\(\int \frac {1}{(d+e x)^{7/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2080]

Optimal result

Integrand size = 39, antiderivative size = 519 \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {1001 c^4 d^4 \sqrt {d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {1001 c^4 d^4 e}{128 \left (c d^2-a e^2\right )^6 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3003 c^5 d^5 e \sqrt {d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3003 c^5 d^5 e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{128 \left (c d^2-a e^2\right )^{15/2}} \]

[Out]

1/5/(-a*e^2+c*d^2)/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+13/40*c*d/(-a*e^2+c*d^2)^2/(e*x+d)^(5
/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+143/240*c^2*d^2/(-a*e^2+c*d^2)^3/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(3/2)+3003/128*c^5*d^5*e^(3/2)*arctan(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c
*d^2)^(1/2)/(e*x+d)^(1/2))/(-a*e^2+c*d^2)^(15/2)+429/320*c^3*d^3/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(3/2)/(e*x+d)^(1/2)-1001/320*c^4*d^4*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3
/2)-1001/128*c^4*d^4*e/(-a*e^2+c*d^2)^6/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3003/128*c^5*d^5
*e*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {686, 680, 674, 211} \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {3003 c^5 d^5 e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{128 \left (c d^2-a e^2\right )^{15/2}}+\frac {3003 c^5 d^5 e \sqrt {d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {1001 c^4 d^4 e}{128 \sqrt {d+e x} \left (c d^2-a e^2\right )^6 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {1001 c^4 d^4 \sqrt {d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {429 c^3 d^3}{320 \sqrt {d+e x} \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {143 c^2 d^2}{240 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {13 c d}{40 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {1}{5 (d+e x)^{7/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[In]

Int[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

1/(5*(c*d^2 - a*e^2)*(d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (13*c*d)/(40*(c*d^2 - a*
e^2)^2*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (143*c^2*d^2)/(240*(c*d^2 - a*e^2)^3*(
d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (429*c^3*d^3)/(320*(c*d^2 - a*e^2)^4*Sqrt[d +
e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (1001*c^4*d^4*Sqrt[d + e*x])/(320*(c*d^2 - a*e^2)^5*(a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (1001*c^4*d^4*e)/(128*(c*d^2 - a*e^2)^6*Sqrt[d + e*x]*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3003*c^5*d^5*e*Sqrt[d + e*x])/(128*(c*d^2 - a*e^2)^7*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2]) + (3003*c^5*d^5*e^(3/2)*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/
(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(128*(c*d^2 - a*e^2)^(15/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 680

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*c*d - b*e)*(d + e
*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(
b^2 - 4*a*c))), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ[0, m, 1] && IntegerQ[2*p]

Rule 686

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*((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, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(13 c d) \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{10 \left (c d^2-a e^2\right )} \\ & = \frac {1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {\left (143 c^2 d^2\right ) \int \frac {1}{(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}{80 \left (c d^2-a e^2\right )^2} \\ & = \frac {1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {\left (429 c^3 d^3\right ) \int \frac {1}{\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}{160 \left (c d^2-a e^2\right )^3} \\ & = \frac {1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt {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 (3003 c^4 d^4\right ) \int \frac {\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}{640 \left (c d^2-a e^2\right )^4} \\ & = \frac {1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt {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 {1001 c^4 d^4 \sqrt {d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {\left (1001 c^4 d^4 e\right ) \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{128 \left (c d^2-a e^2\right )^5} \\ & = \frac {1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt {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 {1001 c^4 d^4 \sqrt {d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {1001 c^4 d^4 e}{128 \left (c d^2-a e^2\right )^6 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (3003 c^5 d^5 e\right ) \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{256 \left (c d^2-a e^2\right )^6} \\ & = \frac {1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt {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 {1001 c^4 d^4 \sqrt {d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {1001 c^4 d^4 e}{128 \left (c d^2-a e^2\right )^6 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3003 c^5 d^5 e \sqrt {d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (3003 c^5 d^5 e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 \left (c d^2-a e^2\right )^7} \\ & = \frac {1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt {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 {1001 c^4 d^4 \sqrt {d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {1001 c^4 d^4 e}{128 \left (c d^2-a e^2\right )^6 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3003 c^5 d^5 e \sqrt {d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (3003 c^5 d^5 e^3\right ) \text {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{128 \left (c d^2-a e^2\right )^7} \\ & = \frac {1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt {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 {1001 c^4 d^4 \sqrt {d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {1001 c^4 d^4 e}{128 \left (c d^2-a e^2\right )^6 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3003 c^5 d^5 e \sqrt {d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3003 c^5 d^5 e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{128 \left (c d^2-a e^2\right )^{15/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.08 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {c^5 d^5 (d+e x)^{5/2} \left (\frac {(a e+c d x) \left (384 a^6 e^{12}-48 a^5 c d e^{10} (61 d+13 e x)+8 a^4 c^2 d^2 e^8 \left (1253 d^2+676 d e x+143 e^2 x^2\right )-2 a^3 c^3 d^3 e^6 \left (10535 d^3+11557 d^2 e x+6149 d e^2 x^2+1287 e^3 x^3\right )+3 a^2 c^4 d^4 e^4 \left (11865 d^4+26390 d^3 e x+28028 d^2 e^2 x^2+14586 d e^3 x^3+3003 e^4 x^4\right )+2 a c^5 d^5 e^2 \left (12160 d^5+96395 d^4 e x+232375 d^3 e^2 x^2+260403 d^2 e^3 x^3+141141 d e^4 x^4+30030 e^5 x^5\right )+c^6 d^6 \left (-1280 d^6+16640 d^5 e x+137995 d^4 e^2 x^2+338910 d^3 e^3 x^3+384384 d^2 e^4 x^4+210210 d e^5 x^5+45045 e^6 x^6\right )\right )}{c^5 d^5 \left (c d^2-a e^2\right )^7 (d+e x)^5}+\frac {45045 e^{3/2} (a e+c d x)^{5/2} \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{15/2}}\right )}{1920 ((a e+c d x) (d+e x))^{5/2}} \]

[In]

Integrate[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(c^5*d^5*(d + e*x)^(5/2)*(((a*e + c*d*x)*(384*a^6*e^12 - 48*a^5*c*d*e^10*(61*d + 13*e*x) + 8*a^4*c^2*d^2*e^8*(
1253*d^2 + 676*d*e*x + 143*e^2*x^2) - 2*a^3*c^3*d^3*e^6*(10535*d^3 + 11557*d^2*e*x + 6149*d*e^2*x^2 + 1287*e^3
*x^3) + 3*a^2*c^4*d^4*e^4*(11865*d^4 + 26390*d^3*e*x + 28028*d^2*e^2*x^2 + 14586*d*e^3*x^3 + 3003*e^4*x^4) + 2
*a*c^5*d^5*e^2*(12160*d^5 + 96395*d^4*e*x + 232375*d^3*e^2*x^2 + 260403*d^2*e^3*x^3 + 141141*d*e^4*x^4 + 30030
*e^5*x^5) + c^6*d^6*(-1280*d^6 + 16640*d^5*e*x + 137995*d^4*e^2*x^2 + 338910*d^3*e^3*x^3 + 384384*d^2*e^4*x^4
+ 210210*d*e^5*x^5 + 45045*e^6*x^6)))/(c^5*d^5*(c*d^2 - a*e^2)^7*(d + e*x)^5) + (45045*e^(3/2)*(a*e + c*d*x)^(
5/2)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(15/2)))/(1920*((a*e + c*d*x)*(d
+ e*x))^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1542\) vs. \(2(463)=926\).

Time = 2.80 (sec) , antiderivative size = 1543, normalized size of antiderivative = 2.97

method	result	size
default	\(\text {Expression too large to display}\)	\(1543\)

[In]

int(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/1920*((c*d*x+a*e)*(e*x+d))^(1/2)*(-60060*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d^5*e^7*x^5-192790*((a*e^2-c*d^2)*e)^
(1/2)*a*c^5*d^9*e^3*x+2928*((a*e^2-c*d^2)*e)^(1/2)*a^5*c*d^2*e^10-10024*((a*e^2-c*d^2)*e)^(1/2)*a^4*c^2*d^4*e^
8+21070*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^3*d^6*e^6-35595*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^4*d^8*e^4+45045*arctanh(e*
(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^5*d^5*e^8*x^5*(c*d*x+a*e)^(1/2)+225225*arctanh(e*(c*d*x+a*e)^(1
/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^5*d^6*e^7*x^4*(c*d*x+a*e)^(1/2)+450450*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*
d^2)*e)^(1/2))*a*c^5*d^7*e^6*x^3*(c*d*x+a*e)^(1/2)+450450*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))
*a*c^5*d^8*e^5*x^2*(c*d*x+a*e)^(1/2)+225225*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^5*d^9*e^4
*x*(c*d*x+a*e)^(1/2)-282282*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d^6*e^6*x^4-520806*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d^7
*e^5*x^3-464750*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d^8*e^4*x^2-45045*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^6*e^6*x^6+2574*(
(a*e^2-c*d^2)*e)^(1/2)*a^3*c^3*d^3*e^9*x^3-43758*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^4*d^5*e^7*x^3-1144*((a*e^2-c*d^
2)*e)^(1/2)*a^4*c^2*d^2*e^10*x^2+12298*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^3*d^4*e^8*x^2-9009*((a*e^2-c*d^2)*e)^(1/2
)*a^2*c^4*d^4*e^8*x^4+1280*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^12-384*((a*e^2-c*d^2)*e)^(1/2)*a^6*e^12+23114*((a*e^2
-c*d^2)*e)^(1/2)*a^3*c^3*d^5*e^7*x-79170*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^4*d^7*e^5*x-210210*((a*e^2-c*d^2)*e)^(1
/2)*c^6*d^7*e^5*x^5-384384*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^8*e^4*x^4-338910*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^9*e^3*
x^3-137995*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^10*e^2*x^2-16640*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^11*e*x-24320*((a*e^2-c
*d^2)*e)^(1/2)*a*c^5*d^10*e^2-84084*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^4*d^6*e^6*x^2+624*((a*e^2-c*d^2)*e)^(1/2)*a^
5*c*d*e^11*x-5408*((a*e^2-c*d^2)*e)^(1/2)*a^4*c^2*d^3*e^9*x+450450*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*
e)^(1/2))*c^6*d^9*e^4*x^3*(c*d*x+a*e)^(1/2)+225225*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^6*d^
10*e^3*x^2*(c*d*x+a*e)^(1/2)+45045*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^6*d^11*e^2*x*(c*d*x+
a*e)^(1/2)+45045*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^5*d^10*e^3*(c*d*x+a*e)^(1/2)+45045*a
rctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^6*d^6*e^7*x^6*(c*d*x+a*e)^(1/2)+225225*arctanh(e*(c*d*x+
a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^6*d^7*e^6*x^5*(c*d*x+a*e)^(1/2)+450450*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e
^2-c*d^2)*e)^(1/2))*c^6*d^8*e^5*x^4*(c*d*x+a*e)^(1/2))/(e*x+d)^(11/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^7/((a*e^2-c*
d^2)*e)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1926 vs. \(2 (463) = 926\).

Time = 6.25 (sec) , antiderivative size = 3874, normalized size of antiderivative = 7.46 \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/3840*(45045*(c^7*d^7*e^7*x^8 + a^2*c^5*d^11*e^3 + 2*(3*c^7*d^8*e^6 + a*c^6*d^6*e^8)*x^7 + (15*c^7*d^9*e^5 +
12*a*c^6*d^7*e^7 + a^2*c^5*d^5*e^9)*x^6 + 2*(10*c^7*d^10*e^4 + 15*a*c^6*d^8*e^6 + 3*a^2*c^5*d^6*e^8)*x^5 + 5*
(3*c^7*d^11*e^3 + 8*a*c^6*d^9*e^5 + 3*a^2*c^5*d^7*e^7)*x^4 + 2*(3*c^7*d^12*e^2 + 15*a*c^6*d^10*e^4 + 10*a^2*c^
5*d^8*e^6)*x^3 + (c^7*d^13*e + 12*a*c^6*d^11*e^3 + 15*a^2*c^5*d^9*e^5)*x^2 + 2*(a*c^6*d^12*e^2 + 3*a^2*c^5*d^1
0*e^4)*x)*sqrt(-e/(c*d^2 - a*e^2))*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 + 2*sqrt(c*d*e*x^2 + a*d*
e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*
(45045*c^6*d^6*e^6*x^6 - 1280*c^6*d^12 + 24320*a*c^5*d^10*e^2 + 35595*a^2*c^4*d^8*e^4 - 21070*a^3*c^3*d^6*e^6
+ 10024*a^4*c^2*d^4*e^8 - 2928*a^5*c*d^2*e^10 + 384*a^6*e^12 + 30030*(7*c^6*d^7*e^5 + 2*a*c^5*d^5*e^7)*x^5 + 3
003*(128*c^6*d^8*e^4 + 94*a*c^5*d^6*e^6 + 3*a^2*c^4*d^4*e^8)*x^4 + 858*(395*c^6*d^9*e^3 + 607*a*c^5*d^7*e^5 +
51*a^2*c^4*d^5*e^7 - 3*a^3*c^3*d^3*e^9)*x^3 + 143*(965*c^6*d^10*e^2 + 3250*a*c^5*d^8*e^4 + 588*a^2*c^4*d^6*e^6
- 86*a^3*c^3*d^4*e^8 + 8*a^4*c^2*d^2*e^10)*x^2 + 26*(640*c^6*d^11*e + 7415*a*c^5*d^9*e^3 + 3045*a^2*c^4*d^7*e
^5 - 889*a^3*c^3*d^5*e^7 + 208*a^4*c^2*d^3*e^9 - 24*a^5*c*d*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*
x)*sqrt(e*x + d))/(a^2*c^7*d^20*e^2 - 7*a^3*c^6*d^18*e^4 + 21*a^4*c^5*d^16*e^6 - 35*a^5*c^4*d^14*e^8 + 35*a^6*
c^3*d^12*e^10 - 21*a^7*c^2*d^10*e^12 + 7*a^8*c*d^8*e^14 - a^9*d^6*e^16 + (c^9*d^16*e^6 - 7*a*c^8*d^14*e^8 + 21
*a^2*c^7*d^12*e^10 - 35*a^3*c^6*d^10*e^12 + 35*a^4*c^5*d^8*e^14 - 21*a^5*c^4*d^6*e^16 + 7*a^6*c^3*d^4*e^18 - a
^7*c^2*d^2*e^20)*x^8 + 2*(3*c^9*d^17*e^5 - 20*a*c^8*d^15*e^7 + 56*a^2*c^7*d^13*e^9 - 84*a^3*c^6*d^11*e^11 + 70
*a^4*c^5*d^9*e^13 - 28*a^5*c^4*d^7*e^15 + 4*a^7*c^2*d^3*e^19 - a^8*c*d*e^21)*x^7 + (15*c^9*d^18*e^4 - 93*a*c^8
*d^16*e^6 + 232*a^2*c^7*d^14*e^8 - 280*a^3*c^6*d^12*e^10 + 126*a^4*c^5*d^10*e^12 + 70*a^5*c^4*d^8*e^14 - 112*a
^6*c^3*d^6*e^16 + 48*a^7*c^2*d^4*e^18 - 5*a^8*c*d^2*e^20 - a^9*e^22)*x^6 + 2*(10*c^9*d^19*e^3 - 55*a*c^8*d^17*
e^5 + 108*a^2*c^7*d^15*e^7 - 56*a^3*c^6*d^13*e^9 - 112*a^4*c^5*d^11*e^11 + 210*a^5*c^4*d^9*e^13 - 140*a^6*c^3*
d^7*e^15 + 32*a^7*c^2*d^5*e^17 + 6*a^8*c*d^3*e^19 - 3*a^9*d*e^21)*x^5 + 5*(3*c^9*d^20*e^2 - 13*a*c^8*d^18*e^4
+ 10*a^2*c^7*d^16*e^6 + 42*a^3*c^6*d^14*e^8 - 112*a^4*c^5*d^12*e^10 + 112*a^5*c^4*d^10*e^12 - 42*a^6*c^3*d^8*e
^14 - 10*a^7*c^2*d^6*e^16 + 13*a^8*c*d^4*e^18 - 3*a^9*d^2*e^20)*x^4 + 2*(3*c^9*d^21*e - 6*a*c^8*d^19*e^3 - 32*
a^2*c^7*d^17*e^5 + 140*a^3*c^6*d^15*e^7 - 210*a^4*c^5*d^13*e^9 + 112*a^5*c^4*d^11*e^11 + 56*a^6*c^3*d^9*e^13 -
108*a^7*c^2*d^7*e^15 + 55*a^8*c*d^5*e^17 - 10*a^9*d^3*e^19)*x^3 + (c^9*d^22 + 5*a*c^8*d^20*e^2 - 48*a^2*c^7*d
^18*e^4 + 112*a^3*c^6*d^16*e^6 - 70*a^4*c^5*d^14*e^8 - 126*a^5*c^4*d^12*e^10 + 280*a^6*c^3*d^10*e^12 - 232*a^7
*c^2*d^8*e^14 + 93*a^8*c*d^6*e^16 - 15*a^9*d^4*e^18)*x^2 + 2*(a*c^8*d^21*e - 4*a^2*c^7*d^19*e^3 + 28*a^4*c^5*d
^15*e^7 - 70*a^5*c^4*d^13*e^9 + 84*a^6*c^3*d^11*e^11 - 56*a^7*c^2*d^9*e^13 + 20*a^8*c*d^7*e^15 - 3*a^9*d^5*e^1
7)*x), 1/1920*(45045*(c^7*d^7*e^7*x^8 + a^2*c^5*d^11*e^3 + 2*(3*c^7*d^8*e^6 + a*c^6*d^6*e^8)*x^7 + (15*c^7*d^9
*e^5 + 12*a*c^6*d^7*e^7 + a^2*c^5*d^5*e^9)*x^6 + 2*(10*c^7*d^10*e^4 + 15*a*c^6*d^8*e^6 + 3*a^2*c^5*d^6*e^8)*x^
5 + 5*(3*c^7*d^11*e^3 + 8*a*c^6*d^9*e^5 + 3*a^2*c^5*d^7*e^7)*x^4 + 2*(3*c^7*d^12*e^2 + 15*a*c^6*d^10*e^4 + 10*
a^2*c^5*d^8*e^6)*x^3 + (c^7*d^13*e + 12*a*c^6*d^11*e^3 + 15*a^2*c^5*d^9*e^5)*x^2 + 2*(a*c^6*d^12*e^2 + 3*a^2*c
^5*d^10*e^4)*x)*sqrt(e/(c*d^2 - a*e^2))*arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sq
rt(e*x + d)*sqrt(e/(c*d^2 - a*e^2))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) + (45045*c^6*d^6*e^6*x^6 -
1280*c^6*d^12 + 24320*a*c^5*d^10*e^2 + 35595*a^2*c^4*d^8*e^4 - 21070*a^3*c^3*d^6*e^6 + 10024*a^4*c^2*d^4*e^8 -
2928*a^5*c*d^2*e^10 + 384*a^6*e^12 + 30030*(7*c^6*d^7*e^5 + 2*a*c^5*d^5*e^7)*x^5 + 3003*(128*c^6*d^8*e^4 + 94
*a*c^5*d^6*e^6 + 3*a^2*c^4*d^4*e^8)*x^4 + 858*(395*c^6*d^9*e^3 + 607*a*c^5*d^7*e^5 + 51*a^2*c^4*d^5*e^7 - 3*a^
3*c^3*d^3*e^9)*x^3 + 143*(965*c^6*d^10*e^2 + 3250*a*c^5*d^8*e^4 + 588*a^2*c^4*d^6*e^6 - 86*a^3*c^3*d^4*e^8 + 8
*a^4*c^2*d^2*e^10)*x^2 + 26*(640*c^6*d^11*e + 7415*a*c^5*d^9*e^3 + 3045*a^2*c^4*d^7*e^5 - 889*a^3*c^3*d^5*e^7
+ 208*a^4*c^2*d^3*e^9 - 24*a^5*c*d*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^
7*d^20*e^2 - 7*a^3*c^6*d^18*e^4 + 21*a^4*c^5*d^16*e^6 - 35*a^5*c^4*d^14*e^8 + 35*a^6*c^3*d^12*e^10 - 21*a^7*c^
2*d^10*e^12 + 7*a^8*c*d^8*e^14 - a^9*d^6*e^16 + (c^9*d^16*e^6 - 7*a*c^8*d^14*e^8 + 21*a^2*c^7*d^12*e^10 - 35*a
^3*c^6*d^10*e^12 + 35*a^4*c^5*d^8*e^14 - 21*a^5*c^4*d^6*e^16 + 7*a^6*c^3*d^4*e^18 - a^7*c^2*d^2*e^20)*x^8 + 2*
(3*c^9*d^17*e^5 - 20*a*c^8*d^15*e^7 + 56*a^2*c^7*d^13*e^9 - 84*a^3*c^6*d^11*e^11 + 70*a^4*c^5*d^9*e^13 - 28*a^
5*c^4*d^7*e^15 + 4*a^7*c^2*d^3*e^19 - a^8*c*d*e^21)*x^7 + (15*c^9*d^18*e^4 - 93*a*c^8*d^16*e^6 + 232*a^2*c^7*d
^14*e^8 - 280*a^3*c^6*d^12*e^10 + 126*a^4*c^5*d^10*e^12 + 70*a^5*c^4*d^8*e^14 - 112*a^6*c^3*d^6*e^16 + 48*a^7*
c^2*d^4*e^18 - 5*a^8*c*d^2*e^20 - a^9*e^22)*x^6 + 2*(10*c^9*d^19*e^3 - 55*a*c^8*d^17*e^5 + 108*a^2*c^7*d^15*e^
7 - 56*a^3*c^6*d^13*e^9 - 112*a^4*c^5*d^11*e^11 + 210*a^5*c^4*d^9*e^13 - 140*a^6*c^3*d^7*e^15 + 32*a^7*c^2*d^5
*e^17 + 6*a^8*c*d^3*e^19 - 3*a^9*d*e^21)*x^5 + 5*(3*c^9*d^20*e^2 - 13*a*c^8*d^18*e^4 + 10*a^2*c^7*d^16*e^6 + 4
2*a^3*c^6*d^14*e^8 - 112*a^4*c^5*d^12*e^10 + 112*a^5*c^4*d^10*e^12 - 42*a^6*c^3*d^8*e^14 - 10*a^7*c^2*d^6*e^16
+ 13*a^8*c*d^4*e^18 - 3*a^9*d^2*e^20)*x^4 + 2*(3*c^9*d^21*e - 6*a*c^8*d^19*e^3 - 32*a^2*c^7*d^17*e^5 + 140*a^
3*c^6*d^15*e^7 - 210*a^4*c^5*d^13*e^9 + 112*a^5*c^4*d^11*e^11 + 56*a^6*c^3*d^9*e^13 - 108*a^7*c^2*d^7*e^15 + 5
5*a^8*c*d^5*e^17 - 10*a^9*d^3*e^19)*x^3 + (c^9*d^22 + 5*a*c^8*d^20*e^2 - 48*a^2*c^7*d^18*e^4 + 112*a^3*c^6*d^1
6*e^6 - 70*a^4*c^5*d^14*e^8 - 126*a^5*c^4*d^12*e^10 + 280*a^6*c^3*d^10*e^12 - 232*a^7*c^2*d^8*e^14 + 93*a^8*c*
d^6*e^16 - 15*a^9*d^4*e^18)*x^2 + 2*(a*c^8*d^21*e - 4*a^2*c^7*d^19*e^3 + 28*a^4*c^5*d^15*e^7 - 70*a^5*c^4*d^13
*e^9 + 84*a^6*c^3*d^11*e^11 - 56*a^7*c^2*d^9*e^13 + 20*a^8*c*d^7*e^15 - 3*a^9*d^5*e^17)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(7/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1056 vs. \(2 (463) = 926\).

Time = 0.68 (sec) , antiderivative size = 1056, normalized size of antiderivative = 2.03 \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/1920*(45045*c^5*d^5*e*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))/((c^7*d^14*abs(e
) - 7*a*c^6*d^12*e^2*abs(e) + 21*a^2*c^5*d^10*e^4*abs(e) - 35*a^3*c^4*d^8*e^6*abs(e) + 35*a^4*c^3*d^6*e^8*abs(
e) - 21*a^5*c^2*d^4*e^10*abs(e) + 7*a^6*c*d^2*e^12*abs(e) - a^7*e^14*abs(e))*sqrt(c*d^2*e - a*e^3)) - 1280*(c^
6*d^7*e^2 - a*c^5*d^5*e^4 - 18*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^5*d^5*e)/((c^7*d^14*abs(e) - 7*a*c^6*d^12
*e^2*abs(e) + 21*a^2*c^5*d^10*e^4*abs(e) - 35*a^3*c^4*d^8*e^6*abs(e) + 35*a^4*c^3*d^6*e^8*abs(e) - 21*a^5*c^2*
d^4*e^10*abs(e) + 7*a^6*c*d^2*e^12*abs(e) - a^7*e^14*abs(e))*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)) + (355
95*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^9*d^13*e^5 - 142380*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^8
*d^11*e^7 + 213570*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^7*d^9*e^9 - 142380*sqrt((e*x + d)*c*d*e - c*d
^2*e + a*e^3)*a^3*c^6*d^7*e^11 + 35595*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^4*c^5*d^5*e^13 + 121310*((e*x
+ d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^8*d^11*e^4 - 363930*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^7*d^9
*e^6 + 363930*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^6*d^7*e^8 - 121310*((e*x + d)*c*d*e - c*d^2*e +
a*e^3)^(3/2)*a^3*c^5*d^5*e^10 + 160384*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^7*d^9*e^3 - 320768*((e*x +
d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^6*d^7*e^5 + 160384*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*c^5*d^5
*e^7 + 96290*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^6*d^7*e^2 - 96290*((e*x + d)*c*d*e - c*d^2*e + a*e^3)
^(7/2)*a*c^5*d^5*e^4 + 22005*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*c^5*d^5*e)/((c^7*d^14*abs(e) - 7*a*c^6*
d^12*e^2*abs(e) + 21*a^2*c^5*d^10*e^4*abs(e) - 35*a^3*c^4*d^8*e^6*abs(e) + 35*a^4*c^3*d^6*e^8*abs(e) - 21*a^5*
c^2*d^4*e^10*abs(e) + 7*a^6*c*d^2*e^12*abs(e) - a^7*e^14*abs(e))*(e*x + d)^5*c^5*d^5*e^5))*e^2

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \]

[In]

int(1/((d + e*x)^(7/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)

[Out]

int(1/((d + e*x)^(7/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)), x)