\(\int \frac {(f+g x)^4 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x}} \, dx\) [679]

Optimal result

Integrand size = 46, antiderivative size = 336 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=-\frac {128 (c d f-a e g)^3 \left (2 a e^2 g-c d (5 e f-3 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3465 c^5 d^5 e (d+e x)^{3/2}}+\frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1155 c^4 d^4 e \sqrt {d+e x}}+\frac {32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}} \]

[Out]

-128/3465*(-a*e*g+c*d*f)^3*(2*a*e^2*g-c*d*(-3*d*g+5*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^5/d^5/e/(e
*x+d)^(3/2)+32/231*(-a*e*g+c*d*f)^2*(g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3/(e*x+d)^(3/2)+16
/99*(-a*e*g+c*d*f)*(g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2/(e*x+d)^(3/2)+2/11*(g*x+f)^4*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/(e*x+d)^(3/2)+128/1155*g*(-a*e*g+c*d*f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(3/2)/c^4/d^4/e/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {884, 808, 662} \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=-\frac {128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{3465 c^5 d^5 e (d+e x)^{3/2}}+\frac {128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3}{1155 c^4 d^4 e \sqrt {d+e x}}+\frac {32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{99 c^2 d^2 (d+e x)^{3/2}}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}} \]

[In]

Int[((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]

[Out]

(-128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3465
*c^5*d^5*e*(d + e*x)^(3/2)) + (128*g*(c*d*f - a*e*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(1155*c^
4*d^4*e*Sqrt[d + e*x]) + (32*(c*d*f - a*e*g)^2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(231
*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d*f - a*e*g)*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(99
*c^2*d^2*(d + e*x)^(3/2)) + (2*(f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(11*c*d*(d + e*x)^(3
/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 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])

Rule 884

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(-e)*(d + e*x)^(m - 1)*(f + g*x)^n*((a + b*x + c*x^2)^(p + 1)/(c*(m - n - 1))), x] - Dist[n*((c*e*f + c*d
*g - b*e*g)/(c*e*(m - n - 1))), Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b,
c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Int
egerQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])

Rubi steps \begin{align*} \text {integral}& = \frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac {(8 (c d f-a e g)) \int \frac {(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{11 c d} \\ & = \frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac {\left (16 (c d f-a e g)^2\right ) \int \frac {(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{33 c^2 d^2} \\ & = \frac {32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac {\left (64 (c d f-a e g)^3\right ) \int \frac {(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{231 c^3 d^3} \\ & = \frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1155 c^4 d^4 e \sqrt {d+e x}}+\frac {32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac {\left (64 (c d f-a e g)^3 \left (5 f-\frac {3 d g}{e}-\frac {2 a e g}{c d}\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{1155 c^3 d^3} \\ & = \frac {128 (c d f-a e g)^3 \left (5 f-\frac {3 d g}{e}-\frac {2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3465 c^4 d^4 (d+e x)^{3/2}}+\frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1155 c^4 d^4 e \sqrt {d+e x}}+\frac {32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.58 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (11 f+3 g x)+48 a^2 c^2 d^2 e^2 g^2 \left (33 f^2+22 f g x+5 g^2 x^2\right )-8 a c^3 d^3 e g \left (231 f^3+297 f^2 g x+165 f g^2 x^2+35 g^3 x^3\right )+c^4 d^4 \left (1155 f^4+2772 f^3 g x+2970 f^2 g^2 x^2+1540 f g^3 x^3+315 g^4 x^4\right )\right )}{3465 c^5 d^5 (d+e x)^{3/2}} \]

[In]

Integrate[((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*(11*f + 3*g*x) + 48*a^2*c^2*d^2*e^2*g
^2*(33*f^2 + 22*f*g*x + 5*g^2*x^2) - 8*a*c^3*d^3*e*g*(231*f^3 + 297*f^2*g*x + 165*f*g^2*x^2 + 35*g^3*x^3) + c^
4*d^4*(1155*f^4 + 2772*f^3*g*x + 2970*f^2*g^2*x^2 + 1540*f*g^3*x^3 + 315*g^4*x^4)))/(3465*c^5*d^5*(d + e*x)^(3
/2))

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.81

method	result	size
default	\(\frac {2 \left (c d x +a e \right ) \left (315 g^{4} x^{4} c^{4} d^{4}-280 a \,c^{3} d^{3} e \,g^{4} x^{3}+1540 c^{4} d^{4} f \,g^{3} x^{3}+240 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-1320 a \,c^{3} d^{3} e f \,g^{3} x^{2}+2970 c^{4} d^{4} f^{2} g^{2} x^{2}-192 a^{3} c d \,e^{3} g^{4} x +1056 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -2376 a \,c^{3} d^{3} e \,f^{2} g^{2} x +2772 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-704 a^{3} c d \,e^{3} f \,g^{3}+1584 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-1848 a \,c^{3} d^{3} e \,f^{3} g +1155 f^{4} c^{4} d^{4}\right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{3465 c^{5} d^{5} \sqrt {e x +d}}\)	\(273\)
gosper	\(\frac {2 \left (c d x +a e \right ) \left (315 g^{4} x^{4} c^{4} d^{4}-280 a \,c^{3} d^{3} e \,g^{4} x^{3}+1540 c^{4} d^{4} f \,g^{3} x^{3}+240 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-1320 a \,c^{3} d^{3} e f \,g^{3} x^{2}+2970 c^{4} d^{4} f^{2} g^{2} x^{2}-192 a^{3} c d \,e^{3} g^{4} x +1056 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -2376 a \,c^{3} d^{3} e \,f^{2} g^{2} x +2772 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-704 a^{3} c d \,e^{3} f \,g^{3}+1584 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-1848 a \,c^{3} d^{3} e \,f^{3} g +1155 f^{4} c^{4} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3465 c^{5} d^{5} \sqrt {e x +d}}\)	\(283\)

[In]

int((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3465*(c*d*x+a*e)*(315*c^4*d^4*g^4*x^4-280*a*c^3*d^3*e*g^4*x^3+1540*c^4*d^4*f*g^3*x^3+240*a^2*c^2*d^2*e^2*g^4
*x^2-1320*a*c^3*d^3*e*f*g^3*x^2+2970*c^4*d^4*f^2*g^2*x^2-192*a^3*c*d*e^3*g^4*x+1056*a^2*c^2*d^2*e^2*f*g^3*x-23
76*a*c^3*d^3*e*f^2*g^2*x+2772*c^4*d^4*f^3*g*x+128*a^4*e^4*g^4-704*a^3*c*d*e^3*f*g^3+1584*a^2*c^2*d^2*e^2*f^2*g
^2-1848*a*c^3*d^3*e*f^3*g+1155*c^4*d^4*f^4)*((c*d*x+a*e)*(e*x+d))^(1/2)/c^5/d^5/(e*x+d)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.12 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, c^{5} d^{5} g^{4} x^{5} + 1155 \, a c^{4} d^{4} e f^{4} - 1848 \, a^{2} c^{3} d^{3} e^{2} f^{3} g + 1584 \, a^{3} c^{2} d^{2} e^{3} f^{2} g^{2} - 704 \, a^{4} c d e^{4} f g^{3} + 128 \, a^{5} e^{5} g^{4} + 35 \, {\left (44 \, c^{5} d^{5} f g^{3} + a c^{4} d^{4} e g^{4}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{5} f^{2} g^{2} + 22 \, a c^{4} d^{4} e f g^{3} - 4 \, a^{2} c^{3} d^{3} e^{2} g^{4}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{5} f^{3} g + 99 \, a c^{4} d^{4} e f^{2} g^{2} - 44 \, a^{2} c^{3} d^{3} e^{2} f g^{3} + 8 \, a^{3} c^{2} d^{2} e^{3} g^{4}\right )} x^{2} + {\left (1155 \, c^{5} d^{5} f^{4} + 924 \, a c^{4} d^{4} e f^{3} g - 792 \, a^{2} c^{3} d^{3} e^{2} f^{2} g^{2} + 352 \, a^{3} c^{2} d^{2} e^{3} f g^{3} - 64 \, a^{4} c d e^{4} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3465 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]

[In]

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*c^5*d^5*g^4*x^5 + 1155*a*c^4*d^4*e*f^4 - 1848*a^2*c^3*d^3*e^2*f^3*g + 1584*a^3*c^2*d^2*e^3*f^2*g^2
- 704*a^4*c*d*e^4*f*g^3 + 128*a^5*e^5*g^4 + 35*(44*c^5*d^5*f*g^3 + a*c^4*d^4*e*g^4)*x^4 + 10*(297*c^5*d^5*f^2
*g^2 + 22*a*c^4*d^4*e*f*g^3 - 4*a^2*c^3*d^3*e^2*g^4)*x^3 + 6*(462*c^5*d^5*f^3*g + 99*a*c^4*d^4*e*f^2*g^2 - 44*
a^2*c^3*d^3*e^2*f*g^3 + 8*a^3*c^2*d^2*e^3*g^4)*x^2 + (1155*c^5*d^5*f^4 + 924*a*c^4*d^4*e*f^3*g - 792*a^2*c^3*d
^3*e^2*f^2*g^2 + 352*a^3*c^2*d^2*e^3*f*g^3 - 64*a^4*c*d*e^4*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
)*sqrt(e*x + d)/(c^5*d^5*e*x + c^5*d^6)

Sympy [F]

\[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{4}}{\sqrt {d + e x}}\, dx \]

[In]

integrate((g*x+f)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))*(f + g*x)**4/sqrt(d + e*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.95 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (c d x + a e\right )}^{\frac {3}{2}} f^{4}}{3 \, c d} + \frac {8 \, {\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt {c d x + a e} f^{3} g}{15 \, c^{2} d^{2}} + \frac {4 \, {\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{2} g^{2}}{35 \, c^{3} d^{3}} + \frac {8 \, {\left (35 \, c^{4} d^{4} x^{4} + 5 \, a c^{3} d^{3} e x^{3} - 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 8 \, a^{3} c d e^{3} x - 16 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f g^{3}}{315 \, c^{4} d^{4}} + \frac {2 \, {\left (315 \, c^{5} d^{5} x^{5} + 35 \, a c^{4} d^{4} e x^{4} - 40 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 48 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 64 \, a^{4} c d e^{4} x + 128 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} g^{4}}{3465 \, c^{5} d^{5}} \]

[In]

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/3*(c*d*x + a*e)^(3/2)*f^4/(c*d) + 8/15*(3*c^2*d^2*x^2 + a*c*d*e*x - 2*a^2*e^2)*sqrt(c*d*x + a*e)*f^3*g/(c^2*
d^2) + 4/35*(15*c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 - 4*a^2*c*d*e^2*x + 8*a^3*e^3)*sqrt(c*d*x + a*e)*f^2*g^2/(c^3*
d^3) + 8/315*(35*c^4*d^4*x^4 + 5*a*c^3*d^3*e*x^3 - 6*a^2*c^2*d^2*e^2*x^2 + 8*a^3*c*d*e^3*x - 16*a^4*e^4)*sqrt(
c*d*x + a*e)*f*g^3/(c^4*d^4) + 2/3465*(315*c^5*d^5*x^5 + 35*a*c^4*d^4*e*x^4 - 40*a^2*c^3*d^3*e^2*x^3 + 48*a^3*
c^2*d^2*e^3*x^2 - 64*a^4*c*d*e^4*x + 128*a^5*e^5)*sqrt(c*d*x + a*e)*g^4/(c^5*d^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1107 vs. \(2 (306) = 612\).

Time = 0.32 (sec) , antiderivative size = 1107, normalized size of antiderivative = 3.29 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/3465*(1155*f^4*((sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^2*e + a*e^3)*a*e^2)/(c*d) + ((e*x + d)*c*d*e - c*d
^2*e + a*e^3)^(3/2)/(c*d*e))*abs(e)/e^2 + 198*f^2*g^2*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e +
a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)/(c^3*d^3*e^2
) + (35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3
+ 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))/(c^3*d^3*e^5))*abs(e)/e^2 - 44*f*g^3*((35*sqrt(-c*d^2*e + a*e
^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sqrt(-c*d^
2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)/(c^4*d^4*e^3) + (105*((e*x + d)*c*d*e - c*d^2*
e + a*e^3)^(3/2)*a^3*e^9 - 189*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((e*x + d)*c*d*e - c*d^
2*e + a*e^3)^(7/2)*a*e^3 - 35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))/(c^4*d^4*e^7))*abs(e)/e^2 + g^4*((315
*sqrt(-c*d^2*e + a*e^3)*c^5*d^10 - 35*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^8*e^2 - 40*sqrt(-c*d^2*e + a*e^3)*a^2*c^3
*d^6*e^4 - 48*sqrt(-c*d^2*e + a*e^3)*a^3*c^2*d^4*e^6 - 64*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*e^8 - 128*sqrt(-c*d
^2*e + a*e^3)*a^5*e^10)/(c^5*d^5*e^4) + (1155*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772*((e*x
+ d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^3*e^9 + 2970*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*e^6 - 1540*((
e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*a*e^3 + 315*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))/(c^5*d^5*e^9))
*abs(e)/e^2 - 924*f^3*g*((3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^
2*e + a*e^3)*a^2*e^4)/(c^2*d^2) + (5*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((e*x + d)*c*d*e - c*
d^2*e + a*e^3)^(5/2))/(c^2*d^2*e^2))*abs(e)/e^3)/e

Mupad [B] (verification not implemented)

Time = 12.36 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.03 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^4\,x^5}{11}+\frac {256\,a^5\,e^5\,g^4-1408\,a^4\,c\,d\,e^4\,f\,g^3+3168\,a^3\,c^2\,d^2\,e^3\,f^2\,g^2-3696\,a^2\,c^3\,d^3\,e^2\,f^3\,g+2310\,a\,c^4\,d^4\,e\,f^4}{3465\,c^5\,d^5}+\frac {x\,\left (-128\,a^4\,c\,d\,e^4\,g^4+704\,a^3\,c^2\,d^2\,e^3\,f\,g^3-1584\,a^2\,c^3\,d^3\,e^2\,f^2\,g^2+1848\,a\,c^4\,d^4\,e\,f^3\,g+2310\,c^5\,d^5\,f^4\right )}{3465\,c^5\,d^5}+\frac {4\,g\,x^2\,\left (8\,a^3\,e^3\,g^3-44\,a^2\,c\,d\,e^2\,f\,g^2+99\,a\,c^2\,d^2\,e\,f^2\,g+462\,c^3\,d^3\,f^3\right )}{1155\,c^3\,d^3}+\frac {4\,g^2\,x^3\,\left (-4\,a^2\,e^2\,g^2+22\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,c^2\,d^2}+\frac {2\,g^3\,x^4\,\left (a\,e\,g+44\,c\,d\,f\right )}{99\,c\,d}\right )}{\sqrt {d+e\,x}} \]

[In]

int(((f + g*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^(1/2),x)

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*g^4*x^5)/11 + (256*a^5*e^5*g^4 + 2310*a*c^4*d^4*e*f^4 - 369
6*a^2*c^3*d^3*e^2*f^3*g - 1408*a^4*c*d*e^4*f*g^3 + 3168*a^3*c^2*d^2*e^3*f^2*g^2)/(3465*c^5*d^5) + (x*(2310*c^5
*d^5*f^4 - 128*a^4*c*d*e^4*g^4 + 704*a^3*c^2*d^2*e^3*f*g^3 + 1848*a*c^4*d^4*e*f^3*g - 1584*a^2*c^3*d^3*e^2*f^2
*g^2))/(3465*c^5*d^5) + (4*g*x^2*(8*a^3*e^3*g^3 + 462*c^3*d^3*f^3 + 99*a*c^2*d^2*e*f^2*g - 44*a^2*c*d*e^2*f*g^
2))/(1155*c^3*d^3) + (4*g^2*x^3*(297*c^2*d^2*f^2 - 4*a^2*e^2*g^2 + 22*a*c*d*e*f*g))/(693*c^2*d^2) + (2*g^3*x^4
*(a*e*g + 44*c*d*f))/(99*c*d)))/(d + e*x)^(1/2)