3.2282 \(\int (d+e x)^{5/2} (a+b x+c x^2)^3 \, dx\)
Optimal. Leaf size=286 \[ \frac {2 c (d+e x)^{15/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{13/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{13 e^7}+\frac {6 (d+e x)^{11/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^7}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^3}{7 e^7}-\frac {6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \]
[Out]
2/7*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(7/2)/e^7-2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(9/2)/e^7+6/11*(a*e
^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(11/2)/e^7-2/13*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-
2*c*e*(-3*a*e+5*b*d))*(e*x+d)^(13/2)/e^7+2/5*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(15/2)/e^7-6/17*c^
2*(-b*e+2*c*d)*(e*x+d)^(17/2)/e^7+2/19*c^3*(e*x+d)^(19/2)/e^7
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Rubi [A] time = 0.18, antiderivative size = 286, 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} \[ \frac {2 c (d+e x)^{15/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{13/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{13 e^7}+\frac {6 (d+e x)^{11/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^7}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^3}{7 e^7}-\frac {6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]
[Out]
(2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(7/2))/(7*e^7) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(
9/2))/(3*e^7) + (6*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(11/2))/(11*e^7
) - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(13/2))/(13*e^7) + (2*c*(5*c^2*d
^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(15/2))/(5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(17/2))/(17*e^7)
+ (2*c^3*(d + e*x)^(19/2))/(19*e^7)
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 )^3 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{e^6}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{e^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{9/2}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{11/2}}{e^6}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (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-b d e+a e^2\right )^3 (d+e x)^{7/2}}{7 e^7}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{9/2}}{3 e^7}+\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{11 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{13/2}}{13 e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{15/2}}{5 e^7}-\frac {6 c^2 (2 c d-b e) (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 = 0.87, size = 321, normalized size = 1.12 \[ \frac {2 \left ((d+e x)^{7/2} (a+x (b+c x))^3-\frac {2 (d+e x)^{9/2} \left (-646 c e^2 \left (65 a^2 e^2 (2 d-9 e x)-15 a b e \left (8 d^2-36 d e x+99 e^2 x^2\right )+2 b^2 \left (16 d^3-72 d^2 e x+198 d e^2 x^2-429 e^3 x^3\right )\right )+1615 b e^3 \left (143 a^2 e^2+26 a b e (9 e x-2 d)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+19 c^2 e \left (68 a e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+5 b \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )-10 c^3 \left (256 d^5-1152 d^4 e x+3168 d^3 e^2 x^2-6864 d^2 e^3 x^3+12870 d e^4 x^4-21879 e^5 x^5\right )\right )}{692835 e^6}\right )}{7 e} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]
[Out]
(2*((d + e*x)^(7/2)*(a + x*(b + c*x))^3 - (2*(d + e*x)^(9/2)*(-10*c^3*(256*d^5 - 1152*d^4*e*x + 3168*d^3*e^2*x
^2 - 6864*d^2*e^3*x^3 + 12870*d*e^4*x^4 - 21879*e^5*x^5) + 1615*b*e^3*(143*a^2*e^2 + 26*a*b*e*(-2*d + 9*e*x) +
b^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2)) - 646*c*e^2*(65*a^2*e^2*(2*d - 9*e*x) - 15*a*b*e*(8*d^2 - 36*d*e*x + 99*
e^2*x^2) + 2*b^2*(16*d^3 - 72*d^2*e*x + 198*d*e^2*x^2 - 429*e^3*x^3)) + 19*c^2*e*(68*a*e*(-16*d^3 + 72*d^2*e*x
- 198*d*e^2*x^2 + 429*e^3*x^3) + 5*b*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^
4))))/(692835*e^6)))/(7*e)
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fricas [B] time = 0.88, size = 727, normalized size = 2.54 \[ \frac {2 \, {\left (255255 \, c^{3} e^{9} x^{9} + 5120 \, c^{3} d^{9} - 24320 \, b c^{2} d^{8} e - 461890 \, a^{2} b d^{4} e^{5} + 692835 \, a^{3} d^{3} e^{6} + 41344 \, {\left (b^{2} c + a c^{2}\right )} d^{7} e^{2} - 25840 \, {\left (b^{3} + 6 \, a b c\right )} d^{6} e^{3} + 167960 \, {\left (a b^{2} + a^{2} c\right )} d^{5} e^{4} + 45045 \, {\left (13 \, c^{3} d e^{8} + 19 \, b c^{2} e^{9}\right )} x^{8} + 3003 \, {\left (115 \, c^{3} d^{2} e^{7} + 665 \, b c^{2} d e^{8} + 323 \, {\left (b^{2} c + a c^{2}\right )} e^{9}\right )} x^{7} + 231 \, {\left (5 \, c^{3} d^{3} e^{6} + 5225 \, b c^{2} d^{2} e^{7} + 10013 \, {\left (b^{2} c + a c^{2}\right )} d e^{8} + 1615 \, {\left (b^{3} + 6 \, a b c\right )} e^{9}\right )} x^{6} - 63 \, {\left (20 \, c^{3} d^{4} e^{5} - 95 \, b c^{2} d^{3} e^{6} - 22933 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{7} - 14535 \, {\left (b^{3} + 6 \, a b c\right )} d e^{8} - 20995 \, {\left (a b^{2} + a^{2} c\right )} e^{9}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{5} e^{4} - 190 \, b c^{2} d^{4} e^{5} + 46189 \, a^{2} b e^{9} + 323 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{6} + 17119 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{7} + 96577 \, {\left (a b^{2} + a^{2} c\right )} d e^{8}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{6} e^{3} - 1520 \, b c^{2} d^{5} e^{4} - 877591 \, a^{2} b d e^{8} - 138567 \, a^{3} e^{9} + 2584 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{5} - 1615 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{6} - 474487 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{7}\right )} x^{3} + 3 \, {\left (640 \, c^{3} d^{7} e^{2} - 3040 \, b c^{2} d^{6} e^{3} + 1154725 \, a^{2} b d^{2} e^{7} + 692835 \, a^{3} d e^{8} + 5168 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{4} - 3230 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{5} + 20995 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{6}\right )} x^{2} - {\left (2560 \, c^{3} d^{8} e - 12160 \, b c^{2} d^{7} e^{2} - 230945 \, a^{2} b d^{3} e^{6} - 2078505 \, a^{3} d^{2} e^{7} + 20672 \, {\left (b^{2} c + a c^{2}\right )} d^{6} e^{3} - 12920 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} e^{4} + 83980 \, {\left (a b^{2} + a^{2} c\right )} d^{4} e^{5}\right )} x\right )} \sqrt {e x + d}}{4849845 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="fricas")
[Out]
2/4849845*(255255*c^3*e^9*x^9 + 5120*c^3*d^9 - 24320*b*c^2*d^8*e - 461890*a^2*b*d^4*e^5 + 692835*a^3*d^3*e^6 +
41344*(b^2*c + a*c^2)*d^7*e^2 - 25840*(b^3 + 6*a*b*c)*d^6*e^3 + 167960*(a*b^2 + a^2*c)*d^5*e^4 + 45045*(13*c^
3*d*e^8 + 19*b*c^2*e^9)*x^8 + 3003*(115*c^3*d^2*e^7 + 665*b*c^2*d*e^8 + 323*(b^2*c + a*c^2)*e^9)*x^7 + 231*(5*
c^3*d^3*e^6 + 5225*b*c^2*d^2*e^7 + 10013*(b^2*c + a*c^2)*d*e^8 + 1615*(b^3 + 6*a*b*c)*e^9)*x^6 - 63*(20*c^3*d^
4*e^5 - 95*b*c^2*d^3*e^6 - 22933*(b^2*c + a*c^2)*d^2*e^7 - 14535*(b^3 + 6*a*b*c)*d*e^8 - 20995*(a*b^2 + a^2*c)
*e^9)*x^5 + 35*(40*c^3*d^5*e^4 - 190*b*c^2*d^4*e^5 + 46189*a^2*b*e^9 + 323*(b^2*c + a*c^2)*d^3*e^6 + 17119*(b^
3 + 6*a*b*c)*d^2*e^7 + 96577*(a*b^2 + a^2*c)*d*e^8)*x^4 - 5*(320*c^3*d^6*e^3 - 1520*b*c^2*d^5*e^4 - 877591*a^2
*b*d*e^8 - 138567*a^3*e^9 + 2584*(b^2*c + a*c^2)*d^4*e^5 - 1615*(b^3 + 6*a*b*c)*d^3*e^6 - 474487*(a*b^2 + a^2*
c)*d^2*e^7)*x^3 + 3*(640*c^3*d^7*e^2 - 3040*b*c^2*d^6*e^3 + 1154725*a^2*b*d^2*e^7 + 692835*a^3*d*e^8 + 5168*(b
^2*c + a*c^2)*d^5*e^4 - 3230*(b^3 + 6*a*b*c)*d^4*e^5 + 20995*(a*b^2 + a^2*c)*d^3*e^6)*x^2 - (2560*c^3*d^8*e -
12160*b*c^2*d^7*e^2 - 230945*a^2*b*d^3*e^6 - 2078505*a^3*d^2*e^7 + 20672*(b^2*c + a*c^2)*d^6*e^3 - 12920*(b^3
+ 6*a*b*c)*d^5*e^4 + 83980*(a*b^2 + a^2*c)*d^4*e^5)*x)*sqrt(e*x + d)/e^7
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giac [B] time = 0.36, size = 3066, normalized size = 10.72 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")
[Out]
2/4849845*(4849845*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*d^3*e^(-1) + 969969*(3*(x*e + d)^(5/2) - 10*(x*
e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*d^3*e^(-2) + 969969*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 1
5*sqrt(x*e + d)*d^2)*a^2*c*d^3*e^(-2) + 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)*b^3*d^3*e^(-3) + 831402*(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*c*d^3*e^(-3) + 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)*b^2*c*d^3*e^(-4) + 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) + 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)*b*c^2*d^3*e^(-5) + 1615*(231*(x*e + d)^(13/2) - 16
38*(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) + 2909907*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)
*d + 15*sqrt(x*e + d)*d^2)*a^2*b*d^2*e^(-1) + 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*b^2*d^2*e^(-2) + 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) + 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)*b^3*d^2*e^(-3) + 277134*(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*c*d^2*e^(-3) + 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)*b^2*c*d^2*e^(-4) + 62985*(63*(x*e + d)^(1
1/2) - 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) + 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)*b*c^2*d^2*e^(-5) + 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*sq
rt(x*e + d)*d)*a^3*d^2 + 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*b*d*e^(-1) + 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*b^2*d*e^(-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) + 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)*b^3*d*e^(-3) + 125970*(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*c*d*e^(
-3) + 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)*b^2*c*d*e^(-4) + 14535*(23
1*(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) + 6783*(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)*b*c^2*d*e^(-5) + 133*(64
35*(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 + 8
50850*(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 + 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^2*b*e^(-1) + 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*b^2*e^(-2) +
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) + 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)*b^3*e^(-3) + 9690*(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*b*c*e^(-3) + 2261*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 250
25*(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 - 6
435*sqrt(x*e + d)*d^7)*b^2*c*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)*a*c^2*e^(-4) + 133*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 2
35620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(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)*b*c^2*e^(-5) + 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 = 495, normalized size = 1.73 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (255255 c^{3} x^{6} e^{6}+855855 b \,c^{2} e^{6} x^{5}-180180 c^{3} d \,e^{5} x^{5}+969969 a \,c^{2} e^{6} x^{4}+969969 b^{2} c \,e^{6} x^{4}-570570 b \,c^{2} d \,e^{5} x^{4}+120120 c^{3} d^{2} e^{4} x^{4}+2238390 a b c \,e^{6} x^{3}-596904 a \,c^{2} d \,e^{5} x^{3}+373065 b^{3} e^{6} x^{3}-596904 b^{2} c d \,e^{5} x^{3}+351120 b \,c^{2} d^{2} e^{4} x^{3}-73920 c^{3} d^{3} e^{3} x^{3}+1322685 a^{2} c \,e^{6} x^{2}+1322685 a \,b^{2} e^{6} x^{2}-1220940 a b c d \,e^{5} x^{2}+325584 a \,c^{2} d^{2} e^{4} x^{2}-203490 b^{3} d \,e^{5} x^{2}+325584 b^{2} c \,d^{2} e^{4} x^{2}-191520 b \,c^{2} d^{3} e^{3} x^{2}+40320 c^{3} d^{4} e^{2} x^{2}+1616615 a^{2} b \,e^{6} x -587860 a^{2} c d \,e^{5} x -587860 a \,b^{2} d \,e^{5} x +542640 a b c \,d^{2} e^{4} x -144704 a \,c^{2} d^{3} e^{3} x +90440 b^{3} d^{2} e^{4} x -144704 b^{2} c \,d^{3} e^{3} x +85120 b \,c^{2} d^{4} e^{2} x -17920 c^{3} d^{5} e x +692835 a^{3} e^{6}-461890 a^{2} b d \,e^{5}+167960 a^{2} c \,d^{2} e^{4}+167960 a \,b^{2} d^{2} e^{4}-155040 a b c \,d^{3} e^{3}+41344 a \,c^{2} d^{4} e^{2}-25840 b^{3} d^{3} e^{3}+41344 b^{2} c \,d^{4} e^{2}-24320 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{4849845 e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x)
[Out]
2/4849845*(e*x+d)^(7/2)*(255255*c^3*e^6*x^6+855855*b*c^2*e^6*x^5-180180*c^3*d*e^5*x^5+969969*a*c^2*e^6*x^4+969
969*b^2*c*e^6*x^4-570570*b*c^2*d*e^5*x^4+120120*c^3*d^2*e^4*x^4+2238390*a*b*c*e^6*x^3-596904*a*c^2*d*e^5*x^3+3
73065*b^3*e^6*x^3-596904*b^2*c*d*e^5*x^3+351120*b*c^2*d^2*e^4*x^3-73920*c^3*d^3*e^3*x^3+1322685*a^2*c*e^6*x^2+
1322685*a*b^2*e^6*x^2-1220940*a*b*c*d*e^5*x^2+325584*a*c^2*d^2*e^4*x^2-203490*b^3*d*e^5*x^2+325584*b^2*c*d^2*e
^4*x^2-191520*b*c^2*d^3*e^3*x^2+40320*c^3*d^4*e^2*x^2+1616615*a^2*b*e^6*x-587860*a^2*c*d*e^5*x-587860*a*b^2*d*
e^5*x+542640*a*b*c*d^2*e^4*x-144704*a*c^2*d^3*e^3*x+90440*b^3*d^2*e^4*x-144704*b^2*c*d^3*e^3*x+85120*b*c^2*d^4
*e^2*x-17920*c^3*d^5*e*x+692835*a^3*e^6-461890*a^2*b*d*e^5+167960*a^2*c*d^2*e^4+167960*a*b^2*d^2*e^4-155040*a*
b*c*d^3*e^3+41344*a*c^2*d^4*e^2-25840*b^3*d^3*e^3+41344*b^2*c*d^4*e^2-24320*b*c^2*d^5*e+5120*c^3*d^6)/e^7
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maxima [A] time = 0.95, size = 407, normalized size = 1.42 \[ \frac {2 \, {\left (255255 \, {\left (e x + d\right )}^{\frac {19}{2}} c^{3} - 855855 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 373065 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 1322685 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 1616615 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 692835 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{4849845 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="maxima")
[Out]
2/4849845*(255255*(e*x + d)^(19/2)*c^3 - 855855*(2*c^3*d - b*c^2*e)*(e*x + d)^(17/2) + 969969*(5*c^3*d^2 - 5*b
*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(15/2) - 373065*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*
e^2 - (b^3 + 6*a*b*c)*e^3)*(e*x + d)^(13/2) + 1322685*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2
- (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^(11/2) - 1616615*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e
^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d)^(9/2) + 692835*(
c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a
*b^2 + a^2*c)*d^2*e^4)*(e*x + d)^(7/2))/e^7
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mupad [B] time = 0.10, size = 297, normalized size = 1.04 \[ \frac {{\left (d+e\,x\right )}^{11/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{11\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {{\left (d+e\,x\right )}^{15/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{15\,e^7}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{7\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{13\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{3\,e^7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x)
[Out]
((d + e*x)^(11/2)*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^4 - 6*b^3*d*e^3 + 36*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2
- 60*b*c^2*d^3*e - 36*a*b*c*d*e^3))/(11*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d
+ e*x)^(17/2))/(17*e^7) + ((d + e*x)^(15/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(15*e^7)
+ (2*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e)^3)/(7*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(13/2)*(b^2*e^2 + 10*c^2*
d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(13*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(9/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(3*e^7)
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sympy [A] time = 94.43, size = 2363, normalized size = 8.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**(5/2)*(c*x**2+b*x+a)**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*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*a**2*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 + 6*a**2*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 + 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*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 + 12*a*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 + 6*a*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
+ 12*a*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 + 24*a*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 + 12*a*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 + 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*b**
3*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 + 4*b**3*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 + 2*b**3*(-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 + 6*b**2
*c*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*b**2*c*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*b**2*
c*(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 + 6*b*c**2*d**2*(-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**6 + 12*b*c**2*d*(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**6 + 6*b*c**2*(-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**6 + 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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