3.6.25 \(\int (d+e x)^{5/2} (a+c x^2)^3 \, dx\)

Optimal. Leaf size=204 \[ \frac {2 c^2 (d+e x)^{15/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac {8 c^2 d (d+e x)^{13/2} \left (3 a e^2+5 c d^2\right )}{13 e^7}+\frac {6 c (d+e x)^{11/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{11 e^7}-\frac {4 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )^2}{3 e^7}+\frac {2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^3}{7 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7}-\frac {12 c^3 d (d+e x)^{17/2}}{17 e^7} \]

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Rubi [A]  time = 0.10, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {697} \begin {gather*} \frac {2 c^2 (d+e x)^{15/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac {8 c^2 d (d+e x)^{13/2} \left (3 a e^2+5 c d^2\right )}{13 e^7}+\frac {6 c (d+e x)^{11/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{11 e^7}-\frac {4 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )^2}{3 e^7}+\frac {2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^3}{7 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7}-\frac {12 c^3 d (d+e x)^{17/2}}{17 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(a + c*x^2)^3,x]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(7/2))/(7*e^7) - (4*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(9/2))/(3*e^7) + (6*c*(c*d^
2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(13/2))/(13*e
^7) + (2*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(15/2))/(5*e^7) - (12*c^3*d*(d + e*x)^(17/2))/(17*e^7) + (2*c^3*(d +
e*x)^(19/2))/(19*e^7)

Rule 697

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

Rubi steps

\begin {align*} \int (d+e x)^{5/2} \left (a+c x^2\right )^3 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3 (d+e x)^{5/2}}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{e^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{11/2}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{13/2}}{e^6}-\frac {6 c^3 d (d+e x)^{15/2}}{e^6}+\frac {c^3 (d+e x)^{17/2}}{e^6}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{7/2}}{7 e^7}-\frac {4 c d \left (c d^2+a e^2\right )^2 (d+e x)^{9/2}}{3 e^7}+\frac {6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{13/2}}{13 e^7}+\frac {2 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{15/2}}{5 e^7}-\frac {12 c^3 d (d+e x)^{17/2}}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 188, normalized size = 0.92 \begin {gather*} \frac {2 \left (\frac {1}{5} c^2 (d+e x)^{15/2} \left (a e^2+5 c d^2\right )-\frac {4}{13} c^2 d (d+e x)^{13/2} \left (3 a e^2+5 c d^2\right )+\frac {3}{11} c (d+e x)^{11/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )-\frac {2}{3} c d (d+e x)^{9/2} \left (a e^2+c d^2\right )^2+\frac {1}{7} (d+e x)^{7/2} \left (a e^2+c d^2\right )^3+\frac {1}{19} c^3 (d+e x)^{19/2}-\frac {6}{17} c^3 d (d+e x)^{17/2}\right )}{e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(((c*d^2 + a*e^2)^3*(d + e*x)^(7/2))/7 - (2*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(9/2))/3 + (3*c*(c*d^2 + a*e^2)
*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/11 - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(13/2))/13 + (c^2*(5*c*d^2 +
a*e^2)*(d + e*x)^(15/2))/5 - (6*c^3*d*(d + e*x)^(17/2))/17 + (c^3*(d + e*x)^(19/2))/19))/e^7

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IntegrateAlgebraic [A]  time = 0.12, size = 240, normalized size = 1.18 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (692835 a^3 e^6+2078505 a^2 c d^2 e^4-3233230 a^2 c d e^4 (d+e x)+1322685 a^2 c e^4 (d+e x)^2+2078505 a c^2 d^4 e^2-6466460 a c^2 d^3 e^2 (d+e x)+7936110 a c^2 d^2 e^2 (d+e x)^2-4476780 a c^2 d e^2 (d+e x)^3+969969 a c^2 e^2 (d+e x)^4+692835 c^3 d^6-3233230 c^3 d^5 (d+e x)+6613425 c^3 d^4 (d+e x)^2-7461300 c^3 d^3 (d+e x)^3+4849845 c^3 d^2 (d+e x)^4-1711710 c^3 d (d+e x)^5+255255 c^3 (d+e x)^6\right )}{4849845 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(692835*c^3*d^6 + 2078505*a*c^2*d^4*e^2 + 2078505*a^2*c*d^2*e^4 + 692835*a^3*e^6 - 3233230*
c^3*d^5*(d + e*x) - 6466460*a*c^2*d^3*e^2*(d + e*x) - 3233230*a^2*c*d*e^4*(d + e*x) + 6613425*c^3*d^4*(d + e*x
)^2 + 7936110*a*c^2*d^2*e^2*(d + e*x)^2 + 1322685*a^2*c*e^4*(d + e*x)^2 - 7461300*c^3*d^3*(d + e*x)^3 - 447678
0*a*c^2*d*e^2*(d + e*x)^3 + 4849845*c^3*d^2*(d + e*x)^4 + 969969*a*c^2*e^2*(d + e*x)^4 - 1711710*c^3*d*(d + e*
x)^5 + 255255*c^3*(d + e*x)^6))/(4849845*e^7)

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fricas [B]  time = 0.40, size = 356, normalized size = 1.75 \begin {gather*} \frac {2 \, {\left (255255 \, c^{3} e^{9} x^{9} + 585585 \, c^{3} d e^{8} x^{8} + 5120 \, c^{3} d^{9} + 41344 \, a c^{2} d^{7} e^{2} + 167960 \, a^{2} c d^{5} e^{4} + 692835 \, a^{3} d^{3} e^{6} + 3003 \, {\left (115 \, c^{3} d^{2} e^{7} + 323 \, a c^{2} e^{9}\right )} x^{7} + 231 \, {\left (5 \, c^{3} d^{3} e^{6} + 10013 \, a c^{2} d e^{8}\right )} x^{6} - 63 \, {\left (20 \, c^{3} d^{4} e^{5} - 22933 \, a c^{2} d^{2} e^{7} - 20995 \, a^{2} c e^{9}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{5} e^{4} + 323 \, a c^{2} d^{3} e^{6} + 96577 \, a^{2} c d e^{8}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{6} e^{3} + 2584 \, a c^{2} d^{4} e^{5} - 474487 \, a^{2} c d^{2} e^{7} - 138567 \, a^{3} e^{9}\right )} x^{3} + 3 \, {\left (640 \, c^{3} d^{7} e^{2} + 5168 \, a c^{2} d^{5} e^{4} + 20995 \, a^{2} c d^{3} e^{6} + 692835 \, a^{3} d e^{8}\right )} x^{2} - {\left (2560 \, c^{3} d^{8} e + 20672 \, a c^{2} d^{6} e^{3} + 83980 \, a^{2} c d^{4} e^{5} - 2078505 \, a^{3} d^{2} e^{7}\right )} x\right )} \sqrt {e x + d}}{4849845 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*(255255*c^3*e^9*x^9 + 585585*c^3*d*e^8*x^8 + 5120*c^3*d^9 + 41344*a*c^2*d^7*e^2 + 167960*a^2*c*d^5*e
^4 + 692835*a^3*d^3*e^6 + 3003*(115*c^3*d^2*e^7 + 323*a*c^2*e^9)*x^7 + 231*(5*c^3*d^3*e^6 + 10013*a*c^2*d*e^8)
*x^6 - 63*(20*c^3*d^4*e^5 - 22933*a*c^2*d^2*e^7 - 20995*a^2*c*e^9)*x^5 + 35*(40*c^3*d^5*e^4 + 323*a*c^2*d^3*e^
6 + 96577*a^2*c*d*e^8)*x^4 - 5*(320*c^3*d^6*e^3 + 2584*a*c^2*d^4*e^5 - 474487*a^2*c*d^2*e^7 - 138567*a^3*e^9)*
x^3 + 3*(640*c^3*d^7*e^2 + 5168*a*c^2*d^5*e^4 + 20995*a^2*c*d^3*e^6 + 692835*a^3*d*e^8)*x^2 - (2560*c^3*d^8*e
+ 20672*a*c^2*d^6*e^3 + 83980*a^2*c*d^4*e^5 - 2078505*a^3*d^2*e^7)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.26, size = 1222, normalized size = 5.99

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*(969969*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*d^3*e^(-2) + 46189*(
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*c^2*d^3*e^(-4) + 1615*(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)*c^3*
d^3*e^(-6) + 1247103*(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*c*d^2*e^(-2) + 62985*(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*c^2*d^2*e^(-4) + 2261*(429*(x*e + d)^(15/
2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*
d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^3*d^2*e^(-6) + 4849845
*sqrt(x*e + d)*a^3*d^3 + 4849845*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*d^2 + 138567*(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^2*c*d*e^(
-2) + 14535*(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)*a*c^2*d*e^(-4) + 133*(6435
*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850
850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7
+ 109395*sqrt(x*e + d)*d^8)*c^3*d*e^(-6) + 969969*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d
)*d^2)*a^3*d + 20995*(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^2*c*e^(-2) + 2261*(429*(x*e + d)^(15/2) - 3465*
(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 2702
7*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*a*c^2*e^(-4) + 21*(12155*(x*e + d)
^(19/2) - 122265*(x*e + d)^(17/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*
e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 +
692835*(x*e + d)^(3/2)*d^8 - 230945*sqrt(x*e + d)*d^9)*c^3*e^(-6) + 138567*(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^3)*e^(-1)

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maple [A]  time = 0.05, size = 205, normalized size = 1.00 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (255255 c^{3} x^{6} e^{6}-180180 c^{3} d \,e^{5} x^{5}+969969 a \,c^{2} e^{6} x^{4}+120120 c^{3} d^{2} e^{4} x^{4}-596904 a \,c^{2} d \,e^{5} x^{3}-73920 c^{3} d^{3} e^{3} x^{3}+1322685 a^{2} c \,e^{6} x^{2}+325584 a \,c^{2} d^{2} e^{4} x^{2}+40320 c^{3} d^{4} e^{2} x^{2}-587860 a^{2} c d \,e^{5} x -144704 a \,c^{2} d^{3} e^{3} x -17920 c^{3} d^{5} e x +692835 e^{6} a^{3}+167960 a^{2} c \,d^{2} e^{4}+41344 a \,c^{2} d^{4} e^{2}+5120 c^{3} d^{6}\right )}{4849845 e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*(e*x+d)^(7/2)*(255255*c^3*e^6*x^6-180180*c^3*d*e^5*x^5+969969*a*c^2*e^6*x^4+120120*c^3*d^2*e^4*x^4-5
96904*a*c^2*d*e^5*x^3-73920*c^3*d^3*e^3*x^3+1322685*a^2*c*e^6*x^2+325584*a*c^2*d^2*e^4*x^2+40320*c^3*d^4*e^2*x
^2-587860*a^2*c*d*e^5*x-144704*a*c^2*d^3*e^3*x-17920*c^3*d^5*e*x+692835*a^3*e^6+167960*a^2*c*d^2*e^4+41344*a*c
^2*d^4*e^2+5120*c^3*d^6)/e^7

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maxima [A]  time = 1.40, size = 209, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (255255 \, {\left (e x + d\right )}^{\frac {19}{2}} c^{3} - 1711710 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} d + 969969 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 1492260 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 1322685 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 3233230 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 692835 \, {\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 (e x + d\right )}^{\frac {7}{2}}\right )}}{4849845 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*(255255*(e*x + d)^(19/2)*c^3 - 1711710*(e*x + d)^(17/2)*c^3*d + 969969*(5*c^3*d^2 + a*c^2*e^2)*(e*x
+ d)^(15/2) - 1492260*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(e*x + d)^(13/2) + 1322685*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^
2*c*e^4)*(e*x + d)^(11/2) - 3233230*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(9/2) + 692835*(c^3*d^
6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(e*x + d)^(7/2))/e^7

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mupad [B]  time = 0.07, size = 187, normalized size = 0.92 \begin {gather*} \frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{11\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {12\,c^3\,d\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}-\frac {4\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3*(d + e*x)^(5/2),x)

[Out]

((30*c^3*d^2 + 6*a*c^2*e^2)*(d + e*x)^(15/2))/(15*e^7) + ((d + e*x)^(11/2)*(30*c^3*d^4 + 6*a^2*c*e^4 + 36*a*c^
2*d^2*e^2))/(11*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*e^7) + (2*(a*e^2 + c*d^2)^3*(d + e*x)^(7/2))/(7*e^7) - ((4
0*c^3*d^3 + 24*a*c^2*d*e^2)*(d + e*x)^(13/2))/(13*e^7) - (12*c^3*d*(d + e*x)^(17/2))/(17*e^7) - (4*c*d*(a*e^2
+ c*d^2)^2*(d + e*x)^(9/2))/(3*e^7)

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sympy [A]  time = 32.81, size = 945, normalized size = 4.63

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**3*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
6*a**2*c*d**2*(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*c*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**3 + 6*a**2*
c*(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**3 + 6*a*c**2*d**2*(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 + 12*a*c**2*d*(-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 + 6*a*c**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)*
*(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(
15/2)/15)/e**5 + 2*c**3*d**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7
- 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15
)/e**7 + 4*c**3*d*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d
+ e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d
+ e*x)**(17/2)/17)/e**7 + 2*c**3*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/
2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e
*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**7

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