3.17.30 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^9} \, dx\)
Optimal. Leaf size=171 \[ \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 (d+e x)^7 \left (c d^2-a e^2\right )^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^8 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 171, normalized size of antiderivative = 1.00,
number of steps used = 3, 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 {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}}{693 (d+e x)^7 \left (c d^2-a e^2\right )^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^8 \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}}{11 (d+e x)^9 \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)^9,x]
[Out]
(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^9) + (8*c*d*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(99*(c*d^2 - a*e^2)^2*(d + e*x)^8) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(7/2))/(693*(c*d^2 - a*e^2)^3*(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)^9} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac {(4 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)^8} \, dx}{11 \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}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 \left (c d^2-a e^2\right )^2 (d+e x)^8}+\frac {\left (8 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)^7} \, dx}{99 \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}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 \left (c d^2-a e^2\right )^2 (d+e x)^8}+\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}}{693 \left (c d^2-a e^2\right )^3 (d+e x)^7}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 104, normalized size = 0.61 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (63 a^2 e^4-14 a c d e^2 (11 d+2 e x)+c^2 d^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )\right )}{693 (d+e x)^6 \left (c d^2-a e^2\right )^3} \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)^9,x]
[Out]
(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(63*a^2*e^4 - 14*a*c*d*e^2*(11*d + 2*e*x) + c^2*d^2*(99*d^2 +
44*d*e*x + 8*e^2*x^2)))/(693*(c*d^2 - a*e^2)^3*(d + e*x)^6)
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IntegrateAlgebraic [F] time = 180.04, 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)^9,x]
[Out]
$Aborted
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fricas [B] time = 32.84, size = 569, normalized size = 3.33 \begin {gather*} \frac {2 \, {\left (8 \, c^{5} d^{5} e^{2} x^{5} + 99 \, a^{3} c^{2} d^{4} e^{3} - 154 \, a^{4} c d^{2} e^{5} + 63 \, a^{5} e^{7} + 4 \, {\left (11 \, c^{5} d^{6} e - a c^{4} d^{4} e^{3}\right )} x^{4} + {\left (99 \, c^{5} d^{7} - 22 \, a c^{4} d^{5} e^{2} + 3 \, a^{2} c^{3} d^{3} e^{4}\right )} x^{3} + {\left (297 \, a c^{4} d^{6} e - 330 \, a^{2} c^{3} d^{4} e^{3} + 113 \, a^{3} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (297 \, a^{2} c^{3} d^{5} e^{2} - 418 \, a^{3} c^{2} d^{3} e^{4} + 161 \, a^{4} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{693 \, {\left (c^{3} d^{12} - 3 \, a c^{2} d^{10} e^{2} + 3 \, a^{2} c d^{8} e^{4} - a^{3} d^{6} e^{6} + {\left (c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}\right )} x^{6} + 6 \, {\left (c^{3} d^{7} e^{5} - 3 \, a c^{2} d^{5} e^{7} + 3 \, a^{2} c d^{3} e^{9} - a^{3} d e^{11}\right )} x^{5} + 15 \, {\left (c^{3} d^{8} e^{4} - 3 \, a c^{2} d^{6} e^{6} + 3 \, a^{2} c d^{4} e^{8} - a^{3} d^{2} e^{10}\right )} x^{4} + 20 \, {\left (c^{3} d^{9} e^{3} - 3 \, a c^{2} d^{7} e^{5} + 3 \, a^{2} c d^{5} e^{7} - a^{3} d^{3} e^{9}\right )} x^{3} + 15 \, {\left (c^{3} d^{10} e^{2} - 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{6} e^{6} - a^{3} d^{4} e^{8}\right )} x^{2} + 6 \, {\left (c^{3} d^{11} e - 3 \, a c^{2} d^{9} e^{3} + 3 \, a^{2} c d^{7} e^{5} - a^{3} d^{5} e^{7}\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)^9,x, algorithm="fricas")
[Out]
2/693*(8*c^5*d^5*e^2*x^5 + 99*a^3*c^2*d^4*e^3 - 154*a^4*c*d^2*e^5 + 63*a^5*e^7 + 4*(11*c^5*d^6*e - a*c^4*d^4*e
^3)*x^4 + (99*c^5*d^7 - 22*a*c^4*d^5*e^2 + 3*a^2*c^3*d^3*e^4)*x^3 + (297*a*c^4*d^6*e - 330*a^2*c^3*d^4*e^3 + 1
13*a^3*c^2*d^2*e^5)*x^2 + (297*a^2*c^3*d^5*e^2 - 418*a^3*c^2*d^3*e^4 + 161*a^4*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)/(c^3*d^12 - 3*a*c^2*d^10*e^2 + 3*a^2*c*d^8*e^4 - a^3*d^6*e^6 + (c^3*d^6*e^6 - 3*a*c^2
*d^4*e^8 + 3*a^2*c*d^2*e^10 - a^3*e^12)*x^6 + 6*(c^3*d^7*e^5 - 3*a*c^2*d^5*e^7 + 3*a^2*c*d^3*e^9 - a^3*d*e^11)
*x^5 + 15*(c^3*d^8*e^4 - 3*a*c^2*d^6*e^6 + 3*a^2*c*d^4*e^8 - a^3*d^2*e^10)*x^4 + 20*(c^3*d^9*e^3 - 3*a*c^2*d^7
*e^5 + 3*a^2*c*d^5*e^7 - a^3*d^3*e^9)*x^3 + 15*(c^3*d^10*e^2 - 3*a*c^2*d^8*e^4 + 3*a^2*c*d^6*e^6 - a^3*d^4*e^8
)*x^2 + 6*(c^3*d^11*e - 3*a*c^2*d^9*e^3 + 3*a^2*c*d^7*e^5 - a^3*d^5*e^7)*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)^9,x, algorithm="giac")
[Out]
Timed out
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maple [A] time = 0.06, size = 146, normalized size = 0.85 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (8 c^{2} d^{2} e^{2} x^{2}-28 a c d \,e^{3} x +44 c^{2} d^{3} e x +63 a^{2} e^{4}-154 a c \,d^{2} e^{2}+99 c^{2} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{693 \left (e x +d \right )^{8} \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 )} \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)^9,x)
[Out]
-2/693*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-28*a*c*d*e^3*x+44*c^2*d^3*e*x+63*a^2*e^4-154*a*c*d^2*e^2+99*c^2*d^4)*(c*
d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/(e*x+d)^8/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)
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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)^9,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 = 7.21, size = 4096, normalized size = 23.95
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)^9,x)
[Out]
(((d*((8*c^5*d^6)/(99*e*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) - (4*c^4*d^4*(19*a*e^2 - 15*c*d^2))/(99*e*(a*
e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e))))/e + (4*c^3*d^3*(33*a^2*e^4 + 16*c^2*d^4 - 47*a*c*d^2*e^2))/(99*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 - (((2*a^3*e
^4)/(11*a*e^3 - 11*c*d^2*e) - (d*((d*((2*c^3*d^4)/(11*a*e^3 - 11*c*d^2*e) - (6*a*c^2*d^2*e^2)/(11*a*e^3 - 11*c
*d^2*e)))/e + (6*a^2*c*d*e^3)/(11*a*e^3 - 11*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)^6 + (((d*((16*c^6*d^7)/(693*e*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (8*c^5*d^5*(29*a*e^2 - 25*c*d^2
))/(693*e*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e + (8*c^4*d^4*(466*a^2*e^4 + 331*c^2*d^4 - 787*a*c*d^2*e
^2))/(3465*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 + (((6*c^4*d^5 + 22*a*c^3*d^3*e^2)/(77*e^2*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) - (4*c^4*d^5)/(11*e^2*
(a*e^2 - c*d^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 + (((188*c^
5*d^6 - 148*a*c^4*d^4*e^2)/(495*e^2*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (8*c^5*d^6)/(99*e^2*(a*e^2 - c*
d^2)^2*(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*((d*((64*c^8*
d^9)/(10395*e*(a*e^2 - c*d^2)^6) - (64*c^7*d^7*(23*a*e^2 - 20*c*d^2))/(10395*e*(a*e^2 - c*d^2)^6)))/e + (64*c^
6*d^6*(218*a^2*e^4 + 175*c^2*d^4 - 390*a*c*d^2*e^2))/(10395*e^2*(a*e^2 - c*d^2)^6)))/e - (64*a*c^5*d^5*(196*a^
2*e^4 + 175*c^2*d^4 - 370*a*c*d^2*e^2))/(10395*e*(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) - (((16*c^6*d^7)/(693*e^3*(a*e^2 - c*d^2)^4) + (8*c^5*d^5*(497*a*e^2 - 527*c*d^2))/(10395*e^3*
(a*e^2 - c*d^2)^4))*(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)/(11*e*(a*e^2
- c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (2*c^3*d^3*(7*a*e^2 - 3*c*d^2))/(11*e*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e))
))/e + (8*c^4*d^6 - 34*a*c^3*d^4*e^2 + 38*a^2*c^2*d^2*e^4)/(33*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 + (((4992*c^8*d^11 - 5600*a*c^7*d^9*e^2 - 4960*a^2*c^6
*d^7*e^4 + 5632*a^3*c^5*d^5*e^6)/(10395*e^3*(a*e^2 - c*d^2)^6) - (d*((d*((64*c^8*d^9)/(10395*e*(a*e^2 - c*d^2)
^6) - (32*c^7*d^7*(43*a*e^2 - 37*c*d^2))/(10395*e*(a*e^2 - c*d^2)^6)))/e + (32*c^6*d^6*(373*a^2*e^4 + 293*c^2*
d^4 - 660*a*c*d^2*e^2))/(10395*e^2*(a*e^2 - c*d^2)^6)))/e)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d +
e*x) + (((d*((d*((8*c^5*d^6)/(99*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)) - (8*c^4*d^4*(14*a*e^2 - 11*c*d^2))
/(99*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e + (8*c^3*d^3*(66*a^2*e^4 + 41*c^2*d^4 - 104*a*c*d^2*e^2))/(9
9*e*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e - (8*a*c^2*d^2*(53*a^2*e^4 + 41*c^2*d^4 - 93*a*c*d^2*e^2))/(9
9*(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 + (((d*
((d*((16*c^6*d^7)/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)) - (16*c^5*d^5*(6*a*e^2 - 5*c*d^2))/(231*(a*e^2
- c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e + (16*c^4*d^4*(122*a^2*e^4 + 89*c^2*d^4 - 208*a*c*d^2*e^2))/(693*e*(a*e
^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e - (16*a*c^3*d^3*(105*a^2*e^4 + 89*c^2*d^4 - 193*a*c*d^2*e^2))/(693*(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*((d*
((32*c^7*d^8)/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (32*c^6*d^6*(7*a*e^2 - 6*c*d^2))/(1155*(a*e^2 -
c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e + (32*c^5*d^5*(176*a^2*e^4 + 137*c^2*d^4 - 310*a*c*d^2*e^2))/(3465*e*(a*e
^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e - (32*a*c^4*d^4*(156*a^2*e^4 + 137*c^2*d^4 - 292*a*c*d^2*e^2))/(3465*
(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 + (((d*((
32*c^7*d^8)/(3465*e^2*(a*e^2 - c*d^2)^5) - (16*c^6*d^6*(37*a*e^2 - 33*c*d^2))/(3465*e^2*(a*e^2 - c*d^2)^5)))/e
+ (16*c^5*d^5*(542*a^2*e^4 + 437*c^2*d^4 - 973*a*c*d^2*e^2))/(10395*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*((d*((4*c^4*d^5)/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e)) - (
2*c^3*d^3*(7*a*e^2 - c*d^2))/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*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))/(11*e*(a*e^2 - c*d^2)*(9*a*e^3 - 9*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)/(11*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)/(99*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e)) - (4*c^4*d^4*(19*a*e^2 - 13*c*d^2))/(99
*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e + (4*c^3*d^3*(51*a^2*e^4 + 19*c^2*d^4 - 64*a*c*d^2*e^2))/(99*e*(
a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e - (52*c^5*d^8 - 80*a*c^4*d^6*e^2 - 48*a^2*c^3*d^4*e^4 + 84*a^3*c^2
*d^2*e^6)/(99*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 - (((d*((d*((16*c^6*d^7)/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e)) - (8*c^5*d^5*(29*a*e^2 - 23*c*
d^2))/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e + (16*c^4*d^4*(73*a^2*e^4 + 47*c^2*d^4 - 117*a*c*d^2*e
^2))/(693*e*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e - (424*c^6*d^9 - 520*a*c^5*d^7*e^2 - 416*a^2*c^4*d^5*
e^4 + 528*a^3*c^3*d^3*e^6)/(693*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*((d*((32*c^7*d^8)/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (16*c^6*d
^6*(37*a*e^2 - 31*c*d^2))/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e + (32*c^5*d^5*(131*a^2*e^4 + 97*c
^2*d^4 - 225*a*c*d^2*e^2))/(3465*e*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e - (1680*c^7*d^10 - 1936*a*c^6*
d^8*e^2 - 1664*a^2*c^5*d^6*e^4 + 1952*a^3*c^4*d^4*e^6)/(3465*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 + (((d*((d*((4*c^4*d^5)/(11*(a*e^2 - c*d^2)*(9*a*e^3 -
9*c*d^2*e)) - (12*c^3*d^3*(3*a*e^2 - 2*c*d^2))/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e))))/e + (12*c^2*d^2*(
7*a^2*e^4 + 2*c^2*d^4 - 8*a*c*d^2*e^2))/(11*e*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e))))/e - (4*a*c*d*(13*a^2*e^
4 + 6*c^2*d^4 - 18*a*c*d^2*e^2))/(11*(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 + (1976*c^5*d^5*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(10395*e^3*(a*e^2 - c
*d^2)^3*(d + e*x)) + (52*c^4*d^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(385*e^2*(a*e^2 - c*d^2)*(3*a*
e^3 - 3*c*d^2*e)*(d + e*x)^2)
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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)**9,x)
[Out]
Timed out
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