3.540 \(\int \frac{(a+c x^2)^{3/2}}{(d+e x)^3} \, dx\)
Optimal. Leaf size=161 \[ -\frac{3 c^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^4}-\frac{3 c \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^4 \sqrt{a e^2+c d^2}}+\frac{3 c \sqrt{a+c x^2} (2 d+e x)}{2 e^3 (d+e x)}-\frac{\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]
[Out]
(3*c*(2*d + e*x)*Sqrt[a + c*x^2])/(2*e^3*(d + e*x)) - (a + c*x^2)^(3/2)/(2*e*(d + e*x)^2) - (3*c^(3/2)*d*ArcTa
nh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/e^4 - (3*c*(2*c*d^2 + a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[
a + c*x^2])])/(2*e^4*Sqrt[c*d^2 + a*e^2])
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Rubi [A] time = 0.123799, antiderivative size = 161, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used =
{733, 813, 844, 217, 206, 725} \[ -\frac{3 c^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^4}-\frac{3 c \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^4 \sqrt{a e^2+c d^2}}+\frac{3 c \sqrt{a+c x^2} (2 d+e x)}{2 e^3 (d+e x)}-\frac{\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
Int[(a + c*x^2)^(3/2)/(d + e*x)^3,x]
[Out]
(3*c*(2*d + e*x)*Sqrt[a + c*x^2])/(2*e^3*(d + e*x)) - (a + c*x^2)^(3/2)/(2*e*(d + e*x)^2) - (3*c^(3/2)*d*ArcTa
nh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/e^4 - (3*c*(2*c*d^2 + a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[
a + c*x^2])])/(2*e^4*Sqrt[c*d^2 + a*e^2])
Rule 733
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 1)), x] - Dist[(2*c*p)/(e*(m + 1)), Int[x*(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c,
d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m +
2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 813
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 844
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IGtQ[m, 0]
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 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])
Rule 725
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx &=-\frac{\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}+\frac{(3 c) \int \frac{x \sqrt{a+c x^2}}{(d+e x)^2} \, dx}{2 e}\\ &=\frac{3 c (2 d+e x) \sqrt{a+c x^2}}{2 e^3 (d+e x)}-\frac{\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac{(3 c) \int \frac{-2 a e+4 c d x}{(d+e x) \sqrt{a+c x^2}} \, dx}{4 e^3}\\ &=\frac{3 c (2 d+e x) \sqrt{a+c x^2}}{2 e^3 (d+e x)}-\frac{\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac{\left (3 c^2 d\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{e^4}+\frac{\left (3 c \left (2 c d^2+a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{2 e^4}\\ &=\frac{3 c (2 d+e x) \sqrt{a+c x^2}}{2 e^3 (d+e x)}-\frac{\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac{\left (3 c^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{e^4}-\frac{\left (3 c \left (2 c d^2+a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d^2+a e^2-x^2} \, dx,x,\frac{a e-c d x}{\sqrt{a+c x^2}}\right )}{2 e^4}\\ &=\frac{3 c (2 d+e x) \sqrt{a+c x^2}}{2 e^3 (d+e x)}-\frac{\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac{3 c^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^4}-\frac{3 c \left (2 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{2 e^4 \sqrt{c d^2+a e^2}}\\ \end{align*}
Mathematica [A] time = 0.219271, size = 189, normalized size = 1.17 \[ \frac{-6 c^{3/2} d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\frac{e \sqrt{a+c x^2} \left (c \left (6 d^2+9 d e x+2 e^2 x^2\right )-a e^2\right )}{(d+e x)^2}-\frac{3 c \left (a e^2+2 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}+\frac{3 c \left (a e^2+2 c d^2\right ) \log (d+e x)}{\sqrt{a e^2+c d^2}}}{2 e^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + c*x^2)^(3/2)/(d + e*x)^3,x]
[Out]
((e*Sqrt[a + c*x^2]*(-(a*e^2) + c*(6*d^2 + 9*d*e*x + 2*e^2*x^2)))/(d + e*x)^2 + (3*c*(2*c*d^2 + a*e^2)*Log[d +
e*x])/Sqrt[c*d^2 + a*e^2] - 6*c^(3/2)*d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - (3*c*(2*c*d^2 + a*e^2)*Log[a*e -
c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/Sqrt[c*d^2 + a*e^2])/(2*e^4)
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Maple [B] time = 0.208, size = 2117, normalized size = 13.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^2+a)^(3/2)/(e*x+d)^3,x)
[Out]
-1/2/e/(a*e^2+c*d^2)/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)+1/2*c*d/(a*e^2+c*d^2)^2/(
d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)+1/2/e*c^2*d^2/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/
e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-3/4/e^2*c^3*d^3/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/
e^2)^(1/2)*x-9/4/e^2*c^(5/2)*d^3/(a*e^2+c*d^2)^2*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a
*e^2+c*d^2)/e^2)^(1/2))*a+3/2/e*c^2*d^2/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*
a+3/2/e^3*c^3*d^4/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)-3/2/e^4*c^(7/2)*d^5/(a
*e^2+c*d^2)^2*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))-3/2/e*c^2*d
^2/(a*e^2+c*d^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/
2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a^2-3/e^3*c^3*d^4/(a*e^2+c*d^2)^2/((a*e^2+c
*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e
+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a-3/2/e^5*c^4*d^6/(a*e^2+c*d^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^
2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)
)/(d/e+x))-1/2*c^2*d/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x-3/4*c^2*d/(a*e^2+
c*d^2)^2*a*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-3/4*c^(3/2)*d/(a*e^2+c*d^2)^2*a^2*ln((-c*d/
e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))+1/2/e/(a*e^2+c*d^2)*c*(c*(d/e+x)^2
-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-3/4/e^2/(a*e^2+c*d^2)*c^2*d*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^
2)/e^2)^(1/2)*x-9/4/e^2/(a*e^2+c*d^2)*c^(3/2)*d*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*
e^2+c*d^2)/e^2)^(1/2))*a+3/2/e/(a*e^2+c*d^2)*c*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a+3/2/e^3
/(a*e^2+c*d^2)*c^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*d^2-3/2/e^4/(a*e^2+c*d^2)*c^(5/2)*d^3
*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))-3/2/e/(a*e^2+c*d^2)*c/((
a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d
/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a^2-3/e^3/(a*e^2+c*d^2)*c^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e
^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2
))/(d/e+x))*a*d^2-3/2/e^5/(a*e^2+c*d^2)*c^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+
2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*d^4
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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((c*x^2+a)^(3/2)/(e*x+d)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 7.3965, size = 3198, normalized size = 19.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(3/2)/(e*x+d)^3,x, algorithm="fricas")
[Out]
[1/4*(6*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 2*(c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(c)*log(-2
*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*
c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x
^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(6*c^2*d^4*e + 5*a*c*
d^2*e^3 - a^2*e^5 + 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 + 9*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4
+ a*d^2*e^6 + (c*d^2*e^6 + a*e^8)*x^2 + 2*(c*d^3*e^5 + a*d*e^7)*x), 1/4*(12*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3
*e^2 + a*c*d*e^4)*x^2 + 2*(c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + 3*(2*c^2*
d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d^2 + a*e^2)*log((2*
a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 +
a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5 + 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 + 9
*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4 + a*d^2*e^6 + (c*d^2*e^6 + a*e^8)*x^2 + 2*(c*d^3*e^5
+ a*d*e^7)*x), -1/2*(3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*c^2*d^3*e + a*c*d*e^3)
*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 + (c^2*d
^2 + a*c*e^2)*x^2)) - 3*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 2*(c^2*d^4*e + a*c*d^2*e^3)*x
)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - (6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5 + 2*(c^2*d^
2*e^3 + a*c*e^5)*x^2 + 9*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4 + a*d^2*e^6 + (c*d^2*e^6 + a
*e^8)*x^2 + 2*(c*d^3*e^5 + a*d*e^7)*x), -1/2*(3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(
2*c^2*d^3*e + a*c*d*e^3)*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*
c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) - 6*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 2*(c^
2*d^4*e + a*c*d^2*e^3)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5
+ 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 + 9*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c*d^4*e^4 + a*d^2*e^6 + (c
*d^2*e^6 + a*e^8)*x^2 + 2*(c*d^3*e^5 + a*d*e^7)*x)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**2+a)**(3/2)/(e*x+d)**3,x)
[Out]
Integral((a + c*x**2)**(3/2)/(d + e*x)**3, x)
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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((c*x^2+a)^(3/2)/(e*x+d)^3,x, algorithm="giac")
[Out]
Exception raised: TypeError