Integrand size = 42, antiderivative size = 218 \[ \int (d+e x) (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {(2 c d-b e) (8 c e f+2 c d g-5 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^3 e}+\frac {(5 b e g-8 c (e f+d g)-6 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 c^2 e^2}+\frac {(2 c d-b e)^3 (8 c e f+2 c d g-5 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{64 c^{7/2} e^2} \] Output:
1/64*(-b*e+2*c*d)*(-5*b*e*g+2*c*d*g+8*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2 *x-c*e^2*x^2)^(1/2)/c^3/e+1/24*(5*b*e*g-8*c*(d*g+e*f)-6*c*e*g*x)*(d*(-b*e+ c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^2/e^2+1/64*(-b*e+2*c*d)^3*(-5*b*e*g+2*c*d* g+8*c*e*f)*arctan(c^(1/2)*(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/ c^(7/2)/e^2
Time = 0.89 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.23 \[ \int (d+e x) (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {(2 c d-b e)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {\sqrt {c} \left (15 b^3 e^3 g-2 b^2 c e^2 (12 e f+38 d g+5 e g x)-8 c^3 \left (8 d^3 g-4 d e^2 x (3 f+2 g x)-2 e^3 x^2 (4 f+3 g x)+d^2 e (8 f+3 g x)\right )+4 b c^2 e \left (29 d^2 g+2 e^2 x (2 f+g x)+2 d e (14 f+5 g x)\right )\right )}{(2 c d-b e)^3}-\frac {3 (8 c e f+2 c d g-5 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{\sqrt {d+e x} \sqrt {-b e+c (d-e x)}}\right )}{192 c^{7/2} e^2} \] Input:
Integrate[(d + e*x)*(f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]
Output:
((2*c*d - b*e)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((Sqrt[c]*(15*b^3* e^3*g - 2*b^2*c*e^2*(12*e*f + 38*d*g + 5*e*g*x) - 8*c^3*(8*d^3*g - 4*d*e^2 *x*(3*f + 2*g*x) - 2*e^3*x^2*(4*f + 3*g*x) + d^2*e*(8*f + 3*g*x)) + 4*b*c^ 2*e*(29*d^2*g + 2*e^2*x*(2*f + g*x) + 2*d*e*(14*f + 5*g*x))))/(2*c*d - b*e )^3 - (3*(8*c*e*f + 2*c*d*g - 5*b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqr t[c]*Sqrt[d + e*x])])/(Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)])))/(192*c^ (7/2)*e^2)
Time = 0.69 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1225, 1087, 1092, 217}
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.
| \(\displaystyle \int (d+e x) (f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2} \, dx\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {(2 c d-b e) (-5 b e g+2 c d g+8 c e f) \int \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{16 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-8 c (d g+e f)-6 c e g x)}{24 c^2 e^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(2 c d-b e) (-5 b e g+2 c d g+8 c e f) \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-8 c (d g+e f)-6 c e g x)}{24 c^2 e^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {(2 c d-b e) (-5 b e g+2 c d g+8 c e f) \left (\frac {(2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-8 c (d g+e f)-6 c e g x)}{24 c^2 e^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right ) (-5 b e g+2 c d g+8 c e f)}{16 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-8 c (d g+e f)-6 c e g x)}{24 c^2 e^2}\) |
Input:
Int[(d + e*x)*(f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]
Output:
((5*b*e*g - 8*c*(e*f + d*g) - 6*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2* x^2)^(3/2))/(24*c^2*e^2) + ((2*c*d - b*e)*(8*c*e*f + 2*c*d*g - 5*b*e*g)*(( (b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c) + ((2*c*d - b *e)^2*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e ^2*x^2])])/(8*c^(3/2)*e)))/(16*c^2*e)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(723\) vs. \(2(202)=404\).
Time = 2.25 (sec) , antiderivative size = 724, normalized size of antiderivative = 3.32
| method | result | size |
| default | \(d f \left (-\frac {\left (-2 c \,e^{2} x -b \,e^{2}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{4 c \,e^{2}}-\frac {\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )+\left (d g +e f \right ) \left (-\frac {\left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{3 c \,e^{2}}-\frac {b \left (-\frac {\left (-2 c \,e^{2} x -b \,e^{2}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{4 c \,e^{2}}-\frac {\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{2 c}\right )+e g \left (-\frac {x \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{4 c \,e^{2}}-\frac {5 b \left (-\frac {\left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{3 c \,e^{2}}-\frac {b \left (-\frac {\left (-2 c \,e^{2} x -b \,e^{2}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{4 c \,e^{2}}-\frac {\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{2 c}\right )}{8 c}+\frac {\left (-b d e +c \,d^{2}\right ) \left (-\frac {\left (-2 c \,e^{2} x -b \,e^{2}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{4 c \,e^{2}}-\frac {\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{4 c \,e^{2}}\right )\) | \(724\) |
Input:
int((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method=_RETUR NVERBOSE)
Output:
d*f*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)- 1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^( 1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)))+(d*g+e*f)*(-1/3* (-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-1/4*(-2*c*e^2*x-b*e ^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d ^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2* x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))+e*g*(-1/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c* d^2)^(3/2)/c/e^2-5/8*b/c*(-1/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^ 2-1/2*b/c*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^ (1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c* e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))+1/4*(-b*d *e+c*d^2)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c *d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arct an((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (202) = 404\).
Time = 0.19 (sec) , antiderivative size = 825, normalized size of antiderivative = 3.78 \[ \int (d+e x) (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algori thm="fricas")
Output:
[-1/768*(3*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^ 4)*f + (16*c^4*d^4 - 64*b*c^3*d^3*e + 72*b^2*c^2*d^2*e^2 - 32*b^3*c*d*e^3 + 5*b^4*e^4)*g)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b *c*d*e + b^2*e^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*(48*c^4*e^3*g*x^3 + 8*(8*c^4*e^3*f + (8*c^4*d*e^2 + b* c^3*e^3)*g)*x^2 - 8*(8*c^4*d^2*e - 14*b*c^3*d*e^2 + 3*b^2*c^2*e^3)*f - (64 *c^4*d^3 - 116*b*c^3*d^2*e + 76*b^2*c^2*d*e^2 - 15*b^3*c*e^3)*g + 2*(8*(6* c^4*d*e^2 + b*c^3*e^3)*f - (12*c^4*d^2*e - 20*b*c^3*d*e^2 + 5*b^2*c^2*e^3) *g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^4*e^2), -1/384*(3*(8 *(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f + (16*c^ 4*d^4 - 64*b*c^3*d^3*e + 72*b^2*c^2*d^2*e^2 - 32*b^3*c*d*e^3 + 5*b^4*e^4)* g)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) - 2*(48*c^4* e^3*g*x^3 + 8*(8*c^4*e^3*f + (8*c^4*d*e^2 + b*c^3*e^3)*g)*x^2 - 8*(8*c^4*d ^2*e - 14*b*c^3*d*e^2 + 3*b^2*c^2*e^3)*f - (64*c^4*d^3 - 116*b*c^3*d^2*e + 76*b^2*c^2*d*e^2 - 15*b^3*c*e^3)*g + 2*(8*(6*c^4*d*e^2 + b*c^3*e^3)*f - ( 12*c^4*d^2*e - 20*b*c^3*d*e^2 + 5*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e ^2*x + c*d^2 - b*d*e))/(c^4*e^2)]
Leaf count of result is larger than twice the leaf count of optimal. 952 vs. \(2 (209) = 418\).
Time = 2.36 (sec) , antiderivative size = 952, normalized size of antiderivative = 4.37 \[ \int (d+e x) (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)
Output:
Piecewise((sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2)*(e*g*x**3/4 - x* *2*(-b*e**3*g/8 - c*d*e**2*g - c*e**3*f)/(3*c*e**2) - x*(-2*b*d*e**2*g - b *e**3*f - 5*b*(-b*e**3*g/8 - c*d*e**2*g - c*e**3*f)/(6*c) + c*d**2*e*g - c *d*e**2*f - e*g*(-3*b*d*e + 3*c*d**2)/4)/(2*c*e**2) - (-b*d**2*e*g - 2*b*d *e**2*f - 3*b*(-2*b*d*e**2*g - b*e**3*f - 5*b*(-b*e**3*g/8 - c*d*e**2*g - c*e**3*f)/(6*c) + c*d**2*e*g - c*d*e**2*f - e*g*(-3*b*d*e + 3*c*d**2)/4)/( 4*c) + c*d**3*g + c*d**2*e*f + (-2*b*d*e + 2*c*d**2)*(-b*e**3*g/8 - c*d*e* *2*g - c*e**3*f)/(3*c*e**2))/(c*e**2)) + (-b*d**2*e*f - b*(-b*d**2*e*g - 2 *b*d*e**2*f - 3*b*(-2*b*d*e**2*g - b*e**3*f - 5*b*(-b*e**3*g/8 - c*d*e**2* g - c*e**3*f)/(6*c) + c*d**2*e*g - c*d*e**2*f - e*g*(-3*b*d*e + 3*c*d**2)/ 4)/(4*c) + c*d**3*g + c*d**2*e*f + (-2*b*d*e + 2*c*d**2)*(-b*e**3*g/8 - c* d*e**2*g - c*e**3*f)/(3*c*e**2))/(2*c) + c*d**3*f + (-b*d*e + c*d**2)*(-2* b*d*e**2*g - b*e**3*f - 5*b*(-b*e**3*g/8 - c*d*e**2*g - c*e**3*f)/(6*c) + c*d**2*e*g - c*d*e**2*f - e*g*(-3*b*d*e + 3*c*d**2)/4)/(2*c*e**2))*Piecewi se((log(-b*e**2 - 2*c*e**2*x + 2*sqrt(-c*e**2)*sqrt(-b*d*e - b*e**2*x + c* d**2 - c*e**2*x**2))/sqrt(-c*e**2), Ne(b**2*e**2/(4*c) - b*d*e + c*d**2, 0 )), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(-c*e**2*(b/(2*c) + x)**2), True)) , Ne(c*e**2, 0)), (-2*(g*(-b*d*e - b*e**2*x + c*d**2)**(7/2)/(7*b**2*e**3) + (-b*d*e - b*e**2*x + c*d**2)**(5/2)*(b*d*e*g - b*e**2*f - 2*c*d**2*g)/( 5*b**2*e**3) + (-b*d*e - b*e**2*x + c*d**2)**(3/2)*(-b*c*d**3*e*g + b*c...
Exception generated. \[ \int (d+e x) (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algori thm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (202) = 404\).
Time = 0.16 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.88 \[ \int (d+e x) (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {1}{192} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, {\left (6 \, e g x + \frac {8 \, c^{3} e^{5} f + 8 \, c^{3} d e^{4} g + b c^{2} e^{5} g}{c^{3} e^{4}}\right )} x + \frac {48 \, c^{3} d e^{4} f + 8 \, b c^{2} e^{5} f - 12 \, c^{3} d^{2} e^{3} g + 20 \, b c^{2} d e^{4} g - 5 \, b^{2} c e^{5} g}{c^{3} e^{4}}\right )} x - \frac {64 \, c^{3} d^{2} e^{3} f - 112 \, b c^{2} d e^{4} f + 24 \, b^{2} c e^{5} f + 64 \, c^{3} d^{3} e^{2} g - 116 \, b c^{2} d^{2} e^{3} g + 76 \, b^{2} c d e^{4} g - 15 \, b^{3} e^{5} g}{c^{3} e^{4}}\right )} - \frac {{\left (64 \, c^{4} d^{3} e f - 96 \, b c^{3} d^{2} e^{2} f + 48 \, b^{2} c^{2} d e^{3} f - 8 \, b^{3} c e^{4} f + 16 \, c^{4} d^{4} g - 64 \, b c^{3} d^{3} e g + 72 \, b^{2} c^{2} d^{2} e^{2} g - 32 \, b^{3} c d e^{3} g + 5 \, b^{4} e^{4} g\right )} \log \left ({\left | -b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} {\left | e \right |} \right |}\right )}{128 \, \sqrt {-c} c^{3} e {\left | e \right |}} \] Input:
integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algori thm="giac")
Output:
1/192*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(6*e*g*x + (8*c^3*e ^5*f + 8*c^3*d*e^4*g + b*c^2*e^5*g)/(c^3*e^4))*x + (48*c^3*d*e^4*f + 8*b*c ^2*e^5*f - 12*c^3*d^2*e^3*g + 20*b*c^2*d*e^4*g - 5*b^2*c*e^5*g)/(c^3*e^4)) *x - (64*c^3*d^2*e^3*f - 112*b*c^2*d*e^4*f + 24*b^2*c*e^5*f + 64*c^3*d^3*e ^2*g - 116*b*c^2*d^2*e^3*g + 76*b^2*c*d*e^4*g - 15*b^3*e^5*g)/(c^3*e^4)) - 1/128*(64*c^4*d^3*e*f - 96*b*c^3*d^2*e^2*f + 48*b^2*c^2*d*e^3*f - 8*b^3*c *e^4*f + 16*c^4*d^4*g - 64*b*c^3*d^3*e*g + 72*b^2*c^2*d^2*e^2*g - 32*b^3*c *d*e^3*g + 5*b^4*e^4*g)*log(abs(-b*e^2 + 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x ^2 - b*e^2*x + c*d^2 - b*d*e))*sqrt(-c)*abs(e)))/(sqrt(-c)*c^3*e*abs(e))
Time = 7.27 (sec) , antiderivative size = 801, normalized size of antiderivative = 3.67 \[ \int (d+e x) (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=d\,f\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+\frac {5\,b\,e\,g\,\left (\frac {\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )\,\left (b^3\,e^6+4\,b\,c\,e^4\,\left (c\,d^2-b\,d\,e\right )\right )}{16\,{\left (-c\,e^2\right )}^{5/2}}-\frac {\left (8\,c\,e^2\,\left (-c\,d^2+b\,d\,e+c\,e^2\,x^2\right )-3\,b^2\,e^4+2\,b\,c\,e^4\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{24\,c^2\,e^4}\right )}{8\,c}-\frac {d\,g\,\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )\,\left (b^3\,e^6+4\,b\,c\,e^4\,\left (c\,d^2-b\,d\,e\right )\right )}{16\,{\left (-c\,e^2\right )}^{5/2}}-\frac {e\,f\,\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )\,\left (b^3\,e^6+4\,b\,c\,e^4\,\left (c\,d^2-b\,d\,e\right )\right )}{16\,{\left (-c\,e^2\right )}^{5/2}}+\frac {f\,\left (8\,c\,e^2\,\left (-c\,d^2+b\,d\,e+c\,e^2\,x^2\right )-3\,b^2\,e^4+2\,b\,c\,e^4\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{24\,c^2\,e^3}+\frac {g\,\left (c\,d^2-b\,d\,e\right )\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-\frac {\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )\,\left (\frac {b^2\,e^4}{4}+c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )}{2\,{\left (-c\,e^2\right )}^{3/2}}\right )}{4\,c\,e}-\frac {d\,f\,\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )\,\left (\frac {b^2\,e^4}{4}+c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )}{2\,{\left (-c\,e^2\right )}^{3/2}}-\frac {g\,x\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{4\,c\,e}+\frac {d\,g\,\left (8\,c\,e^2\,\left (-c\,d^2+b\,d\,e+c\,e^2\,x^2\right )-3\,b^2\,e^4+2\,b\,c\,e^4\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{24\,c^2\,e^4} \] Input:
int((f + g*x)*(d + e*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2),x)
Output:
d*f*(x/2 + b/(4*c))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + (5*b*e*g *((log(b*e^2 - 2*(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2 *c*e^2*x)*(b^3*e^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) - ((8 *c*e^2*(c*e^2*x^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e ^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(24*c^2*e^4)))/(8*c) - (d*g*log(b*e^2 - 2 *(-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*(b^3*e ^6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) - (e*f*log(b*e^2 - 2* (-c*e^2)^(1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*(b^3*e^ 6 + 4*b*c*e^4*(c*d^2 - b*d*e)))/(16*(-c*e^2)^(5/2)) + (f*(8*c*e^2*(c*e^2*x ^2 - c*d^2 + b*d*e) - 3*b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(24*c^2*e^3) + (g*(c*d^2 - b*d*e)*((x/2 + b/(4*c))*(c*d^ 2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - (log(b*e^2 - 2*(-c*e^2)^(1/2)*(-( d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*((b^2*e^4)/4 + c*e^2*(c*d ^2 - b*d*e)))/(2*(-c*e^2)^(3/2))))/(4*c*e) - (d*f*log(b*e^2 - 2*(-c*e^2)^( 1/2)*(-(d + e*x)*(b*e - c*d + c*e*x))^(1/2) + 2*c*e^2*x)*((b^2*e^4)/4 + c* e^2*(c*d^2 - b*d*e)))/(2*(-c*e^2)^(3/2)) - (g*x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(4*c*e) + (d*g*(8*c*e^2*(c*e^2*x^2 - c*d^2 + b*d*e) - 3 *b^2*e^4 + 2*b*c*e^4*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(24*c ^2*e^4)
Time = 0.49 (sec) , antiderivative size = 1300, normalized size of antiderivative = 5.96 \[ \int (d+e x) (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
Output:
(i*(15*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))* b**5*e**5*g - 126*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c*d*e**4*g - 24*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)* i)/sqrt( - b*e + 2*c*d))*b**4*c*e**5*f + 408*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**2*d**2*e**3*g + 192*sqrt(c)* asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**2*d*e** 4*f - 624*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d ))*b**2*c**3*d**3*e**2*g - 576*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i )/sqrt( - b*e + 2*c*d))*b**2*c**3*d**2*e**3*f + 432*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**4*d**4*e*g + 768*sqrt(c) *asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**4*d**3*e* *2*f - 96*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d ))*c**5*d**5*g - 384*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**5*d**4*e*f + 15*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b *e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3*c*e**3*g - 76*sqrt(d + e*x)*sq rt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2* d*e**2*g - 24*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*e**3*f - 10*sqrt(d + e*x)*sqrt(b*e - 2*c*d)* sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*e**3*g*x + 116*s qrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d -...