3.544 \(\int \frac{(a+c x^2)^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=269 \[ -\frac{a^2 c^3 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac{a c^2 \sqrt{a+c x^2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac{7 c d e \left (a+c x^2\right )^{5/2}}{30 (d+e x)^5 \left (a e^2+c d^2\right )^2}-\frac{c \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]

[Out]

-(a*c^2*(6*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^4*(d + e*x)^2) - (c*(6*c*d^2 - a*
e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(24*(c*d^2 + a*e^2)^3*(d + e*x)^4) - (e*(a + c*x^2)^(5/2))/(6*(c*d^2 + a
*e^2)*(d + e*x)^6) - (7*c*d*e*(a + c*x^2)^(5/2))/(30*(c*d^2 + a*e^2)^2*(d + e*x)^5) - (a^2*c^3*(6*c*d^2 - a*e^
2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(9/2))

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Rubi [A]  time = 0.183595, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {745, 807, 721, 725, 206} \[ -\frac{a^2 c^3 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac{a c^2 \sqrt{a+c x^2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac{7 c d e \left (a+c x^2\right )^{5/2}}{30 (d+e x)^5 \left (a e^2+c d^2\right )^2}-\frac{c \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

-(a*c^2*(6*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^4*(d + e*x)^2) - (c*(6*c*d^2 - a*
e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(24*(c*d^2 + a*e^2)^3*(d + e*x)^4) - (e*(a + c*x^2)^(5/2))/(6*(c*d^2 + a
*e^2)*(d + e*x)^6) - (7*c*d*e*(a + c*x^2)^(5/2))/(30*(c*d^2 + a*e^2)^2*(d + e*x)^5) - (a^2*c^3*(6*c*d^2 - a*e^
2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(9/2))

Rule 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 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]

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{\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx &=-\frac{e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac{c \int \frac{(-6 d+e x) \left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{6 \left (c d^2+a e^2\right )}\\ &=-\frac{e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac{7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}+\frac{\left (c \left (6 c d^2-a e^2\right )\right ) \int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{6 \left (c d^2+a e^2\right )^2}\\ &=-\frac{c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac{e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac{7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}+\frac{\left (a c^2 \left (6 c d^2-a e^2\right )\right ) \int \frac{\sqrt{a+c x^2}}{(d+e x)^3} \, dx}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac{a c^2 \left (6 c d^2-a e^2\right ) (a e-c d x) \sqrt{a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac{c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac{e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac{7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}+\frac{\left (a^2 c^3 \left (6 c d^2-a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{16 \left (c d^2+a e^2\right )^4}\\ &=-\frac{a c^2 \left (6 c d^2-a e^2\right ) (a e-c d x) \sqrt{a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac{c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac{e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac{7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}-\frac{\left (a^2 c^3 \left (6 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 )}{16 \left (c d^2+a e^2\right )^4}\\ &=-\frac{a c^2 \left (6 c d^2-a e^2\right ) (a e-c d x) \sqrt{a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac{c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac{e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac{7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}-\frac{a^2 c^3 \left (6 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 )}{16 \left (c d^2+a e^2\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.649625, size = 358, normalized size = 1.33 \[ \frac{1}{240} \left (-\frac{\sqrt{a+c x^2} \left (-c^2 (d+e x)^4 \left (-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right ) \left (a e^2+c d^2\right )-c^3 d (d+e x)^5 \left (-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right )-2 c^2 d (d+e x)^3 \left (9 a e^2+2 c d^2\right ) \left (a e^2+c d^2\right )^2-104 c d (d+e x) \left (a e^2+c d^2\right )^4+2 c (d+e x)^2 \left (35 a e^2+38 c d^2\right ) \left (a e^2+c d^2\right )^3+40 \left (a e^2+c d^2\right )^5\right )}{e^3 (d+e x)^6 \left (a e^2+c d^2\right )^4}+\frac{15 a^2 c^3 \left (a e^2-6 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{9/2}}+\frac{15 a^2 c^3 \left (6 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{9/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

(-((Sqrt[a + c*x^2]*(40*(c*d^2 + a*e^2)^5 - 104*c*d*(c*d^2 + a*e^2)^4*(d + e*x) + 2*c*(c*d^2 + a*e^2)^3*(38*c*
d^2 + 35*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2)^2*(2*c*d^2 + 9*a*e^2)*(d + e*x)^3 - c^2*(c*d^2 + a*e^2)*
(4*c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^4)*(d + e*x)^4 - c^3*d*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*a^2*e^4)*(d + e
*x)^5))/(e^3*(c*d^2 + a*e^2)^4*(d + e*x)^6)) + (15*a^2*c^3*(6*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(9/
2) + (15*a^2*c^3*(-6*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(9
/2))/240

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Maple [B]  time = 0.26, size = 5087, normalized size = 18.9 \begin{align*} \text{output 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)^7,x)

[Out]

result too large to display

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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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 157.252, size = 5085, normalized size = 18.9 \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)^7,x, algorithm="fricas")

[Out]

[-1/480*(15*(6*a^2*c^4*d^8 - a^3*c^3*d^6*e^2 + (6*a^2*c^4*d^2*e^6 - a^3*c^3*e^8)*x^6 + 6*(6*a^2*c^4*d^3*e^5 -
a^3*c^3*d*e^7)*x^5 + 15*(6*a^2*c^4*d^4*e^4 - a^3*c^3*d^2*e^6)*x^4 + 20*(6*a^2*c^4*d^5*e^3 - a^3*c^3*d^3*e^5)*x
^3 + 15*(6*a^2*c^4*d^6*e^2 - a^3*c^3*d^4*e^4)*x^2 + 6*(6*a^2*c^4*d^7*e - a^3*c^3*d^5*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)*sq
rt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(246*a^2*c^4*d^8*e + 513*a^3*c^3*d^6*e^3 + 433*a^4*c^2*d^4*e^5 +
206*a^5*c*d^2*e^7 + 40*a^6*e^9 - (4*c^6*d^7*e^2 + 32*a*c^5*d^5*e^4 - 53*a^2*c^4*d^3*e^6 - 81*a^3*c^3*d*e^8)*x
^5 - 3*(8*c^6*d^8*e + 64*a*c^5*d^6*e^3 - 76*a^2*c^4*d^4*e^5 - 137*a^3*c^3*d^2*e^7 - 5*a^4*c^2*e^9)*x^4 - 2*(30
*c^6*d^9 + 239*a*c^5*d^7*e^2 - 158*a^2*c^4*d^5*e^4 - 388*a^3*c^3*d^3*e^6 - 21*a^4*c^2*d*e^8)*x^3 - 2*(114*a*c^
5*d^8*e - 423*a^2*c^4*d^6*e^3 - 698*a^3*c^3*d^4*e^5 - 196*a^4*c^2*d^2*e^7 - 35*a^5*c*e^9)*x^2 - 3*(50*a*c^5*d^
9 - 117*a^2*c^4*d^7*e^2 - 221*a^3*c^3*d^5*e^4 - 66*a^4*c^2*d^3*e^6 - 12*a^5*c*d*e^8)*x)*sqrt(c*x^2 + a))/(c^5*
d^16 + 5*a*c^4*d^14*e^2 + 10*a^2*c^3*d^12*e^4 + 10*a^3*c^2*d^10*e^6 + 5*a^4*c*d^8*e^8 + a^5*d^6*e^10 + (c^5*d^
10*e^6 + 5*a*c^4*d^8*e^8 + 10*a^2*c^3*d^6*e^10 + 10*a^3*c^2*d^4*e^12 + 5*a^4*c*d^2*e^14 + a^5*e^16)*x^6 + 6*(c
^5*d^11*e^5 + 5*a*c^4*d^9*e^7 + 10*a^2*c^3*d^7*e^9 + 10*a^3*c^2*d^5*e^11 + 5*a^4*c*d^3*e^13 + a^5*d*e^15)*x^5
+ 15*(c^5*d^12*e^4 + 5*a*c^4*d^10*e^6 + 10*a^2*c^3*d^8*e^8 + 10*a^3*c^2*d^6*e^10 + 5*a^4*c*d^4*e^12 + a^5*d^2*
e^14)*x^4 + 20*(c^5*d^13*e^3 + 5*a*c^4*d^11*e^5 + 10*a^2*c^3*d^9*e^7 + 10*a^3*c^2*d^7*e^9 + 5*a^4*c*d^5*e^11 +
a^5*d^3*e^13)*x^3 + 15*(c^5*d^14*e^2 + 5*a*c^4*d^12*e^4 + 10*a^2*c^3*d^10*e^6 + 10*a^3*c^2*d^8*e^8 + 5*a^4*c*
d^6*e^10 + a^5*d^4*e^12)*x^2 + 6*(c^5*d^15*e + 5*a*c^4*d^13*e^3 + 10*a^2*c^3*d^11*e^5 + 10*a^3*c^2*d^9*e^7 + 5
*a^4*c*d^7*e^9 + a^5*d^5*e^11)*x), -1/240*(15*(6*a^2*c^4*d^8 - a^3*c^3*d^6*e^2 + (6*a^2*c^4*d^2*e^6 - a^3*c^3*
e^8)*x^6 + 6*(6*a^2*c^4*d^3*e^5 - a^3*c^3*d*e^7)*x^5 + 15*(6*a^2*c^4*d^4*e^4 - a^3*c^3*d^2*e^6)*x^4 + 20*(6*a^
2*c^4*d^5*e^3 - a^3*c^3*d^3*e^5)*x^3 + 15*(6*a^2*c^4*d^6*e^2 - a^3*c^3*d^4*e^4)*x^2 + 6*(6*a^2*c^4*d^7*e - a^3
*c^3*d^5*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)) + (246*a^2*c^4*d^8*e + 513*a^3*c^3*d^6*e^3 + 433*a^4*c^2*d^4*e^5 + 206*a^5*c*
d^2*e^7 + 40*a^6*e^9 - (4*c^6*d^7*e^2 + 32*a*c^5*d^5*e^4 - 53*a^2*c^4*d^3*e^6 - 81*a^3*c^3*d*e^8)*x^5 - 3*(8*c
^6*d^8*e + 64*a*c^5*d^6*e^3 - 76*a^2*c^4*d^4*e^5 - 137*a^3*c^3*d^2*e^7 - 5*a^4*c^2*e^9)*x^4 - 2*(30*c^6*d^9 +
239*a*c^5*d^7*e^2 - 158*a^2*c^4*d^5*e^4 - 388*a^3*c^3*d^3*e^6 - 21*a^4*c^2*d*e^8)*x^3 - 2*(114*a*c^5*d^8*e - 4
23*a^2*c^4*d^6*e^3 - 698*a^3*c^3*d^4*e^5 - 196*a^4*c^2*d^2*e^7 - 35*a^5*c*e^9)*x^2 - 3*(50*a*c^5*d^9 - 117*a^2
*c^4*d^7*e^2 - 221*a^3*c^3*d^5*e^4 - 66*a^4*c^2*d^3*e^6 - 12*a^5*c*d*e^8)*x)*sqrt(c*x^2 + a))/(c^5*d^16 + 5*a*
c^4*d^14*e^2 + 10*a^2*c^3*d^12*e^4 + 10*a^3*c^2*d^10*e^6 + 5*a^4*c*d^8*e^8 + a^5*d^6*e^10 + (c^5*d^10*e^6 + 5*
a*c^4*d^8*e^8 + 10*a^2*c^3*d^6*e^10 + 10*a^3*c^2*d^4*e^12 + 5*a^4*c*d^2*e^14 + a^5*e^16)*x^6 + 6*(c^5*d^11*e^5
+ 5*a*c^4*d^9*e^7 + 10*a^2*c^3*d^7*e^9 + 10*a^3*c^2*d^5*e^11 + 5*a^4*c*d^3*e^13 + a^5*d*e^15)*x^5 + 15*(c^5*d
^12*e^4 + 5*a*c^4*d^10*e^6 + 10*a^2*c^3*d^8*e^8 + 10*a^3*c^2*d^6*e^10 + 5*a^4*c*d^4*e^12 + a^5*d^2*e^14)*x^4 +
20*(c^5*d^13*e^3 + 5*a*c^4*d^11*e^5 + 10*a^2*c^3*d^9*e^7 + 10*a^3*c^2*d^7*e^9 + 5*a^4*c*d^5*e^11 + a^5*d^3*e^
13)*x^3 + 15*(c^5*d^14*e^2 + 5*a*c^4*d^12*e^4 + 10*a^2*c^3*d^10*e^6 + 10*a^3*c^2*d^8*e^8 + 5*a^4*c*d^6*e^10 +
a^5*d^4*e^12)*x^2 + 6*(c^5*d^15*e + 5*a*c^4*d^13*e^3 + 10*a^2*c^3*d^11*e^5 + 10*a^3*c^2*d^9*e^7 + 5*a^4*c*d^7*
e^9 + a^5*d^5*e^11)*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 )^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(3/2)/(e*x+d)**7,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**7, x)

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Giac [B]  time = 1.66162, size = 2454, normalized size = 9.12 \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)^7,x, algorithm="giac")

[Out]

-1/8*(6*a^2*c^4*d^2 - a^3*c^3*e^2)*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/
((c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(-c*d^2 - a*e^2)) + 1/120*(38
4*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*d^10*e + 128*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(17/2)*d^11 + 480*(sqrt(c
)*x - sqrt(c*x^2 + a))^8*c^(15/2)*d^9*e^2 + 320*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^7*d^8*e^3 - 384*(sqrt(c)*x -
sqrt(c*x^2 + a))^5*a*c^8*d^10*e - 64*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(15/2)*d^9*e^2 + 1728*(sqrt(c)*x - s
qrt(c*x^2 + a))^7*a*c^7*d^8*e^3 + 1920*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(13/2)*d^7*e^4 + 480*(sqrt(c)*x - s
qrt(c*x^2 + a))^4*a^2*c^(15/2)*d^9*e^2 + 1280*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*d^6*e^5 - 1728*(sqrt(c)*x
- sqrt(c*x^2 + a))^5*a^2*c^7*d^8*e^3 - 8592*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*d^7*e^4 - 9456*(sqrt(
c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*d^6*e^5 - 320*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^7*d^8*e^3 - 7380*(sqrt(c
)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*d^5*e^6 + 3840*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(13/2)*d^7*e^4 - 25
20*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^5*d^4*e^7 + 19056*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^6*d^6*e^5 - 9
90*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(9/2)*d^3*e^8 + 24440*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(11/2)*d
^5*e^6 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(13/2)*d^7*e^4 - 90*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^
4*d^2*e^9 + 20760*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^5*d^4*e^7 - 2960*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c
^6*d^6*e^5 + 8220*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(9/2)*d^3*e^8 - 18720*(sqrt(c)*x - sqrt(c*x^2 + a))^4*
a^4*c^(11/2)*d^5*e^6 + 2530*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^4*d^2*e^9 - 21480*(sqrt(c)*x - sqrt(c*x^2 +
a))^5*a^4*c^5*d^4*e^7 - 48*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^6*d^6*e^5 + 165*(sqrt(c)*x - sqrt(c*x^2 + a))^1
0*a^3*c^(7/2)*d*e^10 - 14860*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(9/2)*d^3*e^8 + 1656*(sqrt(c)*x - sqrt(c*x^
2 + a))^2*a^5*c^(11/2)*d^5*e^6 + 15*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^3*e^11 - 2700*(sqrt(c)*x - sqrt(c*x
^2 + a))^7*a^4*c^4*d^2*e^9 + 12120*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^5*d^4*e^7 - 285*(sqrt(c)*x - sqrt(c*x
^2 + a))^8*a^4*c^(7/2)*d*e^10 + 11640*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(9/2)*d^3*e^8 + 4*a^6*c^(11/2)*d^5
*e^6 + 235*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^3*e^11 + 7020*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^4*d^2*e^9
- 336*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^5*d^4*e^7 + 810*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(7/2)*d*e^10
- 4038*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(9/2)*d^3*e^8 + 390*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^3*e^11
- 2330*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^4*d^2*e^9 - 930*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(7/2)*d*e^1
0 + 28*a^7*c^(9/2)*d^3*e^8 + 390*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^3*e^11 + 882*(sqrt(c)*x - sqrt(c*x^2 +
a))*a^7*c^4*d^2*e^9 + 321*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(7/2)*d*e^10 + 235*(sqrt(c)*x - sqrt(c*x^2 + a
))^3*a^7*c^3*e^11 - 81*a^8*c^(7/2)*d*e^10 + 15*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^3*e^11)/((c^4*d^8*e^4 + 4*a
*c^3*d^6*e^6 + 6*a^2*c^2*d^4*e^8 + 4*a^3*c*d^2*e^10 + a^4*e^12)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c
)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^6)