3.17.12 \(\int \frac {(b+2 c x) (a+b x+c x^2)^3}{(d+e x)^{5/2}} \, dx\) [1612]

Optimal. Leaf size=421 \[ \frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^8 \sqrt {d+e x}}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \sqrt {d+e x}}{e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^8}-\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{5/2}}{e^8}+\frac {6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7/2}}{7 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{9/2}}{9 e^8}+\frac {4 c^4 (d+e x)^{11/2}}{11 e^8} \]

[Out]

2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^8/(e*x+d)^(3/2)+2/3*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20
*c^3*d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d)^(3/2)/e^8-2*c*(-b*e+2*c*d)*(7*c^2
*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^(5/2)/e^8+6/7*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^(
7/2)/e^8-14/9*c^3*(-b*e+2*c*d)*(e*x+d)^(9/2)/e^8+4/11*c^4*(e*x+d)^(11/2)/e^8-2*(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d
^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))/e^8/(e*x+d)^(1/2)-6*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(
-3*a*e+7*b*d))*(e*x+d)^(1/2)/e^8

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Rubi [A]
time = 0.15, antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {785} \begin {gather*} \frac {2 (d+e x)^{3/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^8}+\frac {6 c^2 (d+e x)^{7/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac {2 c (d+e x)^{5/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac {6 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac {14 c^3 (d+e x)^{9/2} (2 c d-b e)}{9 e^8}+\frac {4 c^4 (d+e x)^{11/2}}{11 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^8*(d + e*x)^(3/2)) - (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2
+ 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(e^8*Sqrt[d + e*x]) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2
+ b^2*e^2 - c*e*(7*b*d - 3*a*e))*Sqrt[d + e*x])/e^8 + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e)
- 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(3/2))/(3*e^8) - (2*
c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(5/2))/e^8 + (6*c^2*(14*c^2*d^2 + 3*b^2*
e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(7/2))/(7*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^8) + (4*c^4*
(d + e*x)^(11/2))/(11*e^8)

Rule 785

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

Rubi steps

\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^{5/2}}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^7 (d+e x)^{3/2}}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right )}{e^7 \sqrt {d+e x}}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \sqrt {d+e x}}{e^7}+\frac {5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^{3/2}}{e^7}+\frac {3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{5/2}}{e^7}-\frac {7 c^3 (2 c d-b e) (d+e x)^{7/2}}{e^7}+\frac {2 c^4 (d+e x)^{9/2}}{e^7}\right ) \, dx\\ &=\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^8 \sqrt {d+e x}}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \sqrt {d+e x}}{e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^8}-\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{5/2}}{e^8}+\frac {6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7/2}}{7 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{9/2}}{9 e^8}+\frac {4 c^4 (d+e x)^{11/2}}{11 e^8}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 598, normalized size = 1.42 \begin {gather*} \frac {-28 c^4 \left (2048 d^7+3072 d^6 e x+768 d^5 e^2 x^2-128 d^4 e^3 x^3+48 d^3 e^4 x^4-24 d^2 e^5 x^5+14 d e^6 x^6-9 e^7 x^7\right )-462 b e^4 \left (a^3 e^3+3 a^2 b e^2 (2 d+3 e x)-3 a b^2 e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )+462 c e^3 \left (-2 a^3 e^3 (2 d+3 e x)+9 a^2 b e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+12 a b^2 e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b^3 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )-198 c^2 e^2 \left (14 a^2 e^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )-7 a b e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+3 b^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )+22 c^3 e \left (-18 a e \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )+7 b \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )\right )}{693 e^8 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-28*c^4*(2048*d^7 + 3072*d^6*e*x + 768*d^5*e^2*x^2 - 128*d^4*e^3*x^3 + 48*d^3*e^4*x^4 - 24*d^2*e^5*x^5 + 14*d
*e^6*x^6 - 9*e^7*x^7) - 462*b*e^4*(a^3*e^3 + 3*a^2*b*e^2*(2*d + 3*e*x) - 3*a*b^2*e*(8*d^2 + 12*d*e*x + 3*e^2*x
^2) + b^3*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) + 462*c*e^3*(-2*a^3*e^3*(2*d + 3*e*x) + 9*a^2*b*e^2*(
8*d^2 + 12*d*e*x + 3*e^2*x^2) + 12*a*b^2*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b^3*(128*d^4 + 192
*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4)) - 198*c^2*e^2*(14*a^2*e^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2
*x^2 - e^3*x^3) - 7*a*b*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + 3*b^2*(256*d^5
+ 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5)) + 22*c^3*e*(-18*a*e*(256*d^5 + 384
*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5) + 7*b*(1024*d^6 + 1536*d^5*e*x + 384*d^4
*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*x^6)))/(693*e^8*(d + e*x)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(893\) vs. \(2(395)=790\).
time = 1.16, size = 894, normalized size = 2.12 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^8*(2*a^2*c^2*e^4*(e*x+d)^(3/2)+7/9*b*c^3*e*(e*x+d)^(9/2)+3*a*b^3*e^5*(e*x+d)^(1/2)+6/7*a*c^3*e^2*(e*x+d)^(
7/2)+9/7*b^2*c^2*e^2*(e*x+d)^(7/2)-3*b^4*d*e^4*(e*x+d)^(1/2)+b^3*c*e^3*(e*x+d)^(5/2)-1/3*(a^3*b*e^7-2*a^3*c*d*
e^6-3*a^2*b^2*d*e^6+9*a^2*b*c*d^2*e^5-6*a^2*c^2*d^3*e^4+3*a*b^3*d^2*e^5-12*a*b^2*c*d^3*e^4+15*a*b*c^2*d^4*e^3-
6*a*c^3*d^5*e^2-b^4*d^3*e^4+5*b^3*c*d^4*e^3-9*b^2*c^2*d^5*e^2+7*b*c^3*d^6*e-2*c^4*d^7)/(e*x+d)^(3/2)+2/11*c^4*
(e*x+d)^(11/2)-(2*a^3*c*e^6+3*a^2*b^2*e^6-18*a^2*b*c*d*e^5+18*a^2*c^2*d^2*e^4-6*a*b^3*d*e^5+36*a*b^2*c*d^2*e^4
-60*a*b*c^2*d^3*e^3+30*a*c^3*d^4*e^2+3*b^4*d^2*e^4-20*b^3*c*d^3*e^3+45*b^2*c^2*d^4*e^2-42*b*c^3*d^5*e+14*c^4*d
^6)/(e*x+d)^(1/2)-14*c^4*d^3*(e*x+d)^(5/2)-36*a*b^2*c*d*e^4*(e*x+d)^(1/2)-9*b^2*c^2*d*e^2*(e*x+d)^(5/2)+4*a*b^
2*c*e^4*(e*x+d)^(3/2)+20*a*c^3*d^2*e^2*(e*x+d)^(3/2)-20/3*b^3*c*d*e^3*(e*x+d)^(3/2)-6*b*c^3*d*e*(e*x+d)^(7/2)-
18*a^2*c^2*d*e^4*(e*x+d)^(1/2)-60*a*c^3*d^3*e^2*(e*x+d)^(1/2)+9*a^2*b*c*e^5*(e*x+d)^(1/2)+30*b^2*c^2*d^2*e^2*(
e*x+d)^(3/2)-140/3*b*c^3*d^3*e*(e*x+d)^(3/2)+30*b^3*c*d^2*e^3*(e*x+d)^(1/2)+1/3*b^4*e^4*(e*x+d)^(3/2)+70/3*c^4
*d^4*(e*x+d)^(3/2)-14/9*c^4*d*(e*x+d)^(9/2)+6*c^4*d^2*(e*x+d)^(7/2)-42*c^4*d^5*(e*x+d)^(1/2)+21*b*c^3*d^2*e*(e
*x+d)^(5/2)+3*a*b*c^2*e^3*(e*x+d)^(5/2)-6*a*c^3*d*e^2*(e*x+d)^(5/2)-90*b^2*c^2*d^3*e^2*(e*x+d)^(1/2)+105*b*c^3
*d^4*e*(e*x+d)^(1/2)+90*a*b*c^2*d^2*e^3*(e*x+d)^(1/2)-20*a*b*c^2*d*e^3*(e*x+d)^(3/2))

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Maxima [A]
time = 0.29, size = 684, normalized size = 1.62 \begin {gather*} \frac {2}{693} \, {\left ({\left (126 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{4} - 539 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 297 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + 3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}} - 693 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e - b^{3} c e^{3} - 3 \, a b c^{2} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 231 \, {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4} + 30 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{2} - 20 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d\right )} {\left (x e + d\right )}^{\frac {3}{2}} - 2079 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e - a b^{3} e^{5} - 3 \, a^{2} b c e^{5} + 10 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{3} - 10 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{2} + {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d\right )} \sqrt {x e + d}\right )} e^{\left (-7\right )} + \frac {231 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e + 3 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{5} - 5 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{4} - a^{3} b e^{7} + {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{3} - 3 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d^{2} - 3 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{4} + 3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6} - 20 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{3} + 3 \, {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{2} - 6 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d\right )} {\left (x e + d\right )} + {\left (3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} d\right )} e^{\left (-7\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*((126*(x*e + d)^(11/2)*c^4 - 539*(2*c^4*d - b*c^3*e)*(x*e + d)^(9/2) + 297*(14*c^4*d^2 - 14*b*c^3*d*e +
3*b^2*c^2*e^2 + 2*a*c^3*e^2)*(x*e + d)^(7/2) - 693*(14*c^4*d^3 - 21*b*c^3*d^2*e - b^3*c*e^3 - 3*a*b*c^2*e^3 +
3*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d)*(x*e + d)^(5/2) + 231*(70*c^4*d^4 - 140*b*c^3*d^3*e + b^4*e^4 + 12*a*b^2*c*
e^4 + 6*a^2*c^2*e^4 + 30*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^2 - 20*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d)*(x*e + d)^(3/2)
- 2079*(14*c^4*d^5 - 35*b*c^3*d^4*e - a*b^3*e^5 - 3*a^2*b*c*e^5 + 10*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^3 - 10*(
b^3*c*e^3 + 3*a*b*c^2*e^3)*d^2 + (b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d)*sqrt(x*e + d))*e^(-7) + 231*(2*
c^4*d^7 - 7*b*c^3*d^6*e + 3*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^5 - 5*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d^4 - a^3*b*e^7
+ (b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d^3 - 3*(a*b^3*e^5 + 3*a^2*b*c*e^5)*d^2 - 3*(14*c^4*d^6 - 42*b*c^
3*d^5*e + 15*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^4 + 3*a^2*b^2*e^6 + 2*a^3*c*e^6 - 20*(b^3*c*e^3 + 3*a*b*c^2*e^3)*
d^3 + 3*(b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d^2 - 6*(a*b^3*e^5 + 3*a^2*b*c*e^5)*d)*(x*e + d) + (3*a^2*b
^2*e^6 + 2*a^3*c*e^6)*d)*e^(-7)/(x*e + d)^(3/2))*e^(-1)

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Fricas [A]
time = 1.26, size = 635, normalized size = 1.51 \begin {gather*} -\frac {2 \, {\left (28672 \, c^{4} d^{7} - {\left (126 \, c^{4} x^{7} + 539 \, b c^{3} x^{6} + 297 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{5} + 693 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{4} - 231 \, a^{3} b + 231 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{3} + 2079 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{2} - 693 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x\right )} e^{7} + 2 \, {\left (98 \, c^{4} d x^{6} + 462 \, b c^{3} d x^{5} + 297 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{4} + 924 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{3} + 693 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{2} - 4158 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x + 231 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} e^{6} - 24 \, {\left (14 \, c^{4} d^{2} x^{5} + 77 \, b c^{3} d^{2} x^{4} + 66 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{3} + 462 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{2} - 231 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x + 231 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2}\right )} e^{5} + 16 \, {\left (42 \, c^{4} d^{3} x^{4} + 308 \, b c^{3} d^{3} x^{3} + 594 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{2} - 2772 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x + 231 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3}\right )} e^{4} - 128 \, {\left (14 \, c^{4} d^{4} x^{3} + 231 \, b c^{3} d^{4} x^{2} - 297 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x + 231 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4}\right )} e^{3} + 768 \, {\left (14 \, c^{4} d^{5} x^{2} - 154 \, b c^{3} d^{5} x + 33 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5}\right )} e^{2} + 7168 \, {\left (6 \, c^{4} d^{6} x - 11 \, b c^{3} d^{6}\right )} e\right )} \sqrt {x e + d}}{693 \, {\left (x^{2} e^{10} + 2 \, d x e^{9} + d^{2} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/693*(28672*c^4*d^7 - (126*c^4*x^7 + 539*b*c^3*x^6 + 297*(3*b^2*c^2 + 2*a*c^3)*x^5 + 693*(b^3*c + 3*a*b*c^2)
*x^4 - 231*a^3*b + 231*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*x^3 + 2079*(a*b^3 + 3*a^2*b*c)*x^2 - 693*(3*a^2*b^2 + 2*
a^3*c)*x)*e^7 + 2*(98*c^4*d*x^6 + 462*b*c^3*d*x^5 + 297*(3*b^2*c^2 + 2*a*c^3)*d*x^4 + 924*(b^3*c + 3*a*b*c^2)*
d*x^3 + 693*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*x^2 - 4158*(a*b^3 + 3*a^2*b*c)*d*x + 231*(3*a^2*b^2 + 2*a^3*c)*d)
*e^6 - 24*(14*c^4*d^2*x^5 + 77*b*c^3*d^2*x^4 + 66*(3*b^2*c^2 + 2*a*c^3)*d^2*x^3 + 462*(b^3*c + 3*a*b*c^2)*d^2*
x^2 - 231*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*x + 231*(a*b^3 + 3*a^2*b*c)*d^2)*e^5 + 16*(42*c^4*d^3*x^4 + 308*b
*c^3*d^3*x^3 + 594*(3*b^2*c^2 + 2*a*c^3)*d^3*x^2 - 2772*(b^3*c + 3*a*b*c^2)*d^3*x + 231*(b^4 + 12*a*b^2*c + 6*
a^2*c^2)*d^3)*e^4 - 128*(14*c^4*d^4*x^3 + 231*b*c^3*d^4*x^2 - 297*(3*b^2*c^2 + 2*a*c^3)*d^4*x + 231*(b^3*c + 3
*a*b*c^2)*d^4)*e^3 + 768*(14*c^4*d^5*x^2 - 154*b*c^3*d^5*x + 33*(3*b^2*c^2 + 2*a*c^3)*d^5)*e^2 + 7168*(6*c^4*d
^6*x - 11*b*c^3*d^6)*e)*sqrt(x*e + d)/(x^2*e^10 + 2*d*x*e^9 + d^2*e^8)

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Sympy [A]
time = 98.42, size = 558, normalized size = 1.33 \begin {gather*} \frac {4 c^{4} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{8}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (14 b c^{3} e - 28 c^{4} d\right )}{9 e^{8}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (12 a c^{3} e^{2} + 18 b^{2} c^{2} e^{2} - 84 b c^{3} d e + 84 c^{4} d^{2}\right )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (30 a b c^{2} e^{3} - 60 a c^{3} d e^{2} + 10 b^{3} c e^{3} - 90 b^{2} c^{2} d e^{2} + 210 b c^{3} d^{2} e - 140 c^{4} d^{3}\right )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (12 a^{2} c^{2} e^{4} + 24 a b^{2} c e^{4} - 120 a b c^{2} d e^{3} + 120 a c^{3} d^{2} e^{2} + 2 b^{4} e^{4} - 40 b^{3} c d e^{3} + 180 b^{2} c^{2} d^{2} e^{2} - 280 b c^{3} d^{3} e + 140 c^{4} d^{4}\right )}{3 e^{8}} + \frac {\sqrt {d + e x} \left (18 a^{2} b c e^{5} - 36 a^{2} c^{2} d e^{4} + 6 a b^{3} e^{5} - 72 a b^{2} c d e^{4} + 180 a b c^{2} d^{2} e^{3} - 120 a c^{3} d^{3} e^{2} - 6 b^{4} d e^{4} + 60 b^{3} c d^{2} e^{3} - 180 b^{2} c^{2} d^{3} e^{2} + 210 b c^{3} d^{4} e - 84 c^{4} d^{5}\right )}{e^{8}} - \frac {2 \left (a e^{2} - b d e + c d^{2}\right )^{2} \cdot \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 c^{2} d^{2}\right )}{e^{8} \sqrt {d + e x}} - \frac {2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3}}{3 e^{8} \left (d + e x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*c**4*(d + e*x)**(11/2)/(11*e**8) + (d + e*x)**(9/2)*(14*b*c**3*e - 28*c**4*d)/(9*e**8) + (d + e*x)**(7/2)*(1
2*a*c**3*e**2 + 18*b**2*c**2*e**2 - 84*b*c**3*d*e + 84*c**4*d**2)/(7*e**8) + (d + e*x)**(5/2)*(30*a*b*c**2*e**
3 - 60*a*c**3*d*e**2 + 10*b**3*c*e**3 - 90*b**2*c**2*d*e**2 + 210*b*c**3*d**2*e - 140*c**4*d**3)/(5*e**8) + (d
+ e*x)**(3/2)*(12*a**2*c**2*e**4 + 24*a*b**2*c*e**4 - 120*a*b*c**2*d*e**3 + 120*a*c**3*d**2*e**2 + 2*b**4*e**
4 - 40*b**3*c*d*e**3 + 180*b**2*c**2*d**2*e**2 - 280*b*c**3*d**3*e + 140*c**4*d**4)/(3*e**8) + sqrt(d + e*x)*(
18*a**2*b*c*e**5 - 36*a**2*c**2*d*e**4 + 6*a*b**3*e**5 - 72*a*b**2*c*d*e**4 + 180*a*b*c**2*d**2*e**3 - 120*a*c
**3*d**3*e**2 - 6*b**4*d*e**4 + 60*b**3*c*d**2*e**3 - 180*b**2*c**2*d**3*e**2 + 210*b*c**3*d**4*e - 84*c**4*d*
*5)/e**8 - 2*(a*e**2 - b*d*e + c*d**2)**2*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)/(e**8*sqrt(d
+ e*x)) - 2*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**3/(3*e**8*(d + e*x)**(3/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 972 vs. \(2 (403) = 806\).
time = 2.92, size = 972, normalized size = 2.31 \begin {gather*} \frac {2}{693} \, {\left (126 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{4} e^{80} - 1078 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{4} d e^{80} + 4158 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{4} d^{2} e^{80} - 9702 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{4} d^{3} e^{80} + 16170 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{4} e^{80} - 29106 \, \sqrt {x e + d} c^{4} d^{5} e^{80} + 539 \, {\left (x e + d\right )}^{\frac {9}{2}} b c^{3} e^{81} - 4158 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{3} d e^{81} + 14553 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{3} d^{2} e^{81} - 32340 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{3} d^{3} e^{81} + 72765 \, \sqrt {x e + d} b c^{3} d^{4} e^{81} + 891 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c^{2} e^{82} + 594 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{3} e^{82} - 6237 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{2} d e^{82} - 4158 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{3} d e^{82} + 20790 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{2} e^{82} + 13860 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{3} d^{2} e^{82} - 62370 \, \sqrt {x e + d} b^{2} c^{2} d^{3} e^{82} - 41580 \, \sqrt {x e + d} a c^{3} d^{3} e^{82} + 693 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c e^{83} + 2079 \, {\left (x e + d\right )}^{\frac {5}{2}} a b c^{2} e^{83} - 4620 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c d e^{83} - 13860 \, {\left (x e + d\right )}^{\frac {3}{2}} a b c^{2} d e^{83} + 20790 \, \sqrt {x e + d} b^{3} c d^{2} e^{83} + 62370 \, \sqrt {x e + d} a b c^{2} d^{2} e^{83} + 231 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} e^{84} + 2772 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} c e^{84} + 1386 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c^{2} e^{84} - 2079 \, \sqrt {x e + d} b^{4} d e^{84} - 24948 \, \sqrt {x e + d} a b^{2} c d e^{84} - 12474 \, \sqrt {x e + d} a^{2} c^{2} d e^{84} + 2079 \, \sqrt {x e + d} a b^{3} e^{85} + 6237 \, \sqrt {x e + d} a^{2} b c e^{85}\right )} e^{\left (-88\right )} - \frac {2 \, {\left (42 \, {\left (x e + d\right )} c^{4} d^{6} - 2 \, c^{4} d^{7} - 126 \, {\left (x e + d\right )} b c^{3} d^{5} e + 7 \, b c^{3} d^{6} e + 135 \, {\left (x e + d\right )} b^{2} c^{2} d^{4} e^{2} + 90 \, {\left (x e + d\right )} a c^{3} d^{4} e^{2} - 9 \, b^{2} c^{2} d^{5} e^{2} - 6 \, a c^{3} d^{5} e^{2} - 60 \, {\left (x e + d\right )} b^{3} c d^{3} e^{3} - 180 \, {\left (x e + d\right )} a b c^{2} d^{3} e^{3} + 5 \, b^{3} c d^{4} e^{3} + 15 \, a b c^{2} d^{4} e^{3} + 9 \, {\left (x e + d\right )} b^{4} d^{2} e^{4} + 108 \, {\left (x e + d\right )} a b^{2} c d^{2} e^{4} + 54 \, {\left (x e + d\right )} a^{2} c^{2} d^{2} e^{4} - b^{4} d^{3} e^{4} - 12 \, a b^{2} c d^{3} e^{4} - 6 \, a^{2} c^{2} d^{3} e^{4} - 18 \, {\left (x e + d\right )} a b^{3} d e^{5} - 54 \, {\left (x e + d\right )} a^{2} b c d e^{5} + 3 \, a b^{3} d^{2} e^{5} + 9 \, a^{2} b c d^{2} e^{5} + 9 \, {\left (x e + d\right )} a^{2} b^{2} e^{6} + 6 \, {\left (x e + d\right )} a^{3} c e^{6} - 3 \, a^{2} b^{2} d e^{6} - 2 \, a^{3} c d e^{6} + a^{3} b e^{7}\right )} e^{\left (-8\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(126*(x*e + d)^(11/2)*c^4*e^80 - 1078*(x*e + d)^(9/2)*c^4*d*e^80 + 4158*(x*e + d)^(7/2)*c^4*d^2*e^80 - 9
702*(x*e + d)^(5/2)*c^4*d^3*e^80 + 16170*(x*e + d)^(3/2)*c^4*d^4*e^80 - 29106*sqrt(x*e + d)*c^4*d^5*e^80 + 539
*(x*e + d)^(9/2)*b*c^3*e^81 - 4158*(x*e + d)^(7/2)*b*c^3*d*e^81 + 14553*(x*e + d)^(5/2)*b*c^3*d^2*e^81 - 32340
*(x*e + d)^(3/2)*b*c^3*d^3*e^81 + 72765*sqrt(x*e + d)*b*c^3*d^4*e^81 + 891*(x*e + d)^(7/2)*b^2*c^2*e^82 + 594*
(x*e + d)^(7/2)*a*c^3*e^82 - 6237*(x*e + d)^(5/2)*b^2*c^2*d*e^82 - 4158*(x*e + d)^(5/2)*a*c^3*d*e^82 + 20790*(
x*e + d)^(3/2)*b^2*c^2*d^2*e^82 + 13860*(x*e + d)^(3/2)*a*c^3*d^2*e^82 - 62370*sqrt(x*e + d)*b^2*c^2*d^3*e^82
- 41580*sqrt(x*e + d)*a*c^3*d^3*e^82 + 693*(x*e + d)^(5/2)*b^3*c*e^83 + 2079*(x*e + d)^(5/2)*a*b*c^2*e^83 - 46
20*(x*e + d)^(3/2)*b^3*c*d*e^83 - 13860*(x*e + d)^(3/2)*a*b*c^2*d*e^83 + 20790*sqrt(x*e + d)*b^3*c*d^2*e^83 +
62370*sqrt(x*e + d)*a*b*c^2*d^2*e^83 + 231*(x*e + d)^(3/2)*b^4*e^84 + 2772*(x*e + d)^(3/2)*a*b^2*c*e^84 + 1386
*(x*e + d)^(3/2)*a^2*c^2*e^84 - 2079*sqrt(x*e + d)*b^4*d*e^84 - 24948*sqrt(x*e + d)*a*b^2*c*d*e^84 - 12474*sqr
t(x*e + d)*a^2*c^2*d*e^84 + 2079*sqrt(x*e + d)*a*b^3*e^85 + 6237*sqrt(x*e + d)*a^2*b*c*e^85)*e^(-88) - 2/3*(42
*(x*e + d)*c^4*d^6 - 2*c^4*d^7 - 126*(x*e + d)*b*c^3*d^5*e + 7*b*c^3*d^6*e + 135*(x*e + d)*b^2*c^2*d^4*e^2 + 9
0*(x*e + d)*a*c^3*d^4*e^2 - 9*b^2*c^2*d^5*e^2 - 6*a*c^3*d^5*e^2 - 60*(x*e + d)*b^3*c*d^3*e^3 - 180*(x*e + d)*a
*b*c^2*d^3*e^3 + 5*b^3*c*d^4*e^3 + 15*a*b*c^2*d^4*e^3 + 9*(x*e + d)*b^4*d^2*e^4 + 108*(x*e + d)*a*b^2*c*d^2*e^
4 + 54*(x*e + d)*a^2*c^2*d^2*e^4 - b^4*d^3*e^4 - 12*a*b^2*c*d^3*e^4 - 6*a^2*c^2*d^3*e^4 - 18*(x*e + d)*a*b^3*d
*e^5 - 54*(x*e + d)*a^2*b*c*d*e^5 + 3*a*b^3*d^2*e^5 + 9*a^2*b*c*d^2*e^5 + 9*(x*e + d)*a^2*b^2*e^6 + 6*(x*e + d
)*a^3*c*e^6 - 3*a^2*b^2*d*e^6 - 2*a^3*c*d*e^6 + a^3*b*e^7)*e^(-8)/(x*e + d)^(3/2)

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Mupad [B]
time = 1.96, size = 677, normalized size = 1.61 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{7\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {\frac {4\,c^4\,d^7}{3}-\left (d+e\,x\right )\,\left (4\,a^3\,c\,e^6+6\,a^2\,b^2\,e^6-36\,a^2\,b\,c\,d\,e^5+36\,a^2\,c^2\,d^2\,e^4-12\,a\,b^3\,d\,e^5+72\,a\,b^2\,c\,d^2\,e^4-120\,a\,b\,c^2\,d^3\,e^3+60\,a\,c^3\,d^4\,e^2+6\,b^4\,d^2\,e^4-40\,b^3\,c\,d^3\,e^3+90\,b^2\,c^2\,d^4\,e^2-84\,b\,c^3\,d^5\,e+28\,c^4\,d^6\right )-\frac {2\,a^3\,b\,e^7}{3}+\frac {2\,b^4\,d^3\,e^4}{3}-2\,a\,b^3\,d^2\,e^5+2\,a^2\,b^2\,d\,e^6+4\,a\,c^3\,d^5\,e^2-\frac {10\,b^3\,c\,d^4\,e^3}{3}+4\,a^2\,c^2\,d^3\,e^4+6\,b^2\,c^2\,d^5\,e^2+\frac {4\,a^3\,c\,d\,e^6}{3}-\frac {14\,b\,c^3\,d^6\,e}{3}-10\,a\,b\,c^2\,d^4\,e^3+8\,a\,b^2\,c\,d^3\,e^4-6\,a^2\,b\,c\,d^2\,e^5}{e^8\,{\left (d+e\,x\right )}^{3/2}}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{3\,e^8}+\frac {6\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{e^8}+\frac {2\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(7/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d*e))/(7*e^8) + (4*c^4*(d + e*x)^(11/2
))/(11*e^8) - ((28*c^4*d - 14*b*c^3*e)*(d + e*x)^(9/2))/(9*e^8) + ((4*c^4*d^7)/3 - (d + e*x)*(28*c^4*d^6 + 4*a
^3*c*e^6 + 6*a^2*b^2*e^6 + 6*b^4*d^2*e^4 + 60*a*c^3*d^4*e^2 - 40*b^3*c*d^3*e^3 + 36*a^2*c^2*d^2*e^4 + 90*b^2*c
^2*d^4*e^2 - 12*a*b^3*d*e^5 - 84*b*c^3*d^5*e - 36*a^2*b*c*d*e^5 - 120*a*b*c^2*d^3*e^3 + 72*a*b^2*c*d^2*e^4) -
(2*a^3*b*e^7)/3 + (2*b^4*d^3*e^4)/3 - 2*a*b^3*d^2*e^5 + 2*a^2*b^2*d*e^6 + 4*a*c^3*d^5*e^2 - (10*b^3*c*d^4*e^3)
/3 + 4*a^2*c^2*d^3*e^4 + 6*b^2*c^2*d^5*e^2 + (4*a^3*c*d*e^6)/3 - (14*b*c^3*d^6*e)/3 - 10*a*b*c^2*d^4*e^3 + 8*a
*b^2*c*d^3*e^4 - 6*a^2*b*c*d^2*e^5)/(e^8*(d + e*x)^(3/2)) + ((d + e*x)^(3/2)*(2*b^4*e^4 + 140*c^4*d^4 + 12*a^2
*c^2*e^4 + 120*a*c^3*d^2*e^2 + 180*b^2*c^2*d^2*e^2 + 24*a*b^2*c*e^4 - 280*b*c^3*d^3*e - 40*b^3*c*d*e^3 - 120*a
*b*c^2*d*e^3))/(3*e^8) + (6*(b*e - 2*c*d)*(d + e*x)^(1/2)*(7*c^3*d^4 + a*b^2*e^4 + 3*a^2*c*e^4 - b^3*d*e^3 + 1
0*a*c^2*d^2*e^2 + 8*b^2*c*d^2*e^2 - 14*b*c^2*d^3*e - 10*a*b*c*d*e^3))/e^8 + (2*c*(b*e - 2*c*d)*(d + e*x)^(5/2)
*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e))/e^8

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