3.437 \(\int \frac{x^3 \sqrt{a d e+(c d^2+a e^2) x+c d e x^2}}{d+e x} \, dx\)

Optimal. Leaf size=345 \[ -\frac{\left (-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-15 a^3 e^6-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 c^3 d^3 e^4}+\frac{\left (c d^2-a e^2\right ) \left (9 a^2 c d^2 e^4+5 a^3 e^6+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{9/2}}+\frac{x^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}+\frac{1}{24} x^2 \left (\frac{a}{c d}-\frac{7 d}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \]

[Out]

((a/(c*d) - (7*d)/e^2)*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/24 + (x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(4*e) - ((105*c^3*d^6 - 25*a*c^2*d^4*e^2 - 17*a^2*c*d^2*e^4 - 15*a^3*e^6 - 2*c*d*e*(35*c^2*d^
4 - 6*a*c*d^2*e^2 - 5*a^2*e^4)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(192*c^3*d^3*e^4) + ((c*d^2 - a
*e^2)*(35*c^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqr
t[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(128*c^(7/2)*d^(7/2)*e^(9/2))

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Rubi [A]  time = 0.506088, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {849, 832, 779, 621, 206} \[ -\frac{\left (-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-15 a^3 e^6-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 c^3 d^3 e^4}+\frac{\left (c d^2-a e^2\right ) \left (9 a^2 c d^2 e^4+5 a^3 e^6+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{9/2}}+\frac{x^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}+\frac{1}{24} x^2 \left (\frac{a}{c d}-\frac{7 d}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a/(c*d) - (7*d)/e^2)*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/24 + (x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(4*e) - ((105*c^3*d^6 - 25*a*c^2*d^4*e^2 - 17*a^2*c*d^2*e^4 - 15*a^3*e^6 - 2*c*d*e*(35*c^2*d^
4 - 6*a*c*d^2*e^2 - 5*a^2*e^4)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(192*c^3*d^3*e^4) + ((c*d^2 - a
*e^2)*(35*c^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqr
t[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(128*c^(7/2)*d^(7/2)*e^(9/2))

Rule 849

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

Rule 832

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

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[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{x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx &=\int \frac{x^3 (a e+c d x)}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=\frac{x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}+\frac{\int \frac{x^2 \left (-3 a c d^2 e-\frac{1}{2} c d \left (7 c d^2-a e^2\right ) x\right )}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 c d e}\\ &=\frac{1}{24} \left (\frac{a}{c d}-\frac{7 d}{e^2}\right ) x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac{x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}+\frac{\int \frac{x \left (a c d^2 e \left (7 c d^2-a e^2\right )+\frac{1}{4} c d \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{12 c^2 d^2 e^2}\\ &=\frac{1}{24} \left (\frac{a}{c d}-\frac{7 d}{e^2}\right ) x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac{x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac{\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac{\left (\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right )\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c^3 d^3 e^4}\\ &=\frac{1}{24} \left (\frac{a}{c d}-\frac{7 d}{e^2}\right ) x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac{x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac{\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac{\left (\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right )\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 )}{64 c^3 d^3 e^4}\\ &=\frac{1}{24} \left (\frac{a}{c d}-\frac{7 d}{e^2}\right ) x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac{x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac{\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac{\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right ) \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 )}{128 c^{7/2} d^{7/2} e^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.76984, size = 304, normalized size = 0.88 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{3 \sqrt{c d} \sqrt{c d^2-a e^2} \left (9 a^2 c d^2 e^4+5 a^3 e^6+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )}{\sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\sqrt{c} \sqrt{d} \sqrt{e} \left (a^2 c d e^4 (10 e x-17 d)-15 a^3 e^6+a c^2 d^2 e^2 \left (-25 d^2+12 d e x-8 e^2 x^2\right )+c^3 d^3 \left (-70 d^2 e x+105 d^3+56 d e^2 x^2-48 e^3 x^3\right )\right )\right )}{192 c^{7/2} d^{7/2} e^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-15*a^3*e^6 + a^2*c*d*e^4*(-17*d + 10*e*x) + a*c^2*
d^2*e^2*(-25*d^2 + 12*d*e*x - 8*e^2*x^2) + c^3*d^3*(105*d^3 - 70*d^2*e*x + 56*d*e^2*x^2 - 48*e^3*x^3))) + (3*S
qrt[c*d]*Sqrt[c*d^2 - a*e^2]*(35*c^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*ArcSinh[(Sqrt[c]*Sq
rt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*
d^2 - a*e^2)])))/(192*c^(7/2)*d^(7/2)*e^(9/2))

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Maple [B]  time = 0.07, size = 946, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x)

[Out]

5/32*e/d^2/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2-5/128*e^4/d^3/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^4-d^3/e^4*(c*d*e*(d/e+x)^2+(a*e^2-c*d
^2)*(d/e+x))^(1/2)-13/24/e^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+29/64/e^4*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)-1/2*d^3/e^2*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)
*(d/e+x))^(1/2))/(d*e*c)^(1/2)*a+1/2*d^5/e^4*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+
x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)*c-3/64*d/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)*a^2+1/4/e^2*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/c-
5/24/e/d^2/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+7/16/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+
5/64*e^2/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+43/64/e^2*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)*a-29/128/e^4*d^5*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)+11/32/e^2*a*d^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)+29/32/e^3*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+19/64/d/c^2*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)*a^2-1/32*e^2*a^3/c^2/d*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)

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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(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.60576, size = 1438, normalized size = 4.17 \begin{align*} \left [-\frac{3 \,{\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt{c d e} \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} - 4 \, \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{c d e} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \,{\left (48 \, c^{4} d^{4} e^{4} x^{3} - 105 \, c^{4} d^{7} e + 25 \, a c^{3} d^{5} e^{3} + 17 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} - 8 \,{\left (7 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \,{\left (35 \, c^{4} d^{6} e^{2} - 6 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{768 \, c^{4} d^{4} e^{5}}, -\frac{3 \,{\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt{-c d e} \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{-c d e}}{2 \,{\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} +{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \,{\left (48 \, c^{4} d^{4} e^{4} x^{3} - 105 \, c^{4} d^{7} e + 25 \, a c^{3} d^{5} e^{3} + 17 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} - 8 \,{\left (7 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \,{\left (35 \, c^{4} d^{6} e^{2} - 6 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{384 \, c^{4} d^{4} e^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algorithm="fricas")

[Out]

[-1/768*(3*(35*c^4*d^8 - 20*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 - 5*a^4*e^8)*sqrt(c*d*e)*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 - 4*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(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - 4*(48*c^4*d^4*e^4*x^3 - 105*c^4*d^7*e + 25*a*c
^3*d^5*e^3 + 17*a^2*c^2*d^3*e^5 + 15*a^3*c*d*e^7 - 8*(7*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^2 + 2*(35*c^4*d^6*e^2 -
6*a*c^3*d^4*e^4 - 5*a^2*c^2*d^2*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^5), -1/384*(3
*(35*c^4*d^8 - 20*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 - 5*a^4*e^8)*sqrt(-c*d*e)*arctan(1/2*sqr
t(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(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e
^2 + (c^2*d^3*e + a*c*d*e^3)*x)) - 2*(48*c^4*d^4*e^4*x^3 - 105*c^4*d^7*e + 25*a*c^3*d^5*e^3 + 17*a^2*c^2*d^3*e
^5 + 15*a^3*c*d*e^7 - 8*(7*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^2 + 2*(35*c^4*d^6*e^2 - 6*a*c^3*d^4*e^4 - 5*a^2*c^2*
d^2*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^5)]

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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(x**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: TypeError