3.1954 \(\int \frac{(d+e x)^5}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=302 \[ \frac{7 e (d+e x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2}+\frac{35 e (d+e x) \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3}+\frac{35 e \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c^4 d^4}+\frac{35 \sqrt{e} \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 c^{9/2} d^{9/2}}-\frac{2 (d+e x)^4}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(-2*(d + e*x)^4)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2])/(8*c^4*d^4) + (35*e*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2])/(12*c^3*d^3) + (7*e*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c^2*d^2) + (35*Sqrt[e
]*(c*d^2 - a*e^2)^3*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])])/(16*c^(9/2)*d^(9/2))

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Rubi [A]  time = 0.24369, antiderivative size = 302, 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} \[ \frac{7 e (d+e x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2}+\frac{35 e (d+e x) \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3}+\frac{35 e \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c^4 d^4}+\frac{35 \sqrt{e} \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 c^{9/2} d^{9/2}}-\frac{2 (d+e x)^4}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-2*(d + e*x)^4)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2])/(8*c^4*d^4) + (35*e*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2])/(12*c^3*d^3) + (7*e*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c^2*d^2) + (35*Sqrt[e
]*(c*d^2 - a*e^2)^3*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])])/(16*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)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^4}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{(7 e) \int \frac{(d+e x)^3}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d}\\ &=-\frac{2 (d+e x)^4}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{7 e (d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac{\left (35 e \left (c d^2-a e^2\right )\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}{6 c^2 d^2}\\ &=-\frac{2 (d+e x)^4}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac{7 e (d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac{\left (35 e \left (c d^2-a e^2\right )^2\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}{8 c^3 d^3}\\ &=-\frac{2 (d+e x)^4}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 e \left (c d^2-a e^2\right )^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^4 d^4}+\frac{35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac{7 e (d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac{\left (35 e \left (c d^2-a e^2\right )^3\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 c^4 d^4}\\ &=-\frac{2 (d+e x)^4}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 e \left (c d^2-a e^2\right )^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^4 d^4}+\frac{35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac{7 e (d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac{\left (35 e \left (c d^2-a e^2\right )^3\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 )}{8 c^4 d^4}\\ &=-\frac{2 (d+e x)^4}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 e \left (c d^2-a e^2\right )^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^4 d^4}+\frac{35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac{7 e (d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac{35 \sqrt{e} \left (c d^2-a e^2\right )^3 \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 )}{16 c^{9/2} d^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0713392, size = 100, normalized size = 0.33 \[ -\frac{2 \left (c d^2-a e^2\right )^4 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \, _2F_1\left (-\frac{7}{2},-\frac{1}{2};\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{c^5 d^5 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^4*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*Hypergeometric2F1[-7/2, -1/2, 1/2, (e*(a*e + c*d*x
))/(-(c*d^2) + a*e^2)])/(c^5*d^5*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [B]  time = 0.065, size = 1850, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

2*d^5*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-125/32*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)^(1/2)+23/12*e^3/c*x^3/(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)+1/3*e^4*x^4/d/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/16*e*d^2/c*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)-125/32*d^3/c/(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)-35/32*e^8/d^5/c^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+245/48*e^6/d^3/c^4/
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-35/16*e*d^2/c*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+395/48*e^2
*d/c^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-28/3*e^4/d/c^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+12
5/24*e^2*d/c*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-105/16*e^3/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)*a+5/12*e^2*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)*a+105/16*e^3/c^2*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+385/48*
e^9/d^2/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)*x*a^4-53/24*e^5*d^2/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)*x*a^2-35/16*e^11/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^5+35/12*e^10/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^5-35/16*e^7/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^3+35/16*e^7/d^4/c^4*x/(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)*a^3-7/32*e^8/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^4-
203/24*e^7/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^3+415/48*e^3*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*a+103/32*e^4*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^2-14/3*e^4/d/c^2*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2)*a-16/3*e^6*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^3+35/24*e
^6/d^3/c^3*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-7/12*e^5/d^2/c^2*x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(1/2)*a+105/16*e^5/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^2-105/16*e^5/d^2/c^3*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-125/16*e*d^6*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)*x-35/32*e^12/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^6

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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)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 10.0756, size = 1574, normalized size = 5.21 \begin{align*} \left [\frac{105 \,{\left (a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7} +{\left (c^{4} d^{7} - 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} - a^{3} c d e^{6}\right )} x\right )} \sqrt{\frac{e}{c d}} \log \left (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 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \,{\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d}}\right ) + 4 \,{\left (8 \, c^{3} d^{3} e^{3} x^{3} - 48 \, c^{3} d^{6} + 231 \, a c^{2} d^{4} e^{2} - 280 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \,{\left (19 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (87 \, c^{3} d^{5} e - 98 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{96 \,{\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}, -\frac{105 \,{\left (a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7} +{\left (c^{4} d^{7} - 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} - a^{3} c d e^{6}\right )} x\right )} \sqrt{-\frac{e}{c d}} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-\frac{e}{c d}}}{2 \,{\left (c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )}}\right ) - 2 \,{\left (8 \, c^{3} d^{3} e^{3} x^{3} - 48 \, c^{3} d^{6} + 231 \, a c^{2} d^{4} e^{2} - 280 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \,{\left (19 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (87 \, c^{3} d^{5} e - 98 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{48 \,{\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/96*(105*(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7 + (c^4*d^7 - 3*a*c^3*d^5*e^2 + 3*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)*sqr
t(e/(c*d))) + 4*(8*c^3*d^3*e^3*x^3 - 48*c^3*d^6 + 231*a*c^2*d^4*e^2 - 280*a^2*c*d^2*e^4 + 105*a^3*e^6 + 2*(19*
c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 + (87*c^3*d^5*e - 98*a*c^2*d^3*e^3 + 35*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*x + a*c^4*d^4*e), -1/48*(105*(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2
*e^5 - a^4*e^7 + (c^4*d^7 - 3*a*c^3*d^5*e^2 + 3*a^2*c^2*d^3*e^4 - a^3*c*d*e^6)*x)*sqrt(-e/(c*d))*arctan(1/2*sq
rt(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*(8*c^3*d^3*e^3*x^3 - 48*c^3*d^6 + 231*a*c^2*d^4*e^2 - 280*a^2*c*d^2*e^4 + 105*a^3*e^
6 + 2*(19*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 + (87*c^3*d^5*e - 98*a*c^2*d^3*e^3 + 35*a^2*c*d*e^5)*x)*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*x + a*c^4*d^4*e)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{5}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Integral((d + e*x)**5/((d + e*x)*(a*e + c*d*x))**(3/2), x)

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Giac [B]  time = 1.34287, size = 840, normalized size = 2.78 \begin{align*} \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (c^{5} d^{7} e^{7} - 2 \, a c^{4} d^{5} e^{9} + a^{2} c^{3} d^{3} e^{11}\right )} x}{c^{6} d^{8} e^{3} - 2 \, a c^{5} d^{6} e^{5} + a^{2} c^{4} d^{4} e^{7}} + \frac{23 \, c^{5} d^{8} e^{6} - 53 \, a c^{4} d^{6} e^{8} + 37 \, a^{2} c^{3} d^{4} e^{10} - 7 \, a^{3} c^{2} d^{2} e^{12}}{c^{6} d^{8} e^{3} - 2 \, a c^{5} d^{6} e^{5} + a^{2} c^{4} d^{4} e^{7}}\right )} x + \frac{125 \, c^{5} d^{9} e^{5} - 362 \, a c^{4} d^{7} e^{7} + 384 \, a^{2} c^{3} d^{5} e^{9} - 182 \, a^{3} c^{2} d^{3} e^{11} + 35 \, a^{4} c d e^{13}}{c^{6} d^{8} e^{3} - 2 \, a c^{5} d^{6} e^{5} + a^{2} c^{4} d^{4} e^{7}}\right )} x + \frac{39 \, c^{5} d^{10} e^{4} + 55 \, a c^{4} d^{8} e^{6} - 472 \, a^{2} c^{3} d^{6} e^{8} + 728 \, a^{3} c^{2} d^{4} e^{10} - 455 \, a^{4} c d^{2} e^{12} + 105 \, a^{5} e^{14}}{c^{6} d^{8} e^{3} - 2 \, a c^{5} d^{6} e^{5} + a^{2} c^{4} d^{4} e^{7}}\right )} x - \frac{48 \, c^{5} d^{11} e^{3} - 327 \, a c^{4} d^{9} e^{5} + 790 \, a^{2} c^{3} d^{7} e^{7} - 896 \, a^{3} c^{2} d^{5} e^{9} + 490 \, a^{4} c d^{3} e^{11} - 105 \, a^{5} d e^{13}}{c^{6} d^{8} e^{3} - 2 \, a c^{5} d^{6} e^{5} + a^{2} c^{4} d^{4} e^{7}}}{24 \, \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}} - \frac{35 \,{\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \sqrt{c d} e^{\left (-\frac{1}{2}\right )} \log \left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{16 \, c^{5} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

1/24*(((2*(4*(c^5*d^7*e^7 - 2*a*c^4*d^5*e^9 + a^2*c^3*d^3*e^11)*x/(c^6*d^8*e^3 - 2*a*c^5*d^6*e^5 + a^2*c^4*d^4
*e^7) + (23*c^5*d^8*e^6 - 53*a*c^4*d^6*e^8 + 37*a^2*c^3*d^4*e^10 - 7*a^3*c^2*d^2*e^12)/(c^6*d^8*e^3 - 2*a*c^5*
d^6*e^5 + a^2*c^4*d^4*e^7))*x + (125*c^5*d^9*e^5 - 362*a*c^4*d^7*e^7 + 384*a^2*c^3*d^5*e^9 - 182*a^3*c^2*d^3*e
^11 + 35*a^4*c*d*e^13)/(c^6*d^8*e^3 - 2*a*c^5*d^6*e^5 + a^2*c^4*d^4*e^7))*x + (39*c^5*d^10*e^4 + 55*a*c^4*d^8*
e^6 - 472*a^2*c^3*d^6*e^8 + 728*a^3*c^2*d^4*e^10 - 455*a^4*c*d^2*e^12 + 105*a^5*e^14)/(c^6*d^8*e^3 - 2*a*c^5*d
^6*e^5 + a^2*c^4*d^4*e^7))*x - (48*c^5*d^11*e^3 - 327*a*c^4*d^9*e^5 + 790*a^2*c^3*d^7*e^7 - 896*a^3*c^2*d^5*e^
9 + 490*a^4*c*d^3*e^11 - 105*a^5*d*e^13)/(c^6*d^8*e^3 - 2*a*c^5*d^6*e^5 + a^2*c^4*d^4*e^7))/sqrt(c*d*x^2*e + a
*d*e + (c*d^2 + a*e^2)*x) - 35/16*(c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - a^3*e^7)*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)