3.20.84 \(\int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)
Optimal. Leaf size=210 \[ -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (d+e x)^3 (2 c d-b e)}-\frac {4 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x) (2 c d-b e)^3}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x)^2 (2 c d-b e)^2} \]
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Rubi [A] time = 0.33, antiderivative size = 210, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used =
{792, 658, 650} \begin {gather*} -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (d+e x)^3 (2 c d-b e)}-\frac {4 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x) (2 c d-b e)^3}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x)^2 (2 c d-b e)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(f + g*x)/((d + e*x)^3*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])/(5*e^2*(2*c*d - b*e)*(d + e*x)^3) - (2*(4*c*e*f + 6
*c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(15*e^2*(2*c*d - b*e)^2*(d + e*x)^2) - (4*c*(4*c*
e*f + 6*c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(15*e^2*(2*c*d - b*e)^3*(d + e*x))
Rule 650
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 658
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 792
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)^3 \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}}{5 e^2 (2 c d-b e) (d+e x)^3}+\frac {(4 c e f+6 c d g-5 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}{5 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}}{5 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 e^2 (2 c d-b e)^2 (d+e x)^2}+\frac {(2 c (4 c e f+6 c d g-5 b e g)) \int \frac {1}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{15 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}}{5 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 e^2 (2 c d-b e)^2 (d+e x)^2}-\frac {4 c (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 e^2 (2 c d-b e)^3 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 166, normalized size = 0.79 \begin {gather*} -\frac {2 (b e-c d+c e x) \left (b^2 e^2 (2 d g+3 e f+5 e g x)-2 b c e \left (7 d^2 g+2 d e (4 f+9 g x)+e^2 x (2 f+5 g x)\right )+4 c^2 \left (3 d^3 g+d^2 e (7 f+9 g x)+3 d e^2 x (2 f+g x)+2 e^3 f x^2\right )\right )}{15 e^2 (d+e x)^2 (b e-2 c d)^3 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(f + g*x)/((d + e*x)^3*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)*(b^2*e^2*(3*e*f + 2*d*g + 5*e*g*x) - 2*b*c*e*(7*d^2*g + e^2*x*(2*f + 5*g*x) + 2*d*e
*(4*f + 9*g*x)) + 4*c^2*(3*d^3*g + 2*e^3*f*x^2 + 3*d*e^2*x*(2*f + g*x) + d^2*e*(7*f + 9*g*x))))/(15*e^2*(-2*c*
d + b*e)^3*(d + e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
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IntegrateAlgebraic [B] time = 18.38, size = 4603, normalized size = 21.92 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(f + g*x)/((d + e*x)^3*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]
[Out]
(-2*Sqrt[-(c*e^2)]*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]*(-112*c^5*d^10*e*f + 176*b*c^4*d^9*e^2*f - 104*b^
2*c^3*d^8*e^3*f + 28*b^3*c^2*d^7*e^4*f - 3*b^4*c*d^6*e^5*f - 48*c^5*d^11*g + 104*b*c^4*d^10*e*g - 76*b^2*c^3*d
^9*e^2*g + 22*b^3*c^2*d^8*e^3*g - 2*b^4*c*d^7*e^4*g + 400*c^5*d^9*e^2*f*x - 216*b*c^4*d^8*e^3*f*x - 508*b^2*c^
3*d^7*e^4*f*x + 590*b^3*c^2*d^6*e^5*f*x - 226*b^4*c*d^5*e^6*f*x + 30*b^5*d^4*e^7*f*x + 2640*c^5*d^10*e*g*x - 7
424*b*c^4*d^9*e^2*g*x + 8208*b^2*c^3*d^8*e^3*g*x - 4480*b^3*c^2*d^7*e^4*g*x + 1211*b^4*c*d^6*e^5*g*x - 130*b^5
*d^5*e^6*g*x + 1264*c^5*d^8*e^3*f*x^2 - 2528*b*c^4*d^7*e^4*f*x^2 + 928*b^2*c^3*d^6*e^5*f*x^2 + 816*b^3*c^2*d^5
*e^6*f*x^2 - 643*b^4*c*d^4*e^7*f*x^2 + 120*b^5*d^3*e^8*f*x^2 + 816*c^5*d^9*e^2*g*x^2 - 6872*b*c^4*d^8*e^3*g*x^
2 + 13252*b^2*c^3*d^7*e^4*g*x^2 - 10586*b^3*c^2*d^6*e^5*g*x^2 + 3828*b^4*c*d^5*e^6*g*x^2 - 520*b^5*d^4*e^7*g*x
^2 - 3920*c^5*d^7*e^4*f*x^3 + 1736*b*c^4*d^6*e^5*f*x^3 + 3604*b^2*c^3*d^5*e^6*f*x^3 - 2226*b^3*c^2*d^4*e^7*f*x
^3 - 120*b^4*c*d^3*e^8*f*x^3 + 180*b^5*d^2*e^9*f*x^3 - 16080*c^5*d^8*e^3*g*x^3 + 38704*b*c^4*d^7*e^4*g*x^3 - 3
0664*b^2*c^3*d^6*e^5*g*x^3 + 6556*b^3*c^2*d^5*e^6*g*x^3 + 2115*b^4*c*d^4*e^7*g*x^3 - 780*b^5*d^3*e^8*g*x^3 - 8
32*c^5*d^6*e^5*f*x^4 + 5632*b*c^4*d^5*e^6*f*x^4 + 272*b^2*c^3*d^4*e^7*f*x^4 - 4480*b^3*c^2*d^3*e^8*f*x^4 + 128
0*b^4*c*d^2*e^9*f*x^4 + 120*b^5*d*e^10*f*x^4 - 3648*c^5*d^7*e^4*g*x^4 + 32288*b*c^4*d^6*e^5*g*x^4 - 52832*b^2*
c^3*d^5*e^6*g*x^4 + 29840*b^3*c^2*d^4*e^7*g*x^4 - 4480*b^4*c*d^3*e^8*g*x^4 - 520*b^5*d^2*e^9*g*x^4 + 6784*c^5*
d^5*e^6*f*x^5 - 832*b*c^4*d^4*e^7*f*x^5 - 7328*b^2*c^3*d^3*e^8*f*x^5 + 336*b^3*c^2*d^2*e^9*f*x^5 + 1460*b^4*c*
d*e^10*f*x^5 + 30*b^5*e^11*f*x^5 + 35136*c^5*d^6*e^5*g*x^5 - 63968*b*c^4*d^5*e^6*g*x^5 + 24608*b^2*c^3*d^4*e^7
*g*x^5 + 11344*b^3*c^2*d^3*e^8*g*x^5 - 6060*b^4*c*d^2*e^9*g*x^5 - 130*b^5*d*e^10*g*x^5 - 1280*c^5*d^4*e^7*f*x^
6 - 4608*b*c^4*d^3*e^8*f*x^6 - 4224*b^2*c^3*d^2*e^9*f*x^6 + 4352*b^3*c^2*d*e^10*f*x^6 + 480*b^4*c*e^11*f*x^6 +
3840*c^5*d^5*e^6*g*x^6 - 44672*b*c^4*d^4*e^7*g*x^6 + 54144*b^2*c^3*d^3*e^8*g*x^6 - 14592*b^3*c^2*d^2*e^9*g*x^
6 - 2080*b^4*c*d*e^10*g*x^6 - 3328*c^5*d^3*e^8*f*x^7 - 2944*b*c^4*d^2*e^9*f*x^7 + 4416*b^2*c^3*d*e^10*f*x^7 +
2016*b^3*c^2*e^11*f*x^7 - 31872*c^5*d^4*e^7*g*x^7 + 40064*b*c^4*d^3*e^8*g*x^7 - 1216*b^2*c^3*d^2*e^9*g*x^7 - 8
736*b^3*c^2*d*e^10*g*x^7 + 2048*b*c^4*d*e^10*f*x^8 + 3072*b^2*c^3*e^11*f*x^8 + 18432*b*c^4*d^2*e^9*g*x^8 - 133
12*b^2*c^3*d*e^10*g*x^8 + 1024*c^5*d*e^10*f*x^9 + 1536*b*c^4*e^11*f*x^9 + 9216*c^5*d^2*e^9*g*x^9 - 6656*b*c^4*
d*e^10*g*x^9) - 2*(-32*c^6*d^11*e^2*f + 32*b*c^5*d^10*e^3*f + 88*b^2*c^4*d^9*e^4*f - 160*b^3*c^3*d^8*e^5*f + 1
00*b^4*c^2*d^7*e^6*f - 28*b^5*c*d^6*e^7*f + 3*b^6*d^5*e^8*f - 528*c^6*d^12*e*g + 1768*b*c^5*d^11*e^2*g - 2428*
b^2*c^4*d^10*e^3*g + 1750*b^3*c^3*d^9*e^4*g - 700*b^4*c^2*d^8*e^5*g + 148*b^5*c*d^7*e^6*g - 13*b^6*d^6*e^7*g -
560*c^6*d^10*e^3*f*x + 1240*b*c^5*d^9*e^4*f*x - 860*b^2*c^4*d^8*e^5*f*x + 10*b^3*c^3*d^7*e^6*f*x + 250*b^4*c^
2*d^6*e^7*f*x - 110*b^5*c*d^5*e^8*f*x + 15*b^6*d^4*e^9*f*x - 240*c^6*d^11*e^2*g*x + 1960*b*c^5*d^10*e^3*g*x -
4340*b^2*c^4*d^9*e^4*g*x + 4390*b^3*c^3*d^8*e^5*g*x - 2300*b^4*c^2*d^7*e^6*g*x + 610*b^5*c*d^6*e^7*g*x - 65*b^
6*d^5*e^8*g*x + 1680*c^6*d^9*e^4*f*x^2 - 1400*b*c^5*d^8*e^5*f*x^2 - 1780*b^2*c^4*d^7*e^6*f*x^2 + 2430*b^3*c^3*
d^6*e^7*f*x^2 - 800*b^4*c^2*d^5*e^8*f*x^2 - 10*b^5*c*d^4*e^9*f*x^2 + 30*b^6*d^3*e^10*f*x^2 + 7920*c^6*d^10*e^3
*g*x^2 - 24600*b*c^5*d^9*e^4*g*x^2 + 28780*b^2*c^4*d^8*e^5*g*x^2 - 15130*b^3*c^3*d^7*e^6*g*x^2 + 2850*b^4*c^2*
d^6*e^7*g*x^2 + 310*b^5*c*d^5*e^8*g*x^2 - 130*b^6*d^4*e^9*g*x^2 + 1840*c^6*d^8*e^5*f*x^3 - 6360*b*c^5*d^7*e^6*
f*x^3 + 3100*b^2*c^4*d^6*e^7*f*x^3 + 3230*b^3*c^3*d^5*e^8*f*x^3 - 2830*b^4*c^2*d^4*e^9*f*x^3 + 500*b^5*c*d^3*e
^10*f*x^3 + 30*b^6*d^2*e^11*f*x^3 + 2160*c^6*d^9*e^4*g*x^3 - 23240*b*c^5*d^8*e^5*g*x^3 + 50100*b^2*c^4*d^7*e^6
*g*x^3 - 43430*b^3*c^3*d^6*e^7*g*x^3 + 16280*b^4*c^2*d^5*e^8*g*x^3 - 1900*b^5*c*d^4*e^9*g*x^3 - 130*b^6*d^3*e^
10*g*x^3 - 6960*c^6*d^7*e^6*f*x^4 + 4440*b*c^5*d^6*e^7*f*x^4 + 7980*b^2*c^4*d^5*e^8*f*x^4 - 4950*b^3*c^3*d^4*e
^9*f*x^4 - 1460*b^4*c^2*d^3*e^10*f*x^4 + 880*b^5*c*d^2*e^11*f*x^4 + 15*b^6*d*e^12*f*x^4 - 30240*c^6*d^8*e^5*g*
x^4 + 76560*b*c^5*d^7*e^6*g*x^4 - 56880*b^2*c^4*d^6*e^7*g*x^4 + 300*b^3*c^3*d^5*e^8*g*x^4 + 14010*b^4*c^2*d^4*
e^9*g*x^4 - 3680*b^5*c*d^3*e^10*g*x^4 - 65*b^6*d^2*e^11*g*x^4 - 192*c^6*d^6*e^7*f*x^5 + 9792*b*c^5*d^5*e^8*f*x
^5 - 48*b^2*c^4*d^4*e^9*f*x^5 - 10320*b^3*c^3*d^3*e^10*f*x^5 + 2920*b^4*c^2*d^2*e^11*f*x^5 + 602*b^5*c*d*e^12*
f*x^5 + 3*b^6*e^13*f*x^5 - 5568*c^6*d^7*e^6*g*x^5 + 65568*b*c^5*d^6*e^7*g*x^5 - 115152*b^2*c^4*d^5*e^8*g*x^5 +
66360*b^3*c^3*d^4*e^9*g*x^5 - 7320*b^4*c^2*d^3*e^10*g*x^5 - 2582*b^5*c*d^2*e^11*g*x^5 - 13*b^6*d*e^12*g*x^5 +
8320*c^6*d^5*e^8*f*x^6 - 1600*b*c^5*d^4*e^9*f*x^6 - 12320*b^2*c^4*d^3*e^10*f*x^6 - 240*b^3*c^3*d^2*e^11*f*x^6
+ 3700*b^4*c^2*d*e^12*f*x^6 + 150*b^5*c*e^13*f*x^6 + 49920*c^6*d^6*e^7*g*x^6 - 94880*b*c^5*d^5*e^8*g*x^6 + 26
960*b^2*c^4*d^4*e^9*g*x^6 + 33040*b^3*c^3*d^3*e^10*g*x^6 - 14700*b^4*c^2*d^2*e^11*g*x^6 - 650*b^5*c*d*e^12*g*x
^6 - 1280*c^6*d^4*e^9*f*x^7 - 7040*b*c^5*d^3*e^10*f*x^7 - 6080*b^2*c^4*d^2*e^11*f*x^7 + 7520*b^3*c^3*d*e^12*f*
x^7 + 1200*b^4*c^2*e^13*f*x^7 + 3840*c^6*d^5*e^8*g*x^7 - 67520*b*c^5*d^4*e^9*g*x^7 + 86080*b^2*c^4*d^3*e^10*g*
x^7 - 21920*b^3*c^3*d^2*e^11*g*x^7 - 5200*b^4*c^2*d*e^12*g*x^7 - 3840*c^6*d^3*e^10*f*x^8 - 3200*b*c^5*d^2*e^11
*f*x^8 + 6080*b^2*c^4*d*e^12*f*x^8 + 3360*b^3*c^3*e^13*f*x^8 - 36480*c^6*d^4*e^9*g*x^8 + 48000*b*c^5*d^3*e^10*
g*x^8 + 3520*b^2*c^4*d^2*e^11*g*x^8 - 14560*b^3*c^3*d*e^12*g*x^8 + 2560*b*c^5*d*e^12*f*x^9 + 3840*b^2*c^4*e^13
*f*x^9 + 23040*b*c^5*d^2*e^11*g*x^9 - 16640*b^2*c^4*d*e^12*g*x^9 + 1024*c^6*d*e^12*f*x^10 + 1536*b*c^5*e^13*f*
x^10 + 9216*c^6*d^2*e^11*g*x^10 - 6656*b*c^5*d*e^12*g*x^10))/(15*c*d^4*e^2*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^
2*x^2]*(-160*c^5*d^8*e^2*x + 320*b*c^4*d^7*e^3*x - 240*b^2*c^3*d^6*e^4*x + 80*b^3*c^2*d^5*e^5*x - 10*b^4*c*d^4
*e^6*x + 320*b*c^4*d^6*e^4*x^2 - 480*b^2*c^3*d^5*e^5*x^2 + 240*b^3*c^2*d^4*e^6*x^2 - 40*b^4*c*d^3*e^7*x^2 + 96
0*c^5*d^6*e^4*x^3 - 1600*b*c^4*d^5*e^5*x^3 + 640*b^2*c^3*d^4*e^6*x^3 + 80*b^3*c^2*d^3*e^7*x^3 - 60*b^4*c*d^2*e
^8*x^3 - 1600*b*c^4*d^4*e^6*x^4 + 1760*b^2*c^3*d^3*e^7*x^4 - 400*b^3*c^2*d^2*e^8*x^4 - 40*b^4*c*d*e^9*x^4 - 19
52*c^5*d^4*e^6*x^5 + 2304*b*c^4*d^3*e^7*x^5 + 208*b^2*c^3*d^2*e^8*x^5 - 480*b^3*c^2*d*e^9*x^5 - 10*b^4*c*e^10*
x^5 + 2304*b*c^4*d^2*e^8*x^6 - 1344*b^2*c^3*d*e^9*x^6 - 160*b^3*c^2*e^10*x^6 + 1664*c^5*d^2*e^8*x^7 - 1024*b*c
^4*d*e^9*x^7 - 672*b^2*c^3*e^10*x^7 - 1024*b*c^4*e^10*x^8 - 512*c^5*e^10*x^9) + 15*c*d^4*e^2*Sqrt[-(c*e^2)]*(-
32*c^5*d^10 + 80*b*c^4*d^9*e - 80*b^2*c^3*d^8*e^2 + 40*b^3*c^2*d^7*e^3 - 10*b^4*c*d^6*e^4 + b^5*d^5*e^5 + 80*b
*c^4*d^8*e^2*x - 160*b^2*c^3*d^7*e^3*x + 120*b^3*c^2*d^6*e^4*x - 40*b^4*c*d^5*e^5*x + 5*b^5*d^4*e^6*x + 480*c^
5*d^8*e^2*x^2 - 1120*b*c^4*d^7*e^3*x^2 + 880*b^2*c^3*d^6*e^4*x^2 - 240*b^3*c^2*d^5*e^5*x^2 - 10*b^4*c*d^4*e^6*
x^2 + 10*b^5*d^3*e^7*x^2 - 1120*b*c^4*d^6*e^4*x^3 + 1920*b^2*c^3*d^5*e^5*x^3 - 1040*b^3*c^2*d^4*e^6*x^3 + 160*
b^4*c*d^3*e^7*x^3 + 10*b^5*d^2*e^8*x^3 - 1760*c^5*d^6*e^4*x^4 + 3280*b*c^4*d^5*e^5*x^4 - 1040*b^2*c^3*d^4*e^6*
x^4 - 680*b^3*c^2*d^3*e^7*x^4 + 290*b^4*c*d^2*e^8*x^4 + 5*b^5*d*e^9*x^4 + 3280*b*c^4*d^4*e^6*x^5 - 4000*b^2*c^
3*d^3*e^7*x^5 + 840*b^3*c^2*d^2*e^8*x^5 + 200*b^4*c*d*e^9*x^5 + b^5*e^10*x^5 + 2720*c^5*d^4*e^6*x^6 - 3520*b*c
^4*d^3*e^7*x^6 - 880*b^2*c^3*d^2*e^8*x^6 + 1200*b^3*c^2*d*e^9*x^6 + 50*b^4*c*e^10*x^6 - 3520*b*c^4*d^2*e^8*x^7
+ 2240*b^2*c^3*d*e^9*x^7 + 400*b^3*c^2*e^10*x^7 - 1920*c^5*d^2*e^8*x^8 + 1280*b*c^4*d*e^9*x^8 + 1120*b^2*c^3*
e^10*x^8 + 1280*b*c^4*e^10*x^9 + 512*c^5*e^10*x^10))
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fricas [A] time = 8.45, size = 368, normalized size = 1.75 \begin {gather*} -\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{2} e^{3} f + {\left (6 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} g\right )} x^{2} + {\left (28 \, c^{2} d^{2} e - 16 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + 2 \, {\left (6 \, c^{2} d^{3} - 7 \, b c d^{2} e + b^{2} d e^{2}\right )} g + {\left (4 \, {\left (6 \, c^{2} d e^{2} - b c e^{3}\right )} f + {\left (36 \, c^{2} d^{2} e - 36 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )} g\right )} x\right )}}{15 \, {\left (8 \, c^{3} d^{6} e^{2} - 12 \, b c^{2} d^{5} e^{3} + 6 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5} + {\left (8 \, c^{3} d^{3} e^{5} - 12 \, b c^{2} d^{2} e^{6} + 6 \, b^{2} c d e^{7} - b^{3} e^{8}\right )} x^{3} + 3 \, {\left (8 \, c^{3} d^{4} e^{4} - 12 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} - b^{3} d e^{7}\right )} x^{2} + 3 \, {\left (8 \, c^{3} d^{5} e^{3} - 12 \, b c^{2} d^{4} e^{4} + 6 \, b^{2} c d^{3} e^{5} - b^{3} d^{2} e^{6}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")
[Out]
-2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^2*e^3*f + (6*c^2*d*e^2 - 5*b*c*e^3)*g)*x^2 + (28*c^2*
d^2*e - 16*b*c*d*e^2 + 3*b^2*e^3)*f + 2*(6*c^2*d^3 - 7*b*c*d^2*e + b^2*d*e^2)*g + (4*(6*c^2*d*e^2 - b*c*e^3)*f
+ (36*c^2*d^2*e - 36*b*c*d*e^2 + 5*b^2*e^3)*g)*x)/(8*c^3*d^6*e^2 - 12*b*c^2*d^5*e^3 + 6*b^2*c*d^4*e^4 - b^3*d
^3*e^5 + (8*c^3*d^3*e^5 - 12*b*c^2*d^2*e^6 + 6*b^2*c*d*e^7 - b^3*e^8)*x^3 + 3*(8*c^3*d^4*e^4 - 12*b*c^2*d^3*e^
5 + 6*b^2*c*d^2*e^6 - b^3*d*e^7)*x^2 + 3*(8*c^3*d^5*e^3 - 12*b*c^2*d^4*e^4 + 6*b^2*c*d^3*e^5 - b^3*d^2*e^6)*x)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((-1024*exp(2)*(sqrt(-c*exp(2)*x^2-b*e
xp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^2*d*exp(1)^5+256*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*
d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*g*b^2*d*exp(1)^3+4096*c*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*e
xp(1))-sqrt(-c*exp(2))*x)^3*g*b*d^2*exp(1)^4-1024*c*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-
sqrt(-c*exp(2))*x)^3*g*b*d^2*exp(1)^2-3072*c^2*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c
*exp(2))*x)^3*g*d^3*exp(1)^3+768*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^
3*b^2*f*exp(1)^4-1024*c*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b*f*d*exp
(1)^5-2048*c*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*b*f*d*exp(1)^3+102
4*c^2*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*f*d^2*exp(1)^4+2048*c^2*exp
(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^3*f*d^2*exp(1)^2+2048*sqrt(-c*exp(2)
)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^2*d^2*exp(1)^6-1024*exp(2)*sqrt(-c
*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^2*d^2*exp(1)^4+1280*exp(2)^
2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b^2*d^2*exp(1)^2-409
6*c*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b*d^3*exp(1)^5-409
6*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b*d^3*exp(1
)^3-1024*c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*b*
d^3*exp(1)+2048*c^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*g*d^
4*exp(1)^4+5120*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)
^2*g*d^4*exp(1)^2+2048*c^2*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*e
xp(2))*x)^2*g*d^4-2304*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2
))*x)^2*b^2*f*d*exp(1)^3+3072*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-
c*exp(2))*x)^2*b*f*d^2*exp(1)^4+6144*c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1
))-sqrt(-c*exp(2))*x)^2*b*f*d^2*exp(1)^2-3072*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-
b*d*exp(1))-sqrt(-c*exp(2))*x)^2*f*d^3*exp(1)^3-6144*c^2*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)
*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)^2*f*d^3*exp(1)-1024*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*ex
p(1))-sqrt(-c*exp(2))*x)*g*b^3*d^2*exp(1)^6+768*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt
(-c*exp(2))*x)*g*b^3*d^2*exp(1)^4+256*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2)
)*x)*g*b^3*d^2*exp(1)^2+7168*c*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^
2*d^3*exp(1)^5-4864*c*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b^2*d^3*e
xp(1)^3-11264*c^2*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b*d^4*exp(1)^4+
2048*c^2*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*b*d^4+5120*c^3*exp(2)*
(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*d^5*exp(1)^3+4096*c^3*exp(2)^2*(sqrt(-c*
exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*g*d^5*exp(1)+1280*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*ex
p(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^3*f*d*exp(1)^5-1280*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^
2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b^3*f*d*exp(1)^3+1024*c*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1
))-sqrt(-c*exp(2))*x)*b^2*f*d^2*exp(1)^6-8448*c*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt
(-c*exp(2))*x)*b^2*f*d^2*exp(1)^4+5120*c*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp
(2))*x)*b^2*f*d^2*exp(1)^2-2048*c^2*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)
*b*f*d^3*exp(1)^5+17408*c^2*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b*f*d
^3*exp(1)^3-6144*c^2*exp(2)^3*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*b*f*d^3*exp(
1)+1024*c^3*exp(2)*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*f*d^4*exp(1)^4-10240*c^
3*exp(2)^2*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*f*d^4*exp(1)^2+2048*sqrt(-c*exp
(2))*g*b^3*d^3*exp(1)^7-3072*exp(2)*sqrt(-c*exp(2))*g*b^3*d^3*exp(1)^5+1280*exp(2)^2*sqrt(-c*exp(2))*g*b^3*d^3
*exp(1)^3-256*exp(2)^3*sqrt(-c*exp(2))*g*b^3*d^3*exp(1)-6144*c*sqrt(-c*exp(2))*g*b^2*d^4*exp(1)^6+5120*c*exp(2
)*sqrt(-c*exp(2))*g*b^2*d^4*exp(1)^4+768*c*exp(2)^2*sqrt(-c*exp(2))*g*b^2*d^4*exp(1)^2-512*c*exp(2)^3*sqrt(-c*
exp(2))*g*b^2*d^4+6144*c^2*sqrt(-c*exp(2))*g*b*d^5*exp(1)^5-1024*c^2*exp(2)*sqrt(-c*exp(2))*g*b*d^5*exp(1)^3-2
048*c^2*exp(2)^2*sqrt(-c*exp(2))*g*b*d^5*exp(1)-2048*c^3*sqrt(-c*exp(2))*g*d^6*exp(1)^4-1024*c^3*exp(2)*sqrt(-
c*exp(2))*g*d^6*exp(1)^2-2048*exp(2)*sqrt(-c*exp(2))*b^3*f*d^2*exp(1)^6+2816*exp(2)^2*sqrt(-c*exp(2))*b^3*f*d^
2*exp(1)^4-768*exp(2)^3*sqrt(-c*exp(2))*b^3*f*d^2*exp(1)^2+7168*c*exp(2)*sqrt(-c*exp(2))*b^2*f*d^3*exp(1)^5-79
36*c*exp(2)^2*sqrt(-c*exp(2))*b^2*f*d^3*exp(1)^3+1536*c*exp(2)^3*sqrt(-c*exp(2))*b^2*f*d^3*exp(1)-8192*c^2*exp
(2)*sqrt(-c*exp(2))*b*f*d^4*exp(1)^4+5120*c^2*exp(2)^2*sqrt(-c*exp(2))*b*f*d^4*exp(1)^2+3072*c^3*exp(2)*sqrt(-
c*exp(2))*f*d^5*exp(1)^3)/(2048*b^2*d^2*exp(1)^7-4096*exp(2)*b^2*d^2*exp(1)^5+2048*exp(2)^2*b^2*d^2*exp(1)^3-4
096*c*b*d^3*exp(1)^6+8192*c*exp(2)*b*d^3*exp(1)^4-4096*c*exp(2)^2*b*d^3*exp(1)^2+2048*c^2*d^4*exp(1)^5-4096*c^
2*exp(2)*d^4*exp(1)^3+2048*c^2*exp(2)^2*d^4*exp(1))/((sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*
exp(2))*x)^2*exp(1)-2*sqrt(-c*exp(2))*(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x)*d+b*
d*exp(1)^2-exp(2)*b*d-c*d^2*exp(1))^2+(-8*exp(2)*g*b^2*d*exp(1)^3+2*exp(2)^2*g*b^2*d*exp(1)+32*c*exp(2)*g*b*d^
2*exp(1)^2-8*c*exp(2)^2*g*b*d^2-24*c^2*exp(2)*g*d^3*exp(1)+6*exp(2)^2*b^2*f*exp(1)^2-8*c*exp(2)*b*f*d*exp(1)^3
-16*c*exp(2)^2*b*f*d*exp(1)+8*c^2*exp(2)*f*d^2*exp(1)^2+16*c^2*exp(2)^2*f*d^2)/16/(b^2*d^2*exp(1)^6-2*exp(2)*b
^2*d^2*exp(1)^4+exp(2)^2*b^2*d^2*exp(1)^2-2*c*b*d^3*exp(1)^5+4*c*exp(2)*b*d^3*exp(1)^3-2*c*exp(2)^2*b*d^3*exp(
1)+c^2*d^4*exp(1)^4-2*c^2*exp(2)*d^4*exp(1)^2+c^2*exp(2)^2*d^4)/sqrt(b*d*exp(1)^3-c*d^2*exp(1)^2+c*d^2*exp(2)-
b*d*exp(1)*exp(2))*atan((-d*sqrt(-c*exp(2))+(sqrt(-c*exp(2)*x^2-b*exp(2)*x+c*d^2-b*d*exp(1))-sqrt(-c*exp(2))*x
)*exp(1))/sqrt(b*d*exp(1)^3-c*d^2*exp(1)^2+c*d^2*exp(2)-b*d*exp(1)*exp(2))))
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maple [A] time = 0.06, size = 236, normalized size = 1.12 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-10 b c \,e^{3} g \,x^{2}+12 c^{2} d \,e^{2} g \,x^{2}+8 c^{2} e^{3} f \,x^{2}+5 b^{2} e^{3} g x -36 b c d \,e^{2} g x -4 b c \,e^{3} f x +36 c^{2} d^{2} e g x +24 c^{2} d \,e^{2} f x +2 b^{2} d \,e^{2} g +3 b^{2} e^{3} f -14 b c \,d^{2} e g -16 b c d \,e^{2} f +12 c^{2} d^{3} g +28 c^{2} d^{2} e f \right )}{15 \left (e x +d \right )^{2} \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
[Out]
-2/15*(c*e*x+b*e-c*d)*(-10*b*c*e^3*g*x^2+12*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x^2+5*b^2*e^3*g*x-36*b*c*d*e^2*g*x-4*b
*c*e^3*f*x+36*c^2*d^2*e*g*x+24*c^2*d*e^2*f*x+2*b^2*d*e^2*g+3*b^2*e^3*f-14*b*c*d^2*e*g-16*b*c*d*e^2*f+12*c^2*d^
3*g+28*c^2*d^2*e*f)/(e*x+d)^2/(b^3*e^3-6*b^2*c*d*e^2+12*b*c^2*d^2*e-8*c^3*d^3)/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(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)^3/(-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 details)Is b*e-2*c*d zero or nonzero?
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mupad [B] time = 3.32, size = 471, normalized size = 2.24 \begin {gather*} \frac {\left (\frac {8\,c^2\,d\,g+16\,c^2\,e\,f-8\,b\,c\,e\,g}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {8\,c^2\,d\,g}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x}-\frac {\left (\frac {2\,b\,g}{5\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}-\frac {4\,c\,d\,g}{5\,e\,\left (3\,b\,e^2-6\,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 )}^2}-\frac {\left (\frac {12\,c\,d\,g-12\,b\,e\,g+8\,c\,e\,f}{5\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}+\frac {4\,c\,d\,g}{5\,e\,\left (3\,b\,e^2-6\,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 )}^2}+\frac {\left (\frac {2\,f}{5\,b\,e^2-10\,c\,d\,e}-\frac {2\,d\,g}{e\,\left (5\,b\,e^2-10\,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 )}^3}-\frac {\left (\frac {4\,c\,g\,\left (3\,b\,e-4\,c\,d\right )}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {8\,c^2\,d\,g}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}\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)^3*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)),x)
[Out]
(((8*c^2*d*g + 16*c^2*e*f - 8*b*c*e*g)/(15*e^2*(b*e - 2*c*d)^3) - (8*c^2*d*g)/(15*e^2*(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) - (((2*b*g)/(5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)) - (4*c*d*g)
/(5*e*(3*b*e^2 - 6*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)^2 - (((12*c*d
*g - 12*b*e*g + 8*c*e*f)/(5*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)) + (4*c*d*g)/(5*e*(3*b*e^2 - 6*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)^2 + (((2*f)/(5*b*e^2 - 10*c*d*e) - (2*d*g)/(e*(
5*b*e^2 - 10*c*d*e)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((4*c*g*(3*b*e - 4*c*d))/(15
*e^2*(b*e - 2*c*d)^3) - (8*c^2*d*g)/(15*e^2*(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)
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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 )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)**3/(-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)**3), x)
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