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

Optimal. Leaf size=285 \[ -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (2 c d-b e) (d+e x)^4}-\frac {2 (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 e^2 (2 c d-b e)^2 (d+e x)^3}-\frac {8 c (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^2}-\frac {16 c^2 (6 c e f+8 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^4 (d+e x)} \]

[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)

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Rubi [A]
time = 0.28, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {806, 672, 664} \begin {gather*} -\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)} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {(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*}

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Mathematica [A]
time = 0.24, size = 247, normalized size = 0.87 \begin {gather*} \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))}} \end {gather*}

Antiderivative was successfully verified.

[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))])

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Maple [A]
time = 0.05, size = 533, normalized size = 1.87

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

Verification of antiderivative is not currently implemented for this CAS.

[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]

g/e^4*(-2/5/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+4/5*c*e^2/(-b*e^2+2*c
*d*e)*(-2/3/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)-4/3*c*e^2/(-b*e^2+2*c
*d*e)^2/(x+d/e)*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)))+(-d*g+e*f)/e^5*(-2/7/(-b*e^2+2*c*d*e)/(x+d
/e)^4*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+6/7*c*e^2/(-b*e^2+2*c*d*e)*(-2/5/(-b*e^2+2*c*d*e)/(x+d
/e)^3*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+4/5*c*e^2/(-b*e^2+2*c*d*e)*(-2/3/(-b*e^2+2*c*d*e)/(x+d
/e)^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)-4/3*c*e^2/(-b*e^2+2*c*d*e)^2/(x+d/e)*(-c*e^2*(x+d/e)^2
+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(2*c*d-%e*b>0)', see `assume?`
for more det

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (276) = 552\).
time = 105.68, size = 578, normalized size = 2.03 \begin {gather*} -\frac {2 \, {\left (104 \, c^{3} d^{4} g - {\left (15 \, b^{3} f - 8 \, {\left (6 \, c^{3} f - 7 \, b c^{2} g\right )} x^{3} + 4 \, {\left (6 \, b c^{2} f - 7 \, b^{2} c g\right )} x^{2} - 3 \, {\left (6 \, b^{2} c f - 7 \, b^{3} g\right )} x\right )} e^{4} + 2 \, {\left (32 \, c^{3} d g x^{3} + 54 \, b^{2} c d f - 3 \, b^{3} d g + 32 \, {\left (3 \, c^{3} d f - 4 \, b c^{2} d g\right )} x^{2} - 2 \, {\left (30 \, b c^{2} d f - 41 \, b^{2} c d g\right )} x\right )} e^{3} + 2 \, {\left (128 \, c^{3} d^{2} g x^{2} - 138 \, b c^{2} d^{2} f + 23 \, b^{2} c d^{2} g + 2 \, {\left (78 \, c^{3} d^{2} f - 131 \, b c^{2} d^{2} g\right )} x\right )} e^{2} + 16 \, {\left (26 \, c^{3} d^{3} g x + 18 \, c^{3} d^{3} f - 9 \, b c^{2} d^{3} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}}{105 \, {\left (16 \, c^{4} d^{8} e^{2} + b^{4} x^{4} e^{10} - 4 \, {\left (2 \, b^{3} c d x^{4} - b^{4} d x^{3}\right )} e^{9} + 2 \, {\left (12 \, b^{2} c^{2} d^{2} x^{4} - 16 \, b^{3} c d^{2} x^{3} + 3 \, b^{4} d^{2} x^{2}\right )} e^{8} - 4 \, {\left (8 \, b c^{3} d^{3} x^{4} - 24 \, b^{2} c^{2} d^{3} x^{3} + 12 \, b^{3} c d^{3} x^{2} - b^{4} d^{3} x\right )} e^{7} + {\left (16 \, c^{4} d^{4} x^{4} - 128 \, b c^{3} d^{4} x^{3} + 144 \, b^{2} c^{2} d^{4} x^{2} - 32 \, b^{3} c d^{4} x + b^{4} d^{4}\right )} e^{6} + 8 \, {\left (8 \, c^{4} d^{5} x^{3} - 24 \, b c^{3} d^{5} x^{2} + 12 \, b^{2} c^{2} d^{5} x - b^{3} c d^{5}\right )} e^{5} + 8 \, {\left (12 \, c^{4} d^{6} x^{2} - 16 \, b c^{3} d^{6} x + 3 \, b^{2} c^{2} d^{6}\right )} e^{4} + 32 \, {\left (2 \, c^{4} d^{7} x - b c^{3} d^{7}\right )} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(104*c^3*d^4*g - (15*b^3*f - 8*(6*c^3*f - 7*b*c^2*g)*x^3 + 4*(6*b*c^2*f - 7*b^2*c*g)*x^2 - 3*(6*b^2*c*f
- 7*b^3*g)*x)*e^4 + 2*(32*c^3*d*g*x^3 + 54*b^2*c*d*f - 3*b^3*d*g + 32*(3*c^3*d*f - 4*b*c^2*d*g)*x^2 - 2*(30*b
*c^2*d*f - 41*b^2*c*d*g)*x)*e^3 + 2*(128*c^3*d^2*g*x^2 - 138*b*c^2*d^2*f + 23*b^2*c*d^2*g + 2*(78*c^3*d^2*f -
131*b*c^2*d^2*g)*x)*e^2 + 16*(26*c^3*d^3*g*x + 18*c^3*d^3*f - 9*b*c^2*d^3*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 +
b*x)*e^2)/(16*c^4*d^8*e^2 + b^4*x^4*e^10 - 4*(2*b^3*c*d*x^4 - b^4*d*x^3)*e^9 + 2*(12*b^2*c^2*d^2*x^4 - 16*b^3*
c*d^2*x^3 + 3*b^4*d^2*x^2)*e^8 - 4*(8*b*c^3*d^3*x^4 - 24*b^2*c^2*d^3*x^3 + 12*b^3*c*d^3*x^2 - b^4*d^3*x)*e^7 +
(16*c^4*d^4*x^4 - 128*b*c^3*d^4*x^3 + 144*b^2*c^2*d^4*x^2 - 32*b^3*c*d^4*x + b^4*d^4)*e^6 + 8*(8*c^4*d^5*x^3
- 24*b*c^3*d^5*x^2 + 12*b^2*c^2*d^5*x - b^3*c*d^5)*e^5 + 8*(12*c^4*d^6*x^2 - 16*b*c^3*d^6*x + 3*b^2*c^2*d^6)*e
^4 + 32*(2*c^4*d^7*x - b*c^3*d^7)*e^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 917 vs. \(2 (276) = 552\).
time = 1.99, size = 917, normalized size = 3.22 \begin {gather*} -\frac {2 \, {\left (8 \, c^{3} d^{4} g + 6 \, c^{3} d^{3} f e + 9 \, b c^{2} d^{3} g e - 56 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} c^{2} d^{3} g - 42 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} c^{2} d^{2} f e - 63 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b \sqrt {-c} c d^{2} g e - 168 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} c^{2} d^{2} g + 12 \, b c^{2} d^{2} f e^{2} + 6 \, b^{2} c d^{2} g e^{2} - 126 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} c^{2} d f e + 21 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} b c d g e + 140 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{3} \sqrt {-c} c d g - 84 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b \sqrt {-c} c d f e^{2} + 63 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b^{2} \sqrt {-c} d g e^{2} + 210 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{3} \sqrt {-c} c f e + 105 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{3} b \sqrt {-c} g e + 140 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{4} c g + 15 \, b^{2} c d f e^{3} - 15 \, b^{3} d g e^{3} - 252 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} b c f e^{2} - 21 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{2} b^{2} g e^{2} - 105 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} b^{2} \sqrt {-c} f e^{3} + 15 \, b^{3} f e^{4}\right )} e^{\left (-2\right )}}{105 \, {\left (\sqrt {-c} d + \sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

-2/105*(8*c^3*d^4*g + 6*c^3*d^3*f*e + 9*b*c^2*d^3*g*e - 56*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2
- b*d*e))*sqrt(-c)*c^2*d^3*g - 42*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*sqrt(-c)*c^2*
d^2*f*e - 63*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b*sqrt(-c)*c*d^2*g*e - 168*(sqrt(-c
*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*c^2*d^2*g + 12*b*c^2*d^2*f*e^2 + 6*b^2*c*d^2*g*e^2 - 1
26*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*c^2*d*f*e + 21*(sqrt(-c*e^2)*x - sqrt(-c*x^
2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*b*c*d*g*e + 140*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*
e))^3*sqrt(-c)*c*d*g - 84*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b*sqrt(-c)*c*d*f*e^2 +
63*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b^2*sqrt(-c)*d*g*e^2 + 210*(sqrt(-c*e^2)*x -
sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^3*sqrt(-c)*c*f*e + 105*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2
- b*x*e^2 - b*d*e))^3*b*sqrt(-c)*g*e + 140*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^4*c*g
+ 15*b^2*c*d*f*e^3 - 15*b^3*d*g*e^3 - 252*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*b*c
*f*e^2 - 21*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^2*b^2*g*e^2 - 105*(sqrt(-c*e^2)*x -
sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*b^2*sqrt(-c)*f*e^3 + 15*b^3*f*e^4)*e^(-2)/(sqrt(-c)*d + sqrt(-c*e^
2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))^7

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Mupad [B]
time = 4.75, size = 624, normalized size = 2.19 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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