\(\int \sqrt {d+e x} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\) [2242]

Optimal result

Integrand size = 46, antiderivative size = 267 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {16 (2 c d-b e)^2 (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3465 c^4 e^2 (d+e x)^{5/2}}-\frac {8 (2 c d-b e) (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{693 c^3 e^2 (d+e x)^{3/2}}-\frac {2 (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2} \]

[Out]

-16/3465*(-b*e+2*c*d)^2*(-6*b*e*g+c*d*g+11*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)
-8/693*(-b*e+2*c*d)*(-6*b*e*g+c*d*g+11*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^3/e^2/(e*x+d)^(3/2)-2/9
9*(-6*b*e*g+c*d*g+11*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^2/e^2/(e*x+d)^(1/2)-2/11*g*(d*(-b*e+c*d)-
b*e^2*x-c*e^2*x^2)^(5/2)*(e*x+d)^(1/2)/c/e^2

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 267, 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 = {808, 670, 662} \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{3465 c^4 e^2 (d+e x)^{5/2}}-\frac {8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{3/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{99 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2} \]

[In]

Int[Sqrt[d + e*x]*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(-16*(2*c*d - b*e)^2*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3465*c^4*e^2*(
d + e*x)^(5/2)) - (8*(2*c*d - b*e)*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(
693*c^3*e^2*(d + e*x)^(3/2)) - (2*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9
9*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*c*e^2)

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> 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] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 808

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] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}-\frac {\left (2 \left (\frac {5}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {1}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{11 c e^3} \\ & = -\frac {2 (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}+\frac {(4 (2 c d-b e) (11 c e f+c d g-6 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{99 c^2 e} \\ & = -\frac {8 (2 c d-b e) (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{693 c^3 e^2 (d+e x)^{3/2}}-\frac {2 (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}+\frac {\left (8 (2 c d-b e)^2 (11 c e f+c d g-6 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{693 c^3 e} \\ & = -\frac {16 (2 c d-b e)^2 (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3465 c^4 e^2 (d+e x)^{5/2}}-\frac {8 (2 c d-b e) (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{693 c^3 e^2 (d+e x)^{3/2}}-\frac {2 (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.69 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-48 b^3 e^3 g+8 b^2 c e^2 (11 e f+40 d g+15 e g x)-2 b c^2 e \left (347 d^2 g+5 e^2 x (22 f+21 g x)+d e (286 f+340 g x)\right )+c^3 \left (422 d^3 g+35 e^3 x^2 (11 f+9 g x)+10 d e^2 x (121 f+98 g x)+d^2 e (1177 f+1055 g x)\right )\right )}{3465 c^4 e^2 \sqrt {d+e x}} \]

[In]

Integrate[Sqrt[d + e*x]*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-48*b^3*e^3*g + 8*b^2*c*e^2*(11*e*f + 40*
d*g + 15*e*g*x) - 2*b*c^2*e*(347*d^2*g + 5*e^2*x*(22*f + 21*g*x) + d*e*(286*f + 340*g*x)) + c^3*(422*d^3*g + 3
5*e^3*x^2*(11*f + 9*g*x) + 10*d*e^2*x*(121*f + 98*g*x) + d^2*e*(1177*f + 1055*g*x))))/(3465*c^4*e^2*Sqrt[d + e
*x])

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {2 \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (x c e +b e -c d \right )^{2} \left (-315 g \,e^{3} x^{3} c^{3}+210 b \,c^{2} e^{3} g \,x^{2}-980 c^{3} d \,e^{2} g \,x^{2}-385 c^{3} e^{3} f \,x^{2}-120 b^{2} c \,e^{3} g x +680 b \,c^{2} d \,e^{2} g x +220 b \,c^{2} e^{3} f x -1055 c^{3} d^{2} e g x -1210 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -320 b^{2} c d \,e^{2} g -88 b^{2} c \,e^{3} f +694 b \,c^{2} d^{2} e g +572 b \,c^{2} d \,e^{2} f -422 c^{3} d^{3} g -1177 d^{2} f \,c^{3} e \right )}{3465 \sqrt {e x +d}\, c^{4} e^{2}}\)	\(229\)
gosper	\(-\frac {2 \left (x c e +b e -c d \right ) \left (-315 g \,e^{3} x^{3} c^{3}+210 b \,c^{2} e^{3} g \,x^{2}-980 c^{3} d \,e^{2} g \,x^{2}-385 c^{3} e^{3} f \,x^{2}-120 b^{2} c \,e^{3} g x +680 b \,c^{2} d \,e^{2} g x +220 b \,c^{2} e^{3} f x -1055 c^{3} d^{2} e g x -1210 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -320 b^{2} c d \,e^{2} g -88 b^{2} c \,e^{3} f +694 b \,c^{2} d^{2} e g +572 b \,c^{2} d \,e^{2} f -422 c^{3} d^{3} g -1177 d^{2} f \,c^{3} e \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{3465 c^{4} e^{2} \left (e x +d \right )^{\frac {3}{2}}}\)	\(235\)

[In]

int((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/3465/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(c*e*x+b*e-c*d)^2*(-315*c^3*e^3*g*x^3+210*b*c^2*e^3*g*x^
2-980*c^3*d*e^2*g*x^2-385*c^3*e^3*f*x^2-120*b^2*c*e^3*g*x+680*b*c^2*d*e^2*g*x+220*b*c^2*e^3*f*x-1055*c^3*d^2*e
*g*x-1210*c^3*d*e^2*f*x+48*b^3*e^3*g-320*b^2*c*d*e^2*g-88*b^2*c*e^3*f+694*b*c^2*d^2*e*g+572*b*c^2*d*e^2*f-422*
c^3*d^3*g-1177*c^3*d^2*e*f)/c^4/e^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (243) = 486\).

Time = 0.42 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.89 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (315 \, c^{5} e^{5} g x^{5} + 35 \, {\left (11 \, c^{5} e^{5} f + 2 \, {\left (5 \, c^{5} d e^{4} + 6 \, b c^{4} e^{5}\right )} g\right )} x^{4} + 5 \, {\left (22 \, {\left (4 \, c^{5} d e^{4} + 5 \, b c^{4} e^{5}\right )} f - {\left (118 \, c^{5} d^{2} e^{3} - 214 \, b c^{4} d e^{4} - 3 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} - 3 \, {\left (11 \, {\left (26 \, c^{5} d^{2} e^{3} - 46 \, b c^{4} d e^{4} - b^{2} c^{3} e^{5}\right )} f + 2 \, {\left (118 \, c^{5} d^{3} e^{2} - 101 \, b c^{4} d^{2} e^{3} - 20 \, b^{2} c^{3} d e^{4} + 3 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} + 11 \, {\left (107 \, c^{5} d^{4} e - 266 \, b c^{4} d^{3} e^{2} + 219 \, b^{2} c^{3} d^{2} e^{3} - 68 \, b^{3} c^{2} d e^{4} + 8 \, b^{4} c e^{5}\right )} f + 2 \, {\left (211 \, c^{5} d^{5} - 769 \, b c^{4} d^{4} e + 1065 \, b^{2} c^{3} d^{3} e^{2} - 691 \, b^{3} c^{2} d^{2} e^{3} + 208 \, b^{4} c d e^{4} - 24 \, b^{5} e^{5}\right )} g - {\left (22 \, {\left (52 \, c^{5} d^{3} e^{2} - 39 \, b c^{4} d^{2} e^{3} - 15 \, b^{2} c^{3} d e^{4} + 2 \, b^{3} c^{2} e^{5}\right )} f - {\left (211 \, c^{5} d^{4} e - 558 \, b c^{4} d^{3} e^{2} + 507 \, b^{2} c^{3} d^{2} e^{3} - 184 \, b^{3} c^{2} d e^{4} + 24 \, b^{4} c e^{5}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{3465 \, {\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \]

[In]

integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-2/3465*(315*c^5*e^5*g*x^5 + 35*(11*c^5*e^5*f + 2*(5*c^5*d*e^4 + 6*b*c^4*e^5)*g)*x^4 + 5*(22*(4*c^5*d*e^4 + 5*
b*c^4*e^5)*f - (118*c^5*d^2*e^3 - 214*b*c^4*d*e^4 - 3*b^2*c^3*e^5)*g)*x^3 - 3*(11*(26*c^5*d^2*e^3 - 46*b*c^4*d
*e^4 - b^2*c^3*e^5)*f + 2*(118*c^5*d^3*e^2 - 101*b*c^4*d^2*e^3 - 20*b^2*c^3*d*e^4 + 3*b^3*c^2*e^5)*g)*x^2 + 11
*(107*c^5*d^4*e - 266*b*c^4*d^3*e^2 + 219*b^2*c^3*d^2*e^3 - 68*b^3*c^2*d*e^4 + 8*b^4*c*e^5)*f + 2*(211*c^5*d^5
- 769*b*c^4*d^4*e + 1065*b^2*c^3*d^3*e^2 - 691*b^3*c^2*d^2*e^3 + 208*b^4*c*d*e^4 - 24*b^5*e^5)*g - (22*(52*c^
5*d^3*e^2 - 39*b*c^4*d^2*e^3 - 15*b^2*c^3*d*e^4 + 2*b^3*c^2*e^5)*f - (211*c^5*d^4*e - 558*b*c^4*d^3*e^2 + 507*
b^2*c^3*d^2*e^3 - 184*b^3*c^2*d*e^4 + 24*b^4*c*e^5)*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)

Sympy [F]

\[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \sqrt {d + e x} \left (f + g x\right )\, dx \]

[In]

integrate((e*x+d)**(1/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*sqrt(d + e*x)*(f + g*x), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (243) = 486\).

Time = 0.25 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.88 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (35 \, c^{4} e^{4} x^{4} + 107 \, c^{4} d^{4} - 266 \, b c^{3} d^{3} e + 219 \, b^{2} c^{2} d^{2} e^{2} - 68 \, b^{3} c d e^{3} + 8 \, b^{4} e^{4} + 10 \, {\left (4 \, c^{4} d e^{3} + 5 \, b c^{3} e^{4}\right )} x^{3} - 3 \, {\left (26 \, c^{4} d^{2} e^{2} - 46 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} - 2 \, {\left (52 \, c^{4} d^{3} e - 39 \, b c^{3} d^{2} e^{2} - 15 \, b^{2} c^{2} d e^{3} + 2 \, b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{315 \, {\left (c^{3} e^{2} x + c^{3} d e\right )}} - \frac {2 \, {\left (315 \, c^{5} e^{5} x^{5} + 422 \, c^{5} d^{5} - 1538 \, b c^{4} d^{4} e + 2130 \, b^{2} c^{3} d^{3} e^{2} - 1382 \, b^{3} c^{2} d^{2} e^{3} + 416 \, b^{4} c d e^{4} - 48 \, b^{5} e^{5} + 70 \, {\left (5 \, c^{5} d e^{4} + 6 \, b c^{4} e^{5}\right )} x^{4} - 5 \, {\left (118 \, c^{5} d^{2} e^{3} - 214 \, b c^{4} d e^{4} - 3 \, b^{2} c^{3} e^{5}\right )} x^{3} - 6 \, {\left (118 \, c^{5} d^{3} e^{2} - 101 \, b c^{4} d^{2} e^{3} - 20 \, b^{2} c^{3} d e^{4} + 3 \, b^{3} c^{2} e^{5}\right )} x^{2} + {\left (211 \, c^{5} d^{4} e - 558 \, b c^{4} d^{3} e^{2} + 507 \, b^{2} c^{3} d^{2} e^{3} - 184 \, b^{3} c^{2} d e^{4} + 24 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{3465 \, {\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \]

[In]

integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/315*(35*c^4*e^4*x^4 + 107*c^4*d^4 - 266*b*c^3*d^3*e + 219*b^2*c^2*d^2*e^2 - 68*b^3*c*d*e^3 + 8*b^4*e^4 + 10
*(4*c^4*d*e^3 + 5*b*c^3*e^4)*x^3 - 3*(26*c^4*d^2*e^2 - 46*b*c^3*d*e^3 - b^2*c^2*e^4)*x^2 - 2*(52*c^4*d^3*e - 3
9*b*c^3*d^2*e^2 - 15*b^2*c^2*d*e^3 + 2*b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^3*e^2*x + c^3*d*e
) - 2/3465*(315*c^5*e^5*x^5 + 422*c^5*d^5 - 1538*b*c^4*d^4*e + 2130*b^2*c^3*d^3*e^2 - 1382*b^3*c^2*d^2*e^3 + 4
16*b^4*c*d*e^4 - 48*b^5*e^5 + 70*(5*c^5*d*e^4 + 6*b*c^4*e^5)*x^4 - 5*(118*c^5*d^2*e^3 - 214*b*c^4*d*e^4 - 3*b^
2*c^3*e^5)*x^3 - 6*(118*c^5*d^3*e^2 - 101*b*c^4*d^2*e^3 - 20*b^2*c^3*d*e^4 + 3*b^3*c^2*e^5)*x^2 + (211*c^5*d^4
*e - 558*b*c^4*d^3*e^2 + 507*b^2*c^3*d^2*e^3 - 184*b^3*c^2*d*e^4 + 24*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*(
e*x + d)*g/(c^4*e^3*x + c^4*d*e^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4177 vs. \(2 (243) = 486\).

Time = 0.42 (sec) , antiderivative size = 4177, normalized size of antiderivative = 15.64 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

-2/3465*(1155*c*d^3*f*((-(e*x + d)*c + 2*c*d - b*e)^(3/2)/c - (2*sqrt(2*c*d - b*e)*c*d - sqrt(2*c*d - b*e)*b*e
)/c) - 1155*b*d^2*e*f*((-(e*x + d)*c + 2*c*d - b*e)^(3/2)/c - (2*sqrt(2*c*d - b*e)*c*d - sqrt(2*c*d - b*e)*b*e
)/c) + 33*c*d*e^2*f*((22*sqrt(2*c*d - b*e)*c^3*d^3 - 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b
^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)/(c^3*e^2) - (35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(e
*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e + 35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2 - 42*((e*x + d)*c - 2*c
*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d + 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d
- b*e)*b*e - 15*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^3*e^2)) + 33*b*e^3*f*((22*s
qrt(2*c*d - b*e)*c^3*d^3 - 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b^2*c*d*e^2 - 8*sqrt(2*c*d
- b*e)*b^3*e^3)/(c^3*e^2) - (35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(e*x + d)*c + 2*c*d - b*e)^(
3/2)*b*c*d*e + 35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2 - 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d
)*c + 2*c*d - b*e)*c*d + 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e - 15*((e*x + d)
*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^3*e^2)) - 33*c*d^2*e*g*((22*sqrt(2*c*d - b*e)*c^3*d^3
- 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)/(c^3*e^2
) - (35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e + 35*(-(e*x
+ d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2 - 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d +
42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e - 15*((e*x + d)*c - 2*c*d + b*e)^3*sqrt
(-(e*x + d)*c + 2*c*d - b*e))/(c^3*e^2)) + 66*b*d*e^2*g*((22*sqrt(2*c*d - b*e)*c^3*d^3 - 19*sqrt(2*c*d - b*e)*
b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)/(c^3*e^2) - (35*(-(e*x + d)*c +
2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e + 35*(-(e*x + d)*c + 2*c*d - b*e)^(
3/2)*b^2*e^2 - 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d + 42*((e*x + d)*c - 2*c*d
+ b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e - 15*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d -
b*e))/(c^3*e^2)) - 11*c*e^3*f*((26*sqrt(2*c*d - b*e)*c^4*d^4 + 47*sqrt(2*c*d - b*e)*b*c^3*d^3*e - 78*sqrt(2*c*
d - b*e)*b^2*c^2*d^2*e^2 + 56*sqrt(2*c*d - b*e)*b^3*c*d*e^3 - 16*sqrt(2*c*d - b*e)*b^4*e^4)/(c^4*e^3) + (105*(
-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^3*d^3 - 315*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^2*d^2*e + 315*(-(e*x +
d)*c + 2*c*d - b*e)^(3/2)*b^2*c*d*e^2 - 105*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^3*e^3 - 189*((e*x + d)*c - 2*
c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^2*d^2 + 378*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c +
2*c*d - b*e)*b*c*d*e - 189*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*e^2 - 135*((e*x
+ d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d + 135*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x +
d)*c + 2*c*d - b*e)*b*e - 35*((e*x + d)*c - 2*c*d + b*e)^4*sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^4*e^3)) - 11*c
*d*e^2*g*((26*sqrt(2*c*d - b*e)*c^4*d^4 + 47*sqrt(2*c*d - b*e)*b*c^3*d^3*e - 78*sqrt(2*c*d - b*e)*b^2*c^2*d^2*
e^2 + 56*sqrt(2*c*d - b*e)*b^3*c*d*e^3 - 16*sqrt(2*c*d - b*e)*b^4*e^4)/(c^4*e^3) + (105*(-(e*x + d)*c + 2*c*d
- b*e)^(3/2)*c^3*d^3 - 315*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^2*d^2*e + 315*(-(e*x + d)*c + 2*c*d - b*e)^(
3/2)*b^2*c*d*e^2 - 105*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^3*e^3 - 189*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e
*x + d)*c + 2*c*d - b*e)*c^2*d^2 + 378*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c*d*e
- 189*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*e^2 - 135*((e*x + d)*c - 2*c*d + b*e)
^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d + 135*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b
*e - 35*((e*x + d)*c - 2*c*d + b*e)^4*sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^4*e^3)) - 11*b*e^3*g*((26*sqrt(2*c*
d - b*e)*c^4*d^4 + 47*sqrt(2*c*d - b*e)*b*c^3*d^3*e - 78*sqrt(2*c*d - b*e)*b^2*c^2*d^2*e^2 + 56*sqrt(2*c*d - b
*e)*b^3*c*d*e^3 - 16*sqrt(2*c*d - b*e)*b^4*e^4)/(c^4*e^3) + (105*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^3*d^3 -
315*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^2*d^2*e + 315*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c*d*e^2 - 105*
(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^3*e^3 - 189*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e
)*c^2*d^2 + 378*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c*d*e - 189*((e*x + d)*c - 2*
c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*e^2 - 135*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c +
2*c*d - b*e)*c*d + 135*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e - 35*((e*x + d)*c -
2*c*d + b*e)^4*sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^4*e^3)) + c*e^3*g*((422*sqrt(2*c*d - b*e)*c^5*d^5 - 483*sq
rt(2*c*d - b*e)*b*c^4*d^4*e + 1000*sqrt(2*c*d - b*e)*b^2*c^3*d^3*e^2 - 1072*sqrt(2*c*d - b*e)*b^3*c^2*d^2*e^3
+ 576*sqrt(2*c*d - b*e)*b^4*c*d*e^4 - 128*sqrt(2*c*d - b*e)*b^5*e^5)/(c^5*e^4) - (1155*(-(e*x + d)*c + 2*c*d -
b*e)^(3/2)*c^4*d^4 - 4620*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^3*d^3*e + 6930*(-(e*x + d)*c + 2*c*d - b*e)^
(3/2)*b^2*c^2*d^2*e^2 - 4620*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^3*c*d*e^3 + 1155*(-(e*x + d)*c + 2*c*d - b*e
)^(3/2)*b^4*e^4 - 2772*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d^3 + 8316*((e*x + d
)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*d^2*e - 8316*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(
e*x + d)*c + 2*c*d - b*e)*b^2*c*d*e^2 + 2772*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^
3*e^3 - 2970*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^2*d^2 + 5940*((e*x + d)*c - 2*c*
d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c*d*e - 2970*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2
*c*d - b*e)*b^2*e^2 - 1540*((e*x + d)*c - 2*c*d + b*e)^4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d + 1540*((e*x + d
)*c - 2*c*d + b*e)^4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e - 315*((e*x + d)*c - 2*c*d + b*e)^5*sqrt(-(e*x + d)*
c + 2*c*d - b*e))/(c^5*e^4)) + 231*c*d^2*f*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqr
t(2*c*d - b*e)*b^2*e^2)/c^2 + (5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)
*b*e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e))/c^2) - 462*b*d*e*f*((2*sqrt(2*c*d - b
*e)*c^2*d^2 + 3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c*d - b*e)*b^2*e^2)/c^2 + (5*(-(e*x + d)*c + 2*c*d - b*e)
^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*
c*d - b*e))/c^2) - 231*b*d^2*g*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c*d - b*
e)*b^2*e^2)/c^2 + (5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e
*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e))/c^2) + 231*c*d^3*g*((2*sqrt(2*c*d - b*e)*c^2*d^2
+ 3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c*d - b*e)*b^2*e^2)/c^2 + (5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d -
5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e))/
c^2)/e)/e

Mupad [B] (verification not implemented)

Time = 11.88 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.65 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (12\,b\,e\,g+10\,c\,d\,g+11\,c\,e\,f\right )}{99}+\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (3\,g\,b^2\,e^2+214\,g\,b\,c\,d\,e+110\,f\,b\,c\,e^2-118\,g\,c^2\,d^2+88\,f\,c^2\,d\,e\right )}{693\,c}+\frac {2\,c\,e^2\,g\,x^5\,\sqrt {d+e\,x}}{11}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-36\,g\,b^3\,c^2\,e^5+240\,g\,b^2\,c^3\,d\,e^4+66\,f\,b^2\,c^3\,e^5+1212\,g\,b\,c^4\,d^2\,e^3+3036\,f\,b\,c^4\,d\,e^4-1416\,g\,c^5\,d^3\,e^2-1716\,f\,c^5\,d^2\,e^3\right )}{3465\,c^4\,e^3}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (-48\,g\,b^3\,e^3+320\,g\,b^2\,c\,d\,e^2+88\,f\,b^2\,c\,e^3-694\,g\,b\,c^2\,d^2\,e-572\,f\,b\,c^2\,d\,e^2+422\,g\,c^3\,d^3+1177\,f\,c^3\,d^2\,e\right )}{3465\,c^4\,e^3}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (24\,g\,b^3\,e^3-160\,g\,b^2\,c\,d\,e^2-44\,f\,b^2\,c\,e^3+347\,g\,b\,c^2\,d^2\,e+286\,f\,b\,c^2\,d\,e^2-211\,g\,c^3\,d^3+1144\,f\,c^3\,d^2\,e\right )}{3465\,c^3\,e^2}\right )}{x+\frac {d}{e}} \]

[In]

int((f + g*x)*(d + e*x)^(1/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2),x)

[Out]

-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*e*x^4*(d + e*x)^(1/2)*(12*b*e*g + 10*c*d*g + 11*c*e*f))/99 +
(2*x^3*(d + e*x)^(1/2)*(3*b^2*e^2*g - 118*c^2*d^2*g + 110*b*c*e^2*f + 88*c^2*d*e*f + 214*b*c*d*e*g))/(693*c)
+ (2*c*e^2*g*x^5*(d + e*x)^(1/2))/11 + (x^2*(d + e*x)^(1/2)*(66*b^2*c^3*e^5*f - 36*b^3*c^2*e^5*g - 1716*c^5*d^
2*e^3*f - 1416*c^5*d^3*e^2*g + 3036*b*c^4*d*e^4*f + 1212*b*c^4*d^2*e^3*g + 240*b^2*c^3*d*e^4*g))/(3465*c^4*e^3
) + (2*(b*e - c*d)^2*(d + e*x)^(1/2)*(422*c^3*d^3*g - 48*b^3*e^3*g + 88*b^2*c*e^3*f + 1177*c^3*d^2*e*f - 572*b
*c^2*d*e^2*f - 694*b*c^2*d^2*e*g + 320*b^2*c*d*e^2*g))/(3465*c^4*e^3) + (2*x*(b*e - c*d)*(d + e*x)^(1/2)*(24*b
^3*e^3*g - 211*c^3*d^3*g - 44*b^2*c*e^3*f + 1144*c^3*d^2*e*f + 286*b*c^2*d*e^2*f + 347*b*c^2*d^2*e*g - 160*b^2
*c*d*e^2*g))/(3465*c^3*e^2)))/(x + d/e)