3.17.16 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=231 \[ \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^5 \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )} \]

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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 )^{5/2}}{1155 (d+e x)^5 \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 )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \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 )^{5/2}}{11 (d+e x)^8 \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)^(3/2)/(d + e*x)^8,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^8) + (4*c*d*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(33*(c*d^2 - a*e^2)^2*(d + e*x)^7) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(5/2))/(231*(c*d^2 - a*e^2)^3*(d + e*x)^6) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)
)/(1155*(c*d^2 - a*e^2)^4*(d + e*x)^5)

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 )^{3/2}}{(d+e x)^8} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\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 )^{3/2}}{(d+e x)^7} \, 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 )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\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 )^{3/2}}{(d+e x)^6} \, dx}{33 \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 )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\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 )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\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 )^{3/2}}{(d+e x)^5} \, dx}{231 \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 )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\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 )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\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 )^{5/2}}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^5}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 138, normalized size = 0.60 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{5/2} \left (-105 a^3 e^6+35 a^2 c d e^4 (11 d+2 e x)-5 a c^2 d^2 e^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )+c^3 d^3 \left (231 d^3+198 d^2 e x+88 d e^2 x^2+16 e^3 x^3\right )\right )}{1155 (d+e x)^8 \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)^(3/2)/(d + e*x)^8,x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-105*a^3*e^6 + 35*a^2*c*d*e^4*(11*d + 2*e*x) - 5*a*c^2*d^2*e^2*(99*d^2 + 4
4*d*e*x + 8*e^2*x^2) + c^3*d^3*(231*d^3 + 198*d^2*e*x + 88*d*e^2*x^2 + 16*e^3*x^3)))/(1155*(c*d^2 - a*e^2)^4*(
d + e*x)^8)

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IntegrateAlgebraic [F]  time = 180.07, 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)^(3/2)/(d + e*x)^8,x]

[Out]

$Aborted

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fricas [B]  time = 27.33, size = 699, normalized size = 3.03 \begin {gather*} \frac {2 \, {\left (16 \, c^{5} d^{5} e^{3} x^{5} + 231 \, a^{2} c^{3} d^{6} e^{2} - 495 \, a^{3} c^{2} d^{4} e^{4} + 385 \, a^{4} c d^{2} e^{6} - 105 \, a^{5} e^{8} + 8 \, {\left (11 \, c^{5} d^{6} e^{2} - a c^{4} d^{4} e^{4}\right )} x^{4} + 2 \, {\left (99 \, c^{5} d^{7} e - 22 \, a c^{4} d^{5} e^{3} + 3 \, a^{2} c^{3} d^{3} e^{5}\right )} x^{3} + {\left (231 \, c^{5} d^{8} - 99 \, a c^{4} d^{6} e^{2} + 33 \, a^{2} c^{3} d^{4} e^{4} - 5 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (231 \, a c^{4} d^{7} e - 396 \, a^{2} c^{3} d^{5} e^{3} + 275 \, a^{3} c^{2} d^{3} e^{5} - 70 \, a^{4} c d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{1155 \, {\left (c^{4} d^{14} - 4 \, a c^{3} d^{12} e^{2} + 6 \, a^{2} c^{2} d^{10} e^{4} - 4 \, a^{3} c d^{8} e^{6} + a^{4} d^{6} e^{8} + {\left (c^{4} d^{8} e^{6} - 4 \, a c^{3} d^{6} e^{8} + 6 \, a^{2} c^{2} d^{4} e^{10} - 4 \, a^{3} c d^{2} e^{12} + a^{4} e^{14}\right )} x^{6} + 6 \, {\left (c^{4} d^{9} e^{5} - 4 \, a c^{3} d^{7} e^{7} + 6 \, a^{2} c^{2} d^{5} e^{9} - 4 \, a^{3} c d^{3} e^{11} + a^{4} d e^{13}\right )} x^{5} + 15 \, {\left (c^{4} d^{10} e^{4} - 4 \, a c^{3} d^{8} e^{6} + 6 \, a^{2} c^{2} d^{6} e^{8} - 4 \, a^{3} c d^{4} e^{10} + a^{4} d^{2} e^{12}\right )} x^{4} + 20 \, {\left (c^{4} d^{11} e^{3} - 4 \, a c^{3} d^{9} e^{5} + 6 \, a^{2} c^{2} d^{7} e^{7} - 4 \, a^{3} c d^{5} e^{9} + a^{4} d^{3} e^{11}\right )} x^{3} + 15 \, {\left (c^{4} d^{12} e^{2} - 4 \, a c^{3} d^{10} e^{4} + 6 \, a^{2} c^{2} d^{8} e^{6} - 4 \, a^{3} c d^{6} e^{8} + a^{4} d^{4} e^{10}\right )} x^{2} + 6 \, {\left (c^{4} d^{13} e - 4 \, a c^{3} d^{11} e^{3} + 6 \, a^{2} c^{2} d^{9} e^{5} - 4 \, a^{3} c d^{7} e^{7} + a^{4} d^{5} 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)^(3/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

2/1155*(16*c^5*d^5*e^3*x^5 + 231*a^2*c^3*d^6*e^2 - 495*a^3*c^2*d^4*e^4 + 385*a^4*c*d^2*e^6 - 105*a^5*e^8 + 8*(
11*c^5*d^6*e^2 - a*c^4*d^4*e^4)*x^4 + 2*(99*c^5*d^7*e - 22*a*c^4*d^5*e^3 + 3*a^2*c^3*d^3*e^5)*x^3 + (231*c^5*d
^8 - 99*a*c^4*d^6*e^2 + 33*a^2*c^3*d^4*e^4 - 5*a^3*c^2*d^2*e^6)*x^2 + 2*(231*a*c^4*d^7*e - 396*a^2*c^3*d^5*e^3
+ 275*a^3*c^2*d^3*e^5 - 70*a^4*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^4*d^14 - 4*a*c^3*d^
12*e^2 + 6*a^2*c^2*d^10*e^4 - 4*a^3*c*d^8*e^6 + a^4*d^6*e^8 + (c^4*d^8*e^6 - 4*a*c^3*d^6*e^8 + 6*a^2*c^2*d^4*e
^10 - 4*a^3*c*d^2*e^12 + a^4*e^14)*x^6 + 6*(c^4*d^9*e^5 - 4*a*c^3*d^7*e^7 + 6*a^2*c^2*d^5*e^9 - 4*a^3*c*d^3*e^
11 + a^4*d*e^13)*x^5 + 15*(c^4*d^10*e^4 - 4*a*c^3*d^8*e^6 + 6*a^2*c^2*d^6*e^8 - 4*a^3*c*d^4*e^10 + a^4*d^2*e^1
2)*x^4 + 20*(c^4*d^11*e^3 - 4*a*c^3*d^9*e^5 + 6*a^2*c^2*d^7*e^7 - 4*a^3*c*d^5*e^9 + a^4*d^3*e^11)*x^3 + 15*(c^
4*d^12*e^2 - 4*a*c^3*d^10*e^4 + 6*a^2*c^2*d^8*e^6 - 4*a^3*c*d^6*e^8 + a^4*d^4*e^10)*x^2 + 6*(c^4*d^13*e - 4*a*
c^3*d^11*e^3 + 6*a^2*c^2*d^9*e^5 - 4*a^3*c*d^7*e^7 + a^4*d^5*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)^(3/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.05, 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}+40 a \,c^{2} d^{2} e^{4} x^{2}-88 c^{3} d^{4} e^{2} x^{2}-70 a^{2} c d \,e^{5} x +220 a \,c^{2} d^{3} e^{3} x -198 c^{3} d^{5} e x +105 a^{3} e^{6}-385 a^{2} c \,d^{2} e^{4}+495 a \,c^{2} d^{4} e^{2}-231 c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{1155 \left (e x +d \right )^{7} \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)^(3/2)/(e*x+d)^8,x)

[Out]

-2/1155*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+40*a*c^2*d^2*e^4*x^2-88*c^3*d^4*e^2*x^2-70*a^2*c*d*e^5*x+220*a*c^2*d^
3*e^3*x-198*c^3*d^5*e*x+105*a^3*e^6-385*a^2*c*d^2*e^4+495*a*c^2*d^4*e^2-231*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d^2*
x+a*d*e)^(3/2)/(e*x+d)^7/(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)^(3/2)/(e*x+d)^8,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 = 5.15, size = 2657, normalized size = 11.50

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)^(3/2)/(d + e*x)^8,x)

[Out]

(((d*((4*c^3*d^4)/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e)) - (2*c^2*d^2*(5*a*e^2 - c*d^2))/(11*(a*e^2 - c*d^
2)*(9*a*e^3 - 9*c*d^2*e))))/e + (2*a*c^2*d^3*e^2 - 2*c^3*d^5 + 4*a^2*c*d*e^4)/(11*e*(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 - (((228*c^4*d^5 - 284*a*c^3*d^3*e^2)
/(693*e*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) + (8*c^4*d^5)/(99*e*(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*((16*c^5*d^6)/(693*(a*e^2 - c*d^2)^3*(5*a*e
^3 - 5*c*d^2*e)) - (16*c^4*d^4*(17*a*e^2 - 15*c*d^2))/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d^2*e))))/e + (16*
a*c^3*d^3*e*(16*a*e^2 - 15*c*d^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*((32*c^6*d^7)/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (64*c^5*d
^5*(10*a*e^2 - 9*c*d^2))/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e))))/e + (32*a*c^4*d^4*e*(19*a*e^2 - 18*c
*d^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 - (((22*c^3*d^4 - 58*a*c^2*d^2*e^2)/(99*e*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) + (4*c^3*d^4)/(11*e*(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 - (((32*c^6*d^7)
/(3465*e^2*(a*e^2 - c*d^2)^5) - (16*c^5*d^5*(71*a*e^2 - 65*c*d^2))/(10395*e^2*(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*((88*c^4*d^5 - 104*a*c^3*d^3*e^2)/(99*(a*e^2 - c*d^2)^2*(7
*a*e^3 - 7*c*d^2*e)) + (8*c^4*d^5)/(99*(a*e^2 - c*d^2)^2*(7*a*e^3 - 7*c*d^2*e))))/e + (8*a*c^2*d^2*e*(12*a*e^2
- 11*c*d^2))/(99*(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*((64*c^7*d^8)/(10395*e*(a*e^2 - c*d^2)^6) - (32*c^6*d^6*(41*a*e^2 - 37*c*d^2))/(10395*e*(a*e^2
- c*d^2)^6)))/e + (32*c^5*d^5*(20*a^2*e^4 - 19*c^2*d^4 + a*c*d^2*e^2))/(10395*e^2*(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*((16*c^5*d^6)/(693*(a*e^2 - c*d^2)^3*(5*a*e^3 - 5*c*d
^2*e)) - (8*c^4*d^4*(27*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 + (8*c^3*d^3*(13*
a^2*e^4 - 12*c^2*d^4 + a*c*d^2*e^2))/(693*e*(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^6*d^7)/(3465*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (16*c
^5*d^5*(35*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 + (16*c^4*d^4*(17*a^2*e^4 - 1
6*c^2*d^4 + a*c*d^2*e^2))/(3465*e*(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 - (((2*a^2*e^3)/(11*a*e^3 - 11*c*d^2*e) + (d*((2*c^2*d^3)/(11*a*e^3 - 11*c*d^2*e) - (
4*a*c*d*e^2)/(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 - (((16*c
^5*d^6)/(693*e*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (8*c^4*d^4*(83*a*e^2 - 73*c*d^2))/(3465*e*(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 - (((d*((64*c^7*d^8
)/(10395*e*(a*e^2 - c*d^2)^6) - (128*c^6*d^6*(11*a*e^2 - 10*c*d^2))/(10395*e*(a*e^2 - c*d^2)^6)))/e + (64*a*c^
5*d^5*(21*a*e^2 - 20*c*d^2))/(10395*(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*((24*c^3*d^4 - 32*a*c^2*d^2*e^2)/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e)) + (4*c^3*d^4)/(11*(a*e^2
- c*d^2)*(9*a*e^3 - 9*c*d^2*e))))/e + (4*a*c*d*e*(7*a*e^2 - 6*c*d^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 + (((d*((8*c^4*d^5)/(99*(a*e^2 - c*d^2)^2*(7*a
*e^3 - 7*c*d^2*e)) - (4*c^3*d^3*(17*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*a
*c^3*d^4*e^2 - 28*c^4*d^6 + 32*a^2*c^2*d^2*e^4)/(99*e*(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 - (16*c^5*d^5*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(115
5*e^2*(a*e^2 - c*d^2)^4*(d + e*x)) - (472*c^4*d^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(3465*e*(a*e^
2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)*(d + e*x)^2) - (32*c^3*d^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(
231*e*(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]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{8}}\, dx \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)**(3/2)/(e*x+d)**8,x)

[Out]

Integral(((d + e*x)*(a*e + c*d*x))**(3/2)/(d + e*x)**8, x)

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