3.17.58 \(\int \frac {1}{(d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\)
Optimal. Leaf size=252 \[ \frac {256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {4 c d}{7 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {2}{7 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 252, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used =
{658, 614, 613} \begin {gather*} \frac {256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \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 )^{3/2}}+\frac {4 c d}{7 (d+e x) \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 )^{3/2}}+\frac {2}{7 (d+e x)^2 \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 )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
2/(7*(c*d^2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (4*c*d)/(7*(c*d^2 - a*e^2)^2
*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (32*c^2*d^2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(21*(c*d^
2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (256*c^3*d^3*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(21*
(c*d^2 - a*e^2)^6*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
Rule 613
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 614
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]
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 {1}{(d+e x)^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}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(10 c d) \int \frac {1}{(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}{7 \left (c d^2-a e^2\right )}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 c d}{7 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {\left (16 c^2 d^2\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{7 \left (c d^2-a e^2\right )^2}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 c d}{7 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {32 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{21 \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 )^{3/2}}-\frac {\left (128 c^3 d^3 e\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{21 \left (c d^2-a e^2\right )^4}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 c d}{7 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {32 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{21 \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 )^{3/2}}+\frac {256 c^3 d^3 e \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 259, normalized size = 1.03 \begin {gather*} \frac {2 \left (-3 a^5 e^{10}+3 a^4 c d e^8 (7 d+2 e x)-2 a^3 c^2 d^2 e^6 \left (35 d^2+28 d e x+8 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right )+3 a c^4 d^4 e^2 \left (35 d^4+280 d^3 e x+560 d^2 e^2 x^2+448 d e^3 x^3+128 e^4 x^4\right )+c^5 d^5 \left (-7 d^5+70 d^4 e x+560 d^3 e^2 x^2+1120 d^2 e^3 x^3+896 d e^4 x^4+256 e^5 x^5\right )\right )}{21 (d+e x)^2 \left (c d^2-a e^2\right )^6 ((d+e x) (a e+c d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
(2*(-3*a^5*e^10 + 3*a^4*c*d*e^8*(7*d + 2*e*x) - 2*a^3*c^2*d^2*e^6*(35*d^2 + 28*d*e*x + 8*e^2*x^2) + 6*a^2*c^3*
d^3*e^4*(35*d^3 + 70*d^2*e*x + 56*d*e^2*x^2 + 16*e^3*x^3) + 3*a*c^4*d^4*e^2*(35*d^4 + 280*d^3*e*x + 560*d^2*e^
2*x^2 + 448*d*e^3*x^3 + 128*e^4*x^4) + c^5*d^5*(-7*d^5 + 70*d^4*e*x + 560*d^3*e^2*x^2 + 1120*d^2*e^3*x^3 + 896
*d*e^4*x^4 + 256*e^5*x^5)))/(21*(c*d^2 - a*e^2)^6*(d + e*x)^2*((a*e + c*d*x)*(d + e*x))^(3/2))
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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[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
$Aborted
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fricas [B] time = 57.72, size = 1058, normalized size = 4.20 \begin {gather*} \frac {2 \, {\left (256 \, c^{5} d^{5} e^{5} x^{5} - 7 \, c^{5} d^{10} + 105 \, a c^{4} d^{8} e^{2} + 210 \, a^{2} c^{3} d^{6} e^{4} - 70 \, a^{3} c^{2} d^{4} e^{6} + 21 \, a^{4} c d^{2} e^{8} - 3 \, a^{5} e^{10} + 128 \, {\left (7 \, c^{5} d^{6} e^{4} + 3 \, a c^{4} d^{4} e^{6}\right )} x^{4} + 32 \, {\left (35 \, c^{5} d^{7} e^{3} + 42 \, a c^{4} d^{5} e^{5} + 3 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 16 \, {\left (35 \, c^{5} d^{8} e^{2} + 105 \, a c^{4} d^{6} e^{4} + 21 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \, {\left (35 \, c^{5} d^{9} e + 420 \, a c^{4} d^{7} e^{3} + 210 \, a^{2} c^{3} d^{5} e^{5} - 28 \, a^{3} c^{2} d^{3} e^{7} + 3 \, a^{4} c d e^{9}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{21 \, {\left (a^{2} c^{6} d^{16} e^{2} - 6 \, a^{3} c^{5} d^{14} e^{4} + 15 \, a^{4} c^{4} d^{12} e^{6} - 20 \, a^{5} c^{3} d^{10} e^{8} + 15 \, a^{6} c^{2} d^{8} e^{10} - 6 \, a^{7} c d^{6} e^{12} + a^{8} d^{4} e^{14} + {\left (c^{8} d^{14} e^{4} - 6 \, a c^{7} d^{12} e^{6} + 15 \, a^{2} c^{6} d^{10} e^{8} - 20 \, a^{3} c^{5} d^{8} e^{10} + 15 \, a^{4} c^{4} d^{6} e^{12} - 6 \, a^{5} c^{3} d^{4} e^{14} + a^{6} c^{2} d^{2} e^{16}\right )} x^{6} + 2 \, {\left (2 \, c^{8} d^{15} e^{3} - 11 \, a c^{7} d^{13} e^{5} + 24 \, a^{2} c^{6} d^{11} e^{7} - 25 \, a^{3} c^{5} d^{9} e^{9} + 10 \, a^{4} c^{4} d^{7} e^{11} + 3 \, a^{5} c^{3} d^{5} e^{13} - 4 \, a^{6} c^{2} d^{3} e^{15} + a^{7} c d e^{17}\right )} x^{5} + {\left (6 \, c^{8} d^{16} e^{2} - 28 \, a c^{7} d^{14} e^{4} + 43 \, a^{2} c^{6} d^{12} e^{6} - 6 \, a^{3} c^{5} d^{10} e^{8} - 55 \, a^{4} c^{4} d^{8} e^{10} + 64 \, a^{5} c^{3} d^{6} e^{12} - 27 \, a^{6} c^{2} d^{4} e^{14} + 2 \, a^{7} c d^{2} e^{16} + a^{8} e^{18}\right )} x^{4} + 4 \, {\left (c^{8} d^{17} e - 3 \, a c^{7} d^{15} e^{3} - 2 \, a^{2} c^{6} d^{13} e^{5} + 19 \, a^{3} c^{5} d^{11} e^{7} - 30 \, a^{4} c^{4} d^{9} e^{9} + 19 \, a^{5} c^{3} d^{7} e^{11} - 2 \, a^{6} c^{2} d^{5} e^{13} - 3 \, a^{7} c d^{3} e^{15} + a^{8} d e^{17}\right )} x^{3} + {\left (c^{8} d^{18} + 2 \, a c^{7} d^{16} e^{2} - 27 \, a^{2} c^{6} d^{14} e^{4} + 64 \, a^{3} c^{5} d^{12} e^{6} - 55 \, a^{4} c^{4} d^{10} e^{8} - 6 \, a^{5} c^{3} d^{8} e^{10} + 43 \, a^{6} c^{2} d^{6} e^{12} - 28 \, a^{7} c d^{4} e^{14} + 6 \, a^{8} d^{2} e^{16}\right )} x^{2} + 2 \, {\left (a c^{7} d^{17} e - 4 \, a^{2} c^{6} d^{15} e^{3} + 3 \, a^{3} c^{5} d^{13} e^{5} + 10 \, a^{4} c^{4} d^{11} e^{7} - 25 \, a^{5} c^{3} d^{9} e^{9} + 24 \, a^{6} c^{2} d^{7} e^{11} - 11 \, a^{7} c d^{5} e^{13} + 2 \, a^{8} d^{3} e^{15}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")
[Out]
2/21*(256*c^5*d^5*e^5*x^5 - 7*c^5*d^10 + 105*a*c^4*d^8*e^2 + 210*a^2*c^3*d^6*e^4 - 70*a^3*c^2*d^4*e^6 + 21*a^4
*c*d^2*e^8 - 3*a^5*e^10 + 128*(7*c^5*d^6*e^4 + 3*a*c^4*d^4*e^6)*x^4 + 32*(35*c^5*d^7*e^3 + 42*a*c^4*d^5*e^5 +
3*a^2*c^3*d^3*e^7)*x^3 + 16*(35*c^5*d^8*e^2 + 105*a*c^4*d^6*e^4 + 21*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*x^2 +
2*(35*c^5*d^9*e + 420*a*c^4*d^7*e^3 + 210*a^2*c^3*d^5*e^5 - 28*a^3*c^2*d^3*e^7 + 3*a^4*c*d*e^9)*x)*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a^2*c^6*d^16*e^2 - 6*a^3*c^5*d^14*e^4 + 15*a^4*c^4*d^12*e^6 - 20*a^5*c^3*d^1
0*e^8 + 15*a^6*c^2*d^8*e^10 - 6*a^7*c*d^6*e^12 + a^8*d^4*e^14 + (c^8*d^14*e^4 - 6*a*c^7*d^12*e^6 + 15*a^2*c^6*
d^10*e^8 - 20*a^3*c^5*d^8*e^10 + 15*a^4*c^4*d^6*e^12 - 6*a^5*c^3*d^4*e^14 + a^6*c^2*d^2*e^16)*x^6 + 2*(2*c^8*d
^15*e^3 - 11*a*c^7*d^13*e^5 + 24*a^2*c^6*d^11*e^7 - 25*a^3*c^5*d^9*e^9 + 10*a^4*c^4*d^7*e^11 + 3*a^5*c^3*d^5*e
^13 - 4*a^6*c^2*d^3*e^15 + a^7*c*d*e^17)*x^5 + (6*c^8*d^16*e^2 - 28*a*c^7*d^14*e^4 + 43*a^2*c^6*d^12*e^6 - 6*a
^3*c^5*d^10*e^8 - 55*a^4*c^4*d^8*e^10 + 64*a^5*c^3*d^6*e^12 - 27*a^6*c^2*d^4*e^14 + 2*a^7*c*d^2*e^16 + a^8*e^1
8)*x^4 + 4*(c^8*d^17*e - 3*a*c^7*d^15*e^3 - 2*a^2*c^6*d^13*e^5 + 19*a^3*c^5*d^11*e^7 - 30*a^4*c^4*d^9*e^9 + 19
*a^5*c^3*d^7*e^11 - 2*a^6*c^2*d^5*e^13 - 3*a^7*c*d^3*e^15 + a^8*d*e^17)*x^3 + (c^8*d^18 + 2*a*c^7*d^16*e^2 - 2
7*a^2*c^6*d^14*e^4 + 64*a^3*c^5*d^12*e^6 - 55*a^4*c^4*d^10*e^8 - 6*a^5*c^3*d^8*e^10 + 43*a^6*c^2*d^6*e^12 - 28
*a^7*c*d^4*e^14 + 6*a^8*d^2*e^16)*x^2 + 2*(a*c^7*d^17*e - 4*a^2*c^6*d^15*e^3 + 3*a^3*c^5*d^13*e^5 + 10*a^4*c^4
*d^11*e^7 - 25*a^5*c^3*d^9*e^9 + 24*a^6*c^2*d^7*e^11 - 11*a^7*c*d^5*e^13 + 2*a^8*d^3*e^15)*x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")
[Out]
sage0*x
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maple [A] time = 0.06, size = 412, normalized size = 1.63 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-256 c^{5} d^{5} e^{5} x^{5}-384 a \,c^{4} d^{4} e^{6} x^{4}-896 c^{5} d^{6} e^{4} x^{4}-96 a^{2} c^{3} d^{3} e^{7} x^{3}-1344 a \,c^{4} d^{5} e^{5} x^{3}-1120 c^{5} d^{7} e^{3} x^{3}+16 a^{3} c^{2} d^{2} e^{8} x^{2}-336 a^{2} c^{3} d^{4} e^{6} x^{2}-1680 a \,c^{4} d^{6} e^{4} x^{2}-560 c^{5} d^{8} e^{2} x^{2}-6 a^{4} c d \,e^{9} x +56 a^{3} c^{2} d^{3} e^{7} x -420 a^{2} c^{3} d^{5} e^{5} x -840 a \,c^{4} d^{7} e^{3} x -70 c^{5} d^{9} e x +3 a^{5} e^{10}-21 a^{4} c \,d^{2} e^{8}+70 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}-105 a \,c^{4} d^{8} e^{2}+7 c^{5} d^{10}\right )}{21 \left (e x +d \right ) \left (a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2),x)
[Out]
-2/21*(c*d*x+a*e)*(-256*c^5*d^5*e^5*x^5-384*a*c^4*d^4*e^6*x^4-896*c^5*d^6*e^4*x^4-96*a^2*c^3*d^3*e^7*x^3-1344*
a*c^4*d^5*e^5*x^3-1120*c^5*d^7*e^3*x^3+16*a^3*c^2*d^2*e^8*x^2-336*a^2*c^3*d^4*e^6*x^2-1680*a*c^4*d^6*e^4*x^2-5
60*c^5*d^8*e^2*x^2-6*a^4*c*d*e^9*x+56*a^3*c^2*d^3*e^7*x-420*a^2*c^3*d^5*e^5*x-840*a*c^4*d^7*e^3*x-70*c^5*d^9*e
*x+3*a^5*e^10-21*a^4*c*d^2*e^8+70*a^3*c^2*d^4*e^6-210*a^2*c^3*d^6*e^4-105*a*c^4*d^8*e^2+7*c^5*d^10)/(e*x+d)/(a
^6*e^12-6*a^5*c*d^2*e^10+15*a^4*c^2*d^4*e^8-20*a^3*c^3*d^6*e^6+15*a^2*c^4*d^8*e^4-6*a*c^5*d^10*e^2+c^6*d^12)/(
c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)
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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(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),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 = 3.10, size = 3654, normalized size = 14.50
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)
[Out]
(((d*((12*c^3*d^4*e^4)/(7*(a*e^2 - c*d^2)^3*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5)) -
(2*c^2*d^2*e^4*(19*a*e^2 - 7*c*d^2))/(7*(a*e^2 - c*d^2)^3*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^
2*c*d^2*e^5))))/e + (e^3*(14*c^3*d^5 - 42*a*c^2*d^3*e^2 + 40*a^2*c*d*e^4))/(7*(a*e^2 - c*d^2)^3*(5*a^3*e^7 - 5
*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3
- (((d*((24*c^4*d^5*e^4)/(35*(a*e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e)) - (8*c^3*d^3*e^4*(11*a*e^2 - 5*c*d^2))/
(35*(a*e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e))))/e + (2*c^2*d^2*e^3*(19*a^2*e^4 - 13*c^2*d^4 + 6*a*c*d^2*e^2))/(
35*(a*e^2 - c*d^2)^6*(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
4*c^4*d^5*e^2)/(35*(a*e^2 - c*d^2)^7) - (4*c^3*d^3*e^2*(47*a*e^2 - 29*c*d^2))/(105*(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) - (2*e^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((d
+ e*x)^4*(7*a^3*e^7 - 7*c^3*d^6*e + 21*a*c^2*d^4*e^3 - 21*a^2*c*d^2*e^5)) - ((x*((a*(((a*e^2 + c*d^2)*((8*c^7*
d^7*e^5*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^7*d^7*e^5*
(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (4*c^6*d
^6*e^4*(13*a^2*e^4 - 31*c^2*d^4 + 42*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*
d*e^5)) - (16*a*c^7*d^8*e^6)/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^6*d^6*e
^4*(a*e^2 + c*d^2)*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))
/c + ((a*e^2 + c*d^2)*((a*((8*c^7*d^7*e^5*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3
+ a^2*c*d*e^5)) - (16*c^7*d^7*e^5*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 +
a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((8*c^7*d^7*e^5*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*
(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^7*d^7*e^5*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c
^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (4*c^6*d^6*e^4*(13*a^2*e^4 - 31*c^2*d^4 + 42*a*c*d^2*e^
2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*a*c^7*d^8*e^6)/(35*(a*e^2 - c*d^2
)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^6*d^6*e^4*(a*e^2 + c*d^2)*(17*a*e^2 - 5*c*d^2))/(105*(
a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (8*c^5*d^5*e^3*(74*a^3*e^6 - 35*c^3*
d^6 + 198*a*c^2*d^4*e^2 - 261*a^2*c*d^2*e^4))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^
5)) + (2*c^5*d^5*e^3*(a*e^2 + c*d^2)*(13*a^2*e^4 - 31*c^2*d^4 + 42*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^
5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2*(70*c^6*d^10 - 420*a*c^5*d^8*e^2 + 1026*a^2*c
^4*d^6*e^4 - 1032*a^3*c^3*d^4*e^6 + 332*a^4*c^2*d^2*e^8))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3
+ a^2*c*d*e^5)) + (4*c^4*d^4*e^2*(a*e^2 + c*d^2)*(74*a^3*e^6 - 35*c^3*d^6 + 198*a*c^2*d^4*e^2 - 261*a^2*c*d^2*
e^4))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) + (a*((a*((8*c^7*d^7*e^5*(a*e^2 + c
*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^7*d^7*e^5*(17*a*e^2 - 5*c*d^
2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^
2)*((8*c^7*d^7*e^5*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c
^7*d^7*e^5*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e)
+ (4*c^6*d^6*e^4*(13*a^2*e^4 - 31*c^2*d^4 + 42*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e
^3 + a^2*c*d*e^5)) - (16*a*c^7*d^8*e^6)/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (
8*c^6*d^6*e^4*(a*e^2 + c*d^2)*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*
c*d*e^5))))/(c*d*e) - (8*c^5*d^5*e^3*(74*a^3*e^6 - 35*c^3*d^6 + 198*a*c^2*d^4*e^2 - 261*a^2*c*d^2*e^4))/(105*(
a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^5*d^5*e^3*(a*e^2 + c*d^2)*(13*a^2*e^4 - 3
1*c^2*d^4 + 42*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + (c*d*e*(
a*e^2 + c*d^2)*(70*c^6*d^10 - 420*a*c^5*d^8*e^2 + 1026*a^2*c^4*d^6*e^4 - 1032*a^3*c^3*d^4*e^6 + 332*a^4*c^2*d^
2*e^8))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*
e*x^2)^(1/2))/((a*e + c*d*x)^2*(d + e*x)^2) + ((x*(((a*e^2 + c*d^2)*((24*c^6*d^6*e^4*(a*e^2 + c*d^2))/(35*(a*e
^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (32*c^6*d^6*e^4*(7*a*e^2 - 4*c*d^2))/(35*(a*e^2 -
c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (4*c^5*d^5*e^3*(251*a^2*e^4 + 207*c^2*d^4 -
446*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (48*a*c^6*d^7*e^5)/(35
*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (16*c^5*d^5*e^3*(a*e^2 + c*d^2)*(7*a*e^2 - 4
*c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) + (a*((24*c^6*d^6*e^4*(a*e^2 + c*
d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (32*c^6*d^6*e^4*(7*a*e^2 - 4*c*d^2)
)/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + (2*c^4*d^4*e^2*(a*e^2 + c*d^2)*(251
*a^2*e^4 + 207*c^2*d^4 - 446*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))
*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((a*e + c*d*x)*(d + e*x))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
[Out]
Integral(1/(((d + e*x)*(a*e + c*d*x))**(5/2)*(d + e*x)**2), x)
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