3.23.29 \(\int \frac {f+g x}{(d+e x)^2 (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\) [2229]

Optimal. Leaf size=283 \[ \frac {16 c (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {128 c^2 (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^6 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

16/105*c*(-7*b*e*g+4*c*d*g+10*c*e*f)*(2*c*x+b)/e/(-b*e+2*c*d)^4/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)-2/7*(-d
*g+e*f)/e^2/(-b*e+2*c*d)/(e*x+d)^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)-2/35*(-7*b*e*g+4*c*d*g+10*c*e*f)/e^2
/(-b*e+2*c*d)^2/(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)+128/105*c^2*(-7*b*e*g+4*c*d*g+10*c*e*f)*(2*c*x+
b)/e/(-b*e+2*c*d)^6/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {806, 672, 628, 627} \begin {gather*} \frac {128 c^2 (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^6 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 c (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (-7 b e g+4 c d g+10 c e f)}{35 e^2 (d+e x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*c*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x))/(105*e*(2*c*d - b*e)^4*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2)) - (2*(e*f - d*g))/(7*e^2*(2*c*d - b*e)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(
10*c*e*f + 4*c*d*g - 7*b*e*g))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))
+ (128*c^2*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x))/(105*e*(2*c*d - b*e)^6*Sqrt[d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(10 c e f+4 c d g-7 b e g) \int \frac {1}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{7 e (2 c d-b e)}\\ &=-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(8 c (10 c e f+4 c d g-7 b e g)) \int \frac {1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{35 e (2 c d-b e)^2}\\ &=\frac {16 c (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {\left (64 c^2 (10 c e f+4 c d g-7 b e g)\right ) \int \frac {1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{105 e (2 c d-b e)^4}\\ &=\frac {16 c (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {128 c^2 (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^6 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 468, normalized size = 1.65 \begin {gather*} \frac {-6 b^5 e^5 (5 e f+2 d g+7 e g x)+96 b^2 c^3 e^2 \left (17 d^4 g+d^2 e^2 x (65 f-54 g x)+2 e^4 x^3 (5 f-14 g x)+40 d e^3 x^2 (f-2 g x)+20 d^3 e (3 f+2 g x)\right )-64 c^5 \left (9 d^6 g-40 e^6 f x^5+4 d^2 e^4 x^3 (5 f-8 g x)-6 d^5 e (5 f-3 g x)-16 d e^5 x^4 (5 f+g x)+8 d^3 e^3 x^2 (15 f+g x)+3 d^4 e^2 x (15 f+16 g x)\right )+32 b c^4 e \left (6 d^5 g+8 d e^4 x^3 (45 f-8 g x)+8 e^5 x^4 (15 f-7 g x)-39 d^4 e (5 f-g x)+12 d^3 e^2 x (-5 f+24 g x)+4 d^2 e^3 x^2 (75 f+43 g x)\right )+4 b^4 c e^4 \left (43 d^2 g+e^2 x (15 f+28 g x)+2 d e (45 f+73 g x)\right )-16 b^3 c^2 e^3 \left (88 d^3 g+2 e^3 x^2 (5 f+21 g x)+2 d e^2 x (25 f+86 g x)+d^2 e (115 f+293 g x)\right )}{105 e^2 (-2 c d+b e)^6 (d+e x)^3 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*b^5*e^5*(5*e*f + 2*d*g + 7*e*g*x) + 96*b^2*c^3*e^2*(17*d^4*g + d^2*e^2*x*(65*f - 54*g*x) + 2*e^4*x^3*(5*f
- 14*g*x) + 40*d*e^3*x^2*(f - 2*g*x) + 20*d^3*e*(3*f + 2*g*x)) - 64*c^5*(9*d^6*g - 40*e^6*f*x^5 + 4*d^2*e^4*x^
3*(5*f - 8*g*x) - 6*d^5*e*(5*f - 3*g*x) - 16*d*e^5*x^4*(5*f + g*x) + 8*d^3*e^3*x^2*(15*f + g*x) + 3*d^4*e^2*x*
(15*f + 16*g*x)) + 32*b*c^4*e*(6*d^5*g + 8*d*e^4*x^3*(45*f - 8*g*x) + 8*e^5*x^4*(15*f - 7*g*x) - 39*d^4*e*(5*f
- g*x) + 12*d^3*e^2*x*(-5*f + 24*g*x) + 4*d^2*e^3*x^2*(75*f + 43*g*x)) + 4*b^4*c*e^4*(43*d^2*g + e^2*x*(15*f
+ 28*g*x) + 2*d*e*(45*f + 73*g*x)) - 16*b^3*c^2*e^3*(88*d^3*g + 2*e^3*x^2*(5*f + 21*g*x) + 2*d*e^2*x*(25*f + 8
6*g*x) + d^2*e*(115*f + 293*g*x)))/(105*e^2*(-2*c*d + b*e)^6*(d + e*x)^3*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)
*(-(b*e) + c*(d - e*x))])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(596\) vs. \(2(267)=534\).
time = 0.09, size = 597, normalized size = 2.11 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g/e^2*(-2/5/(-b*e^2+2*c*d*e)/(x+d/e)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)+8/5*c*e^2/(-b*e^2+2*c*d
*e)*(-2/3*(-2*c*e^2*(x+d/e)-b*e^2+2*c*d*e)/(-b*e^2+2*c*d*e)^2/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2
)-16/3*c*e^2/(-b*e^2+2*c*d*e)^4*(-2*c*e^2*(x+d/e)-b*e^2+2*c*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^3*(-2/7/(-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))^(3/2)+10/7*
c*e^2/(-b*e^2+2*c*d*e)*(-2/5/(-b*e^2+2*c*d*e)/(x+d/e)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)+8/5*c*
e^2/(-b*e^2+2*c*d*e)*(-2/3*(-2*c*e^2*(x+d/e)-b*e^2+2*c*d*e)/(-b*e^2+2*c*d*e)^2/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d
*e)*(x+d/e))^(3/2)-16/3*c*e^2/(-b*e^2+2*c*d*e)^4*(-2*c*e^2*(x+d/e)-b*e^2+2*c*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)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 22323 vs. \(2 (272) = 544\).
time = 1.48, size = 22323, normalized size = 78.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(128*(4*c^4*d*g + 10*c^4*f*e - 7*b*c^3*g*e)*sgn(1/(x*e + d))/(64*sqrt(-c)*c^6*d^6*e - 192*b*sqrt(-c)*c^
5*d^5*e^2 + 240*b^2*sqrt(-c)*c^4*d^4*e^3 - 160*b^3*sqrt(-c)*c^3*d^3*e^4 + 60*b^4*sqrt(-c)*c^2*d^2*e^5 - 12*b^5
*sqrt(-c)*c*d*e^6 + b^6*sqrt(-c)*e^7) + (1030792151040*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^36*sqrt(-c +
2*c*d/(x*e + d) - b*e/(x*e + d))*d^37*g*e^6*sgn(1/(x*e + d))^6 - 4329327034368*(c - 2*c*d/(x*e + d) + b*e/(x*e
+ d))^2*c^37*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^37*g*e^6*sgn(1/(x*e + d))^6 + 14431090114560*c^39*s
qrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^37*g*e^6*sgn(1/(x*e + d))^6 - 4810363371520*c^38*(-c + 2*c*d/(x*e
+ d) - b*e/(x*e + d))^(3/2)*d^37*g*e^6*sgn(1/(x*e + d))^6 - 1030792151040*(c - 2*c*d/(x*e + d) + b*e/(x*e + d)
)^3*c^36*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^36*f*e^7*sgn(1/(x*e + d))^6 + 7215545057280*(c - 2*c*d/(
x*e + d) + b*e/(x*e + d))^2*c^37*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^36*f*e^7*sgn(1/(x*e + d))^6 + 72
155450572800*c^39*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^36*f*e^7*sgn(1/(x*e + d))^6 + 24051816857600*c^
38*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^36*f*e^7*sgn(1/(x*e + d))^6 - 18554258718720*b*(c - 2*c*d/(x
*e + d) + b*e/(x*e + d))^3*c^35*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^36*g*e^7*sgn(1/(x*e + d))^6 + 764
84777607168*b*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^36*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^36*g*e
^7*sgn(1/(x*e + d))^6 - 303052892405760*b*c^38*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^36*g*e^7*sgn(1/(x*
e + d))^6 + 76965813944320*b*c^37*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^36*g*e^7*sgn(1/(x*e + d))^6 +
18554258718720*b*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^35*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^35
*f*e^8*sgn(1/(x*e + d))^6 - 129879811031040*b*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^36*sqrt(-c + 2*c*d/(x*
e + d) - b*e/(x*e + d))*d^35*f*e^8*sgn(1/(x*e + d))^6 - 1298798110310400*b*c^38*sqrt(-c + 2*c*d/(x*e + d) - b*
e/(x*e + d))*d^35*f*e^8*sgn(1/(x*e + d))^6 - 432932703436800*b*c^37*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/
2)*d^35*f*e^8*sgn(1/(x*e + d))^6 + 162349763788800*b^2*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^34*sqrt(-c +
2*c*d/(x*e + d) - b*e/(x*e + d))*d^35*g*e^8*sgn(1/(x*e + d))^6 - 655893045706752*b^2*(c - 2*c*d/(x*e + d) + b*
e/(x*e + d))^2*c^35*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^35*g*e^8*sgn(1/(x*e + d))^6 + 305217555922944
0*b^2*c^37*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^35*g*e^8*sgn(1/(x*e + d))^6 - 584459149639680*b^2*c^36
*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^35*g*e^8*sgn(1/(x*e + d))^6 - 162349763788800*b^2*(c - 2*c*d/(
x*e + d) + b*e/(x*e + d))^3*c^34*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^34*f*e^9*sgn(1/(x*e + d))^6 + 11
36448346521600*b^2*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^35*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^3
4*f*e^9*sgn(1/(x*e + d))^6 + 11364483465216000*b^2*c^37*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^34*f*e^9*
sgn(1/(x*e + d))^6 + 3788161155072000*b^2*c^36*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^34*f*e^9*sgn(1/(
x*e + d))^6 - 919981994803200*b^3*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^33*sqrt(-c + 2*c*d/(x*e + d) - b*e
/(x*e + d))*d^34*g*e^9*sgn(1/(x*e + d))^6 + 3636634708869120*b^3*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^34*
sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^34*g*e^9*sgn(1/(x*e + d))^6 - 19698438006374400*b^3*c^36*sqrt(-c
+ 2*c*d/(x*e + d) - b*e/(x*e + d))*d^34*g*e^9*sgn(1/(x*e + d))^6 + 2777984847052800*b^3*c^35*(-c + 2*c*d/(x*e
+ d) - b*e/(x*e + d))^(3/2)*d^34*g*e^9*sgn(1/(x*e + d))^6 + 919981994803200*b^3*(c - 2*c*d/(x*e + d) + b*e/(x*
e + d))^3*c^33*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^33*f*e^10*sgn(1/(x*e + d))^6 - 6439873963622400*b^
3*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^34*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^33*f*e^10*sgn(1/(x
*e + d))^6 - 64398739636224000*b^3*c^36*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^33*f*e^10*sgn(1/(x*e + d)
)^6 - 21466246545408000*b^3*c^35*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^33*f*e^10*sgn(1/(x*e + d))^6 +
3794925728563200*b^4*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^32*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*
d^33*g*e^10*sgn(1/(x*e + d))^6 - 14650713267240960*b^4*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^33*sqrt(-c +
2*c*d/(x*e + d) - b*e/(x*e + d))*d^33*g*e^10*sgn(1/(x*e + d))^6 + 91768203981619200*b^4*c^35*sqrt(-c + 2*c*d/(
x*e + d) - b*e/(x*e + d))*d^33*g*e^10*sgn(1/(x*e + d))^6 - 9123154781798400*b^4*c^34*(-c + 2*c*d/(x*e + d) - b
*e/(x*e + d))^(3/2)*d^33*g*e^10*sgn(1/(x*e + d))^6 - 3794925728563200*b^4*(c - 2*c*d/(x*e + d) + b*e/(x*e + d)
)^3*c^32*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^32*f*e^11*sgn(1/(x*e + d))^6 + 26564480099942400*b^4*(c
- 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^33*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^32*f*e^11*sgn(1/(x*e +
d))^6 + 265644800999424000*b^4*c^35*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^32*f*e^11*sgn(1/(x*e + d))^6
+ 88548266999808000*b^4*c^34*(-c + 2*c*d/(x*e +...

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Mupad [B]
time = 10.35, size = 2500, normalized size = 8.83 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((800*c^6*d^4*g + 558*b^3*c^3*e^4*f - 222*b^4*c^2*e^4*g - 4192*c^6*d^3*e*f + 384*b*c^5*d^3*e*g + 6624*b*c^5*d^
2*e^2*f - 3392*b^2*c^4*d*e^3*f + 1248*b^3*c^3*d*e^3*g - 1984*b^2*c^4*d^2*e^2*g)/(105*e^2*(b*e - 2*c*d)^8) - x*
((b*((b*((8*c^5*e*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^8) + (16*b*c^5*e^2*g)/(105*(b*e - 2*c*d)^8
)))/c - (4*c^4*(40*c^2*d^2*g - 33*b^2*e^2*g + 62*b*c*e^2*f - 88*c^2*d*e*f + 16*b*c*d*e*g))/(105*(b*e - 2*c*d)^
8) + (16*c^5*g*(c*d^2 - b*d*e))/(105*(b*e - 2*c*d)^8)))/c + (44*b^2*c^4*e^4*f - 30*b^3*c^3*e^4*g - 672*c^6*d^2
*e^2*f + 224*c^6*d^3*e*g + 320*b*c^5*d*e^3*f + 160*b*c^5*d^2*e^2*g - 128*b^2*c^4*d*e^3*g)/(105*e^2*(b*e - 2*c*
d)^8) + ((c*d^2 - b*d*e)*((8*c^5*e*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^8) + (16*b*c^5*e^2*g)/(10
5*(b*e - 2*c*d)^8)))/(c*e^2)) + ((c*d^2 - b*d*e)*((b*((8*c^5*e*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*
d)^8) + (16*b*c^5*e^2*g)/(105*(b*e - 2*c*d)^8)))/c - (4*c^4*(40*c^2*d^2*g - 33*b^2*e^2*g + 62*b*c*e^2*f - 88*c
^2*d*e*f + 16*b*c*d*e*g))/(105*(b*e - 2*c*d)^8) + (16*c^5*g*(c*d^2 - b*d*e))/(105*(b*e - 2*c*d)^8)))/(c*e^2))/
(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + (((8*c^2*g*(3*b*e - 4*c*d))/(105*e^2*(b*e - 2*c*d)^6) - (16*c^3*
d*g)/(105*e^2*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((8*c^4*(2*c
*d*g - 7*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^8) + (16*c^5*d*g)/(105*(b*e - 2*c*d)^8)))/e + (76*b^2*c^3*e^2*g
- 176*c^5*d^2*g + 304*c^5*d*e*f - 200*b*c^4*e^2*f + 8*b*c^4*d*e*g)/(105*e*(b*e - 2*c*d)^8)))/e - (2*b*c^2*(13*
b^2*e^2*g - 44*c^2*d^2*g - 44*b*c*e^2*f + 76*c^2*d*e*f + 4*b*c*d*e*g))/(105*e*(b*e - 2*c*d)^8))*(c*d^2 - c*e^2
*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((4*b*c*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (8*c^2*d*g)/
(35*e*(3*b*e^2 - 6*c*d*e)*(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)^2 + (((276*
b^2*c^3*e^3*f - 352*c^5*d^3*g - 88*b^3*c^2*e^3*g + 608*c^5*d^2*e*f - 832*b*c^4*d*e^2*f + 288*b*c^4*d^2*e*g + 1
04*b^2*c^3*d*e^2*g)/(105*e^2*(b*e - 2*c*d)^8) + (d*((d*((16*c^4*(4*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c
*d)^8) + (16*c^5*d*g)/(105*(b*e - 2*c*d)^8)))/e - (272*b*c^4*e^3*f - 148*b^2*c^3*e^3*g - 448*c^5*d*e^2*f + 128
*c^5*d^2*e*g + 160*b*c^4*d*e^2*g)/(105*e^2*(b*e - 2*c*d)^8)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/
(d + e*x) + (((d*((d*((2*c^2*e^3*(5*b*e*g + 2*c*d*g - 6*c*e*f))/(7*(b*e - 2*c*d)^3*(5*b^3*e^6 - 40*c^3*d^3*e^3
+ 60*b*c^2*d^2*e^4 - 30*b^2*c*d*e^5)) - (4*c^3*d*e^3*g)/(7*(b*e - 2*c*d)^3*(5*b^3*e^6 - 40*c^3*d^3*e^3 + 60*b
*c^2*d^2*e^4 - 30*b^2*c*d*e^5))))/e - (e*(8*b^2*c*e^3*g - 38*b*c^2*e^3*f + 52*c^3*d*e^2*f - 48*c^3*d^2*e*g + 2
6*b*c^2*d*e^2*g))/(7*(b*e - 2*c*d)^3*(5*b^3*e^6 - 40*c^3*d^3*e^3 + 60*b*c^2*d^2*e^4 - 30*b^2*c*d*e^5))))/e - (
e*(16*c^3*d^3*g - 16*b^3*e^3*g + 40*b^2*c*e^3*f + 96*c^3*d^2*e*f - 122*b*c^2*d*e^2*f - 48*b*c^2*d^2*e*g + 48*b
^2*c*d*e^2*g))/(7*(b*e - 2*c*d)^3*(5*b^3*e^6 - 40*c^3*d^3*e^3 + 60*b*c^2*d^2*e^4 - 30*b^2*c*d*e^5)))*(c*d^2 -
c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((8*c*g*(2*b*e - 3*c*d))/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2
*c*d)^4) - (8*c^2*d*g)/(35*e*(3*b*e^2 - 6*c*d*e)*(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)^2 - (((d*((d*((24*c^3*e^2*(b*g - c*f))/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^6) - (8*c^4*d*e*g)/(3
5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^6)))/e - (26*b^2*c^2*e^2*g - 96*c^4*d^2*g + 128*c^4*d*e*f - 88*b*c^3*e^2*f
+ 32*b*c^3*d*e*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^6)))/e + (2*b*c*(4*b^2*e^2*g - 24*c^2*d^2*g - 19*b*c*
e^2*f + 32*c^2*d*e*f + 8*b*c*d*e*g))/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*
e^2*x)^(1/2))/(d + e*x)^2 + (((2*e^2*f)/(7*b^3*e^6 - 56*c^3*d^3*e^3 + 84*b*c^2*d^2*e^4 - 42*b^2*c*d*e^5) - (2*
d*e*g)/(7*b^3*e^6 - 56*c^3*d^3*e^3 + 84*b*c^2*d^2*e^4 - 42*b^2*c*d*e^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)
^(1/2))/(d + e*x)^4 - (((2*b*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^3) - (4*c*d*g)/(7*e*(5*b*e^2 - 10*c*d*e)
*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 + ((x*(((e*(b*e - c*d) + c*d*e)*((
(e*(b*e - c*d) + c*d*e)*(((e*(b*e - c*d) + c*d*e)*((2*c^2*(88*c^6*d^2*e^3*g - 82*b^2*c^4*e^5*g + 136*b*c^5*e^5
*f - 176*c^6*d*e^4*f + 32*b*c^5*d*e^4*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - ((e*
(b*e - c*d) + c*d*e)*((16*c^7*e^3*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 -
4*b*c^2*d*e)) + (16*c^7*e^2*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*
d*e)) - (8*b*c^7*e^4*g)/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (8*b*c^6*e^3*(
2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*d*e^2*g*(b
*e - c*d))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (d*(b*e - c*d)*((16*c^7*e^3
*(2*c*d*g - 5*b*e*g + 3*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^7*e^2*g*(e
*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*...

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