3.14.33 \(\int (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=129 \[ -\frac {8 b^3 (d+e x)^{11/2} (b d-a e)}{11 e^5}+\frac {4 b^2 (d+e x)^{9/2} (b d-a e)^2}{3 e^5}-\frac {8 b (d+e x)^{7/2} (b d-a e)^3}{7 e^5}+\frac {2 (d+e x)^{5/2} (b d-a e)^4}{5 e^5}+\frac {2 b^4 (d+e x)^{13/2}}{13 e^5} \]

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Rubi [A]  time = 0.04, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 43} \begin {gather*} -\frac {8 b^3 (d+e x)^{11/2} (b d-a e)}{11 e^5}+\frac {4 b^2 (d+e x)^{9/2} (b d-a e)^2}{3 e^5}-\frac {8 b (d+e x)^{7/2} (b d-a e)^3}{7 e^5}+\frac {2 (d+e x)^{5/2} (b d-a e)^4}{5 e^5}+\frac {2 b^4 (d+e x)^{13/2}}{13 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(5/2))/(5*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^5) + (4*b^2*(b*d - a*e)^2
*(d + e*x)^(9/2))/(3*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(11/2))/(11*e^5) + (2*b^4*(d + e*x)^(13/2))/(13*e^5)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

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])

Rubi steps

\begin {align*} \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (d+e x)^{3/2}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{5/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{7/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{9/2}}{e^4}+\frac {b^4 (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^4 (d+e x)^{5/2}}{5 e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{7/2}}{7 e^5}+\frac {4 b^2 (b d-a e)^2 (d+e x)^{9/2}}{3 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{11/2}}{11 e^5}+\frac {2 b^4 (d+e x)^{13/2}}{13 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 101, normalized size = 0.78 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (-5460 b^3 (d+e x)^3 (b d-a e)+10010 b^2 (d+e x)^2 (b d-a e)^2-8580 b (d+e x) (b d-a e)^3+3003 (b d-a e)^4+1155 b^4 (d+e x)^4\right )}{15015 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(3003*(b*d - a*e)^4 - 8580*b*(b*d - a*e)^3*(d + e*x) + 10010*b^2*(b*d - a*e)^2*(d + e*x)^2
- 5460*b^3*(b*d - a*e)*(d + e*x)^3 + 1155*b^4*(d + e*x)^4))/(15015*e^5)

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IntegrateAlgebraic [A]  time = 0.09, size = 213, normalized size = 1.65 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (3003 a^4 e^4+8580 a^3 b e^3 (d+e x)-12012 a^3 b d e^3+18018 a^2 b^2 d^2 e^2+10010 a^2 b^2 e^2 (d+e x)^2-25740 a^2 b^2 d e^2 (d+e x)-12012 a b^3 d^3 e+25740 a b^3 d^2 e (d+e x)+5460 a b^3 e (d+e x)^3-20020 a b^3 d e (d+e x)^2+3003 b^4 d^4-8580 b^4 d^3 (d+e x)+10010 b^4 d^2 (d+e x)^2+1155 b^4 (d+e x)^4-5460 b^4 d (d+e x)^3\right )}{15015 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(5/2)*(3003*b^4*d^4 - 12012*a*b^3*d^3*e + 18018*a^2*b^2*d^2*e^2 - 12012*a^3*b*d*e^3 + 3003*a^4*e^
4 - 8580*b^4*d^3*(d + e*x) + 25740*a*b^3*d^2*e*(d + e*x) - 25740*a^2*b^2*d*e^2*(d + e*x) + 8580*a^3*b*e^3*(d +
e*x) + 10010*b^4*d^2*(d + e*x)^2 - 20020*a*b^3*d*e*(d + e*x)^2 + 10010*a^2*b^2*e^2*(d + e*x)^2 - 5460*b^4*d*(
d + e*x)^3 + 5460*a*b^3*e*(d + e*x)^3 + 1155*b^4*(d + e*x)^4))/(15015*e^5)

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fricas [B]  time = 0.40, size = 311, normalized size = 2.41 \begin {gather*} \frac {2 \, {\left (1155 \, b^{4} e^{6} x^{6} + 128 \, b^{4} d^{6} - 832 \, a b^{3} d^{5} e + 2288 \, a^{2} b^{2} d^{4} e^{2} - 3432 \, a^{3} b d^{3} e^{3} + 3003 \, a^{4} d^{2} e^{4} + 210 \, {\left (7 \, b^{4} d e^{5} + 26 \, a b^{3} e^{6}\right )} x^{5} + 35 \, {\left (b^{4} d^{2} e^{4} + 208 \, a b^{3} d e^{5} + 286 \, a^{2} b^{2} e^{6}\right )} x^{4} - 20 \, {\left (2 \, b^{4} d^{3} e^{3} - 13 \, a b^{3} d^{2} e^{4} - 715 \, a^{2} b^{2} d e^{5} - 429 \, a^{3} b e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{4} d^{4} e^{2} - 104 \, a b^{3} d^{3} e^{3} + 286 \, a^{2} b^{2} d^{2} e^{4} + 4576 \, a^{3} b d e^{5} + 1001 \, a^{4} e^{6}\right )} x^{2} - 2 \, {\left (32 \, b^{4} d^{5} e - 208 \, a b^{3} d^{4} e^{2} + 572 \, a^{2} b^{2} d^{3} e^{3} - 858 \, a^{3} b d^{2} e^{4} - 3003 \, a^{4} d e^{5}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*b^4*e^6*x^6 + 128*b^4*d^6 - 832*a*b^3*d^5*e + 2288*a^2*b^2*d^4*e^2 - 3432*a^3*b*d^3*e^3 + 3003*a
^4*d^2*e^4 + 210*(7*b^4*d*e^5 + 26*a*b^3*e^6)*x^5 + 35*(b^4*d^2*e^4 + 208*a*b^3*d*e^5 + 286*a^2*b^2*e^6)*x^4 -
20*(2*b^4*d^3*e^3 - 13*a*b^3*d^2*e^4 - 715*a^2*b^2*d*e^5 - 429*a^3*b*e^6)*x^3 + 3*(16*b^4*d^4*e^2 - 104*a*b^3
*d^3*e^3 + 286*a^2*b^2*d^2*e^4 + 4576*a^3*b*d*e^5 + 1001*a^4*e^6)*x^2 - 2*(32*b^4*d^5*e - 208*a*b^3*d^4*e^2 +
572*a^2*b^2*d^3*e^3 - 858*a^3*b*d^2*e^4 - 3003*a^4*d*e^5)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.24, size = 854, normalized size = 6.62

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(60060*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b*d^2*e^(-1) + 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d
)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b^2*d^2*e^(-2) + 5148*(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*b^3*d^2*e^(-3) + 143*(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)*b^4*d^2*e^(-4) + 24024*(3*(x*e + d)
^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*b*d*e^(-1) + 15444*(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^2*b^2*d*e^(-2) + 1144*(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)*a*b^3*d*e^(-3) +
130*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(
x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^4*d*e^(-4) + 45045*sqrt(x*e + d)*a^4*d^2 + 30030*((x*e + d)^(3/2
) - 3*sqrt(x*e + d)*d)*a^4*d + 5148*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sq
rt(x*e + d)*d^3)*a^3*b*e^(-1) + 858*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 42
0*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^2*b^2*e^(-2) + 260*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)
*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*
b^3*e^(-3) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7
/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^4*e^(-4) + 3003*(3*(
x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4)*e^(-1)

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maple [A]  time = 0.05, size = 186, normalized size = 1.44 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (1155 b^{4} e^{4} x^{4}+5460 a \,b^{3} e^{4} x^{3}-840 b^{4} d \,e^{3} x^{3}+10010 a^{2} b^{2} e^{4} x^{2}-3640 a \,b^{3} d \,e^{3} x^{2}+560 b^{4} d^{2} e^{2} x^{2}+8580 a^{3} b \,e^{4} x -5720 a^{2} b^{2} d \,e^{3} x +2080 a \,b^{3} d^{2} e^{2} x -320 b^{4} d^{3} e x +3003 a^{4} e^{4}-3432 a^{3} b d \,e^{3}+2288 a^{2} b^{2} d^{2} e^{2}-832 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{15015 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(e*x+d)^(5/2)*(1155*b^4*e^4*x^4+5460*a*b^3*e^4*x^3-840*b^4*d*e^3*x^3+10010*a^2*b^2*e^4*x^2-3640*a*b^3*
d*e^3*x^2+560*b^4*d^2*e^2*x^2+8580*a^3*b*e^4*x-5720*a^2*b^2*d*e^3*x+2080*a*b^3*d^2*e^2*x-320*b^4*d^3*e*x+3003*
a^4*e^4-3432*a^3*b*d*e^3+2288*a^2*b^2*d^2*e^2-832*a*b^3*d^3*e+128*b^4*d^4)/e^5

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maxima [A]  time = 1.08, size = 181, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (1155 \, {\left (e x + d\right )}^{\frac {13}{2}} b^{4} - 5460 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 10010 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 8580 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 3003 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{15015 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*(e*x + d)^(13/2)*b^4 - 5460*(b^4*d - a*b^3*e)*(e*x + d)^(11/2) + 10010*(b^4*d^2 - 2*a*b^3*d*e +
a^2*b^2*e^2)*(e*x + d)^(9/2) - 8580*(b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*(e*x + d)^(7/2) +
3003*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(e*x + d)^(5/2))/e^5

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mupad [B]  time = 0.04, size = 112, normalized size = 0.87 \begin {gather*} \frac {2\,b^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {4\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^4*(d + e*x)^(13/2))/(13*e^5) - ((8*b^4*d - 8*a*b^3*e)*(d + e*x)^(11/2))/(11*e^5) + (2*(a*e - b*d)^4*(d +
e*x)^(5/2))/(5*e^5) + (4*b^2*(a*e - b*d)^2*(d + e*x)^(9/2))/(3*e^5) + (8*b*(a*e - b*d)^3*(d + e*x)^(7/2))/(7*e
^5)

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sympy [A]  time = 21.72, size = 559, normalized size = 4.33 \begin {gather*} a^{4} d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 a^{4} \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {8 a^{3} b d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {8 a^{3} b \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {12 a^{2} b^{2} d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {12 a^{2} b^{2} \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {8 a b^{3} d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {8 a b^{3} \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} + \frac {2 b^{4} d \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} + \frac {2 b^{4} \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**4*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 8*a**3*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 8*a**3*b*(d**2*(d + e*x)*
*(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 12*a**2*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(
d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*a**2*b**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2
)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 8*a*b**3*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e
*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 8*a*b**3*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*
(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 2*b**4*
d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 +
(d + e*x)**(11/2)/11)/e**5 + 2*b**4*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/
2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5

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