3.20.95 \(\int \frac {(d+e x)^3 (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=177 \[ \frac {g \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{5/2} e^2}-\frac {2 g (d+e x)}{c^2 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^3 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.30, antiderivative size = 177, 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 =
{788, 652, 621, 204} \begin {gather*} -\frac {2 g (d+e x)}{c^2 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {g \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{5/2} e^2}+\frac {2 (d+e x)^3 (-b e g+c d g+c e f)}{3 c e^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[((d + e*x)^3*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
[Out]
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^3)/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) -
(2*g*(d + e*x))/(c^2*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (g*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqr
t[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(c^(5/2)*e^2)
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 621
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 652
Int[((d_.) + (e_.)*(x_))^2*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + b*x +
c*x^2)^(p + 1))/(c*(p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fr
eeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && LtQ[p,
-1]
Rule 788
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {g \int \frac {(d+e x)^2}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{c e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 g (d+e x)}{c^2 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {g \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 g (d+e x)}{c^2 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(2 g) \operatorname {Subst}\left (\int \frac {1}{-4 c e^2-x^2} \, dx,x,\frac {-b e^2-2 c e^2 x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 g (d+e x)}{c^2 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {g \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{5/2} e^2}\\ \end {align*}
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Mathematica [A] time = 1.23, size = 228, normalized size = 1.29 \begin {gather*} \frac {2 \left (-\frac {\sqrt {c} (d+e x) \left (3 b^2 e^2 g+4 b c e g (e x-2 d)+c^2 \left (5 d^2 g-d e (f+7 g x)-e^2 f x\right )\right )}{b e-c d+c e x}-\frac {3 \sqrt {e} g \sqrt {d+e x} (b e-2 c d)^2 \sqrt {\frac {b e-c d+c e x}{b e-2 c d}} \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )}{\sqrt {e (2 c d-b e)}}\right )}{3 c^{5/2} e^2 (b e-2 c d) \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x)^3*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
[Out]
(2*(-((Sqrt[c]*(d + e*x)*(3*b^2*e^2*g + 4*b*c*e*g*(-2*d + e*x) + c^2*(5*d^2*g - e^2*f*x - d*e*(f + 7*g*x))))/(
-(c*d) + b*e + c*e*x)) - (3*Sqrt[e]*(-2*c*d + b*e)^2*g*Sqrt[d + e*x]*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e
)]*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])/Sqrt[e*(2*c*d - b*e)]))/(3*c^(5/2)*e^2*(-2*c
*d + b*e)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
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IntegrateAlgebraic [B] time = 29.22, size = 2956, normalized size = 16.70 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((d + e*x)^3*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
[Out]
(2*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]*(4*b^4*c^5*d^4*e^6*f - 8*b^5*c^4*d^3*e^7*f + 5*b^6*c^3*d^2*e^8*f
- b^7*c^2*d*e^9*f - 20*b^4*c^5*d^5*e^5*g + 72*b^5*c^4*d^4*e^6*g - 101*b^6*c^3*d^3*e^7*g + 69*b^7*c^2*d^2*e^8*g
- 23*b^8*c*d*e^9*g + 3*b^9*e^10*g + 48*b^2*c^7*d^5*e^5*f*x - 120*b^3*c^6*d^4*e^6*f*x + 108*b^4*c^5*d^3*e^7*f*
x - 46*b^5*c^4*d^2*e^8*f*x + 10*b^6*c^3*d*e^9*f*x - b^7*c^2*e^10*f*x - 144*c^9*d^8*e^2*g*x + 720*b*c^8*d^7*e^3
*g*x - 1380*b^2*c^7*d^6*e^4*g*x + 1272*b^3*c^6*d^5*e^5*g*x - 576*b^4*c^5*d^4*e^6*g*x + 158*b^5*c^4*d^3*e^7*g*x
- 94*b^6*c^3*d^2*e^8*g*x + 55*b^7*c^2*d*e^9*g*x - 11*b^8*c*e^10*g*x - 48*b^3*c^6*d^3*e^7*f*x^2 + 44*b^4*c^5*d
^2*e^8*f*x^2 - 20*b^5*c^4*d*e^9*f*x^2 + 5*b^6*c^3*e^10*f*x^2 + 144*b*c^8*d^6*e^4*g*x^2 - 504*b^2*c^7*d^5*e^5*g
*x^2 + 552*b^3*c^6*d^4*e^6*g*x^2 - 76*b^4*c^5*d^3*e^7*g*x^2 - 196*b^5*c^4*d^2*e^8*g*x^2 + 79*b^6*c^3*d*e^9*g*x
^2 - 2*b^7*c^2*e^10*g*x^2 - 112*b^2*c^7*d^3*e^7*f*x^3 + 104*b^3*c^6*d^2*e^8*f*x^3 - 28*b^4*c^5*d*e^9*f*x^3 - 6
*b^5*c^4*e^10*f*x^3 + 336*c^9*d^6*e^4*g*x^3 - 1152*b*c^8*d^5*e^5*g*x^3 + 1180*b^2*c^7*d^4*e^6*g*x^3 - 320*b^3*
c^6*d^3*e^7*g*x^3 + 60*b^4*c^5*d^2*e^8*g*x^3 - 218*b^5*c^4*d*e^9*g*x^3 + 72*b^6*c^3*e^10*g*x^3 + 64*b^3*c^6*d*
e^9*f*x^4 - 16*b^4*c^5*e^10*f*x^4 - 192*b*c^8*d^4*e^6*g*x^4 + 384*b^2*c^7*d^3*e^7*g*x^4 - 64*b^3*c^6*d^2*e^8*g
*x^4 - 176*b^4*c^5*d*e^9*g*x^4 - 32*b^5*c^4*e^10*g*x^4 + 64*b^2*c^7*d*e^9*f*x^5 + 32*b^3*c^6*e^10*f*x^5 - 192*
c^9*d^4*e^6*g*x^5 + 384*b*c^8*d^3*e^7*g*x^5 - 64*b^2*c^7*d^2*e^8*g*x^5 - 32*b^3*c^6*d*e^9*g*x^5 - 128*b^4*c^5*
e^10*g*x^5) + 2*Sqrt[-(c*e^2)]*(16*b^2*c^7*d^7*e^3*f - 64*b^3*c^6*d^6*e^4*f + 100*b^4*c^5*d^5*e^5*f - 76*b^5*c
^4*d^4*e^6*f + 28*b^6*c^3*d^3*e^7*f - 4*b^7*c^2*d^2*e^8*f - 48*c^9*d^10*g + 312*b*c^8*d^9*e*g - 844*b^2*c^7*d^
8*e^2*g + 1234*b^3*c^6*d^7*e^3*g - 1054*b^4*c^5*d^6*e^4*g + 526*b^5*c^4*d^5*e^5*g - 142*b^6*c^3*d^4*e^6*g + 16
*b^7*c^2*d^3*e^7*g - 24*b^3*c^6*d^5*e^5*f*x + 48*b^4*c^5*d^4*e^6*f*x - 30*b^5*c^4*d^3*e^7*f*x + 6*b^6*c^3*d^2*
e^8*f*x + 72*b*c^8*d^8*e^2*g*x - 360*b^2*c^7*d^7*e^3*g*x + 690*b^3*c^6*d^6*e^4*g*x - 576*b^4*c^5*d^5*e^5*g*x +
96*b^5*c^4*d^4*e^6*g*x + 162*b^6*c^3*d^3*e^7*g*x - 102*b^7*c^2*d^2*e^8*g*x + 18*b^8*c*d*e^9*g*x - 96*b^2*c^7*
d^5*e^5*f*x^2 + 216*b^3*c^6*d^4*e^6*f*x^2 - 168*b^4*c^5*d^3*e^7*f*x^2 + 66*b^5*c^4*d^2*e^8*f*x^2 - 12*b^6*c^3*
d*e^9*f*x^2 + 288*c^9*d^8*e^2*g*x^2 - 1368*b*c^8*d^7*e^3*g*x^2 + 2400*b^2*c^7*d^6*e^4*g*x^2 - 1890*b^3*c^6*d^5
*e^5*g*x^2 + 654*b^4*c^5*d^4*e^6*g*x^2 - 246*b^5*c^4*d^3*e^7*g*x^2 + 276*b^6*c^3*d^2*e^8*g*x^2 - 132*b^7*c^2*d
*e^9*g*x^2 + 18*b^8*c*e^10*g*x^2 + 120*b^3*c^6*d^3*e^7*f*x^3 - 128*b^4*c^5*d^2*e^8*f*x^3 + 54*b^5*c^4*d*e^9*f*
x^3 - 6*b^6*c^3*e^10*f*x^3 - 360*b*c^8*d^6*e^4*g*x^3 + 1224*b^2*c^7*d^5*e^5*g*x^3 - 1266*b^3*c^6*d^4*e^6*g*x^3
+ 196*b^4*c^5*d^3*e^7*g*x^3 + 258*b^5*c^4*d^2*e^8*g*x^3 - 6*b^6*c^3*d*e^9*g*x^3 - 30*b^7*c^2*e^10*g*x^3 + 144
*b^2*c^7*d^3*e^7*f*x^4 - 120*b^3*c^6*d^2*e^8*f*x^4 - 12*b^4*c^5*d*e^9*f*x^4 + 18*b^5*c^4*e^10*f*x^4 - 432*c^9*
d^6*e^4*g*x^4 + 1440*b*c^8*d^5*e^5*g*x^4 - 1332*b^2*c^7*d^4*e^6*g*x^4 + 192*b^3*c^6*d^3*e^7*g*x^4 - 84*b^4*c^5
*d^2*e^8*g*x^4 + 366*b^5*c^4*d*e^9*g*x^4 - 72*b^6*c^3*e^10*g*x^4 - 96*b^3*c^6*d*e^9*f*x^5 + 288*b*c^8*d^4*e^6*
g*x^5 - 576*b^2*c^7*d^3*e^7*g*x^5 + 96*b^3*c^6*d^2*e^8*g*x^5 + 192*b^4*c^5*d*e^9*g*x^5 + 96*b^5*c^4*e^10*g*x^5
- 64*b^2*c^7*d*e^9*f*x^6 - 32*b^3*c^6*e^10*f*x^6 + 192*c^9*d^4*e^6*g*x^6 - 384*b*c^8*d^3*e^7*g*x^6 + 64*b^2*c
^7*d^2*e^8*g*x^6 + 32*b^3*c^6*d*e^9*g*x^6 + 128*b^4*c^5*e^10*g*x^6))/(3*b^4*c^2*e^7*Sqrt[-(c*e^2)]*Sqrt[c*d^2
- b*d*e - b*e^2*x - c*e^2*x^2]*(-24*c^5*d^4*x + 72*b*c^4*d^3*e*x - 78*b^2*c^3*d^2*e^2*x + 36*b^3*c^2*d*e^3*x -
6*b^4*c*e^4*x + 24*b*c^4*d^2*e^2*x^2 - 36*b^2*c^3*d*e^3*x^2 + 12*b^3*c^2*e^4*x^2 + 56*c^5*d^2*e^2*x^3 - 80*b*
c^4*d*e^3*x^3 + 18*b^2*c^3*e^4*x^3 - 32*b*c^4*e^4*x^4 - 32*c^5*e^4*x^5) + 3*b^4*c^2*e^7*(8*c^6*d^6 - 36*b*c^5*
d^5*e + 66*b^2*c^4*d^4*e^2 - 63*b^3*c^3*d^3*e^3 + 33*b^4*c^2*d^2*e^4 - 9*b^5*c*d*e^5 + b^6*e^6 - 12*b*c^5*d^4*
e^2*x + 36*b^2*c^4*d^3*e^3*x - 39*b^3*c^3*d^2*e^4*x + 18*b^4*c^2*d*e^5*x - 3*b^5*c*e^6*x - 48*c^6*d^4*e^2*x^2
+ 132*b*c^5*d^3*e^3*x^2 - 120*b^2*c^4*d^2*e^4*x^2 + 39*b^3*c^3*d*e^5*x^2 - 3*b^4*c^2*e^6*x^2 + 60*b*c^5*d^2*e^
4*x^3 - 84*b^2*c^4*d*e^5*x^3 + 23*b^3*c^3*e^6*x^3 + 72*c^6*d^2*e^4*x^4 - 96*b*c^5*d*e^5*x^4 + 6*b^2*c^4*e^6*x^
4 - 48*b*c^5*e^6*x^5 - 32*c^6*e^6*x^6)) + (g*ArcTan[(2*Sqrt[c]*Sqrt[-(c*e^2)]*x)/(b*e) - (2*Sqrt[c]*Sqrt[c*d^2
- b*d*e - b*e^2*x - c*e^2*x^2])/(b*e)])/(c^(5/2)*e^2) + (Sqrt[-(c*e^2)]*g*Log[4*c^2*d^2 - 4*b*c*d*e + b^2*e^2
- 4*b*c*e^2*x - 8*c^2*e^2*x^2 - 8*c*Sqrt[-(c*e^2)]*x*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]])/(2*c^3*e^3)
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fricas [B] time = 2.39, size = 785, normalized size = 4.44 \begin {gather*} \left [-\frac {3 \, {\left ({\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g x^{2} - 2 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} g x + {\left (2 \, c^{3} d^{3} - 5 \, b c^{2} d^{2} e + 4 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} g\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (c^{3} d e f - {\left (5 \, c^{3} d^{2} - 8 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} g + {\left (c^{3} e^{2} f + {\left (7 \, c^{3} d e - 4 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{6 \, {\left (2 \, c^{6} d^{3} e^{2} - 5 \, b c^{5} d^{2} e^{3} + 4 \, b^{2} c^{4} d e^{4} - b^{3} c^{3} e^{5} + {\left (2 \, c^{6} d e^{4} - b c^{5} e^{5}\right )} x^{2} - 2 \, {\left (2 \, c^{6} d^{2} e^{3} - 3 \, b c^{5} d e^{4} + b^{2} c^{4} e^{5}\right )} x\right )}}, -\frac {3 \, {\left ({\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g x^{2} - 2 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} g x + {\left (2 \, c^{3} d^{3} - 5 \, b c^{2} d^{2} e + 4 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, {\left (c^{3} d e f - {\left (5 \, c^{3} d^{2} - 8 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} g + {\left (c^{3} e^{2} f + {\left (7 \, c^{3} d e - 4 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{3 \, {\left (2 \, c^{6} d^{3} e^{2} - 5 \, b c^{5} d^{2} e^{3} + 4 \, b^{2} c^{4} d e^{4} - b^{3} c^{3} e^{5} + {\left (2 \, c^{6} d e^{4} - b c^{5} e^{5}\right )} x^{2} - 2 \, {\left (2 \, c^{6} d^{2} e^{3} - 3 \, b c^{5} d e^{4} + b^{2} c^{4} e^{5}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")
[Out]
[-1/6*(3*((2*c^3*d*e^2 - b*c^2*e^3)*g*x^2 - 2*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*g*x + (2*c^3*d^3 - 5*b
*c^2*d^2*e + 4*b^2*c*d*e^2 - b^3*e^3)*g)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^
2*e^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*(c^3*d*e*f - (5*c^3*d^2 - 8
*b*c^2*d*e + 3*b^2*c*e^2)*g + (c^3*e^2*f + (7*c^3*d*e - 4*b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 -
b*d*e))/(2*c^6*d^3*e^2 - 5*b*c^5*d^2*e^3 + 4*b^2*c^4*d*e^4 - b^3*c^3*e^5 + (2*c^6*d*e^4 - b*c^5*e^5)*x^2 - 2*
(2*c^6*d^2*e^3 - 3*b*c^5*d*e^4 + b^2*c^4*e^5)*x), -1/3*(3*((2*c^3*d*e^2 - b*c^2*e^3)*g*x^2 - 2*(2*c^3*d^2*e -
3*b*c^2*d*e^2 + b^2*c*e^3)*g*x + (2*c^3*d^3 - 5*b*c^2*d^2*e + 4*b^2*c*d*e^2 - b^3*e^3)*g)*sqrt(c)*arctan(1/2*s
qrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e
)) - 2*(c^3*d*e*f - (5*c^3*d^2 - 8*b*c^2*d*e + 3*b^2*c*e^2)*g + (c^3*e^2*f + (7*c^3*d*e - 4*b*c^2*e^2)*g)*x)*s
qrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(2*c^6*d^3*e^2 - 5*b*c^5*d^2*e^3 + 4*b^2*c^4*d*e^4 - b^3*c^3*e^5 +
(2*c^6*d*e^4 - b*c^5*e^5)*x^2 - 2*(2*c^6*d^2*e^3 - 3*b*c^5*d*e^4 + b^2*c^4*e^5)*x)]
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giac [B] time = 0.89, size = 841, normalized size = 4.75 \begin {gather*} \frac {\sqrt {-c e^{2}} g e^{\left (-3\right )} \log \left ({\left | -2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt {-c e^{2}} b \right |}\right )}{c^{3}} + \frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left ({\left ({\left (\frac {{\left (56 \, c^{5} d^{4} g e^{4} + 8 \, c^{5} d^{3} f e^{5} - 116 \, b c^{4} d^{3} g e^{5} - 12 \, b c^{4} d^{2} f e^{6} + 90 \, b^{2} c^{3} d^{2} g e^{6} + 6 \, b^{2} c^{3} d f e^{7} - 31 \, b^{3} c^{2} d g e^{7} - b^{3} c^{2} f e^{8} + 4 \, b^{4} c g e^{8}\right )} x}{16 \, c^{6} d^{4} e^{3} - 32 \, b c^{5} d^{3} e^{4} + 24 \, b^{2} c^{4} d^{2} e^{5} - 8 \, b^{3} c^{3} d e^{6} + b^{4} c^{2} e^{7}} + \frac {3 \, {\left (24 \, c^{5} d^{5} g e^{3} + 8 \, c^{5} d^{4} f e^{4} - 36 \, b c^{4} d^{4} g e^{4} - 12 \, b c^{4} d^{3} f e^{5} + 10 \, b^{2} c^{3} d^{3} g e^{5} + 6 \, b^{2} c^{3} d^{2} f e^{6} + 9 \, b^{3} c^{2} d^{2} g e^{6} - b^{3} c^{2} d f e^{7} - 6 \, b^{4} c d g e^{7} + b^{5} g e^{8}\right )}}{16 \, c^{6} d^{4} e^{3} - 32 \, b c^{5} d^{3} e^{4} + 24 \, b^{2} c^{4} d^{2} e^{5} - 8 \, b^{3} c^{3} d e^{6} + b^{4} c^{2} e^{7}}\right )} x - \frac {3 \, {\left (8 \, c^{5} d^{6} g e^{2} - 8 \, c^{5} d^{5} f e^{3} - 44 \, b c^{4} d^{5} g e^{3} + 12 \, b c^{4} d^{4} f e^{4} + 70 \, b^{2} c^{3} d^{4} g e^{4} - 6 \, b^{2} c^{3} d^{3} f e^{5} - 49 \, b^{3} c^{2} d^{3} g e^{5} + b^{3} c^{2} d^{2} f e^{6} + 16 \, b^{4} c d^{2} g e^{6} - 2 \, b^{5} d g e^{7}\right )}}{16 \, c^{6} d^{4} e^{3} - 32 \, b c^{5} d^{3} e^{4} + 24 \, b^{2} c^{4} d^{2} e^{5} - 8 \, b^{3} c^{3} d e^{6} + b^{4} c^{2} e^{7}}\right )} x - \frac {40 \, c^{5} d^{7} g e - 8 \, c^{5} d^{6} f e^{2} - 124 \, b c^{4} d^{6} g e^{2} + 12 \, b c^{4} d^{5} f e^{3} + 150 \, b^{2} c^{3} d^{5} g e^{3} - 6 \, b^{2} c^{3} d^{4} f e^{4} - 89 \, b^{3} c^{2} d^{4} g e^{4} + b^{3} c^{2} d^{3} f e^{5} + 26 \, b^{4} c d^{3} g e^{5} - 3 \, b^{5} d^{2} g e^{6}}{16 \, c^{6} d^{4} e^{3} - 32 \, b c^{5} d^{3} e^{4} + 24 \, b^{2} c^{4} d^{2} e^{5} - 8 \, b^{3} c^{3} d e^{6} + b^{4} c^{2} e^{7}}\right )}}{3 \, {\left (c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")
[Out]
sqrt(-c*e^2)*g*e^(-3)*log(abs(-2*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*c - sqrt(-c*e^2
)*b))/c^3 + 2/3*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*((((56*c^5*d^4*g*e^4 + 8*c^5*d^3*f*e^5 - 116*b*c^4*
d^3*g*e^5 - 12*b*c^4*d^2*f*e^6 + 90*b^2*c^3*d^2*g*e^6 + 6*b^2*c^3*d*f*e^7 - 31*b^3*c^2*d*g*e^7 - b^3*c^2*f*e^8
+ 4*b^4*c*g*e^8)*x/(16*c^6*d^4*e^3 - 32*b*c^5*d^3*e^4 + 24*b^2*c^4*d^2*e^5 - 8*b^3*c^3*d*e^6 + b^4*c^2*e^7) +
3*(24*c^5*d^5*g*e^3 + 8*c^5*d^4*f*e^4 - 36*b*c^4*d^4*g*e^4 - 12*b*c^4*d^3*f*e^5 + 10*b^2*c^3*d^3*g*e^5 + 6*b^
2*c^3*d^2*f*e^6 + 9*b^3*c^2*d^2*g*e^6 - b^3*c^2*d*f*e^7 - 6*b^4*c*d*g*e^7 + b^5*g*e^8)/(16*c^6*d^4*e^3 - 32*b*
c^5*d^3*e^4 + 24*b^2*c^4*d^2*e^5 - 8*b^3*c^3*d*e^6 + b^4*c^2*e^7))*x - 3*(8*c^5*d^6*g*e^2 - 8*c^5*d^5*f*e^3 -
44*b*c^4*d^5*g*e^3 + 12*b*c^4*d^4*f*e^4 + 70*b^2*c^3*d^4*g*e^4 - 6*b^2*c^3*d^3*f*e^5 - 49*b^3*c^2*d^3*g*e^5 +
b^3*c^2*d^2*f*e^6 + 16*b^4*c*d^2*g*e^6 - 2*b^5*d*g*e^7)/(16*c^6*d^4*e^3 - 32*b*c^5*d^3*e^4 + 24*b^2*c^4*d^2*e^
5 - 8*b^3*c^3*d*e^6 + b^4*c^2*e^7))*x - (40*c^5*d^7*g*e - 8*c^5*d^6*f*e^2 - 124*b*c^4*d^6*g*e^2 + 12*b*c^4*d^5
*f*e^3 + 150*b^2*c^3*d^5*g*e^3 - 6*b^2*c^3*d^4*f*e^4 - 89*b^3*c^2*d^4*g*e^4 + b^3*c^2*d^3*f*e^5 + 26*b^4*c*d^3
*g*e^5 - 3*b^5*d^2*g*e^6)/(16*c^6*d^4*e^3 - 32*b*c^5*d^3*e^4 + 24*b^2*c^4*d^2*e^5 - 8*b^3*c^3*d*e^6 + b^4*c^2*
e^7))/(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e)^2
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maple [B] time = 0.08, size = 3485, normalized size = 19.69 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
[Out]
8*b*c*e^5/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2*f+104/3*b*c*e^4/
(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3*g+6*e^6*g*b^3/c/(-b^2*e^4+
4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d+1/2*b^2/c*e^4/(-b^2*e^4+4*b*c*d*e^3-4*
c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d*f-3/4*e^4*g*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^
2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d+3*e^3*g*b^2/c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2+1/3*e*g*x^3/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+1/48*e*g*b^3/c^4/(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-1/12*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(3/2)*e^5*x*f-11/24*g*b^2/c^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d-1/e*g/c^2*x/(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(1/2)+1/2/e*g/c^3*b/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+3/2*x/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(3/2)*d*f+1/e*g/c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))+3/4
*b/c^2*x/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*g+3/2*x/c/e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*g+1/(
-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^4*e*g+1/3/(-b^2*e^4+4*b*c*d*e^3
-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^3*e^2*f-1/6*e^7*g*b^5/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c
^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*e*g*b/c^2*x^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+
1/2*e^3*g/c^3*b^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+25/12/e*g*b/c^2/
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2-1/8*e*g*b^2/c^3*x/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+1/48*e^5*g
*b^5/c^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+52/3*b^2*e^4/(-b^2*e^4+4*
b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*g+4*b^2*e^5/(-b^2*e^4+4*b*c*d*e^3-4*c^2*
d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*f+1/4*b/c^2*x*e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*f
-1/24*b^4/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e^5*f+1/3*b^4/c^2*e^
7/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-1/24*b^2/c^3*e/(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(3/2)*f-5/3/e^2/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^3*g+1/3/e/c/(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(3/2)*d^2*f+3*x^2/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*g+5/12/c^2/(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)*b*d*f+x^2/c*e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*f+2/3*b^3/c*e^7/(-b^2*e^4+4*b*c*d*e^3-4*c^
2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*f+1/4*b^3/c^2*e^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*f-4*b^2*e^6/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2)*x*d*f-24*e^5*g*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2)*x*d^2+e^3*g/c^2*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+3/2*e
^3*g*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2-8*e^3*c/(-b^2*e^4
+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^4*g-2*b^3/c*e^6/(-b^2*e^4+4*b*c*d*e^3
-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-12*e^5*g*b^3/c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^
2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2-1/3*e^7*g*b^4/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x-3/8*e^4*g*b^4/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(3/2)*d+3*e^6*g*b^4/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*d+1/24*e^5*g*b^4/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x-13/6*
b^2/c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^3*e^2*g-1/2*b^2/c/(-b^2*e^
4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*e^3*f-13/3*b/(-b^2*e^4+4*b*c*d*e^3-4*c
^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^3*e^2*g-b/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2*e^3*f-8/3*e^4*c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(1/2)*b*d^3*f+2*c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x
*d^4*e*g+2/3*c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^3*e^2*f-16*e^3*
c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4*g-16/3*e^4*c^2/(-b^2*e
^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3*f
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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((e*x+d)^3*(g*x+f)/(-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(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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}} \,d 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)^(5/2),x)
[Out]
int(((f + g*x)*(d + e*x)^3)/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{3} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**3*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
[Out]
Integral((d + e*x)**3*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(5/2), x)
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