3.2036 \(\int (d+e x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx\)

Optimal. Leaf size=295 \[ \frac {256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac {32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt {d+e x}}+\frac {16 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d} \]

[Out]

256/15015*(-a*e^2+c*d^2)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^5/d^5/(e*x+d)^(5/2)+128/3003*(-a*e^2+c*d^
2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^4/d^4/(e*x+d)^(3/2)+2/13*(e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(5/2)/c/d+32/429*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3/(e*x+d)^(1/2)+16/1
43*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*(e*x+d)^(1/2)/c^2/d^2

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Rubi [A]  time = 0.30, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac {256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac {32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt {d+e x}}+\frac {16 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(256*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(15015*c^5*d^5*(d + e*x)^(5/2)) + (128*(
c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3003*c^4*d^4*(d + e*x)^(3/2)) + (32*(c*d^2 -
a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(429*c^3*d^3*Sqrt[d + e*x]) + (16*(c*d^2 - a*e^2)*Sqrt
[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(143*c^2*d^2) + (2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d)

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

\begin {align*} \int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx &=\frac {2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{13 d}\\ &=\frac {16 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d}+\frac {\left (48 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{143 d^2}\\ &=\frac {32 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt {d+e x}}+\frac {16 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d}+\frac {\left (64 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{429 d^3}\\ &=\frac {128 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac {32 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt {d+e x}}+\frac {16 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d}+\frac {\left (128 \left (d^2-\frac {a e^2}{c}\right )^4\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3003 d^4}\\ &=\frac {256 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac {32 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt {d+e x}}+\frac {16 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 187, normalized size = 0.63 \[ \frac {2 ((d+e x) (a e+c d x))^{5/2} \left (128 a^4 e^8-64 a^3 c d e^6 (13 d+5 e x)+16 a^2 c^2 d^2 e^4 \left (143 d^2+130 d e x+35 e^2 x^2\right )-8 a c^3 d^3 e^2 \left (429 d^3+715 d^2 e x+455 d e^2 x^2+105 e^3 x^3\right )+c^4 d^4 \left (3003 d^4+8580 d^3 e x+10010 d^2 e^2 x^2+5460 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 c^5 d^5 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(128*a^4*e^8 - 64*a^3*c*d*e^6*(13*d + 5*e*x) + 16*a^2*c^2*d^2*e^4*(143*d^2
+ 130*d*e*x + 35*e^2*x^2) - 8*a*c^3*d^3*e^2*(429*d^3 + 715*d^2*e*x + 455*d*e^2*x^2 + 105*e^3*x^3) + c^4*d^4*(3
003*d^4 + 8580*d^3*e*x + 10010*d^2*e^2*x^2 + 5460*d*e^3*x^3 + 1155*e^4*x^4)))/(15015*c^5*d^5*(d + e*x)^(5/2))

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fricas [A]  time = 0.70, size = 392, normalized size = 1.33 \[ \frac {2 \, {\left (1155 \, c^{6} d^{6} e^{4} x^{6} + 3003 \, a^{2} c^{4} d^{8} e^{2} - 3432 \, a^{3} c^{3} d^{6} e^{4} + 2288 \, a^{4} c^{2} d^{4} e^{6} - 832 \, a^{5} c d^{2} e^{8} + 128 \, a^{6} e^{10} + 210 \, {\left (26 \, c^{6} d^{7} e^{3} + 7 \, a c^{5} d^{5} e^{5}\right )} x^{5} + 35 \, {\left (286 \, c^{6} d^{8} e^{2} + 208 \, a c^{5} d^{6} e^{4} + a^{2} c^{4} d^{4} e^{6}\right )} x^{4} + 20 \, {\left (429 \, c^{6} d^{9} e + 715 \, a c^{5} d^{7} e^{3} + 13 \, a^{2} c^{4} d^{5} e^{5} - 2 \, a^{3} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \, {\left (1001 \, c^{6} d^{10} + 4576 \, a c^{5} d^{8} e^{2} + 286 \, a^{2} c^{4} d^{6} e^{4} - 104 \, a^{3} c^{3} d^{4} e^{6} + 16 \, a^{4} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \, {\left (3003 \, a c^{5} d^{9} e + 858 \, a^{2} c^{4} d^{7} e^{3} - 572 \, a^{3} c^{3} d^{5} e^{5} + 208 \, a^{4} c^{2} d^{3} e^{7} - 32 \, a^{5} c d e^{9}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{15015 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*c^6*d^6*e^4*x^6 + 3003*a^2*c^4*d^8*e^2 - 3432*a^3*c^3*d^6*e^4 + 2288*a^4*c^2*d^4*e^6 - 832*a^5*c
*d^2*e^8 + 128*a^6*e^10 + 210*(26*c^6*d^7*e^3 + 7*a*c^5*d^5*e^5)*x^5 + 35*(286*c^6*d^8*e^2 + 208*a*c^5*d^6*e^4
+ a^2*c^4*d^4*e^6)*x^4 + 20*(429*c^6*d^9*e + 715*a*c^5*d^7*e^3 + 13*a^2*c^4*d^5*e^5 - 2*a^3*c^3*d^3*e^7)*x^3
+ 3*(1001*c^6*d^10 + 4576*a*c^5*d^8*e^2 + 286*a^2*c^4*d^6*e^4 - 104*a^3*c^3*d^4*e^6 + 16*a^4*c^2*d^2*e^8)*x^2
+ 2*(3003*a*c^5*d^9*e + 858*a^2*c^4*d^7*e^3 - 572*a^3*c^3*d^5*e^5 + 208*a^4*c^2*d^3*e^7 - 32*a^5*c*d*e^9)*x)*s
qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^5*d^5*e*x + c^5*d^6)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 243, normalized size = 0.82 \[ \frac {2 \left (c d x +a e \right ) \left (1155 c^{4} d^{4} e^{4} x^{4}-840 a \,c^{3} d^{3} e^{5} x^{3}+5460 c^{4} d^{5} e^{3} x^{3}+560 a^{2} c^{2} d^{2} e^{6} x^{2}-3640 a \,c^{3} d^{4} e^{4} x^{2}+10010 c^{4} d^{6} e^{2} x^{2}-320 a^{3} c d \,e^{7} x +2080 a^{2} c^{2} d^{3} e^{5} x -5720 a \,c^{3} d^{5} e^{3} x +8580 c^{4} d^{7} e x +128 a^{4} e^{8}-832 a^{3} c \,d^{2} e^{6}+2288 a^{2} c^{2} d^{4} e^{4}-3432 a \,c^{3} d^{6} e^{2}+3003 c^{4} d^{8}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{15015 \left (e x +d \right )^{\frac {3}{2}} c^{5} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(c*d*x+a*e)*(1155*c^4*d^4*e^4*x^4-840*a*c^3*d^3*e^5*x^3+5460*c^4*d^5*e^3*x^3+560*a^2*c^2*d^2*e^6*x^2-3
640*a*c^3*d^4*e^4*x^2+10010*c^4*d^6*e^2*x^2-320*a^3*c*d*e^7*x+2080*a^2*c^2*d^3*e^5*x-5720*a*c^3*d^5*e^3*x+8580
*c^4*d^7*e*x+128*a^4*e^8-832*a^3*c*d^2*e^6+2288*a^2*c^2*d^4*e^4-3432*a*c^3*d^6*e^2+3003*c^4*d^8)*(c*d*e*x^2+a*
e^2*x+c*d^2*x+a*d*e)^(3/2)/c^5/d^5/(e*x+d)^(3/2)

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maxima [A]  time = 1.52, size = 373, normalized size = 1.26 \[ \frac {2 \, {\left (1155 \, c^{6} d^{6} e^{4} x^{6} + 3003 \, a^{2} c^{4} d^{8} e^{2} - 3432 \, a^{3} c^{3} d^{6} e^{4} + 2288 \, a^{4} c^{2} d^{4} e^{6} - 832 \, a^{5} c d^{2} e^{8} + 128 \, a^{6} e^{10} + 210 \, {\left (26 \, c^{6} d^{7} e^{3} + 7 \, a c^{5} d^{5} e^{5}\right )} x^{5} + 35 \, {\left (286 \, c^{6} d^{8} e^{2} + 208 \, a c^{5} d^{6} e^{4} + a^{2} c^{4} d^{4} e^{6}\right )} x^{4} + 20 \, {\left (429 \, c^{6} d^{9} e + 715 \, a c^{5} d^{7} e^{3} + 13 \, a^{2} c^{4} d^{5} e^{5} - 2 \, a^{3} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \, {\left (1001 \, c^{6} d^{10} + 4576 \, a c^{5} d^{8} e^{2} + 286 \, a^{2} c^{4} d^{6} e^{4} - 104 \, a^{3} c^{3} d^{4} e^{6} + 16 \, a^{4} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \, {\left (3003 \, a c^{5} d^{9} e + 858 \, a^{2} c^{4} d^{7} e^{3} - 572 \, a^{3} c^{3} d^{5} e^{5} + 208 \, a^{4} c^{2} d^{3} e^{7} - 32 \, a^{5} c d e^{9}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{15015 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*c^6*d^6*e^4*x^6 + 3003*a^2*c^4*d^8*e^2 - 3432*a^3*c^3*d^6*e^4 + 2288*a^4*c^2*d^4*e^6 - 832*a^5*c
*d^2*e^8 + 128*a^6*e^10 + 210*(26*c^6*d^7*e^3 + 7*a*c^5*d^5*e^5)*x^5 + 35*(286*c^6*d^8*e^2 + 208*a*c^5*d^6*e^4
+ a^2*c^4*d^4*e^6)*x^4 + 20*(429*c^6*d^9*e + 715*a*c^5*d^7*e^3 + 13*a^2*c^4*d^5*e^5 - 2*a^3*c^3*d^3*e^7)*x^3
+ 3*(1001*c^6*d^10 + 4576*a*c^5*d^8*e^2 + 286*a^2*c^4*d^6*e^4 - 104*a^3*c^3*d^4*e^6 + 16*a^4*c^2*d^2*e^8)*x^2
+ 2*(3003*a*c^5*d^9*e + 858*a^2*c^4*d^7*e^3 - 572*a^3*c^3*d^5*e^5 + 208*a^4*c^2*d^3*e^7 - 32*a^5*c*d*e^9)*x)*s
qrt(c*d*x + a*e)*(e*x + d)/(c^5*d^5*e*x + c^5*d^6)

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mupad [B]  time = 1.35, size = 424, normalized size = 1.44 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,e^2\,x^5\,\left (26\,c\,d^2+7\,a\,e^2\right )\,\sqrt {d+e\,x}}{143}+\frac {2\,c\,d\,e^3\,x^6\,\sqrt {d+e\,x}}{13}+\frac {8\,x^3\,\sqrt {d+e\,x}\,\left (-2\,a^3\,e^6+13\,a^2\,c\,d^2\,e^4+715\,a\,c^2\,d^4\,e^2+429\,c^3\,d^6\right )}{3003\,c^2\,d^2}+\frac {\sqrt {d+e\,x}\,\left (256\,a^6\,e^{10}-1664\,a^5\,c\,d^2\,e^8+4576\,a^4\,c^2\,d^4\,e^6-6864\,a^3\,c^3\,d^6\,e^4+6006\,a^2\,c^4\,d^8\,e^2\right )}{15015\,c^5\,d^5\,e}+\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (a^2\,e^4+208\,a\,c\,d^2\,e^2+286\,c^2\,d^4\right )}{429\,c\,d}+\frac {x^2\,\sqrt {d+e\,x}\,\left (96\,a^4\,c^2\,d^2\,e^8-624\,a^3\,c^3\,d^4\,e^6+1716\,a^2\,c^4\,d^6\,e^4+27456\,a\,c^5\,d^8\,e^2+6006\,c^6\,d^{10}\right )}{15015\,c^5\,d^5\,e}+\frac {4\,a\,x\,\sqrt {d+e\,x}\,\left (-32\,a^4\,e^8+208\,a^3\,c\,d^2\,e^6-572\,a^2\,c^2\,d^4\,e^4+858\,a\,c^3\,d^6\,e^2+3003\,c^4\,d^8\right )}{15015\,c^4\,d^4}\right )}{x+\frac {d}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((4*e^2*x^5*(7*a*e^2 + 26*c*d^2)*(d + e*x)^(1/2))/143 + (2*c*d*
e^3*x^6*(d + e*x)^(1/2))/13 + (8*x^3*(d + e*x)^(1/2)*(429*c^3*d^6 - 2*a^3*e^6 + 715*a*c^2*d^4*e^2 + 13*a^2*c*d
^2*e^4))/(3003*c^2*d^2) + ((d + e*x)^(1/2)*(256*a^6*e^10 - 1664*a^5*c*d^2*e^8 + 6006*a^2*c^4*d^8*e^2 - 6864*a^
3*c^3*d^6*e^4 + 4576*a^4*c^2*d^4*e^6))/(15015*c^5*d^5*e) + (2*e*x^4*(d + e*x)^(1/2)*(a^2*e^4 + 286*c^2*d^4 + 2
08*a*c*d^2*e^2))/(429*c*d) + (x^2*(d + e*x)^(1/2)*(6006*c^6*d^10 + 27456*a*c^5*d^8*e^2 + 1716*a^2*c^4*d^6*e^4
- 624*a^3*c^3*d^4*e^6 + 96*a^4*c^2*d^2*e^8))/(15015*c^5*d^5*e) + (4*a*x*(d + e*x)^(1/2)*(3003*c^4*d^8 - 32*a^4
*e^8 + 858*a*c^3*d^6*e^2 + 208*a^3*c*d^2*e^6 - 572*a^2*c^2*d^4*e^4))/(15015*c^4*d^4)))/(x + d/e)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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