3.21.79 \(\int \frac {1}{(d+e x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2079]

Optimal. Leaf size=457 \[ \frac {1}{4 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {11 c d}{24 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {77 c^3 d^3 \sqrt {d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {385 c^3 d^3 e}{64 \left (c d^2-a e^2\right )^5 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {1155 c^4 d^4 e \sqrt {d+e x}}{64 \left (c d^2-a e^2\right )^6 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {1155 c^4 d^4 e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{64 \left (c d^2-a e^2\right )^{13/2}} \]

[Out]

1/4/(-a*e^2+c*d^2)/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+11/24*c*d/(-a*e^2+c*d^2)^2/(e*x+d)^(3
/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1155/64*c^4*d^4*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)^(13/2)+33/32*c^2*d^2/(-a*e^2+c*d^2)^3/(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(1/2)-77/32*c^3*d^3*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(3/2)-385/64*c^3*d^3*e/(-a*e^2+c*d^2)^5/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1
155/64*c^4*d^4*e*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {686, 680, 674, 211} \begin {gather*} \frac {1155 c^4 d^4 e^{3/2} \text {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 )}{64 \left (c d^2-a e^2\right )^{13/2}}+\frac {1155 c^4 d^4 e \sqrt {d+e x}}{64 \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 {385 c^3 d^3 e}{64 \sqrt {d+e x} \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {77 c^3 d^3 \sqrt {d+e x}}{32 \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 {33 c^2 d^2}{32 \sqrt {d+e x} \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 {11 c d}{24 (d+e x)^{3/2} \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}{4 (d+e x)^{5/2} \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)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

1/(4*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (11*c*d)/(24*(c*d^2 - a*
e^2)^2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (33*c^2*d^2)/(32*(c*d^2 - a*e^2)^3*Sqr
t[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (77*c^3*d^3*Sqrt[d + e*x])/(32*(c*d^2 - a*e^2)^4*(
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (385*c^3*d^3*e)/(64*(c*d^2 - a*e^2)^5*Sqrt[d + e*x]*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (1155*c^4*d^4*e*Sqrt[d + e*x])/(64*(c*d^2 - a*e^2)^6*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2]) + (1155*c^4*d^4*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])])/(64*(c*d^2 - a*e^2)^(13/2))

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*} \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 &=\frac {1}{4 \left (c d^2-a e^2\right ) (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 {(11 c d) \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}{8 \left (c d^2-a e^2\right )}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (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 {11 c d}{24 \left (c d^2-a e^2\right )^2 (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 (33 c^2 d^2\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}{16 \left (c d^2-a e^2\right )^2}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (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 {11 c d}{24 \left (c d^2-a e^2\right )^2 (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 {33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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 (231 c^3 d^3\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}{64 \left (c d^2-a e^2\right )^3}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (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 {11 c d}{24 \left (c d^2-a e^2\right )^2 (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 {33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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 {77 c^3 d^3 \sqrt {d+e x}}{32 \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 (385 c^3 d^3 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}{64 \left (c d^2-a e^2\right )^4}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (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 {11 c d}{24 \left (c d^2-a e^2\right )^2 (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 {33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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 {77 c^3 d^3 \sqrt {d+e x}}{32 \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 {385 c^3 d^3 e}{64 \left (c d^2-a e^2\right )^5 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (1155 c^4 d^4 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}{128 \left (c d^2-a e^2\right )^5}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (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 {11 c d}{24 \left (c d^2-a e^2\right )^2 (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 {33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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 {77 c^3 d^3 \sqrt {d+e x}}{32 \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 {385 c^3 d^3 e}{64 \left (c d^2-a e^2\right )^5 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {1155 c^4 d^4 e \sqrt {d+e x}}{64 \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}}+\frac {\left (1155 c^4 d^4 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}{128 \left (c d^2-a e^2\right )^6}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (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 {11 c d}{24 \left (c d^2-a e^2\right )^2 (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 {33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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 {77 c^3 d^3 \sqrt {d+e x}}{32 \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 {385 c^3 d^3 e}{64 \left (c d^2-a e^2\right )^5 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {1155 c^4 d^4 e \sqrt {d+e x}}{64 \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}}+\frac {\left (1155 c^4 d^4 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 )}{64 \left (c d^2-a e^2\right )^6}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (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 {11 c d}{24 \left (c d^2-a e^2\right )^2 (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 {33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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 {77 c^3 d^3 \sqrt {d+e x}}{32 \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 {385 c^3 d^3 e}{64 \left (c d^2-a e^2\right )^5 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {1155 c^4 d^4 e \sqrt {d+e x}}{64 \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}}+\frac {1155 c^4 d^4 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 )}{64 \left (c d^2-a e^2\right )^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 1.57, size = 361, normalized size = 0.79 \begin {gather*} \frac {c^4 d^4 (d+e x)^{5/2} \left (-\frac {(a e+c d x) \left (48 a^5 e^{10}-8 a^4 c d e^8 (41 d+11 e x)+2 a^3 c^2 d^2 e^6 \left (515 d^2+374 d e x+99 e^2 x^2\right )-3 a^2 c^3 d^3 e^4 \left (765 d^3+1265 d^2 e x+891 d e^2 x^2+231 e^3 x^3\right )-2 a c^4 d^4 e^2 \left (1024 d^4+6391 d^3 e x+11484 d^2 e^2 x^2+8547 d e^3 x^3+2310 e^4 x^4\right )+c^5 d^5 \left (128 d^5-1408 d^4 e x-9207 d^3 e^2 x^2-16863 d^2 e^3 x^3-12705 d e^4 x^4-3465 e^5 x^5\right )\right )}{c^4 d^4 \left (c d^2-a e^2\right )^6 (d+e x)^4}+\frac {3465 e^{3/2} (a e+c d x)^{5/2} \tan ^{-1}\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 )^{13/2}}\right )}{192 ((a e+c d x) (d+e x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^4*d^4*(d + e*x)^(5/2)*(-(((a*e + c*d*x)*(48*a^5*e^10 - 8*a^4*c*d*e^8*(41*d + 11*e*x) + 2*a^3*c^2*d^2*e^6*(5
15*d^2 + 374*d*e*x + 99*e^2*x^2) - 3*a^2*c^3*d^3*e^4*(765*d^3 + 1265*d^2*e*x + 891*d*e^2*x^2 + 231*e^3*x^3) -
2*a*c^4*d^4*e^2*(1024*d^4 + 6391*d^3*e*x + 11484*d^2*e^2*x^2 + 8547*d*e^3*x^3 + 2310*e^4*x^4) + c^5*d^5*(128*d
^5 - 1408*d^4*e*x - 9207*d^3*e^2*x^2 - 16863*d^2*e^3*x^3 - 12705*d*e^4*x^4 - 3465*e^5*x^5)))/(c^4*d^4*(c*d^2 -
a*e^2)^6*(d + e*x)^4)) + (3465*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)^(13/2)))/(192*((a*e + c*d*x)*(d + e*x))^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1214\) vs. \(2(407)=814\).
time = 0.88, size = 1215, normalized size = 2.66

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-4620 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{4} d^{4} e^{6} x^{4}-17094 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{4} d^{5} e^{5} x^{3}-22968 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{4} d^{6} e^{4} x^{2}-12782 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{4} d^{7} e^{3} x -3465 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{5} d^{5} e^{5} x^{5}+3465 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{5} e^{6} x^{5} \sqrt {c d x +a e}-328 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{4} c \,d^{2} e^{8}+1030 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} c^{2} d^{4} e^{6}-2295 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{3} d^{6} e^{4}-693 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{3} d^{3} e^{7} x^{3}+198 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} c^{2} d^{2} e^{8} x^{2}-2673 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{3} d^{4} e^{6} x^{2}-88 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{4} c d \,e^{9} x +748 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} c^{2} d^{3} e^{7} x -3795 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{3} d^{5} e^{5} x +13860 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{6} e^{5} x^{4} \sqrt {c d x +a e}-12705 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{5} d^{6} e^{4} x^{4}-16863 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{5} d^{7} e^{3} x^{3}-9207 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{5} d^{8} e^{2} x^{2}-1408 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{5} d^{9} e x +128 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{5} d^{10}+48 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{5} e^{10}-2048 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{4} d^{8} e^{2}+3465 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{9} e^{2} x \sqrt {c d x +a e}+3465 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{4} d^{4} e^{7} x^{4} \sqrt {c d x +a e}+13860 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{4} d^{5} e^{6} x^{3} \sqrt {c d x +a e}+20790 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{4} d^{6} e^{5} x^{2} \sqrt {c d x +a e}+13860 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{4} d^{7} e^{4} x \sqrt {c d x +a e}+3465 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{4} d^{8} e^{3} \sqrt {c d x +a e}+13860 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{8} e^{3} x^{2} \sqrt {c d x +a e}+20790 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{7} e^{4} x^{3} \sqrt {c d x +a e}\right )}{192 \left (e x +d \right )^{\frac {9}{2}} \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right )^{6} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(1215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*((c*d*x+a*e)*(e*x+d))^(1/2)*(-4620*((a*e^2-c*d^2)*e)^(1/2)*a*c^4*d^4*e^6*x^4-17094*((a*e^2-c*d^2)*e)^(1
/2)*a*c^4*d^5*e^5*x^3-22968*((a*e^2-c*d^2)*e)^(1/2)*a*c^4*d^6*e^4*x^2-12782*((a*e^2-c*d^2)*e)^(1/2)*a*c^4*d^7*
e^3*x-3465*((a*e^2-c*d^2)*e)^(1/2)*c^5*d^5*e^5*x^5+3465*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c
^5*d^9*e^2*x*(c*d*x+a*e)^(1/2)+3465*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^4*d^4*e^7*x^4*(c*
d*x+a*e)^(1/2)+13860*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^4*d^5*e^6*x^3*(c*d*x+a*e)^(1/2)+
20790*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^4*d^6*e^5*x^2*(c*d*x+a*e)^(1/2)+13860*arctanh(e
*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^4*d^7*e^4*x*(c*d*x+a*e)^(1/2)-328*((a*e^2-c*d^2)*e)^(1/2)*a^4*
c*d^2*e^8+1030*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^2*d^4*e^6-2295*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^3*d^6*e^4-693*((a*e^
2-c*d^2)*e)^(1/2)*a^2*c^3*d^3*e^7*x^3+198*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^2*d^2*e^8*x^2-2673*((a*e^2-c*d^2)*e)^(
1/2)*a^2*c^3*d^4*e^6*x^2-88*((a*e^2-c*d^2)*e)^(1/2)*a^4*c*d*e^9*x+748*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^2*d^3*e^7*
x-3795*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^3*d^5*e^5*x-12705*((a*e^2-c*d^2)*e)^(1/2)*c^5*d^6*e^4*x^4-16863*((a*e^2-c
*d^2)*e)^(1/2)*c^5*d^7*e^3*x^3-9207*((a*e^2-c*d^2)*e)^(1/2)*c^5*d^8*e^2*x^2-1408*((a*e^2-c*d^2)*e)^(1/2)*c^5*d
^9*e*x+128*((a*e^2-c*d^2)*e)^(1/2)*c^5*d^10+48*((a*e^2-c*d^2)*e)^(1/2)*a^5*e^10-2048*((a*e^2-c*d^2)*e)^(1/2)*a
*c^4*d^8*e^2+3465*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^4*d^8*e^3*(c*d*x+a*e)^(1/2)+3465*ar
ctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^5*d^5*e^6*x^5*(c*d*x+a*e)^(1/2)+13860*arctanh(e*(c*d*x+a*
e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^5*d^6*e^5*x^4*(c*d*x+a*e)^(1/2)+20790*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-
c*d^2)*e)^(1/2))*c^5*d^7*e^4*x^3*(c*d*x+a*e)^(1/2)+13860*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*
c^5*d^8*e^3*x^2*(c*d*x+a*e)^(1/2))/(e*x+d)^(9/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^6/((a*e^2-c*d^2)*e)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1513 vs. \(2 (413) = 826\).
time = 5.20, size = 3064, normalized size = 6.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/384*(3465*(c^6*d^11*x^2*e + a^2*c^4*d^4*x^5*e^8 + (2*a*c^5*d^5*x^6 + 5*a^2*c^4*d^5*x^4)*e^7 + (c^6*d^6*x^7
+ 10*a*c^5*d^6*x^5 + 10*a^2*c^4*d^6*x^3)*e^6 + 5*(c^6*d^7*x^6 + 4*a*c^5*d^7*x^4 + 2*a^2*c^4*d^7*x^2)*e^5 + 5*(
2*c^6*d^8*x^5 + 4*a*c^5*d^8*x^3 + a^2*c^4*d^8*x)*e^4 + (10*c^6*d^9*x^4 + 10*a*c^5*d^9*x^2 + a^2*c^4*d^9)*e^3 +
(5*c^6*d^10*x^3 + 2*a*c^5*d^10*x)*e^2)*sqrt(-e/(c*d^2 - a*e^2))*log((c*d^3 - 2*a*x*e^3 - 2*sqrt(c*d^2*x + a*x
*e^2 + (c*d*x^2 + a*d)*e)*(c*d^2 - a*e^2)*sqrt(x*e + d)*sqrt(-e/(c*d^2 - a*e^2)) - (c*d*x^2 + 2*a*d)*e^2)/(x^2
*e^2 + 2*d*x*e + d^2)) + 2*(1408*c^5*d^9*x*e - 128*c^5*d^10 + 88*a^4*c*d*x*e^9 - 48*a^5*e^10 - 2*(99*a^3*c^2*d
^2*x^2 - 164*a^4*c*d^2)*e^8 + 11*(63*a^2*c^3*d^3*x^3 - 68*a^3*c^2*d^3*x)*e^7 + (4620*a*c^4*d^4*x^4 + 2673*a^2*
c^3*d^4*x^2 - 1030*a^3*c^2*d^4)*e^6 + 33*(105*c^5*d^5*x^5 + 518*a*c^4*d^5*x^3 + 115*a^2*c^3*d^5*x)*e^5 + 3*(42
35*c^5*d^6*x^4 + 7656*a*c^4*d^6*x^2 + 765*a^2*c^3*d^6)*e^4 + 77*(219*c^5*d^7*x^3 + 166*a*c^4*d^7*x)*e^3 + (920
7*c^5*d^8*x^2 + 2048*a*c^4*d^8)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^8*d^19*x^2
+ a^8*x^5*e^19 + (2*a^7*c*d*x^6 + 5*a^8*d*x^4)*e^18 + (a^6*c^2*d^2*x^7 + 4*a^7*c*d^2*x^5 + 10*a^8*d^2*x^3)*e^1
7 - (7*a^6*c^2*d^3*x^6 + 10*a^7*c*d^3*x^4 - 10*a^8*d^3*x^2)*e^16 - (6*a^5*c^3*d^4*x^7 + 35*a^6*c^2*d^4*x^5 + 4
0*a^7*c*d^4*x^3 - 5*a^8*d^4*x)*e^15 - (35*a^6*c^2*d^5*x^4 + 50*a^7*c*d^5*x^2 - a^8*d^5)*e^14 + (15*a^4*c^4*d^6
*x^7 + 70*a^5*c^3*d^6*x^5 + 35*a^6*c^2*d^6*x^3 - 28*a^7*c*d^6*x)*e^13 + (35*a^4*c^4*d^7*x^6 + 140*a^5*c^3*d^7*
x^4 + 91*a^6*c^2*d^7*x^2 - 6*a^7*c*d^7)*e^12 - (20*a^3*c^5*d^8*x^7 + 35*a^4*c^4*d^8*x^5 - 70*a^5*c^3*d^8*x^3 -
63*a^6*c^2*d^8*x)*e^11 - (70*a^3*c^5*d^9*x^6 + 175*a^4*c^4*d^9*x^4 + 56*a^5*c^3*d^9*x^2 - 15*a^6*c^2*d^9)*e^1
0 + (15*a^2*c^6*d^10*x^7 - 56*a^3*c^5*d^10*x^5 - 175*a^4*c^4*d^10*x^3 - 70*a^5*c^3*d^10*x)*e^9 + (63*a^2*c^6*d
^11*x^6 + 70*a^3*c^5*d^11*x^4 - 35*a^4*c^4*d^11*x^2 - 20*a^5*c^3*d^11)*e^8 - (6*a*c^7*d^12*x^7 - 91*a^2*c^6*d^
12*x^5 - 140*a^3*c^5*d^12*x^3 - 35*a^4*c^4*d^12*x)*e^7 - (28*a*c^7*d^13*x^6 - 35*a^2*c^6*d^13*x^4 - 70*a^3*c^5
*d^13*x^2 - 15*a^4*c^4*d^13)*e^6 + (c^8*d^14*x^7 - 50*a*c^7*d^14*x^5 - 35*a^2*c^6*d^14*x^3)*e^5 + (5*c^8*d^15*
x^6 - 40*a*c^7*d^15*x^4 - 35*a^2*c^6*d^15*x^2 - 6*a^3*c^5*d^15)*e^4 + (10*c^8*d^16*x^5 - 10*a*c^7*d^16*x^3 - 7
*a^2*c^6*d^16*x)*e^3 + (10*c^8*d^17*x^4 + 4*a*c^7*d^17*x^2 + a^2*c^6*d^17)*e^2 + (5*c^8*d^18*x^3 + 2*a*c^7*d^1
8*x)*e), 1/192*(3465*(c^6*d^11*x^2*e + a^2*c^4*d^4*x^5*e^8 + (2*a*c^5*d^5*x^6 + 5*a^2*c^4*d^5*x^4)*e^7 + (c^6*
d^6*x^7 + 10*a*c^5*d^6*x^5 + 10*a^2*c^4*d^6*x^3)*e^6 + 5*(c^6*d^7*x^6 + 4*a*c^5*d^7*x^4 + 2*a^2*c^4*d^7*x^2)*e
^5 + 5*(2*c^6*d^8*x^5 + 4*a*c^5*d^8*x^3 + a^2*c^4*d^8*x)*e^4 + (10*c^6*d^9*x^4 + 10*a*c^5*d^9*x^2 + a^2*c^4*d^
9)*e^3 + (5*c^6*d^10*x^3 + 2*a*c^5*d^10*x)*e^2)*arctan(-sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(c*d^2
- a*e^2)*sqrt(x*e + d)*e^(1/2)/(c*d^2*x*e + a*x*e^3 + (c*d*x^2 + a*d)*e^2))*e^(1/2)/sqrt(c*d^2 - a*e^2) + (14
08*c^5*d^9*x*e - 128*c^5*d^10 + 88*a^4*c*d*x*e^9 - 48*a^5*e^10 - 2*(99*a^3*c^2*d^2*x^2 - 164*a^4*c*d^2)*e^8 +
11*(63*a^2*c^3*d^3*x^3 - 68*a^3*c^2*d^3*x)*e^7 + (4620*a*c^4*d^4*x^4 + 2673*a^2*c^3*d^4*x^2 - 1030*a^3*c^2*d^4
)*e^6 + 33*(105*c^5*d^5*x^5 + 518*a*c^4*d^5*x^3 + 115*a^2*c^3*d^5*x)*e^5 + 3*(4235*c^5*d^6*x^4 + 7656*a*c^4*d^
6*x^2 + 765*a^2*c^3*d^6)*e^4 + 77*(219*c^5*d^7*x^3 + 166*a*c^4*d^7*x)*e^3 + (9207*c^5*d^8*x^2 + 2048*a*c^4*d^8
)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^8*d^19*x^2 + a^8*x^5*e^19 + (2*a^7*c*d*x^
6 + 5*a^8*d*x^4)*e^18 + (a^6*c^2*d^2*x^7 + 4*a^7*c*d^2*x^5 + 10*a^8*d^2*x^3)*e^17 - (7*a^6*c^2*d^3*x^6 + 10*a^
7*c*d^3*x^4 - 10*a^8*d^3*x^2)*e^16 - (6*a^5*c^3*d^4*x^7 + 35*a^6*c^2*d^4*x^5 + 40*a^7*c*d^4*x^3 - 5*a^8*d^4*x)
*e^15 - (35*a^6*c^2*d^5*x^4 + 50*a^7*c*d^5*x^2 - a^8*d^5)*e^14 + (15*a^4*c^4*d^6*x^7 + 70*a^5*c^3*d^6*x^5 + 35
*a^6*c^2*d^6*x^3 - 28*a^7*c*d^6*x)*e^13 + (35*a^4*c^4*d^7*x^6 + 140*a^5*c^3*d^7*x^4 + 91*a^6*c^2*d^7*x^2 - 6*a
^7*c*d^7)*e^12 - (20*a^3*c^5*d^8*x^7 + 35*a^4*c^4*d^8*x^5 - 70*a^5*c^3*d^8*x^3 - 63*a^6*c^2*d^8*x)*e^11 - (70*
a^3*c^5*d^9*x^6 + 175*a^4*c^4*d^9*x^4 + 56*a^5*c^3*d^9*x^2 - 15*a^6*c^2*d^9)*e^10 + (15*a^2*c^6*d^10*x^7 - 56*
a^3*c^5*d^10*x^5 - 175*a^4*c^4*d^10*x^3 - 70*a^5*c^3*d^10*x)*e^9 + (63*a^2*c^6*d^11*x^6 + 70*a^3*c^5*d^11*x^4
- 35*a^4*c^4*d^11*x^2 - 20*a^5*c^3*d^11)*e^8 - (6*a*c^7*d^12*x^7 - 91*a^2*c^6*d^12*x^5 - 140*a^3*c^5*d^12*x^3
- 35*a^4*c^4*d^12*x)*e^7 - (28*a*c^7*d^13*x^6 - 35*a^2*c^6*d^13*x^4 - 70*a^3*c^5*d^13*x^2 - 15*a^4*c^4*d^13)*e
^6 + (c^8*d^14*x^7 - 50*a*c^7*d^14*x^5 - 35*a^2*c^6*d^14*x^3)*e^5 + (5*c^8*d^15*x^6 - 40*a*c^7*d^15*x^4 - 35*a
^2*c^6*d^15*x^2 - 6*a^3*c^5*d^15)*e^4 + (10*c^8*d^16*x^5 - 10*a*c^7*d^16*x^3 - 7*a^2*c^6*d^16*x)*e^3 + (10*c^8
*d^17*x^4 + 4*a*c^7*d^17*x^2 + a^2*c^6*d^17)*e^2 + (5*c^8*d^18*x^3 + 2*a*c^7*d^18*x)*e)]

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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 )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [A]
time = 4.10, size = 777, normalized size = 1.70 \begin {gather*} \frac {1}{192} \, {\left (\frac {3465 \, c^{4} d^{4} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {128 \, {\left (c^{5} d^{6} e^{2} - a c^{4} d^{4} e^{4} - 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{4} d^{4} e\right )}}{{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {{\left (2295 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{7} d^{10} e^{4} - 6885 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{6} d^{8} e^{6} + 5855 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{6} d^{8} e^{3} + 6885 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{5} d^{6} e^{8} - 11710 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{5} d^{6} e^{5} + 5153 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{5} d^{6} e^{2} - 2295 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{4} d^{4} e^{10} + 5855 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{4} d^{4} e^{7} - 5153 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{4} d^{4} e^{4} + 1545 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{4} d^{4} e\right )} e^{\left (-4\right )}}{{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} {\left (x e + d\right )}^{4} c^{4} d^{4}}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3465*c^4*d^4*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e/((c^6*d^12 - 6*a*c
^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*sqrt
(c*d^2*e - a*e^3)) - 128*(c^5*d^6*e^2 - a*c^4*d^4*e^4 - 15*((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^4*d^4*e)/((c^
6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 +
a^6*e^12)*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)) + (2295*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^7*d^10*
e^4 - 6885*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*c^6*d^8*e^6 + 5855*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3
/2)*c^6*d^8*e^3 + 6885*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^5*d^6*e^8 - 11710*((x*e + d)*c*d*e - c*d^
2*e + a*e^3)^(3/2)*a*c^5*d^6*e^5 + 5153*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^5*d^6*e^2 - 2295*sqrt((x*e
+ d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^4*d^4*e^10 + 5855*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^4*d^4*e
^7 - 5153*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^4*d^4*e^4 + 1545*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(
7/2)*c^4*d^4*e)*e^(-4)/((c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^
4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*(x*e + d)^4*c^4*d^4))*e

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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