Integrand size = 46, antiderivative size = 265 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 (2 c d-b e)^2 (c e f+c d g-b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^4 e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c^4 e^2 (d+e x)^{3/2}}-\frac {2 (c e f+5 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^4 e^2 (d+e x)^{5/2}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^4 e^2 (d+e x)^{7/2}} \] Output:
-2*(-b*e+2*c*d)^2*(-b*e*g+c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1 /2)/c^4/e^2/(e*x+d)^(1/2)+2/3*(-b*e+2*c*d)*(-3*b*e*g+4*c*d*g+2*c*e*f)*(d*( -b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^4/e^2/(e*x+d)^(3/2)-2/5*(-3*b*e*g+5*c *d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^4/e^2/(e*x+d)^(5/2)+2 /7*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^4/e^2/(e*x+d)^(7/2)
Time = 0.20 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.68 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {2 \sqrt {d+e x} (-c d+b e+c e x) \left (-48 b^3 e^3 g+8 b^2 c e^2 (7 e f+32 d g+3 e g x)-2 b c^2 e \left (219 d^2 g+e^2 x (14 f+9 g x)+2 d e (63 f+26 g x)\right )+c^3 \left (230 d^3 g+3 e^3 x^2 (7 f+5 g x)+2 d e^2 x (49 f+30 g x)+d^2 e (301 f+115 g x)\right )\right )}{105 c^4 e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[((d + e*x)^(5/2)*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2 *x^2],x]
Output:
(2*Sqrt[d + e*x]*(-(c*d) + b*e + c*e*x)*(-48*b^3*e^3*g + 8*b^2*c*e^2*(7*e* f + 32*d*g + 3*e*g*x) - 2*b*c^2*e*(219*d^2*g + e^2*x*(14*f + 9*g*x) + 2*d* e*(63*f + 26*g*x)) + c^3*(230*d^3*g + 3*e^3*x^2*(7*f + 5*g*x) + 2*d*e^2*x* (49*f + 30*g*x) + d^2*e*(301*f + 115*g*x))))/(105*c^4*e^2*Sqrt[(d + e*x)*( -(b*e) + c*(d - e*x))])
Time = 0.84 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1221, 1128, 1128, 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.
| \(\displaystyle \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(-6 b e g+5 c d g+7 c e f) \int \frac {(d+e x)^{5/2}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{7 c e}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-6 b e g+5 c d g+7 c e f) \left (\frac {4 (2 c d-b e) \int \frac {(d+e x)^{3/2}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{5 c}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e}\right )}{7 c e}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-6 b e g+5 c d g+7 c e f) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{3 c}-\frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{5 c}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e}\right )}{7 c e}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {\left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c^2 e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{5 c}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e}\right ) (-6 b e g+5 c d g+7 c e f)}{7 c e}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}\) |
Input:
Int[((d + e*x)^(5/2)*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2], x]
Output:
(-2*g*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(7*c*e^2) + ((7*c*e*f + 5*c*d*g - 6*b*e*g)*((-2*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(5*c*e) + (4*(2*c*d - b*e)*((-4*(2*c*d - b*e)*Sqrt [d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*c^2*e*Sqrt[d + e*x]) - (2*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*c*e)))/(5*c)))/(7*c* e)
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_)*((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*(m + 2*p + 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[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 = 1.86 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.81
| method | result | size |
| default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (-15 e^{3} g \,x^{3} c^{3}+18 b \,c^{2} e^{3} g \,x^{2}-60 c^{3} d \,e^{2} g \,x^{2}-21 c^{3} e^{3} f \,x^{2}-24 b^{2} c \,e^{3} g x +104 b \,c^{2} d \,e^{2} g x +28 b \,c^{2} e^{3} f x -115 c^{3} d^{2} e g x -98 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -256 b^{2} c d \,e^{2} g -56 b^{2} c \,e^{3} f +438 b \,c^{2} d^{2} e g +252 b \,c^{2} d \,e^{2} f -230 c^{3} d^{3} g -301 d^{2} f \,c^{3} e \right )}{105 \sqrt {e x +d}\, c^{4} e^{2}}\) | \(215\) |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-15 e^{3} g \,x^{3} c^{3}+18 b \,c^{2} e^{3} g \,x^{2}-60 c^{3} d \,e^{2} g \,x^{2}-21 c^{3} e^{3} f \,x^{2}-24 b^{2} c \,e^{3} g x +104 b \,c^{2} d \,e^{2} g x +28 b \,c^{2} e^{3} f x -115 c^{3} d^{2} e g x -98 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -256 b^{2} c d \,e^{2} g -56 b^{2} c \,e^{3} f +438 b \,c^{2} d^{2} e g +252 b \,c^{2} d \,e^{2} f -230 c^{3} d^{3} g -301 d^{2} f \,c^{3} e \right ) \sqrt {e x +d}}{105 c^{4} e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\) | \(235\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-15 e^{3} g \,x^{3} c^{3}+18 b \,c^{2} e^{3} g \,x^{2}-60 c^{3} d \,e^{2} g \,x^{2}-21 c^{3} e^{3} f \,x^{2}-24 b^{2} c \,e^{3} g x +104 b \,c^{2} d \,e^{2} g x +28 b \,c^{2} e^{3} f x -115 c^{3} d^{2} e g x -98 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -256 b^{2} c d \,e^{2} g -56 b^{2} c \,e^{3} f +438 b \,c^{2} d^{2} e g +252 b \,c^{2} d \,e^{2} f -230 c^{3} d^{3} g -301 d^{2} f \,c^{3} e \right ) \sqrt {e x +d}}{105 c^{4} e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\) | \(235\) |
Input:
int((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method= _RETURNVERBOSE)
Output:
2/105/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-15*c^3*e^3*g*x^3+18 *b*c^2*e^3*g*x^2-60*c^3*d*e^2*g*x^2-21*c^3*e^3*f*x^2-24*b^2*c*e^3*g*x+104* b*c^2*d*e^2*g*x+28*b*c^2*e^3*f*x-115*c^3*d^2*e*g*x-98*c^3*d*e^2*f*x+48*b^3 *e^3*g-256*b^2*c*d*e^2*g-56*b^2*c*e^3*f+438*b*c^2*d^2*e*g+252*b*c^2*d*e^2* f-230*c^3*d^3*g-301*c^3*d^2*e*f)/c^4/e^2
Time = 0.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.89 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 \, {\left (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f + 2 \, {\left (10 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (43 \, c^{3} d^{2} e - 36 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + 2 \, {\left (115 \, c^{3} d^{3} - 219 \, b c^{2} d^{2} e + 128 \, b^{2} c d e^{2} - 24 \, b^{3} e^{3}\right )} g + {\left (14 \, {\left (7 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f + {\left (115 \, c^{3} d^{2} e - 104 \, b c^{2} d e^{2} + 24 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{105 \, {\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \] Input:
integrate((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")
Output:
-2/105*(15*c^3*e^3*g*x^3 + 3*(7*c^3*e^3*f + 2*(10*c^3*d*e^2 - 3*b*c^2*e^3) *g)*x^2 + 7*(43*c^3*d^2*e - 36*b*c^2*d*e^2 + 8*b^2*c*e^3)*f + 2*(115*c^3*d ^3 - 219*b*c^2*d^2*e + 128*b^2*c*d*e^2 - 24*b^3*e^3)*g + (14*(7*c^3*d*e^2 - 2*b*c^2*e^3)*f + (115*c^3*d^2*e - 104*b*c^2*d*e^2 + 24*b^2*c*e^3)*g)*x)* sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^4*e^3*x + c^4* d*e^2)
\[ \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}} \left (f + g x\right )}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \] Input:
integrate((e*x+d)**(5/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/ 2),x)
Output:
Integral((d + e*x)**(5/2)*(f + g*x)/sqrt(-(d + e*x)*(b*e - c*d + c*e*x)), x)
Time = 0.08 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.20 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {2 \, {\left (3 \, c^{3} e^{3} x^{3} - 43 \, c^{3} d^{3} + 79 \, b c^{2} d^{2} e - 44 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + {\left (11 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} + {\left (29 \, c^{3} d^{2} e - 18 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} x\right )} f}{15 \, \sqrt {-c e x + c d - b e} c^{3} e} + \frac {2 \, {\left (15 \, c^{4} e^{4} x^{4} - 230 \, c^{4} d^{4} + 668 \, b c^{3} d^{3} e - 694 \, b^{2} c^{2} d^{2} e^{2} + 304 \, b^{3} c d e^{3} - 48 \, b^{4} e^{4} + 3 \, {\left (15 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + {\left (55 \, c^{4} d^{2} e^{2} - 26 \, b c^{3} d e^{3} + 6 \, b^{2} c^{2} e^{4}\right )} x^{2} + {\left (115 \, c^{4} d^{3} e - 219 \, b c^{3} d^{2} e^{2} + 128 \, b^{2} c^{2} d e^{3} - 24 \, b^{3} c e^{4}\right )} x\right )} g}{105 \, \sqrt {-c e x + c d - b e} c^{4} e^{2}} \] Input:
integrate((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="maxima")
Output:
2/15*(3*c^3*e^3*x^3 - 43*c^3*d^3 + 79*b*c^2*d^2*e - 44*b^2*c*d*e^2 + 8*b^3 *e^3 + (11*c^3*d*e^2 - b*c^2*e^3)*x^2 + (29*c^3*d^2*e - 18*b*c^2*d*e^2 + 4 *b^2*c*e^3)*x)*f/(sqrt(-c*e*x + c*d - b*e)*c^3*e) + 2/105*(15*c^4*e^4*x^4 - 230*c^4*d^4 + 668*b*c^3*d^3*e - 694*b^2*c^2*d^2*e^2 + 304*b^3*c*d*e^3 - 48*b^4*e^4 + 3*(15*c^4*d*e^3 - b*c^3*e^4)*x^3 + (55*c^4*d^2*e^2 - 26*b*c^3 *d*e^3 + 6*b^2*c^2*e^4)*x^2 + (115*c^4*d^3*e - 219*b*c^3*d^2*e^2 + 128*b^2 *c^2*d*e^3 - 24*b^3*c*e^4)*x)*g/(sqrt(-c*e*x + c*d - b*e)*c^4*e^2)
Time = 0.33 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 \, {\left (\frac {105 \, {\left (4 \, c^{3} d^{2} e f - 4 \, b c^{2} d e^{2} f + b^{2} c e^{3} f + 4 \, c^{3} d^{3} g - 8 \, b c^{2} d^{2} e g + 5 \, b^{2} c d e^{2} g - b^{3} e^{3} g\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{c^{4}} - \frac {140 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d e f - 70 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c e^{2} f + 280 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d^{2} g - 350 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c d e g + 105 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} e^{2} g - 21 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c e f - 105 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c d g + 63 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b e g - 15 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} g}{c^{4}}\right )}}{105 \, e^{2}} \] Input:
integrate((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")
Output:
-2/105*(105*(4*c^3*d^2*e*f - 4*b*c^2*d*e^2*f + b^2*c*e^3*f + 4*c^3*d^3*g - 8*b*c^2*d^2*e*g + 5*b^2*c*d*e^2*g - b^3*e^3*g)*sqrt(-(e*x + d)*c + 2*c*d - b*e)/c^4 - (140*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d*e*f - 70*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*e^2*f + 280*(-(e*x + d)*c + 2*c*d - b*e)^ (3/2)*c^2*d^2*g - 350*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e*g + 105*( -(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2*g - 21*((e*x + d)*c - 2*c*d + b* e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*e*f - 105*((e*x + d)*c - 2*c*d + b *e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d*g + 63*((e*x + d)*c - 2*c*d + b *e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e*g - 15*((e*x + d)*c - 2*c*d + b *e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*g)/c^4)/e^2
Time = 11.42 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {\left (\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{7\,c}+\frac {\sqrt {d+e\,x}\,\left (-96\,g\,b^3\,e^3+512\,g\,b^2\,c\,d\,e^2+112\,f\,b^2\,c\,e^3-876\,g\,b\,c^2\,d^2\,e-504\,f\,b\,c^2\,d\,e^2+460\,g\,c^3\,d^3+602\,f\,c^3\,d^2\,e\right )}{105\,c^4\,e^3}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (20\,c\,d\,g-6\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^2\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (48\,g\,b^2\,c\,e^3-208\,g\,b\,c^2\,d\,e^2-56\,f\,b\,c^2\,e^3+230\,g\,c^3\,d^2\,e+196\,f\,c^3\,d\,e^2\right )}{105\,c^4\,e^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x+\frac {d}{e}} \] Input:
int(((f + g*x)*(d + e*x)^(5/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2 ),x)
Output:
-(((2*g*x^3*(d + e*x)^(1/2))/(7*c) + ((d + e*x)^(1/2)*(460*c^3*d^3*g - 96* b^3*e^3*g + 112*b^2*c*e^3*f + 602*c^3*d^2*e*f - 504*b*c^2*d*e^2*f - 876*b* c^2*d^2*e*g + 512*b^2*c*d*e^2*g))/(105*c^4*e^3) + (2*x^2*(d + e*x)^(1/2)*( 20*c*d*g - 6*b*e*g + 7*c*e*f))/(35*c^2*e) + (x*(d + e*x)^(1/2)*(48*b^2*c*e ^3*g - 56*b*c^2*e^3*f + 196*c^3*d*e^2*f + 230*c^3*d^2*e*g - 208*b*c^2*d*e^ 2*g))/(105*c^4*e^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(x + d/e )
Time = 0.22 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.75 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {2 \sqrt {-c e x -b e +c d}\, \left (-15 c^{3} e^{3} g \,x^{3}+18 b \,c^{2} e^{3} g \,x^{2}-60 c^{3} d \,e^{2} g \,x^{2}-21 c^{3} e^{3} f \,x^{2}-24 b^{2} c \,e^{3} g x +104 b \,c^{2} d \,e^{2} g x +28 b \,c^{2} e^{3} f x -115 c^{3} d^{2} e g x -98 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -256 b^{2} c d \,e^{2} g -56 b^{2} c \,e^{3} f +438 b \,c^{2} d^{2} e g +252 b \,c^{2} d \,e^{2} f -230 c^{3} d^{3} g -301 c^{3} d^{2} e f \right )}{105 c^{4} e^{2}} \] Input:
int((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
Output:
(2*sqrt( - b*e + c*d - c*e*x)*(48*b**3*e**3*g - 256*b**2*c*d*e**2*g - 56*b **2*c*e**3*f - 24*b**2*c*e**3*g*x + 438*b*c**2*d**2*e*g + 252*b*c**2*d*e** 2*f + 104*b*c**2*d*e**2*g*x + 28*b*c**2*e**3*f*x + 18*b*c**2*e**3*g*x**2 - 230*c**3*d**3*g - 301*c**3*d**2*e*f - 115*c**3*d**2*e*g*x - 98*c**3*d*e** 2*f*x - 60*c**3*d*e**2*g*x**2 - 21*c**3*e**3*f*x**2 - 15*c**3*e**3*g*x**3) )/(105*c**4*e**2)