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

Optimal. Leaf size=189 \[ \frac{5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}+\frac{5 a^2 x \sqrt{a+c x^2} \left (8 c d^2-a e^2\right )}{128 c}+\frac{x \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right )}{48 c}+\frac{5 a x \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right )}{192 c}+\frac{9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c} \]

[Out]

(5*a^2*(8*c*d^2 - a*e^2)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*(8*c*d^2 - a*e^2)*x*(a + c*x^2)^(3/2))/(192*c) + ((
8*c*d^2 - a*e^2)*x*(a + c*x^2)^(5/2))/(48*c) + (9*d*e*(a + c*x^2)^(7/2))/(56*c) + (e*(d + e*x)*(a + c*x^2)^(7/
2))/(8*c) + (5*a^3*(8*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*c^(3/2))

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Rubi [A]  time = 0.087727, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {743, 641, 195, 217, 206} \[ \frac{5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}+\frac{5 a^2 x \sqrt{a+c x^2} \left (8 c d^2-a e^2\right )}{128 c}+\frac{x \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right )}{48 c}+\frac{5 a x \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right )}{192 c}+\frac{9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + c*x^2)^(5/2),x]

[Out]

(5*a^2*(8*c*d^2 - a*e^2)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*(8*c*d^2 - a*e^2)*x*(a + c*x^2)^(3/2))/(192*c) + ((
8*c*d^2 - a*e^2)*x*(a + c*x^2)^(5/2))/(48*c) + (9*d*e*(a + c*x^2)^(7/2))/(56*c) + (e*(d + e*x)*(a + c*x^2)^(7/
2))/(8*c) + (5*a^3*(8*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*c^(3/2))

Rule 743

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))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*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] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 641

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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])

Rubi steps

\begin{align*} \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx &=\frac{e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac{\int \left (8 c d^2-a e^2+9 c d e x\right ) \left (a+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac{9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac{e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac{\left (8 c d^2-a e^2\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac{\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac{e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac{\left (5 a \left (8 c d^2-a e^2\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c}\\ &=\frac{5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac{e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac{\left (5 a^2 \left (8 c d^2-a e^2\right )\right ) \int \sqrt{a+c x^2} \, dx}{64 c}\\ &=\frac{5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt{a+c x^2}}{128 c}+\frac{5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac{e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac{\left (5 a^3 \left (8 c d^2-a e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{128 c}\\ &=\frac{5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt{a+c x^2}}{128 c}+\frac{5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac{e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac{\left (5 a^3 \left (8 c d^2-a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{128 c}\\ &=\frac{5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt{a+c x^2}}{128 c}+\frac{5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac{e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac{5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.122061, size = 162, normalized size = 0.86 \[ \frac{\sqrt{c} \sqrt{a+c x^2} \left (2 a^2 c x \left (924 d^2+1152 d e x+413 e^2 x^2\right )+3 a^3 e (256 d+35 e x)+8 a c^2 x^3 \left (182 d^2+288 d e x+119 e^2 x^2\right )+16 c^3 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )-105 a^3 \left (a e^2-8 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{2688 c^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*Sqrt[a + c*x^2]*(3*a^3*e*(256*d + 35*e*x) + 16*c^3*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 8*a*c^2*x^3
*(182*d^2 + 288*d*e*x + 119*e^2*x^2) + 2*a^2*c*x*(924*d^2 + 1152*d*e*x + 413*e^2*x^2)) - 105*a^3*(-8*c*d^2 + a
*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(2688*c^(3/2))

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Maple [A]  time = 0.056, size = 200, normalized size = 1.1 \begin{align*}{\frac{{e}^{2}x}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{a{e}^{2}x}{48\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}{e}^{2}x}{192\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{e}^{2}{a}^{3}x}{128\,c}\sqrt{c{x}^{2}+a}}-{\frac{5\,{e}^{2}{a}^{4}}{128}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{2\,de}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{d}^{2}x}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a{d}^{2}x}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{d}^{2}x}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,{a}^{3}{d}^{2}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e^2*x*(c*x^2+a)^(7/2)/c-1/48*e^2*a/c*x*(c*x^2+a)^(5/2)-5/192*e^2*a^2/c*x*(c*x^2+a)^(3/2)-5/128*e^2*a^3/c*x
*(c*x^2+a)^(1/2)-5/128*e^2*a^4/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+2/7*d*e*(c*x^2+a)^(7/2)/c+1/6*d^2*x*(c*x^
2+a)^(5/2)+5/24*d^2*a*x*(c*x^2+a)^(3/2)+5/16*d^2*a^2*x*(c*x^2+a)^(1/2)+5/16*d^2*a^3/c^(1/2)*ln(x*c^(1/2)+(c*x^
2+a)^(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((e*x+d)^2*(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.83758, size = 879, normalized size = 4.65 \begin{align*} \left [\frac{105 \,{\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (336 \, c^{4} e^{2} x^{7} + 768 \, c^{4} d e x^{6} + 2304 \, a c^{3} d e x^{4} + 2304 \, a^{2} c^{2} d e x^{2} + 768 \, a^{3} c d e + 56 \,{\left (8 \, c^{4} d^{2} + 17 \, a c^{3} e^{2}\right )} x^{5} + 14 \,{\left (104 \, a c^{3} d^{2} + 59 \, a^{2} c^{2} e^{2}\right )} x^{3} + 21 \,{\left (88 \, a^{2} c^{2} d^{2} + 5 \, a^{3} c e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{5376 \, c^{2}}, -\frac{105 \,{\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (336 \, c^{4} e^{2} x^{7} + 768 \, c^{4} d e x^{6} + 2304 \, a c^{3} d e x^{4} + 2304 \, a^{2} c^{2} d e x^{2} + 768 \, a^{3} c d e + 56 \,{\left (8 \, c^{4} d^{2} + 17 \, a c^{3} e^{2}\right )} x^{5} + 14 \,{\left (104 \, a c^{3} d^{2} + 59 \, a^{2} c^{2} e^{2}\right )} x^{3} + 21 \,{\left (88 \, a^{2} c^{2} d^{2} + 5 \, a^{3} c e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{2688 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/5376*(105*(8*a^3*c*d^2 - a^4*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(336*c^4*e^2*
x^7 + 768*c^4*d*e*x^6 + 2304*a*c^3*d*e*x^4 + 2304*a^2*c^2*d*e*x^2 + 768*a^3*c*d*e + 56*(8*c^4*d^2 + 17*a*c^3*e
^2)*x^5 + 14*(104*a*c^3*d^2 + 59*a^2*c^2*e^2)*x^3 + 21*(88*a^2*c^2*d^2 + 5*a^3*c*e^2)*x)*sqrt(c*x^2 + a))/c^2,
-1/2688*(105*(8*a^3*c*d^2 - a^4*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (336*c^4*e^2*x^7 + 768*c^4
*d*e*x^6 + 2304*a*c^3*d*e*x^4 + 2304*a^2*c^2*d*e*x^2 + 768*a^3*c*d*e + 56*(8*c^4*d^2 + 17*a*c^3*e^2)*x^5 + 14*
(104*a*c^3*d^2 + 59*a^2*c^2*e^2)*x^3 + 21*(88*a^2*c^2*d^2 + 5*a^3*c*e^2)*x)*sqrt(c*x^2 + a))/c^2]

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Sympy [A]  time = 39.1034, size = 539, normalized size = 2.85 \begin{align*} \frac{5 a^{\frac{7}{2}} e^{2} x}{128 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{a^{\frac{5}{2}} d^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{3 a^{\frac{5}{2}} d^{2} x}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{133 a^{\frac{5}{2}} e^{2} x^{3}}{384 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 a^{\frac{3}{2}} c d^{2} x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{127 a^{\frac{3}{2}} c e^{2} x^{5}}{192 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 \sqrt{a} c^{2} d^{2} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{23 \sqrt{a} c^{2} e^{2} x^{7}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{5 a^{4} e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{128 c^{\frac{3}{2}}} + \frac{5 a^{3} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 \sqrt{c}} + 2 a^{2} d e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + 4 a c d e \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + 2 c^{2} d e \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + \frac{c^{3} d^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{c^{3} e^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a**(7/2)*e**2*x/(128*c*sqrt(1 + c*x**2/a)) + a**(5/2)*d**2*x*sqrt(1 + c*x**2/a)/2 + 3*a**(5/2)*d**2*x/(16*sq
rt(1 + c*x**2/a)) + 133*a**(5/2)*e**2*x**3/(384*sqrt(1 + c*x**2/a)) + 35*a**(3/2)*c*d**2*x**3/(48*sqrt(1 + c*x
**2/a)) + 127*a**(3/2)*c*e**2*x**5/(192*sqrt(1 + c*x**2/a)) + 17*sqrt(a)*c**2*d**2*x**5/(24*sqrt(1 + c*x**2/a)
) + 23*sqrt(a)*c**2*e**2*x**7/(48*sqrt(1 + c*x**2/a)) - 5*a**4*e**2*asinh(sqrt(c)*x/sqrt(a))/(128*c**(3/2)) +
5*a**3*d**2*asinh(sqrt(c)*x/sqrt(a))/(16*sqrt(c)) + 2*a**2*d*e*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x
**2)**(3/2)/(3*c), True)) + 4*a*c*d*e*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/
(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + 2*c**2*d*e*Piecewise((8*a**3*sqrt(a + c
*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c
*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, True)) + c**3*d**2*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + c**3*e**2*x**9/
(8*sqrt(a)*sqrt(1 + c*x**2/a))

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Giac [A]  time = 1.32431, size = 257, normalized size = 1.36 \begin{align*} \frac{1}{2688} \,{\left (\frac{768 \, a^{3} d e}{c} +{\left (2 \,{\left (1152 \, a^{2} d e +{\left (4 \,{\left (288 \, a c d e +{\left (6 \,{\left (7 \, c^{2} x e^{2} + 16 \, c^{2} d e\right )} x + \frac{7 \,{\left (8 \, c^{8} d^{2} + 17 \, a c^{7} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac{7 \,{\left (104 \, a c^{7} d^{2} + 59 \, a^{2} c^{6} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac{21 \,{\left (88 \, a^{2} c^{6} d^{2} + 5 \, a^{3} c^{5} e^{2}\right )}}{c^{6}}\right )} x\right )} \sqrt{c x^{2} + a} - \frac{5 \,{\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2688*(768*a^3*d*e/c + (2*(1152*a^2*d*e + (4*(288*a*c*d*e + (6*(7*c^2*x*e^2 + 16*c^2*d*e)*x + 7*(8*c^8*d^2 +
17*a*c^7*e^2)/c^6)*x)*x + 7*(104*a*c^7*d^2 + 59*a^2*c^6*e^2)/c^6)*x)*x + 21*(88*a^2*c^6*d^2 + 5*a^3*c^5*e^2)/c
^6)*x)*sqrt(c*x^2 + a) - 5/128*(8*a^3*c*d^2 - a^4*e^2)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)