3.1640 \(\int \sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=187 \[ -\frac{12 b^5 (d+e x)^{13/2} (b d-a e)}{13 e^7}+\frac{30 b^4 (d+e x)^{11/2} (b d-a e)^2}{11 e^7}-\frac{40 b^3 (d+e x)^{9/2} (b d-a e)^3}{9 e^7}+\frac{30 b^2 (d+e x)^{7/2} (b d-a e)^4}{7 e^7}-\frac{12 b (d+e x)^{5/2} (b d-a e)^5}{5 e^7}+\frac{2 (d+e x)^{3/2} (b d-a e)^6}{3 e^7}+\frac{2 b^6 (d+e x)^{15/2}}{15 e^7} \]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(3/2))/(3*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(5/2))/(5*e^7) + (30*b^2*(b*d - a*e)
^4*(d + e*x)^(7/2))/(7*e^7) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^7) + (30*b^4*(b*d - a*e)^2*(d + e*x)
^(11/2))/(11*e^7) - (12*b^5*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^7) + (2*b^6*(d + e*x)^(15/2))/(15*e^7)

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Rubi [A]  time = 0.0589647, antiderivative size = 187, normalized size of antiderivative = 1., 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} \[ -\frac{12 b^5 (d+e x)^{13/2} (b d-a e)}{13 e^7}+\frac{30 b^4 (d+e x)^{11/2} (b d-a e)^2}{11 e^7}-\frac{40 b^3 (d+e x)^{9/2} (b d-a e)^3}{9 e^7}+\frac{30 b^2 (d+e x)^{7/2} (b d-a e)^4}{7 e^7}-\frac{12 b (d+e x)^{5/2} (b d-a e)^5}{5 e^7}+\frac{2 (d+e x)^{3/2} (b d-a e)^6}{3 e^7}+\frac{2 b^6 (d+e x)^{15/2}}{15 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(3/2))/(3*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(5/2))/(5*e^7) + (30*b^2*(b*d - a*e)
^4*(d + e*x)^(7/2))/(7*e^7) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^7) + (30*b^4*(b*d - a*e)^2*(d + e*x)
^(11/2))/(11*e^7) - (12*b^5*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^7) + (2*b^6*(d + e*x)^(15/2))/(15*e^7)

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 \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 \sqrt{d+e x} \, dx\\ &=\int \left (\frac{(-b d+a e)^6 \sqrt{d+e x}}{e^6}-\frac{6 b (b d-a e)^5 (d+e x)^{3/2}}{e^6}+\frac{15 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{7/2}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{9/2}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{11/2}}{e^6}+\frac{b^6 (d+e x)^{13/2}}{e^6}\right ) \, dx\\ &=\frac{2 (b d-a e)^6 (d+e x)^{3/2}}{3 e^7}-\frac{12 b (b d-a e)^5 (d+e x)^{5/2}}{5 e^7}+\frac{30 b^2 (b d-a e)^4 (d+e x)^{7/2}}{7 e^7}-\frac{40 b^3 (b d-a e)^3 (d+e x)^{9/2}}{9 e^7}+\frac{30 b^4 (b d-a e)^2 (d+e x)^{11/2}}{11 e^7}-\frac{12 b^5 (b d-a e) (d+e x)^{13/2}}{13 e^7}+\frac{2 b^6 (d+e x)^{15/2}}{15 e^7}\\ \end{align*}

Mathematica [A]  time = 0.101242, size = 145, normalized size = 0.78 \[ \frac{2 (d+e x)^{3/2} \left (96525 b^2 (d+e x)^2 (b d-a e)^4-100100 b^3 (d+e x)^3 (b d-a e)^3+61425 b^4 (d+e x)^4 (b d-a e)^2-20790 b^5 (d+e x)^5 (b d-a e)-54054 b (d+e x) (b d-a e)^5+15015 (b d-a e)^6+3003 b^6 (d+e x)^6\right )}{45045 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(15015*(b*d - a*e)^6 - 54054*b*(b*d - a*e)^5*(d + e*x) + 96525*b^2*(b*d - a*e)^4*(d + e*x)^
2 - 100100*b^3*(b*d - a*e)^3*(d + e*x)^3 + 61425*b^4*(b*d - a*e)^2*(d + e*x)^4 - 20790*b^5*(b*d - a*e)*(d + e*
x)^5 + 3003*b^6*(d + e*x)^6))/(45045*e^7)

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Maple [B]  time = 0.047, size = 377, normalized size = 2. \begin{align*}{\frac{6006\,{b}^{6}{x}^{6}{e}^{6}+41580\,{x}^{5}a{b}^{5}{e}^{6}-5544\,{x}^{5}{b}^{6}d{e}^{5}+122850\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-37800\,{x}^{4}a{b}^{5}d{e}^{5}+5040\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+200200\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-109200\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+33600\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-4480\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+193050\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-171600\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+93600\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-28800\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+3840\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+108108\,x{a}^{5}b{e}^{6}-154440\,x{a}^{4}{b}^{2}d{e}^{5}+137280\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-74880\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+23040\,xa{b}^{5}{d}^{4}{e}^{2}-3072\,x{b}^{6}{d}^{5}e+30030\,{a}^{6}{e}^{6}-72072\,{a}^{5}bd{e}^{5}+102960\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-91520\,{b}^{3}{a}^{3}{d}^{3}{e}^{3}+49920\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-15360\,a{b}^{5}{d}^{5}e+2048\,{d}^{6}{b}^{6}}{45045\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(3/2)*(3003*b^6*e^6*x^6+20790*a*b^5*e^6*x^5-2772*b^6*d*e^5*x^5+61425*a^2*b^4*e^6*x^4-18900*a*b
^5*d*e^5*x^4+2520*b^6*d^2*e^4*x^4+100100*a^3*b^3*e^6*x^3-54600*a^2*b^4*d*e^5*x^3+16800*a*b^5*d^2*e^4*x^3-2240*
b^6*d^3*e^3*x^3+96525*a^4*b^2*e^6*x^2-85800*a^3*b^3*d*e^5*x^2+46800*a^2*b^4*d^2*e^4*x^2-14400*a*b^5*d^3*e^3*x^
2+1920*b^6*d^4*e^2*x^2+54054*a^5*b*e^6*x-77220*a^4*b^2*d*e^5*x+68640*a^3*b^3*d^2*e^4*x-37440*a^2*b^4*d^3*e^3*x
+11520*a*b^5*d^4*e^2*x-1536*b^6*d^5*e*x+15015*a^6*e^6-36036*a^5*b*d*e^5+51480*a^4*b^2*d^2*e^4-45760*a^3*b^3*d^
3*e^3+24960*a^2*b^4*d^4*e^2-7680*a*b^5*d^5*e+1024*b^6*d^6)/e^7

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Maxima [B]  time = 1.07513, size = 473, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} b^{6} - 20790 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 61425 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 100100 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 96525 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 54054 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 15015 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*b^6 - 20790*(b^6*d - a*b^5*e)*(e*x + d)^(13/2) + 61425*(b^6*d^2 - 2*a*b^5*d*e +
a^2*b^4*e^2)*(e*x + d)^(11/2) - 100100*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(e*x + d)^(9
/2) + 96525*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*(e*x + d)^(7/2) - 54
054*(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*(e*x + d
)^(5/2) + 15015*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^
5*b*d*e^5 + a^6*e^6)*(e*x + d)^(3/2))/e^7

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Fricas [B]  time = 1.60004, size = 1027, normalized size = 5.49 \begin{align*} \frac{2 \,{\left (3003 \, b^{6} e^{7} x^{7} + 1024 \, b^{6} d^{7} - 7680 \, a b^{5} d^{6} e + 24960 \, a^{2} b^{4} d^{5} e^{2} - 45760 \, a^{3} b^{3} d^{4} e^{3} + 51480 \, a^{4} b^{2} d^{3} e^{4} - 36036 \, a^{5} b d^{2} e^{5} + 15015 \, a^{6} d e^{6} + 231 \,{\left (b^{6} d e^{6} + 90 \, a b^{5} e^{7}\right )} x^{6} - 63 \,{\left (4 \, b^{6} d^{2} e^{5} - 30 \, a b^{5} d e^{6} - 975 \, a^{2} b^{4} e^{7}\right )} x^{5} + 35 \,{\left (8 \, b^{6} d^{3} e^{4} - 60 \, a b^{5} d^{2} e^{5} + 195 \, a^{2} b^{4} d e^{6} + 2860 \, a^{3} b^{3} e^{7}\right )} x^{4} - 5 \,{\left (64 \, b^{6} d^{4} e^{3} - 480 \, a b^{5} d^{3} e^{4} + 1560 \, a^{2} b^{4} d^{2} e^{5} - 2860 \, a^{3} b^{3} d e^{6} - 19305 \, a^{4} b^{2} e^{7}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{5} e^{2} - 960 \, a b^{5} d^{4} e^{3} + 3120 \, a^{2} b^{4} d^{3} e^{4} - 5720 \, a^{3} b^{3} d^{2} e^{5} + 6435 \, a^{4} b^{2} d e^{6} + 18018 \, a^{5} b e^{7}\right )} x^{2} -{\left (512 \, b^{6} d^{6} e - 3840 \, a b^{5} d^{5} e^{2} + 12480 \, a^{2} b^{4} d^{4} e^{3} - 22880 \, a^{3} b^{3} d^{3} e^{4} + 25740 \, a^{4} b^{2} d^{2} e^{5} - 18018 \, a^{5} b d e^{6} - 15015 \, a^{6} e^{7}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*b^6*e^7*x^7 + 1024*b^6*d^7 - 7680*a*b^5*d^6*e + 24960*a^2*b^4*d^5*e^2 - 45760*a^3*b^3*d^4*e^3 +
51480*a^4*b^2*d^3*e^4 - 36036*a^5*b*d^2*e^5 + 15015*a^6*d*e^6 + 231*(b^6*d*e^6 + 90*a*b^5*e^7)*x^6 - 63*(4*b^6
*d^2*e^5 - 30*a*b^5*d*e^6 - 975*a^2*b^4*e^7)*x^5 + 35*(8*b^6*d^3*e^4 - 60*a*b^5*d^2*e^5 + 195*a^2*b^4*d*e^6 +
2860*a^3*b^3*e^7)*x^4 - 5*(64*b^6*d^4*e^3 - 480*a*b^5*d^3*e^4 + 1560*a^2*b^4*d^2*e^5 - 2860*a^3*b^3*d*e^6 - 19
305*a^4*b^2*e^7)*x^3 + 3*(128*b^6*d^5*e^2 - 960*a*b^5*d^4*e^3 + 3120*a^2*b^4*d^3*e^4 - 5720*a^3*b^3*d^2*e^5 +
6435*a^4*b^2*d*e^6 + 18018*a^5*b*e^7)*x^2 - (512*b^6*d^6*e - 3840*a*b^5*d^5*e^2 + 12480*a^2*b^4*d^4*e^3 - 2288
0*a^3*b^3*d^3*e^4 + 25740*a^4*b^2*d^2*e^5 - 18018*a^5*b*d*e^6 - 15015*a^6*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [B]  time = 6.4708, size = 422, normalized size = 2.26 \begin{align*} \frac{2 \left (\frac{b^{6} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{\left (d + e x\right )^{\frac{13}{2}} \left (6 a b^{5} e - 6 b^{6} d\right )}{13 e^{6}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{11 e^{6}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}\right )}{3 e^{6}}\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3*(e*x+d)**(1/2),x)

[Out]

2*(b**6*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(13/2)*(6*a*b**5*e - 6*b**6*d)/(13*e**6) + (d + e*x)**(11/2)*
(15*a**2*b**4*e**2 - 30*a*b**5*d*e + 15*b**6*d**2)/(11*e**6) + (d + e*x)**(9/2)*(20*a**3*b**3*e**3 - 60*a**2*b
**4*d*e**2 + 60*a*b**5*d**2*e - 20*b**6*d**3)/(9*e**6) + (d + e*x)**(7/2)*(15*a**4*b**2*e**4 - 60*a**3*b**3*d*
e**3 + 90*a**2*b**4*d**2*e**2 - 60*a*b**5*d**3*e + 15*b**6*d**4)/(7*e**6) + (d + e*x)**(5/2)*(6*a**5*b*e**5 -
30*a**4*b**2*d*e**4 + 60*a**3*b**3*d**2*e**3 - 60*a**2*b**4*d**3*e**2 + 30*a*b**5*d**4*e - 6*b**6*d**5)/(5*e**
6) + (d + e*x)**(3/2)*(a**6*e**6 - 6*a**5*b*d*e**5 + 15*a**4*b**2*d**2*e**4 - 20*a**3*b**3*d**3*e**3 + 15*a**2
*b**4*d**4*e**2 - 6*a*b**5*d**5*e + b**6*d**6)/(3*e**6))/e

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Giac [B]  time = 1.22188, size = 536, normalized size = 2.87 \begin{align*} \frac{2}{45045} \,{\left (18018 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{5} b e^{\left (-1\right )} + 6435 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a^{4} b^{2} e^{\left (-2\right )} + 2860 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} a^{3} b^{3} e^{\left (-3\right )} + 195 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} a^{2} b^{4} e^{\left (-4\right )} + 30 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5}\right )} a b^{5} e^{\left (-5\right )} +{\left (3003 \,{\left (x e + d\right )}^{\frac{15}{2}} - 20790 \,{\left (x e + d\right )}^{\frac{13}{2}} d + 61425 \,{\left (x e + d\right )}^{\frac{11}{2}} d^{2} - 100100 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{3} + 96525 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{4} - 54054 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{5} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{6}\right )} b^{6} e^{\left (-6\right )} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{6}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(18018*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^5*b*e^(-1) + 6435*(15*(x*e + d)^(7/2) - 42*(x*e + d
)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^4*b^2*e^(-2) + 2860*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x
*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^3*b^3*e^(-3) + 195*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)
*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a^2*b^4*e^(-4) + 30*(693*
(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e
+ d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a*b^5*e^(-5) + (3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d +
61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d
^5 + 15015*(x*e + d)^(3/2)*d^6)*b^6*e^(-6) + 15015*(x*e + d)^(3/2)*a^6)*e^(-1)