Integrand size = 46, antiderivative size = 223 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=-\frac {(c e f-5 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) \sqrt {d+e x}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {(c e f-5 c d g+2 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2 \sqrt {2 c d-b e}} \]
-(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b*e+2*c*d)/(e*x+d )^(5/2)+(2*b*e*g-5*c*d*g+c*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)^(1/2)-(2*b*e*g-5*c *d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)/(e*x+d )^(1/2)
Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.61 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {-e f+3 d g+2 e g x}{d+e x}+\frac {(-c e f+5 c d g-2 b e g) \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{\sqrt {-2 c d+b e} \sqrt {-b e+c (d-e x)}}\right )}{e^2 \sqrt {d+e x}} \]
(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((-(e*f) + 3*d*g + 2*e*g*x)/(d + e *x) + ((-(c*e*f) + 5*c*d*g - 2*b*e*g)*ArcTan[Sqrt[-(b*e) + c*(d - e*x)]/Sq rt[-2*c*d + b*e]])/(Sqrt[-2*c*d + b*e]*Sqrt[-(b*e) + c*(d - e*x)])))/(e^2* Sqrt[d + e*x])
Time = 0.45 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1220, 1131, 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.
| \(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {(2 b e g-5 c d g+c e f) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{3/2}}dx}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^{5/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(2 b e g-5 c d g+c e f) \left ((2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^{5/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle -\frac {(2 b e g-5 c d g+c e f) \left (2 e (2 c d-b e) \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^{5/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\left (\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2 c d-b e} \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e}\right ) (2 b e g-5 c d g+c e f)}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^{5/2} (2 c d-b e)}\) |
-(((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(e^2*(2*c*d - b*e)*(d + e*x)^(5/2))) - ((c*e*f - 5*c*d*g + 2*b*e*g)*((2*Sqrt[d*(c*d - b* e) - b*e^2*x - c*e^2*x^2])/(e*Sqrt[d + e*x]) - (2*Sqrt[2*c*d - b*e]*ArcTan h[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*e*(2*c*d - b*e))
3.23.36.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b *d*e + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne Q[m + 2*p + 1, 0] && IntegerQ[2*p]
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[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Time = 0.36 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.57
| method | result | size |
| default | \(\frac {\sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (-2 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,e^{2} g x +5 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d e g x -\arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c \,e^{2} f x +2 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, e g x -2 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b d e g +5 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c \,d^{2} g -\arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d e f +3 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, d g -\sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, e f \right )}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {-x c e -b e +c d}\, e^{2} \sqrt {b e -2 c d}}\) | \(351\) |
(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/(e*x+d)^(3/2)*(-2*arctan((-c*e*x-b*e+c*d) ^(1/2)/(b*e-2*c*d)^(1/2))*b*e^2*g*x+5*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2 *c*d)^(1/2))*c*d*e*g*x-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c* e^2*f*x+2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*e*g*x-2*arctan((-c*e*x- b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*d*e*g+5*arctan((-c*e*x-b*e+c*d)^(1/2)/ (b*e-2*c*d)^(1/2))*c*d^2*g-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2) )*c*d*e*f+3*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*d*g-(-c*e*x-b*e+c*d)^ (1/2)*(b*e-2*c*d)^(1/2)*e*f)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c*d)^(1/2)
Time = 0.41 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.96 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\left [\frac {{\left (c d^{2} e f + {\left (c e^{3} f - {\left (5 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} - {\left (5 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (c d e^{2} f - {\left (5 \, c d^{2} e - 2 \, b d e^{2}\right )} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (2 \, c d e - b e^{2}\right )} g x - {\left (2 \, c d e - b e^{2}\right )} f + 3 \, {\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt {e x + d}}{2 \, {\left (2 \, c d^{3} e^{2} - b d^{2} e^{3} + {\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x\right )}}, \frac {{\left (c d^{2} e f + {\left (c e^{3} f - {\left (5 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} - {\left (5 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (c d e^{2} f - {\left (5 \, c d^{2} e - 2 \, b d e^{2}\right )} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) + \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (2 \, c d e - b e^{2}\right )} g x - {\left (2 \, c d e - b e^{2}\right )} f + 3 \, {\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt {e x + d}}{2 \, c d^{3} e^{2} - b d^{2} e^{3} + {\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x}\right ] \]
[1/2*((c*d^2*e*f + (c*e^3*f - (5*c*d*e^2 - 2*b*e^3)*g)*x^2 - (5*c*d^3 - 2* b*d^2*e)*g + 2*(c*d*e^2*f - (5*c*d^2*e - 2*b*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*(2*c* d*e - b*e^2)*g*x - (2*c*d*e - b*e^2)*f + 3*(2*c*d^2 - b*d*e)*g)*sqrt(e*x + d))/(2*c*d^3*e^2 - b*d^2*e^3 + (2*c*d*e^4 - b*e^5)*x^2 + 2*(2*c*d^2*e^3 - b*d*e^4)*x), ((c*d^2*e*f + (c*e^3*f - (5*c*d*e^2 - 2*b*e^3)*g)*x^2 - (5*c *d^3 - 2*b*d^2*e)*g + 2*(c*d*e^2*f - (5*c*d^2*e - 2*b*d*e^2)*g)*x)*sqrt(-2 *c*d + b*e)*arctan(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*(2*c*d*e - b*e^2)*g*x - (2*c*d*e - b*e^2 )*f + 3*(2*c*d^2 - b*d*e)*g)*sqrt(e*x + d))/(2*c*d^3*e^2 - b*d^2*e^3 + (2* c*d*e^4 - b*e^5)*x^2 + 2*(2*c*d^2*e^3 - b*d*e^4)*x)]
\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\int { \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Time = 0.36 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.71 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c g - \frac {{\left (c^{2} e f - 5 \, c^{2} d g + 2 \, b c e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} - \frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{2} e f - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{2} d g}{{\left (e x + d\right )} c}}{c e^{2}} \]
(2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*g - (c^2*e*f - 5*c^2*d*g + 2*b*c*e*g )*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/sqrt(-2*c*d + b*e) - (sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^2*e*f - sqrt(-(e*x + d)*c + 2 *c*d - b*e)*c^2*d*g)/((e*x + d)*c))/(c*e^2)
Timed out. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]