3.20.95 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{5/2}} \, dx\) [1995]

Optimal. Leaf size=119 \[ -\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}{3 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{5 e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{7 e^4}+\frac {2 c^3 d^3 (d+e x)^{9/2}}{9 e^4} \]

[Out]

-2/3*(-a*e^2+c*d^2)^3*(e*x+d)^(3/2)/e^4+6/5*c*d*(-a*e^2+c*d^2)^2*(e*x+d)^(5/2)/e^4-6/7*c^2*d^2*(-a*e^2+c*d^2)*
(e*x+d)^(7/2)/e^4+2/9*c^3*d^3*(e*x+d)^(9/2)/e^4

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Rubi [A]
time = 0.04, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {640, 45} \begin {gather*} -\frac {6 c^2 d^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^4}+\frac {6 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^4}-\frac {2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}{3 e^4}+\frac {2 c^3 d^3 (d+e x)^{9/2}}{9 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2))/(3*e^4) + (6*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2))/(5*e^4) - (6*c^2*d^
2*(c*d^2 - a*e^2)*(d + e*x)^(7/2))/(7*e^4) + (2*c^3*d^3*(d + e*x)^(9/2))/(9*e^4)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 640

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

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 111, normalized size = 0.93 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (105 a^3 e^6-63 a^2 c d e^4 (2 d-3 e x)+9 a c^2 d^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )}{315 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(2*(d + e*x)^(3/2)*(105*a^3*e^6 - 63*a^2*c*d*e^4*(2*d - 3*e*x) + 9*a*c^2*d^2*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^
2) + c^3*d^3*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3)))/(315*e^4)

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Maple [A]
time = 0.83, size = 97, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{4}}\)	\(97\)
default	\(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{4}}\)	\(97\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (35 c^{3} d^{3} e^{3} x^{3}+135 a \,c^{2} d^{2} e^{4} x^{2}-30 c^{3} d^{4} e^{2} x^{2}+189 a^{2} c d \,e^{5} x -108 a \,c^{2} d^{3} e^{3} x +24 c^{3} d^{5} e x +105 e^{6} a^{3}-126 e^{4} d^{2} a^{2} c +72 d^{4} e^{2} c^{2} a -16 d^{6} c^{3}\right )}{315 e^{4}}\)	\(131\)
trager	\(\frac {2 \left (35 c^{3} d^{3} e^{4} x^{4}+135 a \,c^{2} d^{2} e^{5} x^{3}+5 c^{3} d^{4} e^{3} x^{3}+189 a^{2} c d \,e^{6} x^{2}+27 a \,c^{2} d^{3} e^{4} x^{2}-6 c^{3} d^{5} e^{2} x^{2}+105 a^{3} e^{7} x +63 a^{2} c \,d^{2} e^{5} x -36 a \,c^{2} d^{4} e^{3} x +8 c^{3} d^{6} e x +105 a^{3} d \,e^{6}-126 a^{2} c \,d^{3} e^{4}+72 a \,c^{2} d^{5} e^{2}-16 c^{3} d^{7}\right ) \sqrt {e x +d}}{315 e^{4}}\)	\(185\)
risch	\(\frac {2 \left (35 c^{3} d^{3} e^{4} x^{4}+135 a \,c^{2} d^{2} e^{5} x^{3}+5 c^{3} d^{4} e^{3} x^{3}+189 a^{2} c d \,e^{6} x^{2}+27 a \,c^{2} d^{3} e^{4} x^{2}-6 c^{3} d^{5} e^{2} x^{2}+105 a^{3} e^{7} x +63 a^{2} c \,d^{2} e^{5} x -36 a \,c^{2} d^{4} e^{3} x +8 c^{3} d^{6} e x +105 a^{3} d \,e^{6}-126 a^{2} c \,d^{3} e^{4}+72 a \,c^{2} d^{5} e^{2}-16 c^{3} d^{7}\right ) \sqrt {e x +d}}{315 e^{4}}\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/e^4*(1/9*c^3*d^3*(e*x+d)^(9/2)+3/7*(a*e^2-c*d^2)*c^2*d^2*(e*x+d)^(7/2)+3/5*(a*e^2-c*d^2)^2*c*d*(e*x+d)^(5/2)
+1/3*(a*e^2-c*d^2)^3*(e*x+d)^(3/2))

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Maxima [A]
time = 0.27, size = 134, normalized size = 1.13 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} d^{3} - 135 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 189 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (x e + d\right )}^{\frac {5}{2}} - 105 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/315*(35*(x*e + d)^(9/2)*c^3*d^3 - 135*(c^3*d^4 - a*c^2*d^2*e^2)*(x*e + d)^(7/2) + 189*(c^3*d^5 - 2*a*c^2*d^3
*e^2 + a^2*c*d*e^4)*(x*e + d)^(5/2) - 105*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*(x*e + d)^(3
/2))*e^(-4)

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Fricas [A]
time = 2.53, size = 173, normalized size = 1.45 \begin {gather*} \frac {2}{315} \, {\left (8 \, c^{3} d^{6} x e - 16 \, c^{3} d^{7} + 105 \, a^{3} x e^{7} + 21 \, {\left (9 \, a^{2} c d x^{2} + 5 \, a^{3} d\right )} e^{6} + 9 \, {\left (15 \, a c^{2} d^{2} x^{3} + 7 \, a^{2} c d^{2} x\right )} e^{5} + {\left (35 \, c^{3} d^{3} x^{4} + 27 \, a c^{2} d^{3} x^{2} - 126 \, a^{2} c d^{3}\right )} e^{4} + {\left (5 \, c^{3} d^{4} x^{3} - 36 \, a c^{2} d^{4} x\right )} e^{3} - 6 \, {\left (c^{3} d^{5} x^{2} - 12 \, a c^{2} d^{5}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/315*(8*c^3*d^6*x*e - 16*c^3*d^7 + 105*a^3*x*e^7 + 21*(9*a^2*c*d*x^2 + 5*a^3*d)*e^6 + 9*(15*a*c^2*d^2*x^3 + 7
*a^2*c*d^2*x)*e^5 + (35*c^3*d^3*x^4 + 27*a*c^2*d^3*x^2 - 126*a^2*c*d^3)*e^4 + (5*c^3*d^4*x^3 - 36*a*c^2*d^4*x)
*e^3 - 6*(c^3*d^5*x^2 - 12*a*c^2*d^5)*e^2)*sqrt(x*e + d)*e^(-4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (110) = 220\).
time = 42.23, size = 644, normalized size = 5.41 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{3} d^{2} e^{3}}{\sqrt {d + e x}} - 4 a^{3} d e^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 2 a^{3} e^{3} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right ) - 6 a^{2} c d^{3} e \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 12 a^{2} c d^{2} e \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right ) - 6 a^{2} c d e \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right ) - \frac {6 a c^{2} d^{4} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {12 a c^{2} d^{3} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} - \frac {6 a c^{2} d^{2} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e} - \frac {2 c^{3} d^{5} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {4 c^{3} d^{4} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {2 c^{3} d^{3} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}}}{e} & \text {for}\: e \neq 0 \\\frac {c^{3} d^{\frac {7}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**(5/2),x)

[Out]

Piecewise(((-2*a**3*d**2*e**3/sqrt(d + e*x) - 4*a**3*d*e**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 2*a**3*e**3*(
d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) - 6*a**2*c*d**3*e*(-d/sqrt(d + e*x) - sqrt(d + e*
x)) - 12*a**2*c*d**2*e*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) - 6*a**2*c*d*e*(-d**3/sqr
t(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5) - 6*a*c**2*d**4*(d**2/sqrt(d + e*
x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 12*a*c**2*d**3*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) +
d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e - 6*a*c**2*d**2*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2
*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e - 2*c**3*d**5*(-d**3/sqrt(d + e*x) - 3*d**2
*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 4*c**3*d**4*(d**4/sqrt(d + e*x) + 4*d**3*sqrt
(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 2*c**3*d**3*(-d**5/s
qrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7
/2)/7 - (d + e*x)**(9/2)/9)/e**3)/e, Ne(e, 0)), (c**3*d**(7/2)*x**4/4, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (100) = 200\).
time = 1.01, size = 356, normalized size = 2.99 \begin {gather*} \frac {2}{315} \, {\left (9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c^{3} d^{4} e^{\left (-3\right )} + 63 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a c^{2} d^{3} e^{\left (-1\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c^{3} d^{3} e^{\left (-3\right )} + 315 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} c d^{2} e + 27 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a c^{2} d^{2} e^{\left (-1\right )} + 63 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} c d e + 315 \, \sqrt {x e + d} a^{3} d e^{3} + 105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} e^{3}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/315*(9*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*c^3*d^4*e^
(-3) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c^2*d^3*e^(-1) + (35*(x*e + d)^(
9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*c^3*
d^3*e^(-3) + 315*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*c*d^2*e + 27*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2
)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*c^2*d^2*e^(-1) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(
3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*d*e + 315*sqrt(x*e + d)*a^3*d*e^3 + 105*((x*e + d)^(3/2) - 3*sqrt(x*e + d
)*d)*a^3*e^3)*e^(-1)

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Mupad [B]
time = 0.07, size = 106, normalized size = 0.89 \begin {gather*} \frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3/(d + e*x)^(5/2),x)

[Out]

(2*(a*e^2 - c*d^2)^3*(d + e*x)^(3/2))/(3*e^4) - ((6*c^3*d^4 - 6*a*c^2*d^2*e^2)*(d + e*x)^(7/2))/(7*e^4) + (2*c
^3*d^3*(d + e*x)^(9/2))/(9*e^4) + (6*c*d*(a*e^2 - c*d^2)^2*(d + e*x)^(5/2))/(5*e^4)

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