### 3.1920 $$\int (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx$$

Optimal. Leaf size=341 $-\frac{7 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{7 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 c d}$

[Out]

(-7*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(512*c^4*d^4*e^
2) + (7*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*c^3*
d^3*e) + (7*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(60*c^2*d^2) + ((d + e*x)*(a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(6*c*d) + (7*(c*d^2 - a*e^2)^6*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sq
rt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(1024*c^(9/2)*d^(9/2)*e^(5/2))

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Rubi [A]  time = 0.263849, antiderivative size = 341, 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 = {670, 640, 612, 621, 206} $-\frac{7 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{7 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 c d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-7*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(512*c^4*d^4*e^
2) + (7*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*c^3*
d^3*e) + (7*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(60*c^2*d^2) + ((d + e*x)*(a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(6*c*d) + (7*(c*d^2 - a*e^2)^6*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sq
rt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(1024*c^(9/2)*d^(9/2)*e^(5/2))

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 612

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

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 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx &=\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac{\left (7 \left (d^2-\frac{a e^2}{c}\right )\right ) \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{12 d}\\ &=\frac{7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac{\left (7 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{24 d^2}\\ &=\frac{7 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}-\frac{\left (7 \left (c d^2-a e^2\right )^4\right ) \int \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}\\ &=-\frac{7 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac{\left (7 \left (c d^2-a e^2\right )^6\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^4 d^4 e^2}\\ &=-\frac{7 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac{\left (7 \left (c d^2-a e^2\right )^6\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 )}{512 c^4 d^4 e^2}\\ &=-\frac{7 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac{7 \left (c d^2-a e^2\right )^6 \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 )}{1024 c^{9/2} d^{9/2} e^{5/2}}\\ \end{align*}

Mathematica [A]  time = 3.22449, size = 328, normalized size = 0.96 $\frac{(a e+c d x)^2 \sqrt{(d+e x) (a e+c d x)} \left (896 c^5 d^5 (d+e x)^2 \left (c d^2-a e^2\right )+560 c^4 d^4 (d+e x) \left (c d^2-a e^2\right )^2+\frac{70 c^3 d^3 \left (c d^2-a e^2\right )^4}{e (a e+c d x)}-\frac{105 c^3 d^3 \left (c d^2-a e^2\right )^5}{e^2 (a e+c d x)^2}+\frac{105 c^{5/2} d^{5/2} \sqrt{c d} \left (c d^2-a e^2\right )^{11/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )}{e^{5/2} (a e+c d x)^{5/2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+280 \left (c^2 d^3-a c d e^2\right )^3+1280 c^6 d^6 (d+e x)^3\right )}{7680 c^7 d^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a*e + c*d*x)^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(280*(c^2*d^3 - a*c*d*e^2)^3 - (105*c^3*d^3*(c*d^2 - a*e^2)^5)/
(e^2*(a*e + c*d*x)^2) + (70*c^3*d^3*(c*d^2 - a*e^2)^4)/(e*(a*e + c*d*x)) + 560*c^4*d^4*(c*d^2 - a*e^2)^2*(d +
e*x) + 896*c^5*d^5*(c*d^2 - a*e^2)*(d + e*x)^2 + 1280*c^6*d^6*(d + e*x)^3 + (105*c^(5/2)*d^(5/2)*Sqrt[c*d]*(c*
d^2 - a*e^2)^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(e^(
5/2)*(a*e + c*d*x)^(5/2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(7680*c^7*d^7)

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Maple [B]  time = 0.053, size = 1302, normalized size = 3.8 \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)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

-21/512*d^6*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-7/192*e^3/d/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2-7/48*e^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(3/2)*x*a+1/6*e*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/d/c+7/64*e*a*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(1/2)*x+105/1024*e^6/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^4-7/512/e^2*d^6*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-7/256*e^4/c^2*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)*a^3+21/512*e^6/d^2/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-7/256*e^2*d^2/c*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-7/192*e*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-7/60*e^2/d^2/c^2
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a+7/192*e^5/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+7/102
4/e^2*d^8*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)+105/1024*e^2*d^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+7/96*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+21/512*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2)*a+7/192/e*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-7/512*e^8/d^4/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2)*a^5-7/256/e*d^5*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+17/60/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(5/2)+7/96*e^4/d^2/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2-7/256*e^7/d^3/c^3*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)*x*a^4-21/128*e^3*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+7/1024*e^10/d^4/c^4*l
n((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^6-21/51
2*e^8/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^5-35/256*e^4*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)*a^3+7/64*e^5*a^3/c^2/d*(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)^2*(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 = 2.18231, size = 2271, normalized size = 6.66 \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)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/30720*(105*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*
a^5*c*d^2*e^10 + a^6*e^12)*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*
(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^11*e + 595*a*c^5*d^9*e^3 + 1686*a^2*c^4*d^7*e^5 - 1386*a^3*c^3*d^5*e^7 + 595
*a^4*c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(37*c^6*d^7*e^5 + 13*a*c^5*d^5*e^7)*x^4 + 16*(387*c^6*d^8*e^4 + 410*
a*c^5*d^6*e^6 + 3*a^2*c^4*d^4*e^8)*x^3 + 8*(377*c^6*d^9*e^3 + 1191*a*c^5*d^7*e^5 + 39*a^2*c^4*d^5*e^7 - 7*a^3*
c^3*d^3*e^9)*x^2 + 2*(35*c^6*d^10*e^2 + 2876*a*c^5*d^8*e^4 + 450*a^2*c^4*d^6*e^6 - 196*a^3*c^3*d^4*e^8 + 35*a^
4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^3), -1/15360*(105*(c^6*d^12 - 6*a*c
^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*sqrt
(-c*d*e)*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(-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*(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^11*e + 595*a*c^5
*d^9*e^3 + 1686*a^2*c^4*d^7*e^5 - 1386*a^3*c^3*d^5*e^7 + 595*a^4*c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(37*c^6*
d^7*e^5 + 13*a*c^5*d^5*e^7)*x^4 + 16*(387*c^6*d^8*e^4 + 410*a*c^5*d^6*e^6 + 3*a^2*c^4*d^4*e^8)*x^3 + 8*(377*c^
6*d^9*e^3 + 1191*a*c^5*d^7*e^5 + 39*a^2*c^4*d^5*e^7 - 7*a^3*c^3*d^3*e^9)*x^2 + 2*(35*c^6*d^10*e^2 + 2876*a*c^5
*d^8*e^4 + 450*a^2*c^4*d^6*e^6 - 196*a^3*c^3*d^4*e^8 + 35*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x))/(c^5*d^5*e^3)]

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

[Out]

Timed out

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Giac [A]  time = 1.26031, size = 671, normalized size = 1.97 \begin{align*} \frac{1}{7680} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c d x e^{3} + \frac{{\left (37 \, c^{6} d^{7} e^{7} + 13 \, a c^{5} d^{5} e^{9}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (387 \, c^{6} d^{8} e^{6} + 410 \, a c^{5} d^{6} e^{8} + 3 \, a^{2} c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (377 \, c^{6} d^{9} e^{5} + 1191 \, a c^{5} d^{7} e^{7} + 39 \, a^{2} c^{4} d^{5} e^{9} - 7 \, a^{3} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (35 \, c^{6} d^{10} e^{4} + 2876 \, a c^{5} d^{8} e^{6} + 450 \, a^{2} c^{4} d^{6} e^{8} - 196 \, a^{3} c^{3} d^{4} e^{10} + 35 \, a^{4} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac{{\left (105 \, c^{6} d^{11} e^{3} - 595 \, a c^{5} d^{9} e^{5} - 1686 \, a^{2} c^{4} d^{7} e^{7} + 1386 \, a^{3} c^{3} d^{5} e^{9} - 595 \, a^{4} c^{2} d^{3} e^{11} + 105 \, a^{5} c d e^{13}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} - \frac{7 \,{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt{c d} e^{\left (-\frac{5}{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 + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{1024 \, c^{5} d^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(10*c*d*x*e^3 + (37*c^6*d^7*e^7 + 13*a*c^5*d^5*
e^9)*e^(-5)/(c^5*d^5))*x + (387*c^6*d^8*e^6 + 410*a*c^5*d^6*e^8 + 3*a^2*c^4*d^4*e^10)*e^(-5)/(c^5*d^5))*x + (3
77*c^6*d^9*e^5 + 1191*a*c^5*d^7*e^7 + 39*a^2*c^4*d^5*e^9 - 7*a^3*c^3*d^3*e^11)*e^(-5)/(c^5*d^5))*x + (35*c^6*d
^10*e^4 + 2876*a*c^5*d^8*e^6 + 450*a^2*c^4*d^6*e^8 - 196*a^3*c^3*d^4*e^10 + 35*a^4*c^2*d^2*e^12)*e^(-5)/(c^5*d
^5))*x - (105*c^6*d^11*e^3 - 595*a*c^5*d^9*e^5 - 1686*a^2*c^4*d^7*e^7 + 1386*a^3*c^3*d^5*e^9 - 595*a^4*c^2*d^3
*e^11 + 105*a^5*c*d*e^13)*e^(-5)/(c^5*d^5)) - 7/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^
3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*sqrt(c*d)*e^(-5/2)*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 + c*d^2*x + a*x*e^2 + a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2)))/
(c^5*d^5)