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

Optimal. Leaf size=295 \[ \frac {256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{45045 c^5 d^5 (d+e x)^{7/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\frac {32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{195 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d} \]

________________________________________________________________________________________

Rubi [A]  time = 0.27, 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} \begin {gather*} \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 )^{7/2}}{45045 c^5 d^5 (d+e x)^{7/2}}+\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 )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\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 )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac {16 \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 )^{7/2}}{195 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),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)^(7/2))/(45045*c^5*d^5*(d + e*x)^(7/2)) + (128*(
c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(6435*c^4*d^4*(d + e*x)^(5/2)) + (32*(c*d^2 -
a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(715*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d^2 - a*e^2)*(a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(195*c^2*d^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(15*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)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )\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 )^{5/2} \, dx}{15 d}\\ &=\frac {16 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx}{65 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 )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 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 )^{5/2}}{(d+e x)^{3/2}} \, dx}{715 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 )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\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 )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 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 )^{5/2}}{(d+e x)^{5/2}} \, dx}{6435 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 )^{7/2}}{45045 c^5 d^5 (d+e x)^{7/2}}+\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 )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\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 )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 197, normalized size = 0.67 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (128 a^4 e^8-64 a^3 c d e^6 (15 d+7 e x)+48 a^2 c^2 d^2 e^4 \left (65 d^2+70 d e x+21 e^2 x^2\right )-8 a c^3 d^3 e^2 \left (715 d^3+1365 d^2 e x+945 d e^2 x^2+231 e^3 x^3\right )+c^4 d^4 \left (6435 d^4+20020 d^3 e x+24570 d^2 e^2 x^2+13860 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 c^5 d^5 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(128*a^4*e^8 - 64*a^3*c*d*e^6*(15*d + 7*e*x) + 48*a^2*c^2*d^2
*e^4*(65*d^2 + 70*d*e*x + 21*e^2*x^2) - 8*a*c^3*d^3*e^2*(715*d^3 + 1365*d^2*e*x + 945*d*e^2*x^2 + 231*e^3*x^3)
+ c^4*d^4*(6435*d^4 + 20020*d^3*e*x + 24570*d^2*e^2*x^2 + 13860*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*c^5*d^5*Sq
rt[d + e*x])

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 467, normalized size = 1.58 \begin {gather*} \frac {2 \, {\left (3003 \, c^{7} d^{7} e^{4} x^{7} + 6435 \, a^{3} c^{4} d^{8} e^{3} - 5720 \, a^{4} c^{3} d^{6} e^{5} + 3120 \, a^{5} c^{2} d^{4} e^{7} - 960 \, a^{6} c d^{2} e^{9} + 128 \, a^{7} e^{11} + 231 \, {\left (60 \, c^{7} d^{8} e^{3} + 31 \, a c^{6} d^{6} e^{5}\right )} x^{6} + 63 \, {\left (390 \, c^{7} d^{9} e^{2} + 540 \, a c^{6} d^{7} e^{4} + 71 \, a^{2} c^{5} d^{5} e^{6}\right )} x^{5} + 35 \, {\left (572 \, c^{7} d^{10} e + 1794 \, a c^{6} d^{8} e^{3} + 636 \, a^{2} c^{5} d^{6} e^{5} + a^{3} c^{4} d^{4} e^{7}\right )} x^{4} + 5 \, {\left (1287 \, c^{7} d^{11} + 10868 \, a c^{6} d^{9} e^{2} + 8814 \, a^{2} c^{5} d^{7} e^{4} + 60 \, a^{3} c^{4} d^{5} e^{6} - 8 \, a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \, {\left (6435 \, a c^{6} d^{10} e + 14300 \, a^{2} c^{5} d^{8} e^{3} + 390 \, a^{3} c^{4} d^{6} e^{5} - 120 \, a^{4} c^{3} d^{4} e^{7} + 16 \, a^{5} c^{2} d^{2} e^{9}\right )} x^{2} + {\left (19305 \, a^{2} c^{5} d^{9} e^{2} + 2860 \, a^{3} c^{4} d^{7} e^{4} - 1560 \, a^{4} c^{3} d^{5} e^{6} + 480 \, a^{5} c^{2} d^{3} e^{8} - 64 \, a^{6} c d e^{10}\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}}{45045 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*c^7*d^7*e^4*x^7 + 6435*a^3*c^4*d^8*e^3 - 5720*a^4*c^3*d^6*e^5 + 3120*a^5*c^2*d^4*e^7 - 960*a^6*c
*d^2*e^9 + 128*a^7*e^11 + 231*(60*c^7*d^8*e^3 + 31*a*c^6*d^6*e^5)*x^6 + 63*(390*c^7*d^9*e^2 + 540*a*c^6*d^7*e^
4 + 71*a^2*c^5*d^5*e^6)*x^5 + 35*(572*c^7*d^10*e + 1794*a*c^6*d^8*e^3 + 636*a^2*c^5*d^6*e^5 + a^3*c^4*d^4*e^7)
*x^4 + 5*(1287*c^7*d^11 + 10868*a*c^6*d^9*e^2 + 8814*a^2*c^5*d^7*e^4 + 60*a^3*c^4*d^5*e^6 - 8*a^4*c^3*d^3*e^8)
*x^3 + 3*(6435*a*c^6*d^10*e + 14300*a^2*c^5*d^8*e^3 + 390*a^3*c^4*d^6*e^5 - 120*a^4*c^3*d^4*e^7 + 16*a^5*c^2*d
^2*e^9)*x^2 + (19305*a^2*c^5*d^9*e^2 + 2860*a^3*c^4*d^7*e^4 - 1560*a^4*c^3*d^5*e^6 + 480*a^5*c^2*d^3*e^8 - 64*
a^6*c*d*e^10)*x)*sqrt(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)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.06, size = 243, normalized size = 0.82 \begin {gather*} \frac {2 \left (c d x +a e \right ) \left (3003 c^{4} d^{4} e^{4} x^{4}-1848 a \,c^{3} d^{3} e^{5} x^{3}+13860 c^{4} d^{5} e^{3} x^{3}+1008 a^{2} c^{2} d^{2} e^{6} x^{2}-7560 a \,c^{3} d^{4} e^{4} x^{2}+24570 c^{4} d^{6} e^{2} x^{2}-448 a^{3} c d \,e^{7} x +3360 a^{2} c^{2} d^{3} e^{5} x -10920 a \,c^{3} d^{5} e^{3} x +20020 c^{4} d^{7} e x +128 a^{4} e^{8}-960 a^{3} c \,d^{2} e^{6}+3120 a^{2} c^{2} d^{4} e^{4}-5720 a \,c^{3} d^{6} e^{2}+6435 c^{4} d^{8}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{45045 \left (e x +d \right )^{\frac {5}{2}} c^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(c*d*x+a*e)*(3003*c^4*d^4*e^4*x^4-1848*a*c^3*d^3*e^5*x^3+13860*c^4*d^5*e^3*x^3+1008*a^2*c^2*d^2*e^6*x^
2-7560*a*c^3*d^4*e^4*x^2+24570*c^4*d^6*e^2*x^2-448*a^3*c*d*e^7*x+3360*a^2*c^2*d^3*e^5*x-10920*a*c^3*d^5*e^3*x+
20020*c^4*d^7*e*x+128*a^4*e^8-960*a^3*c*d^2*e^6+3120*a^2*c^2*d^4*e^4-5720*a*c^3*d^6*e^2+6435*c^4*d^8)*(c*d*e*x
^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/c^5/d^5/(e*x+d)^(5/2)

________________________________________________________________________________________

maxima [A]  time = 1.45, size = 448, normalized size = 1.52 \begin {gather*} \frac {2 \, {\left (3003 \, c^{7} d^{7} e^{4} x^{7} + 6435 \, a^{3} c^{4} d^{8} e^{3} - 5720 \, a^{4} c^{3} d^{6} e^{5} + 3120 \, a^{5} c^{2} d^{4} e^{7} - 960 \, a^{6} c d^{2} e^{9} + 128 \, a^{7} e^{11} + 231 \, {\left (60 \, c^{7} d^{8} e^{3} + 31 \, a c^{6} d^{6} e^{5}\right )} x^{6} + 63 \, {\left (390 \, c^{7} d^{9} e^{2} + 540 \, a c^{6} d^{7} e^{4} + 71 \, a^{2} c^{5} d^{5} e^{6}\right )} x^{5} + 35 \, {\left (572 \, c^{7} d^{10} e + 1794 \, a c^{6} d^{8} e^{3} + 636 \, a^{2} c^{5} d^{6} e^{5} + a^{3} c^{4} d^{4} e^{7}\right )} x^{4} + 5 \, {\left (1287 \, c^{7} d^{11} + 10868 \, a c^{6} d^{9} e^{2} + 8814 \, a^{2} c^{5} d^{7} e^{4} + 60 \, a^{3} c^{4} d^{5} e^{6} - 8 \, a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \, {\left (6435 \, a c^{6} d^{10} e + 14300 \, a^{2} c^{5} d^{8} e^{3} + 390 \, a^{3} c^{4} d^{6} e^{5} - 120 \, a^{4} c^{3} d^{4} e^{7} + 16 \, a^{5} c^{2} d^{2} e^{9}\right )} x^{2} + {\left (19305 \, a^{2} c^{5} d^{9} e^{2} + 2860 \, a^{3} c^{4} d^{7} e^{4} - 1560 \, a^{4} c^{3} d^{5} e^{6} + 480 \, a^{5} c^{2} d^{3} e^{8} - 64 \, a^{6} c d e^{10}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{45045 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*c^7*d^7*e^4*x^7 + 6435*a^3*c^4*d^8*e^3 - 5720*a^4*c^3*d^6*e^5 + 3120*a^5*c^2*d^4*e^7 - 960*a^6*c
*d^2*e^9 + 128*a^7*e^11 + 231*(60*c^7*d^8*e^3 + 31*a*c^6*d^6*e^5)*x^6 + 63*(390*c^7*d^9*e^2 + 540*a*c^6*d^7*e^
4 + 71*a^2*c^5*d^5*e^6)*x^5 + 35*(572*c^7*d^10*e + 1794*a*c^6*d^8*e^3 + 636*a^2*c^5*d^6*e^5 + a^3*c^4*d^4*e^7)
*x^4 + 5*(1287*c^7*d^11 + 10868*a*c^6*d^9*e^2 + 8814*a^2*c^5*d^7*e^4 + 60*a^3*c^4*d^5*e^6 - 8*a^4*c^3*d^3*e^8)
*x^3 + 3*(6435*a*c^6*d^10*e + 14300*a^2*c^5*d^8*e^3 + 390*a^3*c^4*d^6*e^5 - 120*a^4*c^3*d^4*e^7 + 16*a^5*c^2*d
^2*e^9)*x^2 + (19305*a^2*c^5*d^9*e^2 + 2860*a^3*c^4*d^7*e^4 - 1560*a^4*c^3*d^5*e^6 + 480*a^5*c^2*d^3*e^8 - 64*
a^6*c*d*e^10)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^5*d^5*e*x + c^5*d^6)

________________________________________________________________________________________

mupad [B]  time = 1.56, size = 501, normalized size = 1.70 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e\,x^5\,\sqrt {d+e\,x}\,\left (71\,a^2\,e^4+540\,a\,c\,d^2\,e^2+390\,c^2\,d^4\right )}{715}+\frac {2\,x^4\,\sqrt {d+e\,x}\,\left (a^3\,e^6+636\,a^2\,c\,d^2\,e^4+1794\,a\,c^2\,d^4\,e^2+572\,c^3\,d^6\right )}{1287\,c\,d}+\frac {\sqrt {d+e\,x}\,\left (256\,a^7\,e^{11}-1920\,a^6\,c\,d^2\,e^9+6240\,a^5\,c^2\,d^4\,e^7-11440\,a^4\,c^3\,d^6\,e^5+12870\,a^3\,c^4\,d^8\,e^3\right )}{45045\,c^5\,d^5\,e}+\frac {2\,c^2\,d^2\,e^3\,x^7\,\sqrt {d+e\,x}}{15}+\frac {2\,a\,x^2\,\sqrt {d+e\,x}\,\left (16\,a^4\,e^8-120\,a^3\,c\,d^2\,e^6+390\,a^2\,c^2\,d^4\,e^4+14300\,a\,c^3\,d^6\,e^2+6435\,c^4\,d^8\right )}{15015\,c^3\,d^3}+\frac {x^3\,\sqrt {d+e\,x}\,\left (-80\,a^4\,c^3\,d^3\,e^8+600\,a^3\,c^4\,d^5\,e^6+88140\,a^2\,c^5\,d^7\,e^4+108680\,a\,c^6\,d^9\,e^2+12870\,c^7\,d^{11}\right )}{45045\,c^5\,d^5\,e}+\frac {2\,c\,d\,e^2\,x^6\,\left (60\,c\,d^2+31\,a\,e^2\right )\,\sqrt {d+e\,x}}{195}+\frac {2\,a^2\,e\,x\,\sqrt {d+e\,x}\,\left (-64\,a^4\,e^8+480\,a^3\,c\,d^2\,e^6-1560\,a^2\,c^2\,d^4\,e^4+2860\,a\,c^3\,d^6\,e^2+19305\,c^4\,d^8\right )}{45045\,c^4\,d^4}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*e*x^5*(d + e*x)^(1/2)*(71*a^2*e^4 + 390*c^2*d^4 + 540*a*c*d
^2*e^2))/715 + (2*x^4*(d + e*x)^(1/2)*(a^3*e^6 + 572*c^3*d^6 + 1794*a*c^2*d^4*e^2 + 636*a^2*c*d^2*e^4))/(1287*
c*d) + ((d + e*x)^(1/2)*(256*a^7*e^11 - 1920*a^6*c*d^2*e^9 + 12870*a^3*c^4*d^8*e^3 - 11440*a^4*c^3*d^6*e^5 + 6
240*a^5*c^2*d^4*e^7))/(45045*c^5*d^5*e) + (2*c^2*d^2*e^3*x^7*(d + e*x)^(1/2))/15 + (2*a*x^2*(d + e*x)^(1/2)*(1
6*a^4*e^8 + 6435*c^4*d^8 + 14300*a*c^3*d^6*e^2 - 120*a^3*c*d^2*e^6 + 390*a^2*c^2*d^4*e^4))/(15015*c^3*d^3) + (
x^3*(d + e*x)^(1/2)*(12870*c^7*d^11 + 108680*a*c^6*d^9*e^2 + 88140*a^2*c^5*d^7*e^4 + 600*a^3*c^4*d^5*e^6 - 80*
a^4*c^3*d^3*e^8))/(45045*c^5*d^5*e) + (2*c*d*e^2*x^6*(31*a*e^2 + 60*c*d^2)*(d + e*x)^(1/2))/195 + (2*a^2*e*x*(
d + e*x)^(1/2)*(19305*c^4*d^8 - 64*a^4*e^8 + 2860*a*c^3*d^6*e^2 + 480*a^3*c*d^2*e^6 - 1560*a^2*c^2*d^4*e^4))/(
45045*c^4*d^4)))/(x + d/e)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________