Integrand size = 22, antiderivative size = 194 \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )^{3/2}}+\frac {e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^2} \]
-e^4*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/d^2/(a*e^2+ c*d^2)^(3/2)+e*arctanh((c*x^2+a)^(1/2)/a^(1/2))/a^(3/2)/d^2-e/a/d^2/(c*x^2 +a)^(1/2)-1/a/d/x/(c*x^2+a)^(1/2)-2*c*x/a^2/d/(c*x^2+a)^(1/2)+e^2*(c*d*x+a *e)/a/d^2/(a*e^2+c*d^2)/(c*x^2+a)^(1/2)
Time = 0.69 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {-\frac {d \left (a^2 e^2+2 c^2 d^2 x^2+a c \left (d^2+d e x+e^2 x^2\right )\right )}{a^2 \left (c d^2+a e^2\right ) x \sqrt {a+c x^2}}+\frac {2 e^4 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}-\frac {2 e \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{d^2} \]
(-((d*(a^2*e^2 + 2*c^2*d^2*x^2 + a*c*(d^2 + d*e*x + e^2*x^2)))/(a^2*(c*d^2 + a*e^2)*x*Sqrt[a + c*x^2])) + (2*e^4*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[ a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(-(c*d^2) - a*e^2)^(3/2) - (2*e*ArcTa nh[(Sqrt[c]*x - Sqrt[a + c*x^2])/Sqrt[a]])/a^(3/2))/d^2
Time = 0.37 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+c x^2\right )^{3/2} (d+e x)} \, dx\) |
| \(\Big \downarrow \) 617 |
| \(\displaystyle \int \left (\frac {e^2}{d^2 \left (a+c x^2\right )^{3/2} (d+e x)}-\frac {e}{d^2 x \left (a+c x^2\right )^{3/2}}+\frac {1}{d x^2 \left (a+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^2}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}-\frac {e^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}+\frac {e^2 (a e+c d x)}{a d^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}}\) |
-(e/(a*d^2*Sqrt[a + c*x^2])) - 1/(a*d*x*Sqrt[a + c*x^2]) - (2*c*x)/(a^2*d* Sqrt[a + c*x^2]) + (e^2*(a*e + c*d*x))/(a*d^2*(c*d^2 + a*e^2)*Sqrt[a + c*x ^2]) - (e^4*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/ (d^2*(c*d^2 + a*e^2)^(3/2)) + (e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(a^(3/2 )*d^2)
3.4.40.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(360\) vs. \(2(174)=348\).
Time = 0.39 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.86
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+a}}{a^{2} d x}+\frac {e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}} d^{2}}-\frac {c \sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} c +2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{2 a^{2} \left (e \sqrt {-a c}+c d \right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}+\frac {c \sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} c -2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{2 a^{2} \left (e \sqrt {-a c}-c d \right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}+\frac {c \,e^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{2} \left (e \sqrt {-a c}+c d \right ) \left (e \sqrt {-a c}-c d \right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(361\) |
| default | \(\frac {-\frac {1}{a x \sqrt {c \,x^{2}+a}}-\frac {2 c x}{a^{2} \sqrt {c \,x^{2}+a}}}{d}-\frac {e \left (\frac {1}{a \sqrt {c \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{d^{2}}+\frac {e \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 e c d \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{d^{2}}\) | \(403\) |
-1/a^2/d*(c*x^2+a)^(1/2)/x+1/a^(3/2)/d^2*e*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/ 2))/x)-1/2/a^2*c/(e*(-a*c)^(1/2)+c*d)/(x-(-a*c)^(1/2)/c)*((x-(-a*c)^(1/2)/ c)^2*c+2*(-a*c)^(1/2)*(x-(-a*c)^(1/2)/c))^(1/2)+1/2/a^2*c/(e*(-a*c)^(1/2)- c*d)/(x+(-a*c)^(1/2)/c)*((x+(-a*c)^(1/2)/c)^2*c-2*(-a*c)^(1/2)*(x+(-a*c)^( 1/2)/c))^(1/2)+1/d^2*c*e^3/(e*(-a*c)^(1/2)+c*d)/(e*(-a*c)^(1/2)-c*d)/((a*e ^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2/e*c*d*(x+d/e)+2*((a*e^2+c*d ^2)/e^2)^(1/2)*((x+d/e)^2*c-2/e*c*d*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d /e))
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (175) = 350\).
Time = 0.50 (sec) , antiderivative size = 1556, normalized size of antiderivative = 8.02 \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/2*((a^2*c*e^4*x^3 + a^3*e^4*x)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a *c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c* d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + ((c^3*d^4*e + 2*a *c^2*d^2*e^3 + a^2*c*e^5)*x^3 + (a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)* x)*sqrt(a)*log(-(c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a*c^2* d^5 + 2*a^2*c*d^3*e^2 + a^3*d*e^4 + (2*c^3*d^5 + 3*a*c^2*d^3*e^2 + a^2*c*d *e^4)*x^2 + (a*c^2*d^4*e + a^2*c*d^2*e^3)*x)*sqrt(c*x^2 + a))/((a^2*c^3*d^ 6 + 2*a^3*c^2*d^4*e^2 + a^4*c*d^2*e^4)*x^3 + (a^3*c^2*d^6 + 2*a^4*c*d^4*e^ 2 + a^5*d^2*e^4)*x), -1/2*(2*(a^2*c*e^4*x^3 + a^3*e^4*x)*sqrt(-c*d^2 - a*e ^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a ^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) - ((c^3*d^4*e + 2*a*c^2*d^2*e^3 + a^2*c *e^5)*x^3 + (a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)*x)*sqrt(a)*log(-(c*x ^2 + 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(a*c^2*d^5 + 2*a^2*c*d^3*e^ 2 + a^3*d*e^4 + (2*c^3*d^5 + 3*a*c^2*d^3*e^2 + a^2*c*d*e^4)*x^2 + (a*c^2*d ^4*e + a^2*c*d^2*e^3)*x)*sqrt(c*x^2 + a))/((a^2*c^3*d^6 + 2*a^3*c^2*d^4*e^ 2 + a^4*c*d^2*e^4)*x^3 + (a^3*c^2*d^6 + 2*a^4*c*d^4*e^2 + a^5*d^2*e^4)*x), -1/2*(2*((c^3*d^4*e + 2*a*c^2*d^2*e^3 + a^2*c*e^5)*x^3 + (a*c^2*d^4*e + 2 *a^2*c*d^2*e^3 + a^3*e^5)*x)*sqrt(-a)*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - ( a^2*c*e^4*x^3 + a^3*e^4*x)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x ...
\[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
\[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{2}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {2 \, e^{4} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c d^{4} + a d^{2} e^{2}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {\frac {{\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x}{a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}} + \frac {a^{2} c^{2} d^{2} e + a^{3} c e^{3}}{a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}}}{\sqrt {c x^{2} + a}} - \frac {2 \, e \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a d^{2}} + \frac {2 \, \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )} a d} \]
-2*e^4*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c*d^4 + a*d^2*e^2)*sqrt(-c*d^2 - a*e^2)) - ((a*c^3*d^3 + a^2*c^2 *d*e^2)*x/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4) + (a^2*c^2*d^2*e + a^3 *c*e^3)/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4))/sqrt(c*x^2 + a) - 2*e*a rctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a*d^2) + 2*sqrt(c )/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)*a*d)
Timed out. \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]