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

Optimal. Leaf size=166 \[ \frac {2 (d+e x)^{11/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{11 e^5}-\frac {4 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{9 e^5}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2}{7 e^5}-\frac {4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \]

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Rubi [A]  time = 0.09, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {698} \begin {gather*} \frac {2 (d+e x)^{11/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{11 e^5}-\frac {4 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{9 e^5}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2}{7 e^5}-\frac {4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 698

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

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 173, normalized size = 1.04 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (65 e^2 \left (99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-10 c e \left (3 b \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )-13 a e \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(c^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 65*e^2*(9
9*a^2*e^2 + 22*a*b*e*(-2*d + 7*e*x) + b^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 10*c*e*(-13*a*e*(8*d^2 - 28*d*e*x
+ 63*e^2*x^2) + 3*b*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3))))/(45045*e^5)

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IntegrateAlgebraic [A]  time = 0.11, size = 229, normalized size = 1.38 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (6435 a^2 e^4+10010 a b e^3 (d+e x)-12870 a b d e^3+12870 a c d^2 e^2-20020 a c d e^2 (d+e x)+8190 a c e^2 (d+e x)^2+6435 b^2 d^2 e^2-10010 b^2 d e^2 (d+e x)+4095 b^2 e^2 (d+e x)^2-12870 b c d^3 e+30030 b c d^2 e (d+e x)-24570 b c d e (d+e x)^2+6930 b c e (d+e x)^3+6435 c^2 d^4-20020 c^2 d^3 (d+e x)+24570 c^2 d^2 (d+e x)^2-13860 c^2 d (d+e x)^3+3003 c^2 (d+e x)^4\right )}{45045 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(6435*c^2*d^4 - 12870*b*c*d^3*e + 6435*b^2*d^2*e^2 + 12870*a*c*d^2*e^2 - 12870*a*b*d*e^3 +
6435*a^2*e^4 - 20020*c^2*d^3*(d + e*x) + 30030*b*c*d^2*e*(d + e*x) - 10010*b^2*d*e^2*(d + e*x) - 20020*a*c*d*e
^2*(d + e*x) + 10010*a*b*e^3*(d + e*x) + 24570*c^2*d^2*(d + e*x)^2 - 24570*b*c*d*e*(d + e*x)^2 + 4095*b^2*e^2*
(d + e*x)^2 + 8190*a*c*e^2*(d + e*x)^2 - 13860*c^2*d*(d + e*x)^3 + 6930*b*c*e*(d + e*x)^3 + 3003*c^2*(d + e*x)
^4))/(45045*e^5)

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fricas [B]  time = 0.42, size = 365, normalized size = 2.20 \begin {gather*} \frac {2 \, {\left (3003 \, c^{2} e^{7} x^{7} + 128 \, c^{2} d^{7} - 480 \, b c d^{6} e - 2860 \, a b d^{4} e^{3} + 6435 \, a^{2} d^{3} e^{4} + 520 \, {\left (b^{2} + 2 \, a c\right )} d^{5} e^{2} + 231 \, {\left (31 \, c^{2} d e^{6} + 30 \, b c e^{7}\right )} x^{6} + 63 \, {\left (71 \, c^{2} d^{2} e^{5} + 270 \, b c d e^{6} + 65 \, {\left (b^{2} + 2 \, a c\right )} e^{7}\right )} x^{5} + 35 \, {\left (c^{2} d^{3} e^{4} + 318 \, b c d^{2} e^{5} + 286 \, a b e^{7} + 299 \, {\left (b^{2} + 2 \, a c\right )} d e^{6}\right )} x^{4} - 5 \, {\left (8 \, c^{2} d^{4} e^{3} - 30 \, b c d^{3} e^{4} - 5434 \, a b d e^{6} - 1287 \, a^{2} e^{7} - 1469 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{5}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{5} e^{2} - 60 \, b c d^{4} e^{3} + 7150 \, a b d^{2} e^{5} + 6435 \, a^{2} d e^{6} + 65 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{4}\right )} x^{2} - {\left (64 \, c^{2} d^{6} e - 240 \, b c d^{5} e^{2} - 1430 \, a b d^{3} e^{4} - 19305 \, a^{2} d^{2} e^{5} + 260 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*c^2*e^7*x^7 + 128*c^2*d^7 - 480*b*c*d^6*e - 2860*a*b*d^4*e^3 + 6435*a^2*d^3*e^4 + 520*(b^2 + 2*a
*c)*d^5*e^2 + 231*(31*c^2*d*e^6 + 30*b*c*e^7)*x^6 + 63*(71*c^2*d^2*e^5 + 270*b*c*d*e^6 + 65*(b^2 + 2*a*c)*e^7)
*x^5 + 35*(c^2*d^3*e^4 + 318*b*c*d^2*e^5 + 286*a*b*e^7 + 299*(b^2 + 2*a*c)*d*e^6)*x^4 - 5*(8*c^2*d^4*e^3 - 30*
b*c*d^3*e^4 - 5434*a*b*d*e^6 - 1287*a^2*e^7 - 1469*(b^2 + 2*a*c)*d^2*e^5)*x^3 + 3*(16*c^2*d^5*e^2 - 60*b*c*d^4
*e^3 + 7150*a*b*d^2*e^5 + 6435*a^2*d*e^6 + 65*(b^2 + 2*a*c)*d^3*e^4)*x^2 - (64*c^2*d^6*e - 240*b*c*d^5*e^2 - 1
430*a*b*d^3*e^4 - 19305*a^2*d^2*e^5 + 260*(b^2 + 2*a*c)*d^4*e^3)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.27, size = 1498, normalized size = 9.02

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+b*x+a)^2,x, algorithm="giac")

[Out]

2/45045*(30030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*d^3*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(
3/2)*d + 15*sqrt(x*e + d)*d^2)*b^2*d^3*e^(-2) + 6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e +
d)*d^2)*a*c*d^3*e^(-2) + 2574*(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)*b*c*d^3*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)*c^2*d^3*e^(-4) + 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d +
15*sqrt(x*e + d)*d^2)*a*b*d^2*e^(-1) + 3861*(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)*b^2*d^2*e^(-2) + 7722*(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*d^2*e^(-2) + 858*(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*c*d^2*e^(-3) + 195*(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)*c^2*d^2*e^(-4) + 45045*sqrt(x*e + d)*a^2*d^3 + 45045*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*d^2
+ 7722*(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*d*e^(-1
) + 429*(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^2*d*e^(-2) + 858*(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*d*e^(-2) + 390*(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*
c*d*e^(-3) + 45*(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^2*d*e^(-4) + 9009*(3
*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*d + 286*(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*e^(-1) + 65*(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^2*e^(-2) + 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)*a*c*e^(-2) + 30*(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*c*e^(-3) + 7*(429*(x*e + d)^(15/2) - 346
5*(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 - 27
027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^2*e^(-4) + 1287*(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)*e^(-1)

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maple [A]  time = 0.05, size = 194, normalized size = 1.17 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 c^{2} x^{4} e^{4}+6930 b c \,e^{4} x^{3}-1848 c^{2} d \,e^{3} x^{3}+8190 a c \,e^{4} x^{2}+4095 b^{2} e^{4} x^{2}-3780 b c d \,e^{3} x^{2}+1008 c^{2} d^{2} e^{2} x^{2}+10010 a b \,e^{4} x -3640 a c d \,e^{3} x -1820 b^{2} d \,e^{3} x +1680 b c \,d^{2} e^{2} x -448 c^{2} d^{3} e x +6435 a^{2} e^{4}-2860 a b d \,e^{3}+1040 a c \,d^{2} e^{2}+520 b^{2} d^{2} e^{2}-480 b c \,d^{3} e +128 c^{2} d^{4}\right )}{45045 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(e*x+d)^(7/2)*(3003*c^2*e^4*x^4+6930*b*c*e^4*x^3-1848*c^2*d*e^3*x^3+8190*a*c*e^4*x^2+4095*b^2*e^4*x^2-
3780*b*c*d*e^3*x^2+1008*c^2*d^2*e^2*x^2+10010*a*b*e^4*x-3640*a*c*d*e^3*x-1820*b^2*d*e^3*x+1680*b*c*d^2*e^2*x-4
48*c^2*d^3*e*x+6435*a^2*e^4-2860*a*b*d*e^3+1040*a*c*d^2*e^2+520*b^2*d^2*e^2-480*b*c*d^3*e+128*c^2*d^4)/e^5

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maxima [A]  time = 0.83, size = 176, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{2} - 6930 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 4095 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 10010 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{45045 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^2 - 6930*(2*c^2*d - b*c*e)*(e*x + d)^(13/2) + 4095*(6*c^2*d^2 - 6*b*c*d*e + (
b^2 + 2*a*c)*e^2)*(e*x + d)^(11/2) - 10010*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d)
^(9/2) + 6435*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*(e*x + d)^(7/2))/e^5

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mupad [B]  time = 0.84, size = 148, normalized size = 0.89 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{11\,e^5}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{7\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{9\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 41.43, size = 1129, normalized size = 6.80

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+b*x+a)**2,x)

[Out]

a**2*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**2*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
4*a*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 8*a*b*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x
)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 4*a*b*(-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**2 + 4*a*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 + 8*a*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 + 4*a*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 + 2*b**2*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 + 4*b**2*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 + 2*b**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**3 + 4*b*c*d**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**4 + 8*b*c*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**4 + 4*b*c*(-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**4 + 2*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)/1
1)/e**5 + 4*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 + 2*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

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