Integrand size = 46, antiderivative size = 155 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2 (2 c d-b e)^{3/2}} \]
-2*(-d*g+e*f)*arctanh((d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/(-b*e+2*c*d)^ (1/2)/(e*x+d)^(1/2))/e^2/(-b*e+2*c*d)^(3/2)+2*(-b*e*g+c*d*g+c*e*f)*(e*x+d) ^(1/2)/c/e^2/(-b*e+2*c*d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)
Time = 0.19 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (\sqrt {-2 c d+b e} (c e f+c d g-b e g)+c (e f-d g) \sqrt {-b e+c (d-e x)} \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )\right )}{c e^2 (-2 c d+b e)^{3/2} \sqrt {(d+e x) (-b e+c (d-e x))}} \]
(-2*Sqrt[d + e*x]*(Sqrt[-2*c*d + b*e]*(c*e*f + c*d*g - b*e*g) + c*(e*f - d *g)*Sqrt[-(b*e) + c*(d - e*x)]*ArcTan[Sqrt[-(b*e) + c*(d - e*x)]/Sqrt[-2*c *d + b*e]]))/(c*e^2*(-2*c*d + b*e)^(3/2)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e *x))])
Time = 0.34 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1218, 1136, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x} (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {(e f-d g) \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{e (2 c d-b e)}+\frac {2 \sqrt {d+e x} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {2 (e f-d g) \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}}{2 c d-b e}+\frac {2 \sqrt {d+e x} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {d+e x} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2 (2 c d-b e)^{3/2}}\) |
(2*(c*e*f + c*d*g - b*e*g)*Sqrt[d + e*x])/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(e*f - d*g)*ArcTanh[Sqrt[d*(c*d - b*e ) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d + e*x])])/(e^2*(2*c*d - b*e)^(3/2))
3.23.72.3.1 Defintions of rubi rules used
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/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x _Symbol] :> Simp[2*e Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))) I nt[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d , e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 0.37 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.28
| method | result | size |
| default | \(-\frac {2 \left (\arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d g \sqrt {-x c e -b e +c d}-\arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c e f \sqrt {-x c e -b e +c d}+\sqrt {b e -2 c d}\, b e g -\sqrt {b e -2 c d}\, c d g -\sqrt {b e -2 c d}\, c e f \right ) \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}}{\left (b e -2 c d \right )^{\frac {3}{2}} c \,e^{2} \left (x c e +b e -c d \right ) \sqrt {e x +d}}\) | \(199\) |
-2*(arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c*d*g*(-c*e*x-b*e+c*d )^(1/2)-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c*e*f*(-c*e*x-b*e +c*d)^(1/2)+(b*e-2*c*d)^(1/2)*b*e*g-(b*e-2*c*d)^(1/2)*c*d*g-(b*e-2*c*d)^(1 /2)*c*e*f)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/(b*e-2*c*d)^(3/2)/c/e^2/(c*e*x +b*e-c*d)/(e*x+d)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (143) = 286\).
Time = 0.31 (sec) , antiderivative size = 776, normalized size of antiderivative = 5.01 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (c^{2} e^{3} f - c^{2} d e^{2} g\right )} x^{2} - {\left (c^{2} d^{2} e - b c d e^{2}\right )} f + {\left (c^{2} d^{3} - b c d^{2} e\right )} g + {\left (b c e^{3} f - b c d e^{2} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} f + {\left (2 \, c^{2} d^{2} - 3 \, b c d e + b^{2} e^{2}\right )} g\right )} \sqrt {e x + d}}{4 \, c^{4} d^{4} e^{2} - 8 \, b c^{3} d^{3} e^{3} + 5 \, b^{2} c^{2} d^{2} e^{4} - b^{3} c d e^{5} - {\left (4 \, c^{4} d^{2} e^{4} - 4 \, b c^{3} d e^{5} + b^{2} c^{2} e^{6}\right )} x^{2} - {\left (4 \, b c^{3} d^{2} e^{4} - 4 \, b^{2} c^{2} d e^{5} + b^{3} c e^{6}\right )} x}, \frac {2 \, {\left ({\left ({\left (c^{2} e^{3} f - c^{2} d e^{2} g\right )} x^{2} - {\left (c^{2} d^{2} e - b c d e^{2}\right )} f + {\left (c^{2} d^{3} - b c d^{2} e\right )} g + {\left (b c e^{3} f - b c d e^{2} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) + \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} f + {\left (2 \, c^{2} d^{2} - 3 \, b c d e + b^{2} e^{2}\right )} g\right )} \sqrt {e x + d}\right )}}{4 \, c^{4} d^{4} e^{2} - 8 \, b c^{3} d^{3} e^{3} + 5 \, b^{2} c^{2} d^{2} e^{4} - b^{3} c d e^{5} - {\left (4 \, c^{4} d^{2} e^{4} - 4 \, b c^{3} d e^{5} + b^{2} c^{2} e^{6}\right )} x^{2} - {\left (4 \, b c^{3} d^{2} e^{4} - 4 \, b^{2} c^{2} d e^{5} + b^{3} c e^{6}\right )} x}\right ] \]
[-(((c^2*e^3*f - c^2*d*e^2*g)*x^2 - (c^2*d^2*e - b*c*d*e^2)*f + (c^2*d^3 - b*c*d^2*e)*g + (b*c*e^3*f - b*c*d*e^2*g)*x)*sqrt(2*c*d - b*e)*log(-(c*e^2 *x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x + 2*sqrt(-c*e^2*x^2 - b*e^2 *x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((2*c^2*d*e - b*c*e^2 )*f + (2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*g)*sqrt(e*x + d))/(4*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3 + 5*b^2*c^2*d^2*e^4 - b^3*c*d*e^5 - (4*c^4*d^2*e^4 - 4*b* c^3*d*e^5 + b^2*c^2*e^6)*x^2 - (4*b*c^3*d^2*e^4 - 4*b^2*c^2*d*e^5 + b^3*c* e^6)*x), 2*(((c^2*e^3*f - c^2*d*e^2*g)*x^2 - (c^2*d^2*e - b*c*d*e^2)*f + ( c^2*d^3 - b*c*d^2*e)*g + (b*c*e^3*f - b*c*d*e^2*g)*x)*sqrt(-2*c*d + b*e)*a rctan(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*sqrt(e *x + d)/(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)) + sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((2*c^2*d*e - b*c*e^2)*f + (2*c^2*d^2 - 3*b*c*d*e + b^2* e^2)*g)*sqrt(e*x + d))/(4*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3 + 5*b^2*c^2*d^2*e^ 4 - b^3*c*d*e^5 - (4*c^4*d^2*e^4 - 4*b*c^3*d*e^5 + b^2*c^2*e^6)*x^2 - (4*b *c^3*d^2*e^4 - 4*b^2*c^2*d*e^5 + b^3*c*e^6)*x)]
\[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d + e x} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {e x + d} {\left (g x + f\right )}}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (143) = 286\).
Time = 0.30 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (e f - d g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (2 \, c d e^{2} - b e^{3}\right )} \sqrt {-2 \, c d + b e}} - \frac {2 \, {\left (\sqrt {2 \, c d - b e} c e f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - \sqrt {2 \, c d - b e} c d g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) + \sqrt {-2 \, c d + b e} c e f + \sqrt {-2 \, c d + b e} c d g - \sqrt {-2 \, c d + b e} b e g\right )}}{2 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c^{2} d e^{2} - \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b c e^{3}} + \frac {2 \, {\left (c e f + c d g - b e g\right )}}{{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}} \]
2*(e*f - d*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/ ((2*c*d*e^2 - b*e^3)*sqrt(-2*c*d + b*e)) - 2*(sqrt(2*c*d - b*e)*c*e*f*arct an(sqrt(2*c*d - b*e)/sqrt(-2*c*d + b*e)) - sqrt(2*c*d - b*e)*c*d*g*arctan( sqrt(2*c*d - b*e)/sqrt(-2*c*d + b*e)) + sqrt(-2*c*d + b*e)*c*e*f + sqrt(-2 *c*d + b*e)*c*d*g - sqrt(-2*c*d + b*e)*b*e*g)/(2*sqrt(2*c*d - b*e)*sqrt(-2 *c*d + b*e)*c^2*d*e^2 - sqrt(2*c*d - b*e)*sqrt(-2*c*d + b*e)*b*c*e^3) + 2* (c*e*f + c*d*g - b*e*g)/((2*c^2*d*e^2 - b*c*e^3)*sqrt(-(e*x + d)*c + 2*c*d - b*e))
Timed out. \[ \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\sqrt {d+e\,x}}{{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}} \,d x \]