Integrand size = 22, antiderivative size = 168 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=-\frac {\sqrt {a+c x^2}}{2 a d x^2}+\frac {e \sqrt {a+c x^2}}{a d^2 x}+\frac {e^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \sqrt {c d^2+a e^2}}+\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d}-\frac {e^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3} \]
1/2*c*arctanh((c*x^2+a)^(1/2)/a^(1/2))/a^(3/2)/d-e^2*arctanh((c*x^2+a)^(1/ 2)/a^(1/2))/d^3/a^(1/2)+e^3*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^ 2+a)^(1/2))/d^3/(a*e^2+c*d^2)^(1/2)-1/2*(c*x^2+a)^(1/2)/a/d/x^2+e*(c*x^2+a )^(1/2)/a/d^2/x
Time = 0.48 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\frac {\sqrt {a} \left (d \left (c d^2+a e^2\right ) (-d+2 e x) \sqrt {a+c x^2}-4 a e^3 \sqrt {-c d^2-a e^2} x^2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )\right )-2 \left (c^2 d^4-a c d^2 e^2-2 a^2 e^4\right ) x^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d^3 \left (c d^2+a e^2\right ) x^2} \]
(Sqrt[a]*(d*(c*d^2 + a*e^2)*(-d + 2*e*x)*Sqrt[a + c*x^2] - 4*a*e^3*Sqrt[-( c*d^2) - a*e^2]*x^2*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-( c*d^2) - a*e^2]]) - 2*(c^2*d^4 - a*c*d^2*e^2 - 2*a^2*e^4)*x^2*ArcTanh[(Sqr t[c]*x - Sqrt[a + c*x^2])/Sqrt[a]])/(2*a^(3/2)*d^3*(c*d^2 + a*e^2)*x^2)
Time = 0.33 (sec) , antiderivative size = 168, 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.
| \(\displaystyle \int \frac {1}{x^3 \sqrt {a+c x^2} (d+e x)} \, dx\) |
| \(\Big \downarrow \) 617 |
| \(\displaystyle \int \left (-\frac {e^3}{d^3 \sqrt {a+c x^2} (d+e x)}+\frac {e^2}{d^3 x \sqrt {a+c x^2}}-\frac {e}{d^2 x^2 \sqrt {a+c x^2}}+\frac {1}{d x^3 \sqrt {a+c x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d}-\frac {e^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3}+\frac {e^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^3 \sqrt {a e^2+c d^2}}+\frac {e \sqrt {a+c x^2}}{a d^2 x}-\frac {\sqrt {a+c x^2}}{2 a d x^2}\) |
-1/2*Sqrt[a + c*x^2]/(a*d*x^2) + (e*Sqrt[a + c*x^2])/(a*d^2*x) + (e^3*ArcT anh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^3*Sqrt[c*d^2 + a*e^2]) + (c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(3/2)*d) - (e^2*ArcT anh[Sqrt[a + c*x^2]/Sqrt[a]])/(Sqrt[a]*d^3)
3.4.33.3.1 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]
Time = 0.44 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+a}\, \left (-2 e x +d \right )}{2 a \,d^{2} x^{2}}+\frac {\frac {2 e^{2} a \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {\left (-2 e^{2} a +c \,d^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d \sqrt {a}}}{2 a \,d^{2}}\) | \(209\) |
| default | \(\frac {-\frac {\sqrt {c \,x^{2}+a}}{2 a \,x^{2}}+\frac {c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}}{d}-\frac {e^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d^{3} \sqrt {a}}+\frac {e \sqrt {c \,x^{2}+a}}{a \,d^{2} x}+\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{3} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(235\) |
-1/2*(c*x^2+a)^(1/2)*(-2*e*x+d)/a/d^2/x^2+1/2/a/d^2*(2*e^2*a/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))+1 /d*(-2*a*e^2+c*d^2)/a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x))
Time = 0.57 (sec) , antiderivative size = 956, normalized size of antiderivative = 5.69 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\left [\frac {2 \, \sqrt {c d^{2} + a e^{2}} a^{2} e^{3} x^{2} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - {\left (c^{2} d^{4} - a c d^{2} e^{2} - 2 \, a^{2} e^{4}\right )} \sqrt {a} x^{2} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (a c d^{4} + a^{2} d^{2} e^{2} - 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (a^{2} c d^{5} + a^{3} d^{3} e^{2}\right )} x^{2}}, \frac {4 \, \sqrt {-c d^{2} - a e^{2}} a^{2} e^{3} x^{2} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (c^{2} d^{4} - a c d^{2} e^{2} - 2 \, a^{2} e^{4}\right )} \sqrt {a} x^{2} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (a c d^{4} + a^{2} d^{2} e^{2} - 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (a^{2} c d^{5} + a^{3} d^{3} e^{2}\right )} x^{2}}, \frac {\sqrt {c d^{2} + a e^{2}} a^{2} e^{3} x^{2} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - {\left (c^{2} d^{4} - a c d^{2} e^{2} - 2 \, a^{2} e^{4}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (a c d^{4} + a^{2} d^{2} e^{2} - 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c d^{5} + a^{3} d^{3} e^{2}\right )} x^{2}}, \frac {2 \, \sqrt {-c d^{2} - a e^{2}} a^{2} e^{3} x^{2} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (c^{2} d^{4} - a c d^{2} e^{2} - 2 \, a^{2} e^{4}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (a c d^{4} + a^{2} d^{2} e^{2} - 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c d^{5} + a^{3} d^{3} e^{2}\right )} x^{2}}\right ] \]
[1/4*(2*sqrt(c*d^2 + a*e^2)*a^2*e^3*x^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)*sqr t(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) - (c^2*d^4 - a*c*d^2*e^2 - 2*a^2* e^4)*sqrt(a)*x^2*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*( a*c*d^4 + a^2*d^2*e^2 - 2*(a*c*d^3*e + a^2*d*e^3)*x)*sqrt(c*x^2 + a))/((a^ 2*c*d^5 + a^3*d^3*e^2)*x^2), 1/4*(4*sqrt(-c*d^2 - a*e^2)*a^2*e^3*x^2*arcta n(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^2*d^4 - a*c*d^2*e^2 - 2*a^2*e^4)*sqrt(a)*x^ 2*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a*c*d^4 + a^2*d ^2*e^2 - 2*(a*c*d^3*e + a^2*d*e^3)*x)*sqrt(c*x^2 + a))/((a^2*c*d^5 + a^3*d ^3*e^2)*x^2), 1/2*(sqrt(c*d^2 + a*e^2)*a^2*e^3*x^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^2*d^4 - a*c*d^2*e ^2 - 2*a^2*e^4)*sqrt(-a)*x^2*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (a*c*d^4 + a^2*d^2*e^2 - 2*(a*c*d^3*e + a^2*d*e^3)*x)*sqrt(c*x^2 + a))/((a^2*c*d^5 + a^3*d^3*e^2)*x^2), 1/2*(2*sqrt(-c*d^2 - a*e^2)*a^2*e^3*x^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^2*d^4 - a*c*d^2*e^2 - 2*a^2*e^4)*sqrt(-a)*x^2*arctan (sqrt(-a)/sqrt(c*x^2 + a)) - (a*c*d^4 + a^2*d^2*e^2 - 2*(a*c*d^3*e + a^2*d *e^3)*x)*sqrt(c*x^2 + a))/((a^2*c*d^5 + a^3*d^3*e^2)*x^2)]
\[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {1}{x^{3} \sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
\[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} x^{3}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=-c^{\frac {3}{2}} {\left (\frac {2 \, e^{3} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{\sqrt {-c d^{2} - a e^{2}} c^{\frac {3}{2}} d^{3}} + \frac {{\left (c d^{2} - 2 \, a e^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a c^{\frac {3}{2}} d^{3}} - \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} \sqrt {c} d - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a e + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a \sqrt {c} d + 2 \, a^{2} e}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{2} a c d^{2}}\right )} \]
-c^(3/2)*(2*e^3*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt (-c*d^2 - a*e^2))/(sqrt(-c*d^2 - a*e^2)*c^(3/2)*d^3) + (c*d^2 - 2*a*e^2)*a rctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a*c^(3/2)*d^3) - ((sqrt(c)*x - sqrt(c*x^2 + a))^3*sqrt(c)*d - 2*(sqrt(c)*x - sqrt(c*x^2 + a ))^2*a*e + (sqrt(c)*x - sqrt(c*x^2 + a))*a*sqrt(c)*d + 2*a^2*e)/(((sqrt(c) *x - sqrt(c*x^2 + a))^2 - a)^2*a*c*d^2))
Timed out. \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {1}{x^3\,\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]