3.21.38 \(\int \sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx\) [2038]

Optimal. Leaf size=171 \[ \frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 c^3 d^3 (d+e x)^{5/2}}+\frac {8 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}} \]

[Out]

16/315*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3/(e*x+d)^(5/2)+8/63*(-a*e^2+c*d^2)*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^2/d^2/(e*x+d)^(3/2)+2/9*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/(e*x
+d)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662} \begin {gather*} \frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 c^3 d^3 (d+e x)^{5/2}}+\frac {8 \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}}{63 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*c^3*d^3*(d + e*x)^(5/2)) + (8*(c*d^2
- a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(63*c^2*d^2*(d + e*x)^(3/2)) + (2*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(5/2))/(9*c*d*Sqrt[d + e*x])

Rule 662

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*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

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[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

\begin {align*} \int \sqrt {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 &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}}+\frac {\left (4 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{9 d}\\ &=\frac {8 \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}}{63 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{63 d^2}\\ &=\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 c^3 d^3 (d+e x)^{5/2}}+\frac {8 \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}}{63 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 88, normalized size = 0.51 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2} \left (8 a^2 e^4-4 a c d e^2 (9 d+5 e x)+c^2 d^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )\right )}{315 c^3 d^3 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(8*a^2*e^4 - 4*a*c*d*e^2*(9*d + 5*e*x) + c^2*d^2*(63*d^2 + 90*d*e*x + 35*e^
2*x^2)))/(315*c^3*d^3*(d + e*x)^(5/2))

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Maple [A]
time = 0.74, size = 102, normalized size = 0.60

method	result	size
default	\(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{2} \left (35 e^{2} x^{2} c^{2} d^{2}-20 a c d \,e^{3} x +90 c^{2} d^{3} e x +8 a^{2} e^{4}-36 a c \,d^{2} e^{2}+63 c^{2} d^{4}\right )}{315 \sqrt {e x +d}\, c^{3} d^{3}}\)	\(102\)
gosper	\(\frac {2 \left (c d x +a e \right ) \left (35 e^{2} x^{2} c^{2} d^{2}-20 a c d \,e^{3} x +90 c^{2} d^{3} e x +8 a^{2} e^{4}-36 a c \,d^{2} e^{2}+63 c^{2} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{315 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(c*d*x+a*e)^2*(35*c^2*d^2*e^2*x^2-20*a*c*d*e^3*x+90*c^2*d^3*e*
x+8*a^2*e^4-36*a*c*d^2*e^2+63*c^2*d^4)/c^3/d^3

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Maxima [A]
time = 0.30, size = 185, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} d^{4} x^{4} e^{2} + 63 \, a^{2} c^{2} d^{4} e^{2} - 36 \, a^{3} c d^{2} e^{4} + 8 \, a^{4} e^{6} + 10 \, {\left (9 \, c^{4} d^{5} e + 5 \, a c^{3} d^{3} e^{3}\right )} x^{3} + 3 \, {\left (21 \, c^{4} d^{6} + 48 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (63 \, a c^{3} d^{5} e + 9 \, a^{2} c^{2} d^{3} e^{3} - 2 \, a^{3} c d e^{5}\right )} x\right )} \sqrt {c d x + a e} {\left (x e + d\right )}}{315 \, {\left (c^{3} d^{3} x e + c^{3} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*c^4*d^4*x^4*e^2 + 63*a^2*c^2*d^4*e^2 - 36*a^3*c*d^2*e^4 + 8*a^4*e^6 + 10*(9*c^4*d^5*e + 5*a*c^3*d^3*
e^3)*x^3 + 3*(21*c^4*d^6 + 48*a*c^3*d^4*e^2 + a^2*c^2*d^2*e^4)*x^2 + 2*(63*a*c^3*d^5*e + 9*a^2*c^2*d^3*e^3 - 2
*a^3*c*d*e^5)*x)*sqrt(c*d*x + a*e)*(x*e + d)/(c^3*d^3*x*e + c^3*d^4)

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Fricas [A]
time = 2.65, size = 206, normalized size = 1.20 \begin {gather*} \frac {2 \, {\left (63 \, c^{4} d^{6} x^{2} - 4 \, a^{3} c d x e^{5} + 8 \, a^{4} e^{6} + 3 \, {\left (a^{2} c^{2} d^{2} x^{2} - 12 \, a^{3} c d^{2}\right )} e^{4} + 2 \, {\left (25 \, a c^{3} d^{3} x^{3} + 9 \, a^{2} c^{2} d^{3} x\right )} e^{3} + {\left (35 \, c^{4} d^{4} x^{4} + 144 \, a c^{3} d^{4} x^{2} + 63 \, a^{2} c^{2} d^{4}\right )} e^{2} + 18 \, {\left (5 \, c^{4} d^{5} x^{3} + 7 \, a c^{3} d^{5} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{315 \, {\left (c^{3} d^{3} x e + c^{3} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(63*c^4*d^6*x^2 - 4*a^3*c*d*x*e^5 + 8*a^4*e^6 + 3*(a^2*c^2*d^2*x^2 - 12*a^3*c*d^2)*e^4 + 2*(25*a*c^3*d^3
*x^3 + 9*a^2*c^2*d^3*x)*e^3 + (35*c^4*d^4*x^4 + 144*a*c^3*d^4*x^2 + 63*a^2*c^2*d^4)*e^2 + 18*(5*c^4*d^5*x^3 +
7*a*c^3*d^5*x)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c^3*d^3*x*e + c^3*d^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \sqrt {d + e x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1108 vs. \(2 (157) = 314\).
time = 1.27, size = 1108, normalized size = 6.48 \begin {gather*} -\frac {2}{315} \, {\left (21 \, c d^{3} {\left (\frac {{\left (5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} e^{\left (-2\right )}}{c^{2} d^{2}} + \frac {3 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} - \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} - 2 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4}}{c^{2} d^{2}}\right )} e^{\left (-2\right )} - 105 \, a d^{2} {\left (\frac {{\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} e^{\left (-1\right )}}{c d} + \frac {\sqrt {-c d^{2} e + a e^{3}} c d^{2} - \sqrt {-c d^{2} e + a e^{3}} a e^{2}}{c d}\right )} - 6 \, c d^{2} {\left (\frac {{\left (15 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} - 3 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} - 4 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6}\right )} e^{\left (-2\right )}}{c^{3} d^{3}} + \frac {{\left (35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} e^{\left (-5\right )}}{c^{3} d^{3}}\right )} + c d {\left (\frac {{\left (35 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{8} - 5 \, \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{6} e^{2} - 6 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{2} d^{4} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c d^{2} e^{6} - 16 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} e^{8}\right )} e^{\left (-3\right )}}{c^{4} d^{4}} + \frac {{\left (105 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} e^{9} - 189 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} e^{6} + 135 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a e^{3} - 35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}}\right )} e^{\left (-7\right )}}{c^{4} d^{4}}\right )} e + 42 \, a d {\left (\frac {{\left (5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} e^{\left (-2\right )}}{c^{2} d^{2}} + \frac {3 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} - \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} - 2 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4}}{c^{2} d^{2}}\right )} - 3 \, a {\left (\frac {{\left (15 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} - 3 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} - 4 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6}\right )} e^{\left (-2\right )}}{c^{3} d^{3}} + \frac {{\left (35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} e^{\left (-5\right )}}{c^{3} d^{3}}\right )} e^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(21*c*d^3*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(
5/2))*e^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^
2*e + a*e^3)*a^2*e^4)/(c^2*d^2))*e^(-2) - 105*a*d^2*(((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*e^(-1)/(c*d) +
(sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^2*e + a*e^3)*a*e^2)/(c*d)) - 6*c*d^2*((15*sqrt(-c*d^2*e + a*e^3)*c^3
*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e
^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e -
c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3)) + c*d*((35*sqrt
(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4
- 8*sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (105*((x*e +
d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x*e
+ d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4))*e
+ 42*a*d*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*e
^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2*e + a
*e^3)*a^2*e^4)/(c^2*d^2)) - 3*a*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 -
4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*
c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*
d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e^2)*e^(-1)

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Mupad [B]
time = 1.02, size = 230, normalized size = 1.35 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (x^3\,\left (\frac {4\,c\,d^2}{7}+\frac {20\,a\,e^2}{63}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (16\,a^4\,e^6-72\,a^3\,c\,d^2\,e^4+126\,a^2\,c^2\,d^4\,e^2\right )}{315\,c^3\,d^3\,e}+\frac {2\,c\,d\,e\,x^4\,\sqrt {d+e\,x}}{9}+\frac {x^2\,\sqrt {d+e\,x}\,\left (6\,a^2\,c^2\,d^2\,e^4+288\,a\,c^3\,d^4\,e^2+126\,c^4\,d^6\right )}{315\,c^3\,d^3\,e}+\frac {4\,a\,x\,\sqrt {d+e\,x}\,\left (-2\,a^2\,e^4+9\,a\,c\,d^2\,e^2+63\,c^2\,d^4\right )}{315\,c^2\,d^2}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(x^3*((20*a*e^2)/63 + (4*c*d^2)/7)*(d + e*x)^(1/2) + ((d + e*x)
^(1/2)*(16*a^4*e^6 - 72*a^3*c*d^2*e^4 + 126*a^2*c^2*d^4*e^2))/(315*c^3*d^3*e) + (2*c*d*e*x^4*(d + e*x)^(1/2))/
9 + (x^2*(d + e*x)^(1/2)*(126*c^4*d^6 + 288*a*c^3*d^4*e^2 + 6*a^2*c^2*d^2*e^4))/(315*c^3*d^3*e) + (4*a*x*(d +
e*x)^(1/2)*(63*c^2*d^4 - 2*a^2*e^4 + 9*a*c*d^2*e^2))/(315*c^2*d^2)))/(x + d/e)

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