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

Optimal. Leaf size=204 \[ \frac {6 c^2 (d+e x)^{11/2} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac {8 c^2 d (d+e x)^{9/2} \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac {6 c (d+e x)^{7/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {12 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^7}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}{3 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7}-\frac {12 c^3 d (d+e x)^{13/2}}{13 e^7} \]

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Rubi [A]  time = 0.08, antiderivative size = 204, 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 {6 c^2 (d+e x)^{11/2} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac {8 c^2 d (d+e x)^{9/2} \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac {6 c (d+e x)^{7/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {12 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^7}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}{3 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7}-\frac {12 c^3 d (d+e x)^{13/2}}{13 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*(a + c*x^2)^3,x]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(3/2))/(3*e^7) - (12*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2))/(5*e^7) + (6*c*(c*d
^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(7/2))/(7*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(9/2))/(9*e^7)
+ (6*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11*e^7) - (12*c^3*d*(d + e*x)^(13/2))/(13*e^7) + (2*c^3*(d + e*
x)^(15/2))/(15*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 \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3 \sqrt {d+e x}}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{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)^{5/2}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{7/2}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{e^6}-\frac {6 c^3 d (d+e x)^{11/2}}{e^6}+\frac {c^3 (d+e x)^{13/2}}{e^6}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{3 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{5 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)^{7/2}}{7 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{9 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac {12 c^3 d (d+e x)^{13/2}}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 170, normalized size = 0.83 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (15015 a^3 e^6+1287 a^2 c e^4 \left (8 d^2-12 d e x+15 e^2 x^2\right )+39 a c^2 e^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+c^3 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(15015*a^3*e^6 + 1287*a^2*c*e^4*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 39*a*c^2*e^2*(128*d^4 - 1
92*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + c^3*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2
- 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6)))/(45045*e^7)

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IntegrateAlgebraic [A]  time = 0.08, size = 240, normalized size = 1.18 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (15015 a^3 e^6+45045 a^2 c d^2 e^4-54054 a^2 c d e^4 (d+e x)+19305 a^2 c e^4 (d+e x)^2+45045 a c^2 d^4 e^2-108108 a c^2 d^3 e^2 (d+e x)+115830 a c^2 d^2 e^2 (d+e x)^2-60060 a c^2 d e^2 (d+e x)^3+12285 a c^2 e^2 (d+e x)^4+15015 c^3 d^6-54054 c^3 d^5 (d+e x)+96525 c^3 d^4 (d+e x)^2-100100 c^3 d^3 (d+e x)^3+61425 c^3 d^2 (d+e x)^4-20790 c^3 d (d+e x)^5+3003 c^3 (d+e x)^6\right )}{45045 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(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 - 54054*c^3*d^5*
(d + e*x) - 108108*a*c^2*d^3*e^2*(d + e*x) - 54054*a^2*c*d*e^4*(d + e*x) + 96525*c^3*d^4*(d + e*x)^2 + 115830*
a*c^2*d^2*e^2*(d + e*x)^2 + 19305*a^2*c*e^4*(d + e*x)^2 - 100100*c^3*d^3*(d + e*x)^3 - 60060*a*c^2*d*e^2*(d +
e*x)^3 + 61425*c^3*d^2*(d + e*x)^4 + 12285*a*c^2*e^2*(d + e*x)^4 - 20790*c^3*d*(d + e*x)^5 + 3003*c^3*(d + e*x
)^6))/(45045*e^7)

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fricas [A]  time = 0.39, size = 252, normalized size = 1.24 \begin {gather*} \frac {2 \, {\left (3003 \, c^{3} e^{7} x^{7} + 231 \, c^{3} d e^{6} x^{6} + 1024 \, c^{3} d^{7} + 4992 \, a c^{2} d^{5} e^{2} + 10296 \, a^{2} c d^{3} e^{4} + 15015 \, a^{3} d e^{6} - 63 \, {\left (4 \, c^{3} d^{2} e^{5} - 195 \, a c^{2} e^{7}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{3} e^{4} + 39 \, a c^{2} d e^{6}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{4} e^{3} + 312 \, a c^{2} d^{2} e^{5} - 3861 \, a^{2} c e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{5} e^{2} + 624 \, a c^{2} d^{3} e^{4} + 1287 \, a^{2} c d e^{6}\right )} x^{2} - {\left (512 \, c^{3} d^{6} e + 2496 \, a c^{2} d^{4} e^{3} + 5148 \, a^{2} c d^{2} e^{5} - 15015 \, a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{45045 \, 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]

2/45045*(3003*c^3*e^7*x^7 + 231*c^3*d*e^6*x^6 + 1024*c^3*d^7 + 4992*a*c^2*d^5*e^2 + 10296*a^2*c*d^3*e^4 + 1501
5*a^3*d*e^6 - 63*(4*c^3*d^2*e^5 - 195*a*c^2*e^7)*x^5 + 35*(8*c^3*d^3*e^4 + 39*a*c^2*d*e^6)*x^4 - 5*(64*c^3*d^4
*e^3 + 312*a*c^2*d^2*e^5 - 3861*a^2*c*e^7)*x^3 + 3*(128*c^3*d^5*e^2 + 624*a*c^2*d^3*e^4 + 1287*a^2*c*d*e^6)*x^
2 - (512*c^3*d^6*e + 2496*a*c^2*d^4*e^3 + 5148*a^2*c*d^2*e^5 - 15015*a^3*e^7)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.24, size = 498, normalized size = 2.44 \begin {gather*} \frac {2}{45045} \, {\left (9009 \, {\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 d e^{\left (-2\right )} + 429 \, {\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} d e^{\left (-4\right )} + 15 \, {\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} d e^{\left (-6\right )} + 3861 \, {\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} c e^{\left (-2\right )} + 195 \, {\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 )} a c^{2} e^{\left (-4\right )} + 7 \, {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} - 3465 \, {\left (x e + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {x e + d} d^{7}\right )} c^{3} e^{\left (-6\right )} + 45045 \, \sqrt {x e + d} a^{3} d + 15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} 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]

2/45045*(9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*d*e^(-2) + 429*(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*e^(-4) + 15*(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*d*e^(-6) + 3
861*(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*e^(-2) +
195*(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*c^2*e^(-4) + 7*(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*e^(-6) + 45045*sqrt(x*e + d)*a^3*d + 15015*((x*e +
d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3)*e^(-1)

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maple [A]  time = 0.06, size = 205, normalized size = 1.00 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 c^{3} x^{6} e^{6}-2772 c^{3} d \,e^{5} x^{5}+12285 a \,c^{2} e^{6} x^{4}+2520 c^{3} d^{2} e^{4} x^{4}-10920 a \,c^{2} d \,e^{5} x^{3}-2240 c^{3} d^{3} e^{3} x^{3}+19305 a^{2} c \,e^{6} x^{2}+9360 a \,c^{2} d^{2} e^{4} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}-15444 a^{2} c d \,e^{5} x -7488 a \,c^{2} d^{3} e^{3} x -1536 c^{3} d^{5} e x +15015 e^{6} a^{3}+10296 a^{2} c \,d^{2} e^{4}+4992 a \,c^{2} d^{4} e^{2}+1024 c^{3} d^{6}\right )}{45045 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]

2/45045*(e*x+d)^(3/2)*(3003*c^3*e^6*x^6-2772*c^3*d*e^5*x^5+12285*a*c^2*e^6*x^4+2520*c^3*d^2*e^4*x^4-10920*a*c^
2*d*e^5*x^3-2240*c^3*d^3*e^3*x^3+19305*a^2*c*e^6*x^2+9360*a*c^2*d^2*e^4*x^2+1920*c^3*d^4*e^2*x^2-15444*a^2*c*d
*e^5*x-7488*a*c^2*d^3*e^3*x-1536*c^3*d^5*e*x+15015*a^3*e^6+10296*a^2*c*d^2*e^4+4992*a*c^2*d^4*e^2+1024*c^3*d^6
)/e^7

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maxima [A]  time = 1.38, size = 209, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} - 20790 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{3} d + 12285 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 20020 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 19305 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 54054 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, 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="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^3 - 20790*(e*x + d)^(13/2)*c^3*d + 12285*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(1
1/2) - 20020*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(e*x + d)^(9/2) + 19305*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*(e*
x + d)^(7/2) - 54054*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(5/2) + 15015*(c^3*d^6 + 3*a*c^2*d^4*
e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(e*x + d)^(3/2))/e^7

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mupad [B]  time = 0.05, size = 187, normalized size = 0.92 \begin {gather*} \frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{7\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}-\frac {12\,c^3\,d\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {12\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,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)^(11/2))/(11*e^7) + ((d + e*x)^(7/2)*(30*c^3*d^4 + 6*a^2*c*e^4 + 36*a*c^2
*d^2*e^2))/(7*e^7) + (2*c^3*(d + e*x)^(15/2))/(15*e^7) + (2*(a*e^2 + c*d^2)^3*(d + e*x)^(3/2))/(3*e^7) - ((40*
c^3*d^3 + 24*a*c^2*d*e^2)*(d + e*x)^(9/2))/(9*e^7) - (12*c^3*d*(d + e*x)^(13/2))/(13*e^7) - (12*c*d*(a*e^2 + c
*d^2)^2*(d + e*x)^(5/2))/(5*e^7)

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sympy [A]  time = 4.50, size = 265, normalized size = 1.30 \begin {gather*} \frac {2 \left (- \frac {6 c^{3} d \left (d + e x\right )^{\frac {13}{2}}}{13 e^{6}} + \frac {c^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 6 a^{2} c d e^{4} - 12 a c^{2} d^{3} e^{2} - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )}{3 e^{6}}\right )}{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)

[Out]

2*(-6*c**3*d*(d + e*x)**(13/2)/(13*e**6) + c**3*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(11/2)*(3*a*c**2*e**2
+ 15*c**3*d**2)/(11*e**6) + (d + e*x)**(9/2)*(-12*a*c**2*d*e**2 - 20*c**3*d**3)/(9*e**6) + (d + e*x)**(7/2)*(
3*a**2*c*e**4 + 18*a*c**2*d**2*e**2 + 15*c**3*d**4)/(7*e**6) + (d + e*x)**(5/2)*(-6*a**2*c*d*e**4 - 12*a*c**2*
d**3*e**2 - 6*c**3*d**5)/(5*e**6) + (d + e*x)**(3/2)*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c*
*3*d**6)/(3*e**6))/e

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