{\left (b \sqrt {d f}+f a +c d \right ) \left (\frac {4 c \left (b \sqrt {d f}+f a +c d \right )}{f}-\frac {\left (2 c \sqrt {d f}+b f \right )^{2}}{f^{2}}\right ) \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}-\frac {f \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\left (b \sqrt {d f}+f a +c d \right ) \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{2 d}-\frac {\frac {f}{\left (-b \sqrt {d f}+f a +c d \right ) \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}-\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (2 c \left (x +\frac {\sqrt {d f}}{f}\right )+\frac {-2 c \sqrt {d f}+b f}{f}\right )}{\left (-b \sqrt {d f}+f a +c d \right ) \left (\frac {4 c \left (-b \sqrt {d f}+f a +c d \right )}{f}-\frac {\left (-2 c \sqrt {d f}+b f \right )^{2}}{f^{2}}\right ) \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}-\frac {f \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\left (-b \sqrt {d f}+f a +c d \right ) \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{2 d}\) | \(990\) |
1/d*(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))-1/2/d*(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/d* (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*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*((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)^(...
Timed out. \[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=- \int \frac {1}{- a d x \sqrt {a + b x + c x^{2}} + a f x^{3} \sqrt {a + b x + c x^{2}} - b d x^{2} \sqrt {a + b x + c x^{2}} + b f x^{4} \sqrt {a + b x + c x^{2}} - c d x^{3} \sqrt {a + b x + c x^{2}} + c f x^{5} \sqrt {a + b x + c x^{2}}}\, dx \]
-Integral(1/(-a*d*x*sqrt(a + b*x + c*x**2) + a*f*x**3*sqrt(a + b*x + c*x** 2) - b*d*x**2*sqrt(a + b*x + c*x**2) + b*f*x**4*sqrt(a + b*x + c*x**2) - c *d*x**3*sqrt(a + b*x + c*x**2) + c*f*x**5*sqrt(a + b*x + c*x**2)), x)
\[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\int { -\frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (f x^{2} - d\right )} x} \,d x } \]
\[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\int { -\frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (f x^{2} - d\right )} x} \,d x } \]
Timed out. \[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\int \frac {1}{x\,\left (d-f\,x^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]