3.2054 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^2}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=154 \[ -\frac {10 b^4 (d+e x)^{9/2} (b d-a e)}{9 e^6}+\frac {20 b^3 (d+e x)^{7/2} (b d-a e)^2}{7 e^6}-\frac {4 b^2 (d+e x)^{5/2} (b d-a e)^3}{e^6}+\frac {10 b (d+e x)^{3/2} (b d-a e)^4}{3 e^6}-\frac {2 \sqrt {d+e x} (b d-a e)^5}{e^6}+\frac {2 b^5 (d+e x)^{11/2}}{11 e^6} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 43} \[ -\frac {10 b^4 (d+e x)^{9/2} (b d-a e)}{9 e^6}+\frac {20 b^3 (d+e x)^{7/2} (b d-a e)^2}{7 e^6}-\frac {4 b^2 (d+e x)^{5/2} (b d-a e)^3}{e^6}+\frac {10 b (d+e x)^{3/2} (b d-a e)^4}{3 e^6}-\frac {2 \sqrt {d+e x} (b d-a e)^5}{e^6}+\frac {2 b^5 (d+e x)^{11/2}}{11 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.07, size = 123, normalized size = 0.80 \[ \frac {2 \sqrt {d+e x} \left (-385 b^4 (d+e x)^4 (b d-a e)+990 b^3 (d+e x)^3 (b d-a e)^2-1386 b^2 (d+e x)^2 (b d-a e)^3+1155 b (d+e x) (b d-a e)^4-693 (b d-a e)^5+63 b^5 (d+e x)^5\right )}{693 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-693*(b*d - a*e)^5 + 1155*b*(b*d - a*e)^4*(d + e*x) - 1386*b^2*(b*d - a*e)^3*(d + e*x)^2 + 9
90*b^3*(b*d - a*e)^2*(d + e*x)^3 - 385*b^4*(b*d - a*e)*(d + e*x)^4 + 63*b^5*(d + e*x)^5))/(693*e^6)

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fricas [A]  time = 0.90, size = 261, normalized size = 1.69 \[ \frac {2 \, {\left (63 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 1408 \, a b^{4} d^{4} e - 3168 \, a^{2} b^{3} d^{3} e^{2} + 3696 \, a^{3} b^{2} d^{2} e^{3} - 2310 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} - 35 \, {\left (2 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{2} e^{3} - 44 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 88 \, a b^{4} d^{2} e^{3} + 198 \, a^{2} b^{3} d e^{4} - 231 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 704 \, a b^{4} d^{3} e^{2} + 1584 \, a^{2} b^{3} d^{2} e^{3} - 1848 \, a^{3} b^{2} d e^{4} + 1155 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*b^5*e^5*x^5 - 256*b^5*d^5 + 1408*a*b^4*d^4*e - 3168*a^2*b^3*d^3*e^2 + 3696*a^3*b^2*d^2*e^3 - 2310*a^
4*b*d*e^4 + 693*a^5*e^5 - 35*(2*b^5*d*e^4 - 11*a*b^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 44*a*b^4*d*e^4 + 99*a^2*b^
3*e^5)*x^3 - 6*(16*b^5*d^3*e^2 - 88*a*b^4*d^2*e^3 + 198*a^2*b^3*d*e^4 - 231*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e
- 704*a*b^4*d^3*e^2 + 1584*a^2*b^3*d^2*e^3 - 1848*a^3*b^2*d*e^4 + 1155*a^4*b*e^5)*x)*sqrt(e*x + d)/e^6

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giac [B]  time = 0.17, size = 298, normalized size = 1.94 \[ \frac {2}{693} \, {\left (1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{4} b e^{\left (-1\right )} + 462 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{3} b^{2} e^{\left (-2\right )} + 198 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2} b^{3} e^{\left (-3\right )} + 11 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a b^{4} e^{\left (-4\right )} + {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{5} e^{\left (-5\right )} + 693 \, \sqrt {x e + d} a^{5}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*b*e^(-1) + 462*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d
+ 15*sqrt(x*e + d)*d^2)*a^3*b^2*e^(-2) + 198*(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^3*e^(-3) + 11*(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^4*e^(-4) + (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^5*e^(-5) + 693*sqrt(x*e + d)*a^5)*e^(-1)

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maple [B]  time = 0.04, size = 273, normalized size = 1.77 \[ \frac {2 \left (63 b^{5} e^{5} x^{5}+385 a \,b^{4} e^{5} x^{4}-70 b^{5} d \,e^{4} x^{4}+990 a^{2} b^{3} e^{5} x^{3}-440 a \,b^{4} d \,e^{4} x^{3}+80 b^{5} d^{2} e^{3} x^{3}+1386 a^{3} b^{2} e^{5} x^{2}-1188 a^{2} b^{3} d \,e^{4} x^{2}+528 a \,b^{4} d^{2} e^{3} x^{2}-96 b^{5} d^{3} e^{2} x^{2}+1155 a^{4} b \,e^{5} x -1848 a^{3} b^{2} d \,e^{4} x +1584 a^{2} b^{3} d^{2} e^{3} x -704 a \,b^{4} d^{3} e^{2} x +128 b^{5} d^{4} e x +693 a^{5} e^{5}-2310 a^{4} b d \,e^{4}+3696 a^{3} b^{2} d^{2} e^{3}-3168 a^{2} b^{3} d^{3} e^{2}+1408 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \sqrt {e x +d}}{693 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*b^5*e^5*x^5+385*a*b^4*e^5*x^4-70*b^5*d*e^4*x^4+990*a^2*b^3*e^5*x^3-440*a*b^4*d*e^4*x^3+80*b^5*d^2*e^
3*x^3+1386*a^3*b^2*e^5*x^2-1188*a^2*b^3*d*e^4*x^2+528*a*b^4*d^2*e^3*x^2-96*b^5*d^3*e^2*x^2+1155*a^4*b*e^5*x-18
48*a^3*b^2*d*e^4*x+1584*a^2*b^3*d^2*e^3*x-704*a*b^4*d^3*e^2*x+128*b^5*d^4*e*x+693*a^5*e^5-2310*a^4*b*d*e^4+369
6*a^3*b^2*d^2*e^3-3168*a^2*b^3*d^3*e^2+1408*a*b^4*d^4*e-256*b^5*d^5)*(e*x+d)^(1/2)/e^6

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maxima [A]  time = 0.56, size = 259, normalized size = 1.68 \[ \frac {2 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{5} - 385 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 990 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 1386 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 693 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {e x + d}\right )}}{693 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*(e*x + d)^(11/2)*b^5 - 385*(b^5*d - a*b^4*e)*(e*x + d)^(9/2) + 990*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*
e^2)*(e*x + d)^(7/2) - 1386*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(5/2) + 1155*(
b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d)^(3/2) - 693*(b^5*d^5 - 5*
a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(e*x + d))/e^6

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mupad [B]  time = 0.05, size = 137, normalized size = 0.89 \[ \frac {2\,b^5\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,\sqrt {d+e\,x}}{e^6}+\frac {4\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {10\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 84.60, size = 740, normalized size = 4.81 \[ \begin {cases} \frac {- \frac {2 a^{5} d}{\sqrt {d + e x}} - 2 a^{5} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {10 a^{4} b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {10 a^{4} b \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {20 a^{3} b^{2} d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {20 a^{3} b^{2} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {20 a^{2} b^{3} d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {20 a^{2} b^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {10 a b^{4} d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {10 a b^{4} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} - \frac {2 b^{5} d \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{5}} - \frac {2 b^{5} \left (\frac {d^{6}}{\sqrt {d + e x}} + 6 d^{5} \sqrt {d + e x} - 5 d^{4} \left (d + e x\right )^{\frac {3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac {5}{2}} - \frac {15 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {2 d \left (d + e x\right )^{\frac {9}{2}}}{3} - \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}}}{e} & \text {for}\: e \neq 0 \\\frac {\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{3}}{6 b} & \text {otherwise} \end {cases}}{\sqrt {d}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*a**5*d/sqrt(d + e*x) - 2*a**5*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 10*a**4*b*d*(-d/sqrt(d + e*x
) - sqrt(d + e*x))/e - 10*a**4*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 20*a**3*b**
2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 20*a**3*b**2*(-d**3/sqrt(d + e*x) - 3
*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 20*a**2*b**3*d*(-d**3/sqrt(d + e*x) - 3*
d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 20*a**2*b**3*(d**4/sqrt(d + e*x) + 4*d**3
*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 10*a*b**4*d*(d*
*4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/
7)/e**4 - 10*a*b**4*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x
)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 - 2*b**5*d*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d +
e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e*
*5 - 2*b**5*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 1
5*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5)/e, Ne(e, 0)), (Piecewise((a**
5*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**3/(6*b), True))/sqrt(d), True))

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