3.21.46 \(\int (d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [2046]

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

[Out]

256/45045*(-a*e^2+c*d^2)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^5/d^5/(e*x+d)^(7/2)+128/6435*(-a*e^2+c*d^
2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^4/d^4/(e*x+d)^(5/2)+32/715*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(7/2)/c^3/d^3/(e*x+d)^(3/2)+16/195*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d
^2/(e*x+d)^(1/2)+2/15*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*(e*x+d)^(1/2)/c/d

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Rubi [A]
time = 0.17, 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 = {670, 662} \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 662

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 670

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*}

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Mathematica [A]
time = 0.13, size = 197, normalized size = 0.67 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e 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])

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Maple [A]
time = 0.74, size = 235, normalized size = 0.80

method	result	size
default	\(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{3} \left (3003 e^{4} x^{4} c^{4} d^{4}-1848 a \,c^{3} d^{3} e^{5} x^{3}+13860 c^{4} d^{5} e^{3} x^{3}+1008 e^{6} a^{2} x^{2} c^{2} d^{2}-7560 e^{4} a \,x^{2} c^{3} d^{4}+24570 e^{2} x^{2} c^{4} d^{6}-448 a^{3} c d \,e^{7} x +3360 a^{2} c^{2} d^{3} e^{5} x -10920 a \,c^{3} d^{5} e^{3} x +20020 c^{4} d^{7} e x +128 a^{4} e^{8}-960 a^{3} c \,d^{2} e^{6}+3120 a^{2} c^{2} d^{4} e^{4}-5720 a \,c^{3} d^{6} e^{2}+6435 c^{4} d^{8}\right )}{45045 \sqrt {e x +d}\, c^{5} d^{5}}\)	\(235\)
gosper	\(\frac {2 \left (c d x +a e \right ) \left (3003 e^{4} x^{4} c^{4} d^{4}-1848 a \,c^{3} d^{3} e^{5} x^{3}+13860 c^{4} d^{5} e^{3} x^{3}+1008 e^{6} a^{2} x^{2} c^{2} d^{2}-7560 e^{4} a \,x^{2} c^{3} d^{4}+24570 e^{2} x^{2} c^{4} d^{6}-448 a^{3} c d \,e^{7} x +3360 a^{2} c^{2} d^{3} e^{5} x -10920 a \,c^{3} d^{5} e^{3} x +20020 c^{4} d^{7} e x +128 a^{4} e^{8}-960 a^{3} c \,d^{2} e^{6}+3120 a^{2} c^{2} d^{4} e^{4}-5720 a \,c^{3} d^{6} e^{2}+6435 c^{4} d^{8}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{45045 c^{5} d^{5} \left (e x +d \right )^{\frac {5}{2}}}\)	\(243\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(c*d*x+a*e)^3*(3003*c^4*d^4*e^4*x^4-1848*a*c^3*d^3*e^5*x^3+1
3860*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+3
360*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^5/d^5

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Maxima [A]
time = 0.32, size = 426, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (3003 \, c^{7} d^{7} x^{7} e^{4} + 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 (x e + d\right )}}{45045 \, {\left (c^{5} d^{5} x e + 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*x^7*e^4 + 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)*(x*e + d)/(c^5*d^5*x*e + c^5*d^6)

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Fricas [A]
time = 3.05, size = 461, normalized size = 1.56 \begin {gather*} \frac {2 \, {\left (6435 \, c^{7} d^{11} x^{3} - 64 \, a^{6} c d x e^{10} + 128 \, a^{7} e^{11} + 48 \, {\left (a^{5} c^{2} d^{2} x^{2} - 20 \, a^{6} c d^{2}\right )} e^{9} - 40 \, {\left (a^{4} c^{3} d^{3} x^{3} - 12 \, a^{5} c^{2} d^{3} x\right )} e^{8} + 5 \, {\left (7 \, a^{3} c^{4} d^{4} x^{4} - 72 \, a^{4} c^{3} d^{4} x^{2} + 624 \, a^{5} c^{2} d^{4}\right )} e^{7} + 3 \, {\left (1491 \, a^{2} c^{5} d^{5} x^{5} + 100 \, a^{3} c^{4} d^{5} x^{3} - 520 \, a^{4} c^{3} d^{5} x\right )} e^{6} + {\left (7161 \, a c^{6} d^{6} x^{6} + 22260 \, a^{2} c^{5} d^{6} x^{4} + 1170 \, a^{3} c^{4} d^{6} x^{2} - 5720 \, a^{4} c^{3} d^{6}\right )} e^{5} + {\left (3003 \, c^{7} d^{7} x^{7} + 34020 \, a c^{6} d^{7} x^{5} + 44070 \, a^{2} c^{5} d^{7} x^{3} + 2860 \, a^{3} c^{4} d^{7} x\right )} e^{4} + 15 \, {\left (924 \, c^{7} d^{8} x^{6} + 4186 \, a c^{6} d^{8} x^{4} + 2860 \, a^{2} c^{5} d^{8} x^{2} + 429 \, a^{3} c^{4} d^{8}\right )} e^{3} + 65 \, {\left (378 \, c^{7} d^{9} x^{5} + 836 \, a c^{6} d^{9} x^{3} + 297 \, a^{2} c^{5} d^{9} x\right )} e^{2} + 715 \, {\left (28 \, c^{7} d^{10} x^{4} + 27 \, a c^{6} d^{10} x^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{45045 \, {\left (c^{5} d^{5} x e + 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*(6435*c^7*d^11*x^3 - 64*a^6*c*d*x*e^10 + 128*a^7*e^11 + 48*(a^5*c^2*d^2*x^2 - 20*a^6*c*d^2)*e^9 - 40*(
a^4*c^3*d^3*x^3 - 12*a^5*c^2*d^3*x)*e^8 + 5*(7*a^3*c^4*d^4*x^4 - 72*a^4*c^3*d^4*x^2 + 624*a^5*c^2*d^4)*e^7 + 3
*(1491*a^2*c^5*d^5*x^5 + 100*a^3*c^4*d^5*x^3 - 520*a^4*c^3*d^5*x)*e^6 + (7161*a*c^6*d^6*x^6 + 22260*a^2*c^5*d^
6*x^4 + 1170*a^3*c^4*d^6*x^2 - 5720*a^4*c^3*d^6)*e^5 + (3003*c^7*d^7*x^7 + 34020*a*c^6*d^7*x^5 + 44070*a^2*c^5
*d^7*x^3 + 2860*a^3*c^4*d^7*x)*e^4 + 15*(924*c^7*d^8*x^6 + 4186*a*c^6*d^8*x^4 + 2860*a^2*c^5*d^8*x^2 + 429*a^3
*c^4*d^8)*e^3 + 65*(378*c^7*d^9*x^5 + 836*a*c^6*d^9*x^3 + 297*a^2*c^5*d^9*x)*e^2 + 715*(28*c^7*d^10*x^4 + 27*a
*c^6*d^10*x^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c^5*d^5*x*e + c^5*d^6)

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Sympy [F(-1)] Timed out
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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4197 vs. \(2 (271) = 542\).
time = 1.04, size = 4197, normalized size = 14.23 \begin {gather*} \text {Too large to display} \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]

2/45045*(429*c^2*d^6*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*
d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*c*d*e - c*d
^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*d*e - c*d^2
*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e^(-1) - 6006*a*c*d^5*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3
- 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*e^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c
*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2*e + a*e^3)*a^2*e^4)/(c^2*d^2))*e^(-1) - 572*c^2*d^5*((35*sqrt(-c*d
^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*
sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (105*((x*e + d)*c
*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x*e + d)
*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4)) + 1501
5*a^2*d^4*(((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*e^(-1)/(c*d) + (sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^
2*e + a*e^3)*a*e^2)/(c*d))*e + 3432*a*c*d^4*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c
^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35
*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*(
(x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e + 78*c^2*d^4*((315*sqrt(-c*d^2*e + a*e^3)*c^5*d^
10 - 35*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^8*e^2 - 40*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^6*e^4 - 48*sqrt(-c*d^2*e +
a*e^3)*a^3*c^2*d^4*e^6 - 64*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*e^8 - 128*sqrt(-c*d^2*e + a*e^3)*a^5*e^10)*e^(-4)
/(c^5*d^5) + (1155*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772*((x*e + d)*c*d*e - c*d^2*e + a*e^
3)^(5/2)*a^3*e^9 + 2970*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*e^6 - 1540*((x*e + d)*c*d*e - c*d^2*e +
a*e^3)^(9/2)*a*e^3 + 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))*e^(-9)/(c^5*d^5))*e - 1716*a*c*d^3*((35*s
qrt(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*
e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (105*((x*
e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((
x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4)
)*e^2 - 20*c^2*d^3*((693*sqrt(-c*d^2*e + a*e^3)*c^6*d^12 - 63*sqrt(-c*d^2*e + a*e^3)*a*c^5*d^10*e^2 - 70*sqrt(
-c*d^2*e + a*e^3)*a^2*c^4*d^8*e^4 - 80*sqrt(-c*d^2*e + a*e^3)*a^3*c^3*d^6*e^6 - 96*sqrt(-c*d^2*e + a*e^3)*a^4*
c^2*d^4*e^8 - 128*sqrt(-c*d^2*e + a*e^3)*a^5*c*d^2*e^10 - 256*sqrt(-c*d^2*e + a*e^3)*a^6*e^12)*e^(-5)/(c^6*d^6
) + (3003*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^5*e^15 - 9009*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*
a^4*e^12 + 12870*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^3*e^9 - 10010*((x*e + d)*c*d*e - c*d^2*e + a*e^3)
^(9/2)*a^2*e^6 + 4095*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2)*a*e^3 - 693*((x*e + d)*c*d*e - c*d^2*e + a*e^
3)^(13/2))*e^(-11)/(c^6*d^6))*e^2 - 12012*a^2*d^3*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((x*
e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*e^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e +
a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2*e + a*e^3)*a^2*e^4)/(c^2*d^2))*e + 2574*a^2*d^2*((15*sqrt(-c*d^2*e + a*e^3)
*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e +
a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*
e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e^3 + 104*a
*c*d^2*((315*sqrt(-c*d^2*e + a*e^3)*c^5*d^10 - 35*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^8*e^2 - 40*sqrt(-c*d^2*e + a*
e^3)*a^2*c^3*d^6*e^4 - 48*sqrt(-c*d^2*e + a*e^3)*a^3*c^2*d^4*e^6 - 64*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*e^8 - 1
28*sqrt(-c*d^2*e + a*e^3)*a^5*e^10)*e^(-4)/(c^5*d^5) + (1155*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^1
2 - 2772*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^3*e^9 + 2970*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^
2*e^6 - 1540*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*a*e^3 + 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))
*e^(-9)/(c^5*d^5))*e^3 + c^2*d^2*((3003*sqrt(-c*d^2*e + a*e^3)*c^7*d^14 - 231*sqrt(-c*d^2*e + a*e^3)*a*c^6*d^1
2*e^2 - 252*sqrt(-c*d^2*e + a*e^3)*a^2*c^5*d^10*e^4 - 280*sqrt(-c*d^2*e + a*e^3)*a^3*c^4*d^8*e^6 - 320*sqrt(-c
*d^2*e + a*e^3)*a^4*c^3*d^6*e^8 - 384*sqrt(-c*d^2*e + a*e^3)*a^5*c^2*d^4*e^10 - 512*sqrt(-c*d^2*e + a*e^3)*a^6
*c*d^2*e^12 - 1024*sqrt(-c*d^2*e + a*e^3)*a^7*e^14)*e^(-6)/(c^7*d^7) + (15015*((x*e + d)*c*d*e - c*d^2*e + a*e
^3)^(3/2)*a^6*e^18 - 54054*((x*e + d)*c*d*e - c...

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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)

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