\(\int \frac {f+g x}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\) [2215]

Optimal result

Integrand size = 44, antiderivative size = 285 \[ \int \frac {f+g x}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (2 c d-b e) (d+e x)^4}-\frac {2 (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 e^2 (2 c d-b e)^2 (d+e x)^3}-\frac {8 c (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^2}-\frac {16 c^2 (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^4 (d+e x)} \]

[Out]

-2/7*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)/(e*x+d)^4-2/35*(-7*b*e*g+8*c*d*g+6*c*e
*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^3-8/105*c*(-7*b*e*g+8*c*d*g+6*c*e*f)*(d*
(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)^3/(e*x+d)^2-16/105*c^2*(-7*b*e*g+8*c*d*g+6*c*e*f)*(d*(-b*
e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)^4/(e*x+d)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {806, 672, 664} \[ \int \frac {f+g x}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {16 c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x) (2 c d-b e)^4}-\frac {8 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x)^2 (2 c d-b e)^3}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{35 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (d+e x)^4 (2 c d-b e)} \]

[In]

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

[Out]

(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(7*e^2*(2*c*d - b*e)*(d + e*x)^4) - (2*(6*c*e*f + 8
*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^3) - (8*c*(6*c*
e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^3*(d + e*x)^2) - (1
6*c^2*(6*c*e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^4*(d + e
*x))

Rule 664

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^m*((a +
b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b*e))), 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 + 2*p + 2, 0]

Rule 672

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a
+ b*x + c*x^2)^(p + 1)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d -
b*e))), Int[(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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(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] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (2 c d-b e) (d+e x)^4}+\frac {(6 c e f+8 c d g-7 b e g) \int \frac {1}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{7 e (2 c d-b e)} \\ & = -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (2 c d-b e) (d+e x)^4}-\frac {2 (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 e^2 (2 c d-b e)^2 (d+e x)^3}+\frac {(4 c (6 c e f+8 c d g-7 b e g)) \int \frac {1}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{35 e (2 c d-b e)^2} \\ & = -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (2 c d-b e) (d+e x)^4}-\frac {2 (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 e^2 (2 c d-b e)^2 (d+e x)^3}-\frac {8 c (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^2}+\frac {\left (8 c^2 (6 c e f+8 c d g-7 b e g)\right ) \int \frac {1}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{105 e (2 c d-b e)^3} \\ & = -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (2 c d-b e) (d+e x)^4}-\frac {2 (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 e^2 (2 c d-b e)^2 (d+e x)^3}-\frac {8 c (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^2}-\frac {16 c^2 (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^4 (d+e x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.87 \[ \int \frac {f+g x}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {2 (-c d+b e+c e x) \left (-3 b^3 e^3 (5 e f+2 d g+7 e g x)+8 c^3 \left (13 d^4 g+6 e^4 f x^3+8 d e^3 x^2 (3 f+g x)+4 d^3 e (9 f+13 g x)+d^2 e^2 x (39 f+32 g x)\right )+2 b^2 c e^2 \left (23 d^2 g+e^2 x (9 f+14 g x)+d e (54 f+82 g x)\right )-4 b c^2 e \left (36 d^3 g+2 e^3 x^2 (3 f+7 g x)+2 d e^2 x (15 f+32 g x)+d^2 e (69 f+131 g x)\right )\right )}{105 e^2 (-2 c d+b e)^4 (d+e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))}} \]

[In]

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

[Out]

(2*(-(c*d) + b*e + c*e*x)*(-3*b^3*e^3*(5*e*f + 2*d*g + 7*e*g*x) + 8*c^3*(13*d^4*g + 6*e^4*f*x^3 + 8*d*e^3*x^2*
(3*f + g*x) + 4*d^3*e*(9*f + 13*g*x) + d^2*e^2*x*(39*f + 32*g*x)) + 2*b^2*c*e^2*(23*d^2*g + e^2*x*(9*f + 14*g*
x) + d*e*(54*f + 82*g*x)) - 4*b*c^2*e*(36*d^3*g + 2*e^3*x^2*(3*f + 7*g*x) + 2*d*e^2*x*(15*f + 32*g*x) + d^2*e*
(69*f + 131*g*x))))/(105*e^2*(-2*c*d + b*e)^4*(d + e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.30

method	result	size
trager	\(\frac {2 \left (56 b \,c^{2} e^{4} g \,x^{3}-64 c^{3} d \,e^{3} g \,x^{3}-48 c^{3} e^{4} f \,x^{3}-28 b^{2} c \,e^{4} g \,x^{2}+256 b \,c^{2} d \,e^{3} g \,x^{2}+24 b \,c^{2} e^{4} f \,x^{2}-256 c^{3} d^{2} e^{2} g \,x^{2}-192 c^{3} d \,e^{3} f \,x^{2}+21 b^{3} e^{4} g x -164 b^{2} c d \,e^{3} g x -18 b^{2} c \,e^{4} f x +524 b \,c^{2} d^{2} e^{2} g x +120 b \,c^{2} d \,e^{3} f x -416 c^{3} d^{3} e g x -312 c^{3} d^{2} e^{2} f x +6 b^{3} d \,e^{3} g +15 b^{3} e^{4} f -46 b^{2} c \,d^{2} e^{2} g -108 b^{2} c d \,e^{3} f +144 b \,c^{2} d^{3} e g +276 b \,c^{2} d^{2} e^{2} f -104 c^{3} d^{4} g -288 c^{3} d^{3} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{105 \left (b^{4} e^{4}-8 b^{3} c d \,e^{3}+24 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right ) e^{2} \left (e x +d \right )^{4}}\)	\(370\)
gosper	\(-\frac {2 \left (x c e +b e -c d \right ) \left (56 b \,c^{2} e^{4} g \,x^{3}-64 c^{3} d \,e^{3} g \,x^{3}-48 c^{3} e^{4} f \,x^{3}-28 b^{2} c \,e^{4} g \,x^{2}+256 b \,c^{2} d \,e^{3} g \,x^{2}+24 b \,c^{2} e^{4} f \,x^{2}-256 c^{3} d^{2} e^{2} g \,x^{2}-192 c^{3} d \,e^{3} f \,x^{2}+21 b^{3} e^{4} g x -164 b^{2} c d \,e^{3} g x -18 b^{2} c \,e^{4} f x +524 b \,c^{2} d^{2} e^{2} g x +120 b \,c^{2} d \,e^{3} f x -416 c^{3} d^{3} e g x -312 c^{3} d^{2} e^{2} f x +6 b^{3} d \,e^{3} g +15 b^{3} e^{4} f -46 b^{2} c \,d^{2} e^{2} g -108 b^{2} c d \,e^{3} f +144 b \,c^{2} d^{3} e g +276 b \,c^{2} d^{2} e^{2} f -104 c^{3} d^{4} g -288 c^{3} d^{3} e f \right )}{105 \left (e x +d \right )^{3} e^{2} \left (b^{4} e^{4}-8 b^{3} c d \,e^{3}+24 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\)	\(382\)
default	\(\frac {g \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{5 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {4 c \,e^{2} \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 c \,e^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )}\right )}{5 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{4}}+\frac {\left (-d g +e f \right ) \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{7 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{4}}+\frac {6 c \,e^{2} \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{5 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {4 c \,e^{2} \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 c \,e^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )}\right )}{5 \left (-b \,e^{2}+2 c d e \right )}\right )}{7 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{5}}\)	\(533\)

[In]

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

[Out]

2/105*(56*b*c^2*e^4*g*x^3-64*c^3*d*e^3*g*x^3-48*c^3*e^4*f*x^3-28*b^2*c*e^4*g*x^2+256*b*c^2*d*e^3*g*x^2+24*b*c^
2*e^4*f*x^2-256*c^3*d^2*e^2*g*x^2-192*c^3*d*e^3*f*x^2+21*b^3*e^4*g*x-164*b^2*c*d*e^3*g*x-18*b^2*c*e^4*f*x+524*
b*c^2*d^2*e^2*g*x+120*b*c^2*d*e^3*f*x-416*c^3*d^3*e*g*x-312*c^3*d^2*e^2*f*x+6*b^3*d*e^3*g+15*b^3*e^4*f-46*b^2*
c*d^2*e^2*g-108*b^2*c*d*e^3*f+144*b*c^2*d^3*e*g+276*b*c^2*d^2*e^2*f-104*c^3*d^4*g-288*c^3*d^3*e*f)/(b^4*e^4-8*
b^3*c*d*e^3+24*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d^4)/e^2/(e*x+d)^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (269) = 538\).

Time = 40.49 (sec) , antiderivative size = 606, normalized size of antiderivative = 2.13 \[ \int \frac {f+g x}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (8 \, {\left (6 \, c^{3} e^{4} f + {\left (8 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} g\right )} x^{3} + 4 \, {\left (6 \, {\left (8 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} f + {\left (64 \, c^{3} d^{2} e^{2} - 64 \, b c^{2} d e^{3} + 7 \, b^{2} c e^{4}\right )} g\right )} x^{2} + 3 \, {\left (96 \, c^{3} d^{3} e - 92 \, b c^{2} d^{2} e^{2} + 36 \, b^{2} c d e^{3} - 5 \, b^{3} e^{4}\right )} f + 2 \, {\left (52 \, c^{3} d^{4} - 72 \, b c^{2} d^{3} e + 23 \, b^{2} c d^{2} e^{2} - 3 \, b^{3} d e^{3}\right )} g + {\left (6 \, {\left (52 \, c^{3} d^{2} e^{2} - 20 \, b c^{2} d e^{3} + 3 \, b^{2} c e^{4}\right )} f + {\left (416 \, c^{3} d^{3} e - 524 \, b c^{2} d^{2} e^{2} + 164 \, b^{2} c d e^{3} - 21 \, b^{3} e^{4}\right )} g\right )} x\right )}}{105 \, {\left (16 \, c^{4} d^{8} e^{2} - 32 \, b c^{3} d^{7} e^{3} + 24 \, b^{2} c^{2} d^{6} e^{4} - 8 \, b^{3} c d^{5} e^{5} + b^{4} d^{4} e^{6} + {\left (16 \, c^{4} d^{4} e^{6} - 32 \, b c^{3} d^{3} e^{7} + 24 \, b^{2} c^{2} d^{2} e^{8} - 8 \, b^{3} c d e^{9} + b^{4} e^{10}\right )} x^{4} + 4 \, {\left (16 \, c^{4} d^{5} e^{5} - 32 \, b c^{3} d^{4} e^{6} + 24 \, b^{2} c^{2} d^{3} e^{7} - 8 \, b^{3} c d^{2} e^{8} + b^{4} d e^{9}\right )} x^{3} + 6 \, {\left (16 \, c^{4} d^{6} e^{4} - 32 \, b c^{3} d^{5} e^{5} + 24 \, b^{2} c^{2} d^{4} e^{6} - 8 \, b^{3} c d^{3} e^{7} + b^{4} d^{2} e^{8}\right )} x^{2} + 4 \, {\left (16 \, c^{4} d^{7} e^{3} - 32 \, b c^{3} d^{6} e^{4} + 24 \, b^{2} c^{2} d^{5} e^{5} - 8 \, b^{3} c d^{4} e^{6} + b^{4} d^{3} e^{7}\right )} x\right )}} \]

[In]

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

[Out]

-2/105*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(8*(6*c^3*e^4*f + (8*c^3*d*e^3 - 7*b*c^2*e^4)*g)*x^3 + 4*(6*
(8*c^3*d*e^3 - b*c^2*e^4)*f + (64*c^3*d^2*e^2 - 64*b*c^2*d*e^3 + 7*b^2*c*e^4)*g)*x^2 + 3*(96*c^3*d^3*e - 92*b*
c^2*d^2*e^2 + 36*b^2*c*d*e^3 - 5*b^3*e^4)*f + 2*(52*c^3*d^4 - 72*b*c^2*d^3*e + 23*b^2*c*d^2*e^2 - 3*b^3*d*e^3)
*g + (6*(52*c^3*d^2*e^2 - 20*b*c^2*d*e^3 + 3*b^2*c*e^4)*f + (416*c^3*d^3*e - 524*b*c^2*d^2*e^2 + 164*b^2*c*d*e
^3 - 21*b^3*e^4)*g)*x)/(16*c^4*d^8*e^2 - 32*b*c^3*d^7*e^3 + 24*b^2*c^2*d^6*e^4 - 8*b^3*c*d^5*e^5 + b^4*d^4*e^6
+ (16*c^4*d^4*e^6 - 32*b*c^3*d^3*e^7 + 24*b^2*c^2*d^2*e^8 - 8*b^3*c*d*e^9 + b^4*e^10)*x^4 + 4*(16*c^4*d^5*e^5
- 32*b*c^3*d^4*e^6 + 24*b^2*c^2*d^3*e^7 - 8*b^3*c*d^2*e^8 + b^4*d*e^9)*x^3 + 6*(16*c^4*d^6*e^4 - 32*b*c^3*d^5
*e^5 + 24*b^2*c^2*d^4*e^6 - 8*b^3*c*d^3*e^7 + b^4*d^2*e^8)*x^2 + 4*(16*c^4*d^7*e^3 - 32*b*c^3*d^6*e^4 + 24*b^2
*c^2*d^5*e^5 - 8*b^3*c*d^4*e^6 + b^4*d^3*e^7)*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {f+g x}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` f
or more deta

Giac [F(-2)]

Exception generated. \[ \int \frac {f+g x}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{1,[8,4,16,0]%%%}+%%%{-16,[7,5,15,1]%%%}+%%%{112,[6,6
,14,2]%%%}+

Mupad [B] (verification not implemented)

Time = 13.69 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.19 \[ \int \frac {f+g x}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {\left (\frac {40\,c^2\,d\,g+48\,c^2\,e\,f-40\,b\,c\,e\,g}{35\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}-\frac {8\,c^2\,d\,g}{35\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {8\,c\,g\,\left (2\,b\,e-3\,c\,d\right )}{35\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}-\frac {8\,c^2\,d\,g}{35\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {2\,b\,g}{7\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}-\frac {4\,c\,d\,g}{7\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {16\,c\,d\,g-16\,b\,e\,g+12\,c\,e\,f}{7\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}+\frac {4\,c\,d\,g}{7\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {2\,f}{7\,b\,e^2-14\,c\,d\,e}-\frac {2\,d\,g}{e\,\left (7\,b\,e^2-14\,c\,d\,e\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^4}-\frac {\left (\frac {112\,c^3\,d\,g+96\,c^3\,e\,f-112\,b\,c^2\,e\,g}{105\,e^2\,{\left (b\,e-2\,c\,d\right )}^4}+\frac {16\,c^3\,d\,g}{105\,e^2\,{\left (b\,e-2\,c\,d\right )}^4}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x} \]

[In]

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

[Out]

(((40*c^2*d*g + 48*c^2*e*f - 40*b*c*e*g)/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) - (8*c^2*d*g)/(35*e*(3*b*e
^2 - 6*c*d*e)*(b*e - 2*c*d)^2))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((8*c*g*(2*b*e - 3
*c*d))/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) - (8*c^2*d*g)/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2))*(c
*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((2*b*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)) - (4
*c*d*g)/(7*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (
((16*c*d*g - 16*b*e*g + 12*c*e*f)/(7*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)) + (4*c*d*g)/(7*e*(5*b*e^2 - 10*c*d*
e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 + (((2*f)/(7*b*e^2 - 14*c*d*e) - (
2*d*g)/(e*(7*b*e^2 - 14*c*d*e)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^4 - (((112*c^3*d*g + 9
6*c^3*e*f - 112*b*c^2*e*g)/(105*e^2*(b*e - 2*c*d)^4) + (16*c^3*d*g)/(105*e^2*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*
x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)