Integrand size = 46, antiderivative size = 116 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx=-\frac {2 (c e f+c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c^2 e^2 (d+e x)^{3/2}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^2 e^2 (d+e x)^{5/2}} \] Output:
-2/3*(-b*e*g+c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^2/e^2/( e*x+d)^(3/2)+2/5*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^2/e^2/(e*x+d)^ (5/2)
Time = 0.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx=\frac {2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))} (-2 b e g+c (5 e f+2 d g+3 e g x))}{15 c^2 e^2 \sqrt {d+e x}} \] Input:
Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/Sqrt[d + e *x],x]
Output:
(2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-2*b*e*g + c*(5*e*f + 2*d*g + 3*e*g*x)))/(15*c^2*e^2*Sqrt[d + e*x])
Time = 0.55 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1221, 1122}
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.
\(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{\sqrt {d+e x}} \, dx\) |
\(\Big \downarrow \) 1221 |
\(\displaystyle \frac {(-2 b e g-c d g+5 c e f) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}dx}{5 c e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt {d+e x}}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g-c d g+5 c e f)}{15 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt {d+e x}}\) |
Input:
Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/Sqrt[d + e*x],x]
Output:
(-2*(5*c*e*f - c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2 ))/(15*c^2*e^2*(d + e*x)^(3/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^ 2)^(3/2))/(5*c*e^2*Sqrt[d + e*x])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) Int[(d + e*x)^m*(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] && NeQ[m + 2*p + 2, 0]
Time = 2.80 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.61
method | result | size |
default | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-3 c e g x +2 b e g -2 c d g -5 f c e \right ) \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}}{15 c^{2} e^{2} \sqrt {e x +d}}\) | \(71\) |
gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-3 c e g x +2 b e g -2 c d g -5 f c e \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{15 c^{2} e^{2} \sqrt {e x +d}}\) | \(79\) |
orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-3 c e g x +2 b e g -2 c d g -5 f c e \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{15 c^{2} e^{2} \sqrt {e x +d}}\) | \(79\) |
Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(1/2),x,method= _RETURNVERBOSE)
Output:
-2/15*(c*e*x+b*e-c*d)*(-3*c*e*g*x+2*b*e*g-2*c*d*g-5*c*e*f)*(-(e*x+d)*(c*e* x+b*e-c*d))^(1/2)/c^2/e^2/(e*x+d)^(1/2)
Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.21 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (3 \, c^{2} e^{2} g x^{2} - 5 \, {\left (c^{2} d e - b c e^{2}\right )} f - 2 \, {\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} g + {\left (5 \, c^{2} e^{2} f - {\left (c^{2} d e - b c e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15 \, {\left (c^{2} e^{3} x + c^{2} d e^{2}\right )}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")
Output:
2/15*(3*c^2*e^2*g*x^2 - 5*(c^2*d*e - b*c*e^2)*f - 2*(c^2*d^2 - 2*b*c*d*e + b^2*e^2)*g + (5*c^2*e^2*f - (c^2*d*e - b*c*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b *e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^2*e^3*x + c^2*d*e^2)
\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\sqrt {d + e x}}\, dx \] Input:
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**(1/ 2),x)
Output:
Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/sqrt(d + e*x), x)
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (c e x - c d + b e\right )} \sqrt {-c e x + c d - b e} f}{3 \, c e} + \frac {2 \, {\left (3 \, c^{2} e^{2} x^{2} - 2 \, c^{2} d^{2} + 4 \, b c d e - 2 \, b^{2} e^{2} - {\left (c^{2} d e - b c e^{2}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{15 \, c^{2} e^{2}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")
Output:
2/3*(c*e*x - c*d + b*e)*sqrt(-c*e*x + c*d - b*e)*f/(c*e) + 2/15*(3*c^2*e^2 *x^2 - 2*c^2*d^2 + 4*b*c*d*e - 2*b^2*e^2 - (c^2*d*e - b*c*e^2)*x)*sqrt(-c* e*x + c*d - b*e)*g/(c^2*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (104) = 208\).
Time = 0.33 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.27 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx=-\frac {2 \, {\left (15 \, \sqrt {-c e x + c d - b e} c d e f - 15 \, \sqrt {-c e x + c d - b e} b e^{2} f - 5 \, {\left (3 \, \sqrt {-c e x + c d - b e} c d - 3 \, \sqrt {-c e x + c d - b e} b e - {\left (-c e x + c d - b e\right )}^{\frac {3}{2}}\right )} e f + 5 \, {\left (3 \, \sqrt {-c e x + c d - b e} c d - 3 \, \sqrt {-c e x + c d - b e} b e - {\left (-c e x + c d - b e\right )}^{\frac {3}{2}}\right )} d g - \frac {5 \, {\left (3 \, \sqrt {-c e x + c d - b e} c d - 3 \, \sqrt {-c e x + c d - b e} b e - {\left (-c e x + c d - b e\right )}^{\frac {3}{2}}\right )} b e g}{c} - \frac {{\left (15 \, \sqrt {-c e x + c d - b e} c^{2} d^{2} - 30 \, \sqrt {-c e x + c d - b e} b c d e + 15 \, \sqrt {-c e x + c d - b e} b^{2} e^{2} - 10 \, {\left (-c e x + c d - b e\right )}^{\frac {3}{2}} c d + 10 \, {\left (-c e x + c d - b e\right )}^{\frac {3}{2}} b e + 3 \, {\left (c e x - c d + b e\right )}^{2} \sqrt {-c e x + c d - b e}\right )} g}{c}\right )}}{15 \, c e^{2}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")
Output:
-2/15*(15*sqrt(-c*e*x + c*d - b*e)*c*d*e*f - 15*sqrt(-c*e*x + c*d - b*e)*b *e^2*f - 5*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b* e - (-c*e*x + c*d - b*e)^(3/2))*e*f + 5*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*d*g - 5*(3*sq rt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c* d - b*e)^(3/2))*b*e*g/c - (15*sqrt(-c*e*x + c*d - b*e)*c^2*d^2 - 30*sqrt(- c*e*x + c*d - b*e)*b*c*d*e + 15*sqrt(-c*e*x + c*d - b*e)*b^2*e^2 - 10*(-c* e*x + c*d - b*e)^(3/2)*c*d + 10*(-c*e*x + c*d - b*e)^(3/2)*b*e + 3*(c*e*x - c*d + b*e)^2*sqrt(-c*e*x + c*d - b*e))*g/c)/(c*e^2)
Time = 10.95 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.86 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx=\frac {\left (\frac {2\,g\,x^2}{5}+\frac {2\,x\,\left (b\,e\,g-c\,d\,g+5\,c\,e\,f\right )}{15\,c\,e}+\frac {2\,\left (b\,e-c\,d\right )\,\left (2\,c\,d\,g-2\,b\,e\,g+5\,c\,e\,f\right )}{15\,c^2\,e^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{\sqrt {d+e\,x}} \] Input:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^(1/2 ),x)
Output:
(((2*g*x^2)/5 + (2*x*(b*e*g - c*d*g + 5*c*e*f))/(15*c*e) + (2*(b*e - c*d)* (2*c*d*g - 2*b*e*g + 5*c*e*f))/(15*c^2*e^2))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.89 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {-c e x -b e +c d}\, \left (3 c^{2} e^{2} g \,x^{2}+b c \,e^{2} g x -c^{2} d e g x +5 c^{2} e^{2} f x -2 b^{2} e^{2} g +4 b c d e g +5 b c \,e^{2} f -2 c^{2} d^{2} g -5 c^{2} d e f \right )}{15 c^{2} e^{2}} \] Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(1/2),x)
Output:
(2*sqrt( - b*e + c*d - c*e*x)*( - 2*b**2*e**2*g + 4*b*c*d*e*g + 5*b*c*e**2 *f + b*c*e**2*g*x - 2*c**2*d**2*g - 5*c**2*d*e*f - c**2*d*e*g*x + 5*c**2*e **2*f*x + 3*c**2*e**2*g*x**2))/(15*c**2*e**2)