3.2224 \(\int \frac{(d+e x)^4 (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=291 \[ -\frac{2 (d+e x)^2 (-5 b e g+8 c d g+2 c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+8 c d g+2 c e f)}{c^3 e^2 (2 c d-b e)}+\frac{(-5 b e g+8 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{7/2} e^2}+\frac{2 (d+e x)^4 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^4)/(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*(2*c*e*f + 8*c*d*g - 5*b*e*g)*(d + e*x)^2)/(3*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2]) - ((2*c*e*f + 8*c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c^3*e^2*(2*c*d - b*e)) + ((2*
c*e*f + 8*c*d*g - 5*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(2*c
^(7/2)*e^2)

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Rubi [A]  time = 0.428289, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {788, 668, 640, 621, 204} \[ -\frac{2 (d+e x)^2 (-5 b e g+8 c d g+2 c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+8 c d g+2 c e f)}{c^3 e^2 (2 c d-b e)}+\frac{(-5 b e g+8 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{7/2} e^2}+\frac{2 (d+e x)^4 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^4*(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)^4)/(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*(2*c*e*f + 8*c*d*g - 5*b*e*g)*(d + e*x)^2)/(3*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2]) - ((2*c*e*f + 8*c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c^3*e^2*(2*c*d - b*e)) + ((2*
c*e*f + 8*c*d*g - 5*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(2*c
^(7/2)*e^2)

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]

Rule 668

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

Rule 640

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

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 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])

Rubi steps

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

Mathematica [C]  time = 0.277141, size = 139, normalized size = 0.48 \[ \frac{2 (d+e x)^4 \left (\left (\frac{b e-c d+c e x}{b e-2 c d}\right )^{3/2} (-5 b e g+8 c d g+2 c e f) \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{c (d+e x)}{2 c d-b e}\right )-5 (-b e g+c d g+c e f)\right )}{15 c e^2 (b e-2 c d) ((d+e x) (c (d-e x)-b e))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^4*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*(d + e*x)^4*(-5*(c*e*f + c*d*g - b*e*g) + (2*c*e*f + 8*c*d*g - 5*b*e*g)*((-(c*d) + b*e + c*e*x)/(-2*c*d + b
*e))^(3/2)*Hypergeometric2F1[3/2, 5/2, 7/2, (c*(d + e*x))/(2*c*d - b*e)]))/(15*c*e^2*(-2*c*d + b*e)*((d + e*x)
*(-(b*e) + c*(d - e*x)))^(3/2))

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Maple [B]  time = 0.014, size = 5032, normalized size = 17.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(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

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Fricas [A]  time = 28.4054, size = 1840, normalized size = 6.32 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(3*((2*c^3*e^3*f + (8*c^3*d*e^2 - 5*b*c^2*e^3)*g)*x^2 + 2*(c^3*d^2*e - 2*b*c^2*d*e^2 + b^2*c*e^3)*f + (8
*c^3*d^3 - 21*b*c^2*d^2*e + 18*b^2*c*d*e^2 - 5*b^3*e^3)*g - 2*(2*(c^3*d*e^2 - b*c^2*e^3)*f + (8*c^3*d^2*e - 13
*b*c^2*d*e^2 + 5*b^2*c*e^3)*g)*x)*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*(3*c^3*e^2*g*x^2 + 2*(2*c^3*d*e -
3*b*c^2*e^2)*f + (19*c^3*d^2 - 34*b*c^2*d*e + 15*b^2*c*e^2)*g - 2*(4*c^3*e^2*f + (13*c^3*d*e - 10*b*c^2*e^2)*g
)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^6*e^4*x^2 + c^6*d^2*e^2 - 2*b*c^5*d*e^3 + b^2*c^4*e^4 - 2*
(c^6*d*e^3 - b*c^5*e^4)*x), -1/6*(3*((2*c^3*e^3*f + (8*c^3*d*e^2 - 5*b*c^2*e^3)*g)*x^2 + 2*(c^3*d^2*e - 2*b*c^
2*d*e^2 + b^2*c*e^3)*f + (8*c^3*d^3 - 21*b*c^2*d^2*e + 18*b^2*c*d*e^2 - 5*b^3*e^3)*g - 2*(2*(c^3*d*e^2 - b*c^2
*e^3)*f + (8*c^3*d^2*e - 13*b*c^2*d*e^2 + 5*b^2*c*e^3)*g)*x)*sqrt(c)*arctan(1/2*sqrt(-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*(3*c^3*e^2*g*x^2 + 2*(
2*c^3*d*e - 3*b*c^2*e^2)*f + (19*c^3*d^2 - 34*b*c^2*d*e + 15*b^2*c*e^2)*g - 2*(4*c^3*e^2*f + (13*c^3*d*e - 10*
b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^6*e^4*x^2 + c^6*d^2*e^2 - 2*b*c^5*d*e^3 + b^2*
c^4*e^4 - 2*(c^6*d*e^3 - b*c^5*e^4)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{4} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(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)**4*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(5/2), x)

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Giac [B]  time = 1.30143, size = 1507, normalized size = 5.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

-1/3*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*((((3*(16*c^6*d^4*g*e^8 - 32*b*c^5*d^3*g*e^9 + 24*b^2*c^4*d^2*
g*e^10 - 8*b^3*c^3*d*g*e^11 + b^4*c^2*g*e^12)*x/(16*c^7*d^4*e^6 - 32*b*c^6*d^3*e^7 + 24*b^2*c^5*d^2*e^8 - 8*b^
3*c^4*d*e^9 + b^4*c^3*e^10) - 4*(80*c^6*d^5*g*e^7 + 32*c^6*d^4*f*e^8 - 240*b*c^5*d^4*g*e^8 - 64*b*c^5*d^3*f*e^
9 + 280*b^2*c^4*d^3*g*e^9 + 48*b^2*c^4*d^2*f*e^10 - 160*b^3*c^3*d^2*g*e^10 - 16*b^3*c^3*d*f*e^11 + 45*b^4*c^2*
d*g*e^11 + 2*b^4*c^2*f*e^12 - 5*b^5*c*g*e^12)/(16*c^7*d^4*e^6 - 32*b*c^6*d^3*e^7 + 24*b^2*c^5*d^2*e^8 - 8*b^3*
c^4*d*e^9 + b^4*c^3*e^10))*x - 3*(160*c^6*d^6*g*e^6 + 64*c^6*d^5*f*e^7 - 352*b*c^5*d^5*g*e^7 - 96*b*c^5*d^4*f*
e^8 + 224*b^2*c^4*d^4*g*e^8 + 32*b^2*c^4*d^3*f*e^9 + 32*b^3*c^3*d^3*g*e^9 + 16*b^3*c^3*d^2*f*e^10 - 94*b^4*c^2
*d^2*g*e^10 - 12*b^4*c^2*d*f*e^11 + 38*b^5*c*d*g*e^11 + 2*b^5*c*f*e^12 - 5*b^6*g*e^12)/(16*c^7*d^4*e^6 - 32*b*
c^6*d^3*e^7 + 24*b^2*c^5*d^2*e^8 - 8*b^3*c^4*d*e^9 + b^4*c^3*e^10))*x + 6*(32*c^6*d^7*g*e^5 - 192*b*c^5*d^6*g*
e^6 - 32*b*c^5*d^5*f*e^7 + 384*b^2*c^4*d^5*g*e^7 + 64*b^2*c^4*d^4*f*e^8 - 368*b^3*c^3*d^4*g*e^8 - 48*b^3*c^3*d
^3*f*e^9 + 186*b^4*c^2*d^3*g*e^9 + 16*b^4*c^2*d^2*f*e^10 - 48*b^5*c*d^2*g*e^10 - 2*b^5*c*d*f*e^11 + 5*b^6*d*g*
e^11)/(16*c^7*d^4*e^6 - 32*b*c^6*d^3*e^7 + 24*b^2*c^5*d^2*e^8 - 8*b^3*c^4*d*e^9 + b^4*c^3*e^10))*x + (304*c^6*
d^8*g*e^4 + 64*c^6*d^7*f*e^5 - 1152*b*c^5*d^7*g*e^5 - 224*b*c^5*d^6*f*e^6 + 1784*b^2*c^4*d^6*g*e^6 + 288*b^2*c
^4*d^5*f*e^7 - 1448*b^3*c^3*d^5*g*e^7 - 176*b^3*c^3*d^4*f*e^8 + 651*b^4*c^2*d^4*g*e^8 + 52*b^4*c^2*d^3*f*e^9 -
154*b^5*c*d^3*g*e^9 - 6*b^5*c*d^2*f*e^10 + 15*b^6*d^2*g*e^10)/(16*c^7*d^4*e^6 - 32*b*c^6*d^3*e^7 + 24*b^2*c^5
*d^2*e^8 - 8*b^3*c^4*d*e^9 + b^4*c^3*e^10))/(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e)^2 + 1/2*(8*c*d*g + 2*c*f*e -
5*b*g*e)*sqrt(-c*e^2)*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 - sqr
t(-c*e^2)*b))/c^4