\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{\sqrt {d+e x}} \, dx\) [2253]

Optimal result

Integrand size = 46, antiderivative size = 270 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=-\frac {16 (2 c d-b e)^2 (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac {8 (2 c d-b e) (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac {2 (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}} \]

[Out]

-16/9009*(-b*e+2*c*d)^2*(-6*b*e*g-c*d*g+13*c*e*f)*(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)
-8/1287*(-b*e+2*c*d)*(-6*b*e*g-c*d*g+13*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^3/e^2/(e*x+d)^(5/2)-2/
143*(-6*b*e*g-c*d*g+13*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^2/e^2/(e*x+d)^(3/2)-2/13*g*(d*(-b*e+c*d
)-b*e^2*x-c*e^2*x^2)^(7/2)/c/e^2/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 270, 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} \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac {8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{143 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}} \]

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/Sqrt[d + e*x],x]

[Out]

(-16*(2*c*d - b*e)^2*(13*c*e*f - c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9009*c^4*e^2*(
d + e*x)^(7/2)) - (8*(2*c*d - b*e)*(13*c*e*f - c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(
1287*c^3*e^2*(d + e*x)^(5/2)) - (2*(13*c*e*f - c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(
143*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)^(7/2))/(13*c*e^2*Sqrt[d + e*x])

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 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}}-\frac {\left (2 \left (\frac {7}{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 \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx}{13 c e^3} \\ & = -\frac {2 (13 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 )^{7/2}}{143 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 )^{7/2}}{13 c e^2 \sqrt {d+e x}}+\frac {(4 (2 c d-b e) (13 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 )^{5/2}}{(d+e x)^{3/2}} \, dx}{143 c^2 e} \\ & = -\frac {8 (2 c d-b e) (13 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 )^{7/2}}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac {2 (13 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 )^{7/2}}{143 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 )^{7/2}}{13 c e^2 \sqrt {d+e x}}+\frac {\left (8 (2 c d-b e)^2 (13 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 )^{5/2}}{(d+e x)^{5/2}} \, dx}{1287 c^3 e} \\ & = -\frac {16 (2 c d-b e)^2 (13 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 )^{7/2}}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac {8 (2 c d-b e) (13 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 )^{7/2}}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac {2 (13 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 )^{7/2}}{143 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 )^{7/2}}{13 c e^2 \sqrt {d+e x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.68 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-48 b^3 e^3 g+8 b^2 c e^2 (13 e f+44 d g+21 e g x)-2 b c^2 e \left (423 d^2 g+7 e^2 x (26 f+27 g x)+d e (390 f+532 g x)\right )+c^3 \left (542 d^3 g+63 e^3 x^2 (13 f+11 g x)+14 d e^2 x (169 f+144 g x)+d^2 e (1963 f+1897 g x)\right )\right )}{9009 c^4 e^2 \sqrt {d+e x}} \]

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/Sqrt[d + e*x],x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-48*b^3*e^3*g + 8*b^2*c*e^2*(13*e*f + 44*d
*g + 21*e*g*x) - 2*b*c^2*e*(423*d^2*g + 7*e^2*x*(26*f + 27*g*x) + d*e*(390*f + 532*g*x)) + c^3*(542*d^3*g + 63
*e^3*x^2*(13*f + 11*g*x) + 14*d*e^2*x*(169*f + 144*g*x) + d^2*e*(1963*f + 1897*g*x))))/(9009*c^4*e^2*Sqrt[d +
e*x])

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {2 \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (x c e +b e -c d \right )^{3} \left (-693 g \,e^{3} x^{3} c^{3}+378 b \,c^{2} e^{3} g \,x^{2}-2016 c^{3} d \,e^{2} g \,x^{2}-819 c^{3} e^{3} f \,x^{2}-168 b^{2} c \,e^{3} g x +1064 b \,c^{2} d \,e^{2} g x +364 b \,c^{2} e^{3} f x -1897 c^{3} d^{2} e g x -2366 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -352 b^{2} c d \,e^{2} g -104 b^{2} c \,e^{3} f +846 b \,c^{2} d^{2} e g +780 b \,c^{2} d \,e^{2} f -542 c^{3} d^{3} g -1963 d^{2} f \,c^{3} e \right )}{9009 \sqrt {e x +d}\, c^{4} e^{2}}\)	\(229\)
gosper	\(-\frac {2 \left (x c e +b e -c d \right ) \left (-693 g \,e^{3} x^{3} c^{3}+378 b \,c^{2} e^{3} g \,x^{2}-2016 c^{3} d \,e^{2} g \,x^{2}-819 c^{3} e^{3} f \,x^{2}-168 b^{2} c \,e^{3} g x +1064 b \,c^{2} d \,e^{2} g x +364 b \,c^{2} e^{3} f x -1897 c^{3} d^{2} e g x -2366 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -352 b^{2} c d \,e^{2} g -104 b^{2} c \,e^{3} f +846 b \,c^{2} d^{2} e g +780 b \,c^{2} d \,e^{2} f -542 c^{3} d^{3} g -1963 d^{2} f \,c^{3} e \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{9009 c^{4} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\)	\(235\)

[In]

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/9009*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/(e*x+d)^(1/2)*(c*e*x+b*e-c*d)^3*(-693*c^3*e^3*g*x^3+378*b*c^2*e^3*g*x
^2-2016*c^3*d*e^2*g*x^2-819*c^3*e^3*f*x^2-168*b^2*c*e^3*g*x+1064*b*c^2*d*e^2*g*x+364*b*c^2*e^3*f*x-1897*c^3*d^
2*e*g*x-2366*c^3*d*e^2*f*x+48*b^3*e^3*g-352*b^2*c*d*e^2*g-104*b^2*c*e^3*f+846*b*c^2*d^2*e*g+780*b*c^2*d*e^2*f-
542*c^3*d^3*g-1963*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. 675 vs. \(2 (246) = 492\).

Time = 0.33 (sec) , antiderivative size = 675, normalized size of antiderivative = 2.50 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (693 \, c^{6} e^{6} g x^{6} + 63 \, {\left (13 \, c^{6} e^{6} f - {\left (c^{6} d e^{5} - 27 \, b c^{5} e^{6}\right )} g\right )} x^{5} - 7 \, {\left (13 \, {\left (c^{6} d e^{5} - 23 \, b c^{5} e^{6}\right )} f + {\left (296 \, c^{6} d^{2} e^{4} - 280 \, b c^{5} d e^{5} - 159 \, b^{2} c^{4} e^{6}\right )} g\right )} x^{4} - {\left (13 \, {\left (206 \, c^{6} d^{2} e^{4} - 192 \, b c^{5} d e^{5} - 113 \, b^{2} c^{4} e^{6}\right )} f - {\left (206 \, c^{6} d^{3} e^{3} - 3114 \, b c^{5} d^{2} e^{4} + 2893 \, b^{2} c^{4} d e^{5} + 15 \, b^{3} c^{3} e^{6}\right )} g\right )} x^{3} + 3 \, {\left (13 \, {\left (10 \, c^{6} d^{3} e^{3} - 118 \, b c^{5} d^{2} e^{4} + 107 \, b^{2} c^{4} d e^{5} + b^{3} c^{3} e^{6}\right )} f + {\left (683 \, c^{6} d^{4} e^{2} - 1328 \, b c^{5} d^{3} e^{3} + 601 \, b^{2} c^{4} d^{2} e^{4} + 50 \, b^{3} c^{3} d e^{5} - 6 \, b^{4} c^{2} e^{6}\right )} g\right )} x^{2} - 13 \, {\left (151 \, c^{6} d^{5} e - 513 \, b c^{5} d^{4} e^{2} + 641 \, b^{2} c^{4} d^{3} e^{3} - 355 \, b^{3} c^{3} d^{2} e^{4} + 84 \, b^{4} c^{2} d e^{5} - 8 \, b^{5} c e^{6}\right )} f - 2 \, {\left (271 \, c^{6} d^{6} - 1236 \, b c^{5} d^{5} e + 2258 \, b^{2} c^{4} d^{4} e^{2} - 2092 \, b^{3} c^{3} d^{3} e^{3} + 1023 \, b^{4} c^{2} d^{2} e^{4} - 248 \, b^{5} c d e^{5} + 24 \, b^{6} e^{6}\right )} g + {\left (13 \, {\left (271 \, c^{6} d^{4} e^{2} - 512 \, b c^{5} d^{3} e^{3} + 207 \, b^{2} c^{4} d^{2} e^{4} + 38 \, b^{3} c^{3} d e^{5} - 4 \, b^{4} c^{2} e^{6}\right )} f - {\left (271 \, c^{6} d^{5} e - 965 \, b c^{5} d^{4} e^{2} + 1293 \, b^{2} c^{4} d^{3} e^{3} - 799 \, b^{3} c^{3} d^{2} e^{4} + 224 \, b^{4} c^{2} d e^{5} - 24 \, b^{5} c e^{6}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{9009 \, {\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \]

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/9009*(693*c^6*e^6*g*x^6 + 63*(13*c^6*e^6*f - (c^6*d*e^5 - 27*b*c^5*e^6)*g)*x^5 - 7*(13*(c^6*d*e^5 - 23*b*c^5
*e^6)*f + (296*c^6*d^2*e^4 - 280*b*c^5*d*e^5 - 159*b^2*c^4*e^6)*g)*x^4 - (13*(206*c^6*d^2*e^4 - 192*b*c^5*d*e^
5 - 113*b^2*c^4*e^6)*f - (206*c^6*d^3*e^3 - 3114*b*c^5*d^2*e^4 + 2893*b^2*c^4*d*e^5 + 15*b^3*c^3*e^6)*g)*x^3 +
3*(13*(10*c^6*d^3*e^3 - 118*b*c^5*d^2*e^4 + 107*b^2*c^4*d*e^5 + b^3*c^3*e^6)*f + (683*c^6*d^4*e^2 - 1328*b*c^
5*d^3*e^3 + 601*b^2*c^4*d^2*e^4 + 50*b^3*c^3*d*e^5 - 6*b^4*c^2*e^6)*g)*x^2 - 13*(151*c^6*d^5*e - 513*b*c^5*d^4
*e^2 + 641*b^2*c^4*d^3*e^3 - 355*b^3*c^3*d^2*e^4 + 84*b^4*c^2*d*e^5 - 8*b^5*c*e^6)*f - 2*(271*c^6*d^6 - 1236*b
*c^5*d^5*e + 2258*b^2*c^4*d^4*e^2 - 2092*b^3*c^3*d^3*e^3 + 1023*b^4*c^2*d^2*e^4 - 248*b^5*c*d*e^5 + 24*b^6*e^6
)*g + (13*(271*c^6*d^4*e^2 - 512*b*c^5*d^3*e^3 + 207*b^2*c^4*d^2*e^4 + 38*b^3*c^3*d*e^5 - 4*b^4*c^2*e^6)*f - (
271*c^6*d^5*e - 965*b*c^5*d^4*e^2 + 1293*b^2*c^4*d^3*e^3 - 799*b^3*c^3*d^2*e^4 + 224*b^4*c^2*d*e^5 - 24*b^5*c*
e^6)*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]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\sqrt {d + e x}}\, dx \]

[In]

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/sqrt(d + e*x), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (246) = 492\).

Time = 0.25 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.36 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (63 \, c^{5} e^{5} x^{5} - 151 \, c^{5} d^{5} + 513 \, b c^{4} d^{4} e - 641 \, b^{2} c^{3} d^{3} e^{2} + 355 \, b^{3} c^{2} d^{2} e^{3} - 84 \, b^{4} c d e^{4} + 8 \, b^{5} e^{5} - 7 \, {\left (c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} x^{4} - {\left (206 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \, {\left (10 \, c^{5} d^{3} e^{2} - 118 \, b c^{4} d^{2} e^{3} + 107 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} x^{2} + {\left (271 \, c^{5} d^{4} e - 512 \, b c^{4} d^{3} e^{2} + 207 \, b^{2} c^{3} d^{2} e^{3} + 38 \, b^{3} c^{2} d e^{4} - 4 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{693 \, c^{3} e} + \frac {2 \, {\left (693 \, c^{6} e^{6} x^{6} - 542 \, c^{6} d^{6} + 2472 \, b c^{5} d^{5} e - 4516 \, b^{2} c^{4} d^{4} e^{2} + 4184 \, b^{3} c^{3} d^{3} e^{3} - 2046 \, b^{4} c^{2} d^{2} e^{4} + 496 \, b^{5} c d e^{5} - 48 \, b^{6} e^{6} - 63 \, {\left (c^{6} d e^{5} - 27 \, b c^{5} e^{6}\right )} x^{5} - 7 \, {\left (296 \, c^{6} d^{2} e^{4} - 280 \, b c^{5} d e^{5} - 159 \, b^{2} c^{4} e^{6}\right )} x^{4} + {\left (206 \, c^{6} d^{3} e^{3} - 3114 \, b c^{5} d^{2} e^{4} + 2893 \, b^{2} c^{4} d e^{5} + 15 \, b^{3} c^{3} e^{6}\right )} x^{3} + 3 \, {\left (683 \, c^{6} d^{4} e^{2} - 1328 \, b c^{5} d^{3} e^{3} + 601 \, b^{2} c^{4} d^{2} e^{4} + 50 \, b^{3} c^{3} d e^{5} - 6 \, b^{4} c^{2} e^{6}\right )} x^{2} - {\left (271 \, c^{6} d^{5} e - 965 \, b c^{5} d^{4} e^{2} + 1293 \, b^{2} c^{4} d^{3} e^{3} - 799 \, b^{3} c^{3} d^{2} e^{4} + 224 \, b^{4} c^{2} d e^{5} - 24 \, b^{5} c e^{6}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{9009 \, c^{4} e^{2}} \]

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/693*(63*c^5*e^5*x^5 - 151*c^5*d^5 + 513*b*c^4*d^4*e - 641*b^2*c^3*d^3*e^2 + 355*b^3*c^2*d^2*e^3 - 84*b^4*c*d
*e^4 + 8*b^5*e^5 - 7*(c^5*d*e^4 - 23*b*c^4*e^5)*x^4 - (206*c^5*d^2*e^3 - 192*b*c^4*d*e^4 - 113*b^2*c^3*e^5)*x^
3 + 3*(10*c^5*d^3*e^2 - 118*b*c^4*d^2*e^3 + 107*b^2*c^3*d*e^4 + b^3*c^2*e^5)*x^2 + (271*c^5*d^4*e - 512*b*c^4*
d^3*e^2 + 207*b^2*c^3*d^2*e^3 + 38*b^3*c^2*d*e^4 - 4*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*f/(c^3*e) + 2/9009
*(693*c^6*e^6*x^6 - 542*c^6*d^6 + 2472*b*c^5*d^5*e - 4516*b^2*c^4*d^4*e^2 + 4184*b^3*c^3*d^3*e^3 - 2046*b^4*c^
2*d^2*e^4 + 496*b^5*c*d*e^5 - 48*b^6*e^6 - 63*(c^6*d*e^5 - 27*b*c^5*e^6)*x^5 - 7*(296*c^6*d^2*e^4 - 280*b*c^5*
d*e^5 - 159*b^2*c^4*e^6)*x^4 + (206*c^6*d^3*e^3 - 3114*b*c^5*d^2*e^4 + 2893*b^2*c^4*d*e^5 + 15*b^3*c^3*e^6)*x^
3 + 3*(683*c^6*d^4*e^2 - 1328*b*c^5*d^3*e^3 + 601*b^2*c^4*d^2*e^4 + 50*b^3*c^3*d*e^5 - 6*b^4*c^2*e^6)*x^2 - (2
71*c^6*d^5*e - 965*b*c^5*d^4*e^2 + 1293*b^2*c^4*d^3*e^3 - 799*b^3*c^3*d^2*e^4 + 224*b^4*c^2*d*e^5 - 24*b^5*c*e
^6)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^4*e^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6983 vs. \(2 (246) = 492\).

Time = 0.52 (sec) , antiderivative size = 6983, normalized size of antiderivative = 25.86 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-2/45045*(15015*c^2*d^4*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) - 30030*b*c*d^3*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) + 15015*b^2*d^2*e^2*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) + 858*c^2*d^2*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*sq
rt(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)) -
858*b*c*d*e^3*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)) - 429*b^2*e^4*f*((22*sq
rt(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)) + 858*b*c*d^2*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)) - 858*b^2*d*e^3*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)) + 286*b*c*e^4*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*sqr
t(-(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)) - 286*c^2*d^2*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)) + 286*b*c*d*
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 - 1
89*((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)) + 143*b^2*e^4*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)) - 13*c^2*e^4*f*((422*sqrt(2*c*d - b*e)*c^5*d^5 - 4
83*sqrt(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*sq
rt(-(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)) - 26*b*c*e^4*g*((422*sqrt(2*c*d - b*e)*c^5*d^5 - 483*sqrt(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 - 277
2*((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)) + 5*c^2*e^4*g*((542*sqrt(2*c*d - b*e)*c^6*d^6 + 1117*sqrt(2*c*d - b*e)*b*c^5*d^5*e - 2966*sqrt(2*c*d -
b*e)*b^2*c^4*d^4*e^2 + 4144*sqrt(2*c*d - b*e)*b^3*c^3*d^3*e^3 - 3296*sqrt(2*c*d - b*e)*b^4*c^2*d^2*e^4 + 1408
*sqrt(2*c*d - b*e)*b^5*c*d*e^5 - 256*sqrt(2*c*d - b*e)*b^6*e^6)/(c^6*e^5) + (3003*(-(e*x + d)*c + 2*c*d - b*e)
^(3/2)*c^5*d^5 - 15015*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^4*d^4*e + 30030*(-(e*x + d)*c + 2*c*d - b*e)^(3/
2)*b^2*c^3*d^3*e^2 - 30030*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^3*c^2*d^2*e^3 + 15015*(-(e*x + d)*c + 2*c*d -
b*e)^(3/2)*b^4*c*d*e^4 - 3003*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^5*e^5 - 9009*((e*x + d)*c - 2*c*d + b*e)^2*
sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d^4 + 36036*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e
)*b*c^3*d^3*e - 54054*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^2*d^2*e^2 + 36036*(
(e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^3*c*d*e^3 - 9009*((e*x + d)*c - 2*c*d + b*e)^2
*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^4*e^4 - 12870*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*
e)*c^3*d^3 + 38610*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*d^2*e - 38610*((e*x +
d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c*d*e^2 + 12870*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(
-(e*x + d)*c + 2*c*d - b*e)*b^3*e^3 - 10010*((e*x + d)*c - 2*c*d + b*e)^4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^2
*d^2 + 20020*((e*x + d)*c - 2*c*d + b*e)^4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c*d*e - 10010*((e*x + d)*c - 2*c
*d + b*e)^4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*e^2 - 4095*((e*x + d)*c - 2*c*d + b*e)^5*sqrt(-(e*x + d)*c +
2*c*d - b*e)*c*d + 4095*((e*x + d)*c - 2*c*d + b*e)^5*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e - 693*((e*x + d)*c
- 2*c*d + b*e)^6*sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^6*e^5)) - 6006*b*c*d^2*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) + 6006*b^2*d*e^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*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) - 6006*b*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
) + 3003*c^2*d^4*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 + 3003*b^2*d^2*e*g*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 3*s
qrt(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

Mupad [B] (verification not implemented)

Time = 12.33 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.82 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e^2\,x^4\,\left (159\,g\,b^2\,e^2+280\,g\,b\,c\,d\,e+299\,f\,b\,c\,e^2-296\,g\,c^2\,d^2-13\,f\,c^2\,d\,e\right )}{1287}+\frac {2\,c^2\,e^4\,g\,x^6}{13}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\left (-6\,g\,b^3\,e^3+44\,g\,b^2\,c\,d\,e^2+13\,f\,b^2\,c\,e^3+645\,g\,b\,c^2\,d^2\,e+1404\,f\,b\,c^2\,d\,e^2-683\,g\,c^3\,d^3-130\,f\,c^3\,d^2\,e\right )}{3003\,c^2}+\frac {x^3\,\left (30\,g\,b^3\,c^3\,e^6+5786\,g\,b^2\,c^4\,d\,e^5+2938\,f\,b^2\,c^4\,e^6-6228\,g\,b\,c^5\,d^2\,e^4+4992\,f\,b\,c^5\,d\,e^5+412\,g\,c^6\,d^3\,e^3-5356\,f\,c^6\,d^2\,e^4\right )}{9009\,c^4\,e^2}+\frac {2\,c\,e^3\,x^5\,\left (27\,b\,e\,g-c\,d\,g+13\,c\,e\,f\right )}{143}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\left (-48\,g\,b^3\,e^3+352\,g\,b^2\,c\,d\,e^2+104\,f\,b^2\,c\,e^3-846\,g\,b\,c^2\,d^2\,e-780\,f\,b\,c^2\,d\,e^2+542\,g\,c^3\,d^3+1963\,f\,c^3\,d^2\,e\right )}{9009\,c^4\,e^2}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\left (24\,g\,b^3\,e^3-176\,g\,b^2\,c\,d\,e^2-52\,f\,b^2\,c\,e^3+423\,g\,b\,c^2\,d^2\,e+390\,f\,b\,c^2\,d\,e^2-271\,g\,c^3\,d^3+3523\,f\,c^3\,d^2\,e\right )}{9009\,c^3\,e}\right )}{\sqrt {d+e\,x}} \]

[In]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(1/2),x)

[Out]

((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*e^2*x^4*(159*b^2*e^2*g - 296*c^2*d^2*g + 299*b*c*e^2*f - 13*c
^2*d*e*f + 280*b*c*d*e*g))/1287 + (2*c^2*e^4*g*x^6)/13 + (2*x^2*(b*e - c*d)*(13*b^2*c*e^3*f - 683*c^3*d^3*g -
6*b^3*e^3*g - 130*c^3*d^2*e*f + 1404*b*c^2*d*e^2*f + 645*b*c^2*d^2*e*g + 44*b^2*c*d*e^2*g))/(3003*c^2) + (x^3*
(2938*b^2*c^4*e^6*f + 30*b^3*c^3*e^6*g - 5356*c^6*d^2*e^4*f + 412*c^6*d^3*e^3*g + 4992*b*c^5*d*e^5*f - 6228*b*
c^5*d^2*e^4*g + 5786*b^2*c^4*d*e^5*g))/(9009*c^4*e^2) + (2*c*e^3*x^5*(27*b*e*g - c*d*g + 13*c*e*f))/143 + (2*(
b*e - c*d)^3*(542*c^3*d^3*g - 48*b^3*e^3*g + 104*b^2*c*e^3*f + 1963*c^3*d^2*e*f - 780*b*c^2*d*e^2*f - 846*b*c^
2*d^2*e*g + 352*b^2*c*d*e^2*g))/(9009*c^4*e^2) + (2*x*(b*e - c*d)^2*(24*b^3*e^3*g - 271*c^3*d^3*g - 52*b^2*c*e
^3*f + 3523*c^3*d^2*e*f + 390*b*c^2*d*e^2*f + 423*b*c^2*d^2*e*g - 176*b^2*c*d*e^2*g))/(9009*c^3*e)))/(d + e*x)
^(1/2)