3.6.28 \(\int \frac {(a+c x^2)^3}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=200 \[ \frac {2 c^2 (d+e x)^{9/2} \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {8 c^2 d (d+e x)^{7/2} \left (3 a e^2+5 c d^2\right )}{7 e^7}+\frac {6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7}-\frac {4 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}{e^7}+\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )^3}{e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7}-\frac {12 c^3 d (d+e x)^{11/2}}{11 e^7} \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {697} \begin {gather*} \frac {2 c^2 (d+e x)^{9/2} \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {8 c^2 d (d+e x)^{7/2} \left (3 a e^2+5 c d^2\right )}{7 e^7}+\frac {6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7}-\frac {4 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}{e^7}+\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )^3}{e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7}-\frac {12 c^3 d (d+e x)^{11/2}}{11 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/Sqrt[d + e*x],x]

[Out]

(2*(c*d^2 + a*e^2)^3*Sqrt[d + e*x])/e^7 - (4*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2))/e^7 + (6*c*(c*d^2 + a*e^2)
*(5*c*d^2 + a*e^2)*(d + e*x)^(5/2))/(5*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(7/2))/(7*e^7) + (2*c^2*(
5*c*d^2 + a*e^2)*(d + e*x)^(9/2))/(3*e^7) - (12*c^3*d*(d + e*x)^(11/2))/(11*e^7) + (2*c^3*(d + e*x)^(13/2))/(1
3*e^7)

Rule 697

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

Rubi steps

\begin {align*} \int \frac {\left (a+c x^2\right )^3}{\sqrt {d+e x}} \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3}{e^6 \sqrt {d+e x}}-\frac {6 c d \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{5/2}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{e^6}-\frac {6 c^3 d (d+e x)^{9/2}}{e^6}+\frac {c^3 (d+e x)^{11/2}}{e^6}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}{e^7}-\frac {4 c d \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^7}+\frac {6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{7/2}}{7 e^7}+\frac {2 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {12 c^3 d (d+e x)^{11/2}}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 171, normalized size = 0.86 \begin {gather*} \frac {2 \sqrt {d+e x} \left (15015 a^3 e^6+3003 a^2 c e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+143 a c^2 e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{15015 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(15015*a^3*e^6 + 3003*a^2*c*e^4*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 143*a*c^2*e^2*(128*d^4 - 64*d
^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d
^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6)))/(15015*e^7)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.08, size = 240, normalized size = 1.20 \begin {gather*} \frac {2 \sqrt {d+e x} \left (15015 a^3 e^6+45045 a^2 c d^2 e^4-30030 a^2 c d e^4 (d+e x)+9009 a^2 c e^4 (d+e x)^2+45045 a c^2 d^4 e^2-60060 a c^2 d^3 e^2 (d+e x)+54054 a c^2 d^2 e^2 (d+e x)^2-25740 a c^2 d e^2 (d+e x)^3+5005 a c^2 e^2 (d+e x)^4+15015 c^3 d^6-30030 c^3 d^5 (d+e x)+45045 c^3 d^4 (d+e x)^2-42900 c^3 d^3 (d+e x)^3+25025 c^3 d^2 (d+e x)^4-8190 c^3 d (d+e x)^5+1155 c^3 (d+e x)^6\right )}{15015 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + c*x^2)^3/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(15015*c^3*d^6 + 45045*a*c^2*d^4*e^2 + 45045*a^2*c*d^2*e^4 + 15015*a^3*e^6 - 30030*c^3*d^5*(d
+ e*x) - 60060*a*c^2*d^3*e^2*(d + e*x) - 30030*a^2*c*d*e^4*(d + e*x) + 45045*c^3*d^4*(d + e*x)^2 + 54054*a*c^
2*d^2*e^2*(d + e*x)^2 + 9009*a^2*c*e^4*(d + e*x)^2 - 42900*c^3*d^3*(d + e*x)^3 - 25740*a*c^2*d*e^2*(d + e*x)^3
+ 25025*c^3*d^2*(d + e*x)^4 + 5005*a*c^2*e^2*(d + e*x)^4 - 8190*c^3*d*(d + e*x)^5 + 1155*c^3*(d + e*x)^6))/(1
5015*e^7)

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 202, normalized size = 1.01 \begin {gather*} \frac {2 \, {\left (1155 \, c^{3} e^{6} x^{6} - 1260 \, c^{3} d e^{5} x^{5} + 5120 \, c^{3} d^{6} + 18304 \, a c^{2} d^{4} e^{2} + 24024 \, a^{2} c d^{2} e^{4} + 15015 \, a^{3} e^{6} + 35 \, {\left (40 \, c^{3} d^{2} e^{4} + 143 \, a c^{2} e^{6}\right )} x^{4} - 40 \, {\left (40 \, c^{3} d^{3} e^{3} + 143 \, a c^{2} d e^{5}\right )} x^{3} + 3 \, {\left (640 \, c^{3} d^{4} e^{2} + 2288 \, a c^{2} d^{2} e^{4} + 3003 \, a^{2} c e^{6}\right )} x^{2} - 4 \, {\left (640 \, c^{3} d^{5} e + 2288 \, a c^{2} d^{3} e^{3} + 3003 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*c^3*e^6*x^6 - 1260*c^3*d*e^5*x^5 + 5120*c^3*d^6 + 18304*a*c^2*d^4*e^2 + 24024*a^2*c*d^2*e^4 + 15
015*a^3*e^6 + 35*(40*c^3*d^2*e^4 + 143*a*c^2*e^6)*x^4 - 40*(40*c^3*d^3*e^3 + 143*a*c^2*d*e^5)*x^3 + 3*(640*c^3
*d^4*e^2 + 2288*a*c^2*d^2*e^4 + 3003*a^2*c*e^6)*x^2 - 4*(640*c^3*d^5*e + 2288*a*c^2*d^3*e^3 + 3003*a^2*c*d*e^5
)*x)*sqrt(e*x + d)/e^7

________________________________________________________________________________________

giac [A]  time = 0.17, size = 224, normalized size = 1.12 \begin {gather*} \frac {2}{15015} \, {\left (3003 \, {\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^{2} c e^{\left (-2\right )} + 143 \, {\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 c^{2} e^{\left (-4\right )} + 5 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c^{3} e^{\left (-6\right )} + 15015 \, \sqrt {x e + d} a^{3}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*e^(-2) + 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)*
a*c^2*e^(-4) + 5*(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^3*e^(-6) + 15015*sq
rt(x*e + d)*a^3)*e^(-1)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 205, normalized size = 1.02 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (1155 c^{3} x^{6} e^{6}-1260 c^{3} d \,e^{5} x^{5}+5005 a \,c^{2} e^{6} x^{4}+1400 c^{3} d^{2} e^{4} x^{4}-5720 a \,c^{2} d \,e^{5} x^{3}-1600 c^{3} d^{3} e^{3} x^{3}+9009 a^{2} c \,e^{6} x^{2}+6864 a \,c^{2} d^{2} e^{4} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}-12012 a^{2} c d \,e^{5} x -9152 a \,c^{2} d^{3} e^{3} x -2560 c^{3} d^{5} e x +15015 e^{6} a^{3}+24024 a^{2} c \,d^{2} e^{4}+18304 a \,c^{2} d^{4} e^{2}+5120 c^{3} d^{6}\right )}{15015 e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(e*x+d)^(1/2)*(1155*c^3*e^6*x^6-1260*c^3*d*e^5*x^5+5005*a*c^2*e^6*x^4+1400*c^3*d^2*e^4*x^4-5720*a*c^2*
d*e^5*x^3-1600*c^3*d^3*e^3*x^3+9009*a^2*c*e^6*x^2+6864*a*c^2*d^2*e^4*x^2+1920*c^3*d^4*e^2*x^2-12012*a^2*c*d*e^
5*x-9152*a*c^2*d^3*e^3*x-2560*c^3*d^5*e*x+15015*a^3*e^6+24024*a^2*c*d^2*e^4+18304*a*c^2*d^4*e^2+5120*c^3*d^6)/
e^7

________________________________________________________________________________________

maxima [A]  time = 1.38, size = 212, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{3} + \frac {3003 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} c}{e^{2}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a c^{2}}{e^{4}} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(15015*sqrt(e*x + d)*a^3 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*
c/e^2 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 +
315*sqrt(e*x + d)*d^4)*a*c^2/e^4 + 5*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^
2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c
^3/e^6)/e

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 187, normalized size = 0.94 \begin {gather*} \frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{5\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,\sqrt {d+e\,x}}{e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {12\,c^3\,d\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {4\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3/(d + e*x)^(1/2),x)

[Out]

((30*c^3*d^2 + 6*a*c^2*e^2)*(d + e*x)^(9/2))/(9*e^7) + ((d + e*x)^(5/2)*(30*c^3*d^4 + 6*a^2*c*e^4 + 36*a*c^2*d
^2*e^2))/(5*e^7) + (2*c^3*(d + e*x)^(13/2))/(13*e^7) + (2*(a*e^2 + c*d^2)^3*(d + e*x)^(1/2))/e^7 - ((40*c^3*d^
3 + 24*a*c^2*d*e^2)*(d + e*x)^(7/2))/(7*e^7) - (12*c^3*d*(d + e*x)^(11/2))/(11*e^7) - (4*c*d*(a*e^2 + c*d^2)^2
*(d + e*x)^(3/2))/e^7

________________________________________________________________________________________

sympy [A]  time = 60.38, size = 563, normalized size = 2.82 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{3} d}{\sqrt {d + e x}} - 2 a^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {6 a^{2} c 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 {6 a^{2} c \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 {6 a c^{2} 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 {6 a c^{2} \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 c^{3} d \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^{6}} - \frac {2 c^{3} \left (- \frac {d^{7}}{\sqrt {d + e x}} - 7 d^{6} \sqrt {d + e x} + 7 d^{5} \left (d + e x\right )^{\frac {3}{2}} - 7 d^{4} \left (d + e x\right )^{\frac {5}{2}} + 5 d^{3} \left (d + e x\right )^{\frac {7}{2}} - \frac {7 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{3} + \frac {7 d \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{6}}}{e} & \text {for}\: e \neq 0 \\\frac {a^{3} x + a^{2} c x^{3} + \frac {3 a c^{2} x^{5}}{5} + \frac {c^{3} x^{7}}{7}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*a**3*d/sqrt(d + e*x) - 2*a**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 6*a**2*c*d*(d**2/sqrt(d + e*
x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 6*a**2*c*(-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 - 6*a*c**2*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 - 6*a*c**2*(-d**5/sqrt(d + e*x) - 5*d**4*sq
rt(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*c**3*d*(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) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**6 - 2*c**3*(-d**7/sqrt(d
+ e*x) - 7*d**6*sqrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/2) -
7*d**2*(d + e*x)**(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**6)/e, Ne(e, 0)), ((a**3*x + a
**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7)/sqrt(d), True))

________________________________________________________________________________________