3.1964 \(\int \frac{(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=294 \[ -\frac{14 e (d+e x)^3}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{35 e^2 (d+e x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c^3 d^3}+\frac{35 e^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^4 d^4}+\frac{35 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}}-\frac{2 (d+e x)^5}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*(d + e*x)^5)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (14*e*(d + e*x)^3)/(3*c^2*d^2*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e^2*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(
4*c^4*d^4) + (35*e^2*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(6*c^3*d^3) + (35*e^(3/2)*(c*d^2 -
a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2])])/(8*c^(9/2)*d^(9/2))

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Rubi [A]  time = 0.211751, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {668, 670, 640, 621, 206} \[ -\frac{14 e (d+e x)^3}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{35 e^2 (d+e x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c^3 d^3}+\frac{35 e^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^4 d^4}+\frac{35 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}}-\frac{2 (d+e x)^5}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^5)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (14*e*(d + e*x)^3)/(3*c^2*d^2*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e^2*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(
4*c^4*d^4) + (35*e^2*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(6*c^3*d^3) + (35*e^(3/2)*(c*d^2 -
a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2])])/(8*c^(9/2)*d^(9/2))

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 670

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*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{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] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && 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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac{2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{(7 e) \int \frac{(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d}\\ &=-\frac{2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{14 e (d+e x)^3}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (35 e^2\right ) \int \frac{(d+e x)^2}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 c^2 d^2}\\ &=-\frac{2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{14 e (d+e x)^3}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 e^2 (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac{\left (35 e^2 \left (c d^2-a e^2\right )\right ) \int \frac{d+e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 c^3 d^3}\\ &=-\frac{2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{14 e (d+e x)^3}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 e^2 \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac{35 e^2 (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac{\left (35 e^2 \left (c d^2-a e^2\right )^2\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 c^4 d^4}\\ &=-\frac{2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{14 e (d+e x)^3}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 e^2 \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac{35 e^2 (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac{\left (35 e^2 \left (c d^2-a e^2\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d e-x^2} \, dx,x,\frac{c d^2+a e^2+2 c d e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^4 d^4}\\ &=-\frac{2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{14 e (d+e x)^3}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 e^2 \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac{35 e^2 (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac{35 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{c d^2+a e^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0689856, size = 112, normalized size = 0.38 \[ -\frac{2 \left (c d^2-a e^2\right )^3 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left (-\frac{7}{2},-\frac{3}{2};-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{3 c^4 d^4 (a e+c d x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[-7/2, -3/2, -1/2, (e*(a*e + c*d*x))/(-(c
*d^2) + a*e^2)])/(3*c^4*d^4*(a*e + c*d*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

________________________________________________________________________________________

Maple [B]  time = 0.076, size = 4008, normalized size = 13.6 \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)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-35/24*e^3*d/c*x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1165/192*e^6/d^2/c^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(3/2)*a^3-285/16*e^2*d^2/c*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-35/384*e^10/d^6/c^6/(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(3/2)*a^5+637/384*e^8/d^4/c^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+253/48*e*d^9*c^2/
(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1249/128*e^2*d^2/c^2/(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(3/2)*a-765/64*e*d^3/c*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-35/16*e^5/d^3/c^4/(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+35/16*e^7/d^5/c^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+35/8*e^2*d^
4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-165/16*e^5*d^5/(-a^2*e^4+2*a*c*d^
2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+237/16*e^4/c^2*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(3/2)*a-437/384*e^2*d^6/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-185/384
*e^8/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+1/2*e^5*x^5/d/c/(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-35/16*e^3/d/c^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+253/384*d^8*c/(-a^2*
e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+35/16*e*d/c^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2)+625/64*e^4/c^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+17/4*e^4/c*x^4/(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(3/2)+35/8*e^2/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2))/(d*e*c)^(1/2)+35/16*e*d^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-35/8*
e^2/c^2*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/128*d^4/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-115/4*e^
4*d^6*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+35/8*e^10/d^4/c^4/(-a^2
*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4+65/64*e^5*d^3/c/(-a^2*e^4+2*a*c*d^2*
e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2+77/4*e^12/d^2/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^
2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^5-415/64*e^9/d/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^4+265/48*e^7*d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(3/2)*x*a^3-35/24*e^14/d^4/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*x*a^6+265/6*e^8*d^2/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3-415
/8*e^10/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4-115/32*e^3*d^5/(-
a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a-35/4*e^6/c^2/(-a^2*e^4+2*a*c*d^2*e^
2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+65/8*e^6*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+35/8*e^6/d^4/c^4*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^2-35/24*e^7/d^3/c^3*x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(3/2)*a^2+35/16*e^11/d^5/c^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+35/1
6*e^3*d^3/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+35/12*e^5/d/c^2*x^3/(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-261/16*e^11/d/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)*a^5+45/32*e^5/d/c^3*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+35/4*e^4/d^2/c^3*x/(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-35/4*e^4/d^2/c^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a-7/4*e^6/d^2/c^2*x^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*
a-437/48*e^3*d^7*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-3/4*e^3*d/c^2*
x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-35/8*e^5*d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)*a^2+1255/384*e^6*d^2/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(3/2)*a^3-165/128*e^4*d^4/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2-2
61/128*e^10/d^2/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^5-147/16*e^6/d^
2/c^3*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+35/16*e^9/d^3/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-35/8*e^7/d/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)*a^3-35/192*e^13/d^5/c^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3
/2)*x*a^6+77/32*e^11/d^3/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^5+35
/64*e^9/d^5/c^5*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+35/16*e^8/d^4/c^4*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(3/2)*a^3-35/48*e^15/d^5/c^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)*a^7+253/192*e*d^7*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-35/384*e^14/d
^6/c^6/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^7+427/384*e^12/d^4/c^5/(-a^2
*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^6-35/8*e^6/d^4/c^4*x/(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)*a^2-185/48*e^9*d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)*a^4+1255/48*e^7*d^3/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-7/
4*e^7/d^3/c^4*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+427/48*e^13/d^3/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^
4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6+253/24*e^2*d^8*c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x

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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)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 27.4125, size = 1717, normalized size = 5.84 \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)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(105*(a^2*c^2*d^4*e^3 - 2*a^3*c*d^2*e^5 + a^4*e^7 + (c^4*d^6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^2
+ 2*(a*c^3*d^5*e^2 - 2*a^2*c^2*d^3*e^4 + a^3*c*d*e^6)*x)*sqrt(e/(c*d))*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c
*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x + 4*(2*c^2*d^2*e*x + c^2*d^3 + a*c*d*e^2)*sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d))) + 4*(6*c^3*d^3*e^3*x^3 - 8*c^3*d^6 - 56*a*c^2*d^4*e^2 + 175*a^2*c*d^2*
e^4 - 105*a^3*e^6 + 3*(13*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 - 2*(40*c^3*d^5*e - 119*a*c^2*d^3*e^3 + 70*a^2*c*
d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^2*c^4*d^4*e^2), -1/2
4*(105*(a^2*c^2*d^4*e^3 - 2*a^3*c*d^2*e^5 + a^4*e^7 + (c^4*d^6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^2 + 2*
(a*c^3*d^5*e^2 - 2*a^2*c^2*d^3*e^4 + a^3*c*d*e^6)*x)*sqrt(-e/(c*d))*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-e/(c*d))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) - 2*(6*
c^3*d^3*e^3*x^3 - 8*c^3*d^6 - 56*a*c^2*d^4*e^2 + 175*a^2*c*d^2*e^4 - 105*a^3*e^6 + 3*(13*c^3*d^4*e^2 - 7*a*c^2
*d^2*e^4)*x^2 - 2*(40*c^3*d^5*e - 119*a*c^2*d^3*e^3 + 70*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e
^2)*x))/(c^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^2*c^4*d^4*e^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.5907, size = 1391, normalized size = 4.73 \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)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/12*((((3*(2*(c^7*d^11*e^8 - 4*a*c^6*d^9*e^10 + 6*a^2*c^5*d^7*e^12 - 4*a^3*c^4*d^5*e^14 + a^4*c^3*d^3*e^16)*x
/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6*a^2*c^6*d^8*e^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4*d^4*e^11) + (17*c^7*d^12*e
^7 - 75*a*c^6*d^10*e^9 + 130*a^2*c^5*d^8*e^11 - 110*a^3*c^4*d^6*e^13 + 45*a^4*c^3*d^4*e^15 - 7*a^5*c^2*d^2*e^1
7)/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6*a^2*c^6*d^8*e^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4*d^4*e^11))*x + 4*(c^7*d^
13*e^6 + 45*a*c^6*d^11*e^8 - 225*a^2*c^5*d^9*e^10 + 430*a^3*c^4*d^7*e^12 - 405*a^4*c^3*d^5*e^14 + 189*a^5*c^2*
d^3*e^16 - 35*a^6*c*d*e^18)/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6*a^2*c^6*d^8*e^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4
*d^4*e^11))*x - 3*(43*c^7*d^14*e^5 - 305*a*c^6*d^12*e^7 + 825*a^2*c^5*d^10*e^9 - 1075*a^3*c^4*d^8*e^11 + 645*a
^4*c^3*d^6*e^13 - 63*a^5*c^2*d^4*e^15 - 105*a^6*c*d^2*e^17 + 35*a^7*e^19)/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6
*a^2*c^6*d^8*e^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4*d^4*e^11))*x - 6*(16*c^7*d^15*e^4 - 85*a*c^6*d^13*e^6 + 145*a^2
*c^5*d^11*e^8 - 15*a^3*c^4*d^9*e^10 - 250*a^4*c^3*d^7*e^12 + 329*a^5*c^2*d^5*e^14 - 175*a^6*c*d^3*e^16 + 35*a^
7*d*e^18)/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6*a^2*c^6*d^8*e^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4*d^4*e^11))*x - (8
*c^7*d^16*e^3 + 24*a*c^6*d^14*e^5 - 351*a^2*c^5*d^12*e^7 + 1109*a^3*c^4*d^10*e^9 - 1686*a^4*c^3*d^8*e^11 + 138
6*a^5*c^2*d^6*e^13 - 595*a^6*c*d^4*e^15 + 105*a^7*d^2*e^17)/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6*a^2*c^6*d^8*e
^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4*d^4*e^11))/(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2) - 35/8*(c^2*d^4*e^2
- 2*a*c*d^2*e^4 + a^2*e^6)*sqrt(c*d)*e^(-1/2)*log(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt
(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^5*d^5)