3.17.31 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=231 \[ \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^8 \left (c d^2-a e^2\right )^3}+\frac {12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )} \]

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Rubi [A]  time = 0.12, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {658, 650} \begin {gather*} \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^8 \left (c d^2-a e^2\right )^3}+\frac {12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^10,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*(c*d^2 - a*e^2)*(d + e*x)^10) + (12*c*d*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(7/2))/(143*(c*d^2 - a*e^2)^2*(d + e*x)^9) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2)^(7/2))/(429*(c*d^2 - a*e^2)^3*(d + e*x)^8) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7
/2))/(3003*(c*d^2 - a*e^2)^4*(d + e*x)^7)

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), 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 + 2*p + 2, 0]

Rule 658

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

Rubi steps

\begin {align*} \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)^{10}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {(6 c d) \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)^9} \, dx}{13 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac {\left (24 c^2 d^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}}{(d+e x)^8} \, dx}{143 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\frac {\left (16 c^3 d^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)^7} \, dx}{429 \left (c d^2-a e^2\right )^3}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^7}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 148, normalized size = 0.64 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (-231 a^3 e^6+63 a^2 c d e^4 (13 d+2 e x)-7 a c^2 d^2 e^2 \left (143 d^2+52 d e x+8 e^2 x^2\right )+c^3 d^3 \left (429 d^3+286 d^2 e x+104 d e^2 x^2+16 e^3 x^3\right )\right )}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^10,x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-231*a^3*e^6 + 63*a^2*c*d*e^4*(13*d + 2*e*x) - 7*a*c^2*d^2*e
^2*(143*d^2 + 52*d*e*x + 8*e^2*x^2) + c^3*d^3*(429*d^3 + 286*d^2*e*x + 104*d*e^2*x^2 + 16*e^3*x^3)))/(3003*(c*
d^2 - a*e^2)^4*(d + e*x)^7)

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IntegrateAlgebraic [F]  time = 180.06, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^10,x]

[Out]

$Aborted

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fricas [B]  time = 71.50, size = 823, normalized size = 3.56 \begin {gather*} \frac {2 \, {\left (16 \, c^{6} d^{6} e^{3} x^{6} + 429 \, a^{3} c^{3} d^{6} e^{3} - 1001 \, a^{4} c^{2} d^{4} e^{5} + 819 \, a^{5} c d^{2} e^{7} - 231 \, a^{6} e^{9} + 8 \, {\left (13 \, c^{6} d^{7} e^{2} - a c^{5} d^{5} e^{4}\right )} x^{5} + 2 \, {\left (143 \, c^{6} d^{8} e - 26 \, a c^{5} d^{6} e^{3} + 3 \, a^{2} c^{4} d^{4} e^{5}\right )} x^{4} + {\left (429 \, c^{6} d^{9} - 143 \, a c^{5} d^{7} e^{2} + 39 \, a^{2} c^{4} d^{5} e^{4} - 5 \, a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + {\left (1287 \, a c^{5} d^{8} e - 2145 \, a^{2} c^{4} d^{6} e^{3} + 1469 \, a^{3} c^{3} d^{4} e^{5} - 371 \, a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1287 \, a^{2} c^{4} d^{7} e^{2} - 2717 \, a^{3} c^{3} d^{5} e^{4} + 2093 \, a^{4} c^{2} d^{3} e^{6} - 567 \, a^{5} c d e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3003 \, {\left (c^{4} d^{15} - 4 \, a c^{3} d^{13} e^{2} + 6 \, a^{2} c^{2} d^{11} e^{4} - 4 \, a^{3} c d^{9} e^{6} + a^{4} d^{7} e^{8} + {\left (c^{4} d^{8} e^{7} - 4 \, a c^{3} d^{6} e^{9} + 6 \, a^{2} c^{2} d^{4} e^{11} - 4 \, a^{3} c d^{2} e^{13} + a^{4} e^{15}\right )} x^{7} + 7 \, {\left (c^{4} d^{9} e^{6} - 4 \, a c^{3} d^{7} e^{8} + 6 \, a^{2} c^{2} d^{5} e^{10} - 4 \, a^{3} c d^{3} e^{12} + a^{4} d e^{14}\right )} x^{6} + 21 \, {\left (c^{4} d^{10} e^{5} - 4 \, a c^{3} d^{8} e^{7} + 6 \, a^{2} c^{2} d^{6} e^{9} - 4 \, a^{3} c d^{4} e^{11} + a^{4} d^{2} e^{13}\right )} x^{5} + 35 \, {\left (c^{4} d^{11} e^{4} - 4 \, a c^{3} d^{9} e^{6} + 6 \, a^{2} c^{2} d^{7} e^{8} - 4 \, a^{3} c d^{5} e^{10} + a^{4} d^{3} e^{12}\right )} x^{4} + 35 \, {\left (c^{4} d^{12} e^{3} - 4 \, a c^{3} d^{10} e^{5} + 6 \, a^{2} c^{2} d^{8} e^{7} - 4 \, a^{3} c d^{6} e^{9} + a^{4} d^{4} e^{11}\right )} x^{3} + 21 \, {\left (c^{4} d^{13} e^{2} - 4 \, a c^{3} d^{11} e^{4} + 6 \, a^{2} c^{2} d^{9} e^{6} - 4 \, a^{3} c d^{7} e^{8} + a^{4} d^{5} e^{10}\right )} x^{2} + 7 \, {\left (c^{4} d^{14} e - 4 \, a c^{3} d^{12} e^{3} + 6 \, a^{2} c^{2} d^{10} e^{5} - 4 \, a^{3} c d^{8} e^{7} + a^{4} d^{6} e^{9}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^10,x, algorithm="fricas")

[Out]

2/3003*(16*c^6*d^6*e^3*x^6 + 429*a^3*c^3*d^6*e^3 - 1001*a^4*c^2*d^4*e^5 + 819*a^5*c*d^2*e^7 - 231*a^6*e^9 + 8*
(13*c^6*d^7*e^2 - a*c^5*d^5*e^4)*x^5 + 2*(143*c^6*d^8*e - 26*a*c^5*d^6*e^3 + 3*a^2*c^4*d^4*e^5)*x^4 + (429*c^6
*d^9 - 143*a*c^5*d^7*e^2 + 39*a^2*c^4*d^5*e^4 - 5*a^3*c^3*d^3*e^6)*x^3 + (1287*a*c^5*d^8*e - 2145*a^2*c^4*d^6*
e^3 + 1469*a^3*c^3*d^4*e^5 - 371*a^4*c^2*d^2*e^7)*x^2 + (1287*a^2*c^4*d^7*e^2 - 2717*a^3*c^3*d^5*e^4 + 2093*a^
4*c^2*d^3*e^6 - 567*a^5*c*d*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^4*d^15 - 4*a*c^3*d^13*e^2 +
6*a^2*c^2*d^11*e^4 - 4*a^3*c*d^9*e^6 + a^4*d^7*e^8 + (c^4*d^8*e^7 - 4*a*c^3*d^6*e^9 + 6*a^2*c^2*d^4*e^11 - 4*
a^3*c*d^2*e^13 + a^4*e^15)*x^7 + 7*(c^4*d^9*e^6 - 4*a*c^3*d^7*e^8 + 6*a^2*c^2*d^5*e^10 - 4*a^3*c*d^3*e^12 + a^
4*d*e^14)*x^6 + 21*(c^4*d^10*e^5 - 4*a*c^3*d^8*e^7 + 6*a^2*c^2*d^6*e^9 - 4*a^3*c*d^4*e^11 + a^4*d^2*e^13)*x^5
+ 35*(c^4*d^11*e^4 - 4*a*c^3*d^9*e^6 + 6*a^2*c^2*d^7*e^8 - 4*a^3*c*d^5*e^10 + a^4*d^3*e^12)*x^4 + 35*(c^4*d^12
*e^3 - 4*a*c^3*d^10*e^5 + 6*a^2*c^2*d^8*e^7 - 4*a^3*c*d^6*e^9 + a^4*d^4*e^11)*x^3 + 21*(c^4*d^13*e^2 - 4*a*c^3
*d^11*e^4 + 6*a^2*c^2*d^9*e^6 - 4*a^3*c*d^7*e^8 + a^4*d^5*e^10)*x^2 + 7*(c^4*d^14*e - 4*a*c^3*d^12*e^3 + 6*a^2
*c^2*d^10*e^5 - 4*a^3*c*d^8*e^7 + a^4*d^6*e^9)*x)

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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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^10,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.06, size = 217, normalized size = 0.94 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+56 a \,c^{2} d^{2} e^{4} x^{2}-104 c^{3} d^{4} e^{2} x^{2}-126 a^{2} c d \,e^{5} x +364 a \,c^{2} d^{3} e^{3} x -286 c^{3} d^{5} e x +231 a^{3} e^{6}-819 a^{2} c \,d^{2} e^{4}+1001 a \,c^{2} d^{4} e^{2}-429 c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 \left (e x +d \right )^{9} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)/(e*x+d)^10,x)

[Out]

-2/3003*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+56*a*c^2*d^2*e^4*x^2-104*c^3*d^4*e^2*x^2-126*a^2*c*d*e^5*x+364*a*c^2*
d^3*e^3*x-286*c^3*d^5*e*x+231*a^3*e^6-819*a^2*c*d^2*e^4+1001*a*c^2*d^4*e^2-429*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d
^2*x+a*d*e)^(5/2)/(e*x+d)^9/(a^4*e^8-4*a^3*c*d^2*e^6+6*a^2*c^2*d^4*e^4-4*a*c^3*d^6*e^2+c^4*d^8)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 zero or nonzero?

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mupad [B]  time = 9.57, size = 5069, normalized size = 21.94

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^10,x)

[Out]

(((d*((8*c^5*d^6)/(143*e*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)) - (4*c^4*d^4*(21*a*e^2 - 17*c*d^2))/(143*e*(
a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e + (4*c^3*d^3*(110*a^2*e^4 + 53*c^2*d^4 - 157*a*c*d^2*e^2))/(429*e^
2*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 - (((2*
a^3*e^4)/(13*a*e^3 - 13*c*d^2*e) - (d*((d*((2*c^3*d^4)/(13*a*e^3 - 13*c*d^2*e) - (6*a*c^2*d^2*e^2)/(13*a*e^3 -
13*c*d^2*e)))/e + (6*a^2*c*d*e^3)/(13*a*e^3 - 13*c*d^2*e)))/e)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))
/(d + e*x)^7 + (((d*((16*c^6*d^7)/(1287*e*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)) - (8*c^5*d^5*(33*a*e^2 - 29
*c*d^2))/(1287*e*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e + (8*c^4*d^4*(112*a^2*e^4 + 81*c^2*d^4 - 191*a*c
*d^2*e^2))/(1287*e^2*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/
(d + e*x)^3 + (((d*((32*c^7*d^8)/(9009*e*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (16*c^6*d^6*(43*a*e^2 - 39
*c*d^2))/(9009*e*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e + (16*c^5*d^5*(1089*a^2*e^4 + 884*c^2*d^4 - 1963
*a*c*d^2*e^2))/(45045*e^2*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1
/2))/(d + e*x)^2 + (((38*c^4*d^5 + 94*a*c^3*d^3*e^2)/(429*e^2*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (4*c^4*
d^5)/(13*e^2*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^
4 + (((348*c^5*d^6 - 292*a*c^4*d^4*e^2)/(1001*e^2*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) - (8*c^5*d^6)/(143*
e^2*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 + (((
d*((d*((128*c^9*d^10)/(135135*e*(a*e^2 - c*d^2)^7) - (128*c^8*d^8*(10*a*e^2 - 9*c*d^2))/(45045*e*(a*e^2 - c*d^
2)^7)))/e + (128*c^7*d^7*(379*a^2*e^4 + 322*c^2*d^4 - 698*a*c*d^2*e^2))/(135135*e^2*(a*e^2 - c*d^2)^7)))/e - (
128*a*c^6*d^6*(350*a^2*e^4 + 322*c^2*d^4 - 671*a*c*d^2*e^2))/(135135*e*(a*e^2 - c*d^2)^7))*(x*(a*e^2 + c*d^2)
+ a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((d*((128*c^9*d^10)/(135135*e*(a*e^2 - c*d^2)^7) - (64*c^8*d^8*(1
9*a*e^2 - 17*c*d^2))/(45045*e*(a*e^2 - c*d^2)^7)))/e + (128*c^7*d^7*(337*a^2*e^4 + 283*c^2*d^4 - 617*a*c*d^2*e
^2))/(135135*e^2*(a*e^2 - c*d^2)^7)))/e - (64*c^6*d^6*(323*a^3*e^6 + 296*c^3*d^6 - 322*a*c^2*d^4*e^2 - 295*a^2
*c*d^2*e^4))/(135135*e^3*(a*e^2 - c*d^2)^7))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (((d*(
(d*((4*c^4*d^5)/(13*(a*e^2 - c*d^2)*(11*a*e^3 - 11*c*d^2*e)) - (4*c^3*d^3*(10*a*e^2 - 7*c*d^2))/(13*(a*e^2 - c
*d^2)*(11*a*e^3 - 11*c*d^2*e))))/e + (28*c^4*d^6 - 112*a*c^3*d^4*e^2 + 96*a^2*c^2*d^2*e^4)/(13*e*(a*e^2 - c*d^
2)*(11*a*e^3 - 11*c*d^2*e))))/e - (4*a*c*d*(15*a^2*e^4 + 7*c^2*d^4 - 21*a*c*d^2*e^2))/(13*(a*e^2 - c*d^2)*(11*
a*e^3 - 11*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^6 - (((32*c^7*d^8)/(9009*e^3*(a
*e^2 - c*d^2)^5) + (16*c^6*d^6*(511*a*e^2 - 521*c*d^2))/(45045*e^3*(a*e^2 - c*d^2)^5))*(x*(a*e^2 + c*d^2) + a*
d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (((d*((4*c^4*d^5)/(13*e*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e)) - (2*c^3*d^
3*(7*a*e^2 - 3*c*d^2))/(13*e*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e))))/e + (28*c^4*d^6 - 122*a*c^3*d^4*e^2 + 13
8*a^2*c^2*d^2*e^4)/(143*e^2*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1
/2))/(d + e*x)^5 + (((d*((d*((8*c^5*d^6)/(143*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^2*e)) - (8*c^4*d^4*(16*a*e^2
- 13*c*d^2))/(143*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^2*e))))/e + (8*c^3*d^3*(84*a^2*e^4 + 55*c^2*d^4 - 136*a*c
*d^2*e^2))/(143*e*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^2*e))))/e - (8*a*c^2*d^2*(69*a^2*e^4 + 55*c^2*d^4 - 123*a
*c*d^2*e^2))/(143*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d
+ e*x)^5 + (((d*((d*((16*c^6*d^7)/(1287*(a*e^2 - c*d^2)^3*(7*a*e^3 - 7*c*d^2*e)) - (16*c^5*d^5*(7*a*e^2 - 6*c*
d^2))/(429*(a*e^2 - c*d^2)^3*(7*a*e^3 - 7*c*d^2*e))))/e + (16*c^4*d^4*(164*a^2*e^4 + 125*c^2*d^4 - 286*a*c*d^2
*e^2))/(1287*e*(a*e^2 - c*d^2)^3*(7*a*e^3 - 7*c*d^2*e))))/e - (16*a*c^3*d^3*(144*a^2*e^4 + 125*c^2*d^4 - 268*a
*c*d^2*e^2))/(1287*(a*e^2 - c*d^2)^3*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d
+ e*x)^4 + (((d*((d*((32*c^7*d^8)/(9009*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)) - (32*c^6*d^6*(25*a*e^2 - 22
*c*d^2))/(9009*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e))))/e + (32*c^5*d^5*(248*a^2*e^4 + 201*c^2*d^4 - 446*a*c
*d^2*e^2))/(9009*e*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e))))/e - (32*a*c^4*d^4*(224*a^2*e^4 + 201*c^2*d^4 - 4
24*a*c*d^2*e^2))/(9009*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)
)/(d + e*x)^3 + (((d*((d*((64*c^8*d^9)/(45045*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e)) - (64*c^7*d^7*(28*a*e^2
- 25*c*d^2))/(45045*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e))))/e + (64*c^6*d^6*(323*a^2*e^4 + 270*c^2*d^4 - 5
90*a*c*d^2*e^2))/(45045*e*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e))))/e - (64*a*c^5*d^5*(296*a^2*e^4 + 270*c^2*
d^4 - 565*a*c*d^2*e^2))/(45045*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^
2)^(1/2))/(d + e*x)^2 + (((d*((64*c^8*d^9)/(45045*e^2*(a*e^2 - c*d^2)^6) - (32*c^7*d^7*(51*a*e^2 - 47*c*d^2))/
(45045*e^2*(a*e^2 - c*d^2)^6)))/e + (32*c^6*d^6*(1067*a^2*e^4 + 920*c^2*d^4 - 1981*a*c*d^2*e^2))/(135135*e^3*(
a*e^2 - c*d^2)^6))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((d*((4*c^4*d^5)/(13*(a*e^2
- c*d^2)*(11*a*e^3 - 11*c*d^2*e)) - (2*c^3*d^3*(7*a*e^2 - c*d^2))/(13*(a*e^2 - c*d^2)*(11*a*e^3 - 11*c*d^2*e)
)))/e + (2*c^2*d^2*(9*a^2*e^4 + c^2*d^4 - 4*a*c*d^2*e^2))/(13*e*(a*e^2 - c*d^2)*(11*a*e^3 - 11*c*d^2*e))))/e -
(2*c^4*d^7 - 4*a*c^3*d^5*e^2 + 6*a^3*c*d*e^6)/(13*e^2*(a*e^2 - c*d^2)*(11*a*e^3 - 11*c*d^2*e)))*(x*(a*e^2 + c
*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^6 - (((d*((d*((8*c^5*d^6)/(143*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^
2*e)) - (12*c^4*d^4*(7*a*e^2 - 5*c*d^2))/(143*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^2*e))))/e + (8*c^3*d^3*(29*a^
2*e^4 + 11*c^2*d^4 - 37*a*c*d^2*e^2))/(143*e*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^2*e))))/e - (60*c^5*d^8 - 92*a
*c^4*d^6*e^2 - 56*a^2*c^3*d^4*e^4 + 96*a^3*c^2*d^2*e^6)/(143*e^2*(a*e^2 - c*d^2)^2*(9*a*e^3 - 9*c*d^2*e)))*(x*
(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^5 - (((d*((d*((16*c^6*d^7)/(1287*(a*e^2 - c*d^2)^3*(7*a*
e^3 - 7*c*d^2*e)) - (8*c^5*d^5*(11*a*e^2 - 9*c*d^2))/(429*(a*e^2 - c*d^2)^3*(7*a*e^3 - 7*c*d^2*e))))/e + (16*c
^4*d^4*(92*a^2*e^4 + 62*c^2*d^4 - 151*a*c*d^2*e^2))/(1287*e*(a*e^2 - c*d^2)^3*(7*a*e^3 - 7*c*d^2*e))))/e - (55
2*c^6*d^9 - 664*a*c^5*d^7*e^2 - 544*a^2*c^4*d^5*e^4 + 672*a^3*c^3*d^3*e^6)/(1287*e^2*(a*e^2 - c*d^2)^3*(7*a*e^
3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 - (((d*((d*((32*c^7*d^8)/(9009*(a*
e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)) - (16*c^6*d^6*(43*a*e^2 - 37*c*d^2))/(9009*(a*e^2 - c*d^2)^4*(5*a*e^3 -
5*c*d^2*e))))/e + (16*c^5*d^5*(349*a^2*e^4 + 269*c^2*d^4 - 612*a*c*d^2*e^2))/(9009*e*(a*e^2 - c*d^2)^4*(5*a*e^
3 - 5*c*d^2*e))))/e - (2304*c^7*d^10 - 2608*a*c^6*d^8*e^2 - 2288*a^2*c^5*d^6*e^4 + 2624*a^3*c^4*d^4*e^6)/(9009
*e^2*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 - ((
(d*((d*((64*c^8*d^9)/(45045*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e)) - (32*c^7*d^7*(17*a*e^2 - 15*c*d^2))/(150
15*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e))))/e + (32*c^6*d^6*(521*a^2*e^4 + 425*c^2*d^4 - 940*a*c*d^2*e^2))/(
45045*e*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e))))/e - (7168*c^8*d^11 - 7904*a*c^7*d^9*e^2 - 7136*a^2*c^6*d^7*
e^4 + 7936*a^3*c^5*d^5*e^6)/(45045*e^2*(a*e^2 - c*d^2)^5*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e +
c*d*e*x^2)^(1/2))/(d + e*x)^2 - (((16*c^6*d^7)/(1287*e^2*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) + (8*c^5*d^5
*(283*a*e^2 - 293*c*d^2))/(6435*e^2*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d
*e*x^2)^(1/2))/(d + e*x)^2 + (15952*c^6*d^6*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(135135*e^3*(a*e^2
- c*d^2)^4*(d + e*x)) + (88*c^5*d^5*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(455*e^2*(a*e^2 - c*d^2)^2*
(3*a*e^3 - 3*c*d^2*e)*(d + e*x)^2) + (120*c^4*d^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(1001*e^2*(a*
e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)*(d + e*x)^3)

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**10,x)

[Out]

Timed out

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