3.18.47 \(\int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)
Optimal. Leaf size=207 \[ \frac {3 c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{4 \sqrt {e} \left (c d^2-a e^2\right )^{5/2}}+\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 207, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used =
{672, 660, 205} \begin {gather*} \frac {3 c^2 d^2 \tan ^{-1}\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 )}{4 \sqrt {e} \left (c d^2-a e^2\right )^{5/2}}+\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
[Out]
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(2*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) + (3*c*d*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(4*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + (3*c^2*d^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])])/(4*Sqrt[e]*(c*d^2 - a*e^2)^(5/2))
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 660
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 672
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)), I
nt[(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] && EqQ
[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} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {(3 c d) \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 \left (c d^2-a e^2\right )}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (3 c^2 d^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}{8 \left (c d^2-a e^2\right )^2}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (3 c^2 d^2 e\right ) \operatorname {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 )}{4 \left (c d^2-a e^2\right )^2}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {3 c^2 d^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 )}{4 \sqrt {e} \left (c d^2-a e^2\right )^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 81, normalized size = 0.39 \begin {gather*} \frac {2 c^2 d^2 \sqrt {(d+e x) (a e+c d x)} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
[Out]
(2*c^2*d^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[1/2, 3, 3/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])
/((c*d^2 - a*e^2)^3*Sqrt[d + e*x])
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IntegrateAlgebraic [B] time = 65.30, size = 4961, normalized size = 23.97 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
[Out]
(-((-2*c*d^2 + 2*a*e^2 - 3*c*d*(d + e*x))*Sqrt[-((c*d^2*(d + e*x))/e) + a*e*(d + e*x) + (c*d*(d + e*x)^2)/e]*(
c^10*d^20*e - 10*a*c^9*d^18*e^3 + 45*a^2*c^8*d^16*e^5 - 120*a^3*c^7*d^14*e^7 + 210*a^4*c^6*d^12*e^9 - 252*a^5*
c^5*d^10*e^11 + 210*a^6*c^4*d^8*e^13 - 120*a^7*c^3*d^6*e^15 + 45*a^8*c^2*d^4*e^17 - 10*a^9*c*d^2*e^19 + a^10*e
^21 - 200*c^10*d^19*e*(d + e*x) + 1800*a*c^9*d^17*e^3*(d + e*x) - 7200*a^2*c^8*d^15*e^5*(d + e*x) + 16800*a^3*
c^7*d^13*e^7*(d + e*x) - 25200*a^4*c^6*d^11*e^9*(d + e*x) + 25200*a^5*c^5*d^9*e^11*(d + e*x) - 16800*a^6*c^4*d
^7*e^13*(d + e*x) + 7200*a^7*c^3*d^5*e^15*(d + e*x) - 1800*a^8*c^2*d^3*e^17*(d + e*x) + 200*a^9*c*d*e^19*(d +
e*x) + 6600*c^10*d^18*e*(d + e*x)^2 - 52800*a*c^9*d^16*e^3*(d + e*x)^2 + 184800*a^2*c^8*d^14*e^5*(d + e*x)^2 -
369600*a^3*c^7*d^12*e^7*(d + e*x)^2 + 462000*a^4*c^6*d^10*e^9*(d + e*x)^2 - 369600*a^5*c^5*d^8*e^11*(d + e*x)
^2 + 184800*a^6*c^4*d^6*e^13*(d + e*x)^2 - 52800*a^7*c^3*d^4*e^15*(d + e*x)^2 + 6600*a^8*c^2*d^2*e^17*(d + e*x
)^2 - 84480*c^10*d^17*e*(d + e*x)^3 + 591360*a*c^9*d^15*e^3*(d + e*x)^3 - 1774080*a^2*c^8*d^13*e^5*(d + e*x)^3
+ 2956800*a^3*c^7*d^11*e^7*(d + e*x)^3 - 2956800*a^4*c^6*d^9*e^9*(d + e*x)^3 + 1774080*a^5*c^5*d^7*e^11*(d +
e*x)^3 - 591360*a^6*c^4*d^5*e^13*(d + e*x)^3 + 84480*a^7*c^3*d^3*e^15*(d + e*x)^3 + 549120*c^10*d^16*e*(d + e*
x)^4 - 3294720*a*c^9*d^14*e^3*(d + e*x)^4 + 8236800*a^2*c^8*d^12*e^5*(d + e*x)^4 - 10982400*a^3*c^7*d^10*e^7*(
d + e*x)^4 + 8236800*a^4*c^6*d^8*e^9*(d + e*x)^4 - 3294720*a^5*c^5*d^6*e^11*(d + e*x)^4 + 549120*a^6*c^4*d^4*e
^13*(d + e*x)^4 - 2050048*c^10*d^15*e*(d + e*x)^5 + 10250240*a*c^9*d^13*e^3*(d + e*x)^5 - 20500480*a^2*c^8*d^1
1*e^5*(d + e*x)^5 + 20500480*a^3*c^7*d^9*e^7*(d + e*x)^5 - 10250240*a^4*c^6*d^7*e^9*(d + e*x)^5 + 2050048*a^5*
c^5*d^5*e^11*(d + e*x)^5 + 4659200*c^10*d^14*e*(d + e*x)^6 - 18636800*a*c^9*d^12*e^3*(d + e*x)^6 + 27955200*a^
2*c^8*d^10*e^5*(d + e*x)^6 - 18636800*a^3*c^7*d^8*e^7*(d + e*x)^6 + 4659200*a^4*c^6*d^6*e^9*(d + e*x)^6 - 6553
600*c^10*d^13*e*(d + e*x)^7 + 19660800*a*c^9*d^11*e^3*(d + e*x)^7 - 19660800*a^2*c^8*d^9*e^5*(d + e*x)^7 + 655
3600*a^3*c^7*d^7*e^7*(d + e*x)^7 + 5570560*c^10*d^12*e*(d + e*x)^8 - 11141120*a*c^9*d^10*e^3*(d + e*x)^8 + 557
0560*a^2*c^8*d^8*e^5*(d + e*x)^8 - 2621440*c^10*d^11*e*(d + e*x)^9 + 2621440*a*c^9*d^9*e^3*(d + e*x)^9 + 52428
8*c^10*d^10*e*(d + e*x)^10)) - Sqrt[c*d*e]*(-2*c*d^2 + 2*a*e^2 - 3*c*d*(d + e*x))*(-20*c^10*d^20*(d + e*x) + 2
00*a*c^9*d^18*e^2*(d + e*x) - 900*a^2*c^8*d^16*e^4*(d + e*x) + 2400*a^3*c^7*d^14*e^6*(d + e*x) - 4200*a^4*c^6*
d^12*e^8*(d + e*x) + 5040*a^5*c^5*d^10*e^10*(d + e*x) - 4200*a^6*c^4*d^8*e^12*(d + e*x) + 2400*a^7*c^3*d^6*e^1
4*(d + e*x) - 900*a^8*c^2*d^4*e^16*(d + e*x) + 200*a^9*c*d^2*e^18*(d + e*x) - 20*a^10*e^20*(d + e*x) + 1340*c^
10*d^19*(d + e*x)^2 - 12060*a*c^9*d^17*e^2*(d + e*x)^2 + 48240*a^2*c^8*d^15*e^4*(d + e*x)^2 - 112560*a^3*c^7*d
^13*e^6*(d + e*x)^2 + 168840*a^4*c^6*d^11*e^8*(d + e*x)^2 - 168840*a^5*c^5*d^9*e^10*(d + e*x)^2 + 112560*a^6*c
^4*d^7*e^12*(d + e*x)^2 - 48240*a^7*c^3*d^5*e^14*(d + e*x)^2 + 12060*a^8*c^2*d^3*e^16*(d + e*x)^2 - 1340*a^9*c
*d*e^18*(d + e*x)^2 - 26664*c^10*d^18*(d + e*x)^3 + 213312*a*c^9*d^16*e^2*(d + e*x)^3 - 746592*a^2*c^8*d^14*e^
4*(d + e*x)^3 + 1493184*a^3*c^7*d^12*e^6*(d + e*x)^3 - 1866480*a^4*c^6*d^10*e^8*(d + e*x)^3 + 1493184*a^5*c^5*
d^8*e^10*(d + e*x)^3 - 746592*a^6*c^4*d^6*e^12*(d + e*x)^3 + 213312*a^7*c^3*d^4*e^14*(d + e*x)^3 - 26664*a^8*c
^2*d^2*e^16*(d + e*x)^3 + 244992*c^10*d^17*(d + e*x)^4 - 1714944*a*c^9*d^15*e^2*(d + e*x)^4 + 5144832*a^2*c^8*
d^13*e^4*(d + e*x)^4 - 8574720*a^3*c^7*d^11*e^6*(d + e*x)^4 + 8574720*a^4*c^6*d^9*e^8*(d + e*x)^4 - 5144832*a^
5*c^5*d^7*e^10*(d + e*x)^4 + 1714944*a^6*c^4*d^5*e^12*(d + e*x)^4 - 244992*a^7*c^3*d^3*e^14*(d + e*x)^4 - 1244
672*c^10*d^16*(d + e*x)^5 + 7468032*a*c^9*d^14*e^2*(d + e*x)^5 - 18670080*a^2*c^8*d^12*e^4*(d + e*x)^5 + 24893
440*a^3*c^7*d^10*e^6*(d + e*x)^5 - 18670080*a^4*c^6*d^8*e^8*(d + e*x)^5 + 7468032*a^5*c^5*d^6*e^10*(d + e*x)^5
- 1244672*a^6*c^4*d^4*e^12*(d + e*x)^5 + 3820544*c^10*d^15*(d + e*x)^6 - 19102720*a*c^9*d^13*e^2*(d + e*x)^6
+ 38205440*a^2*c^8*d^11*e^4*(d + e*x)^6 - 38205440*a^3*c^7*d^9*e^6*(d + e*x)^6 + 19102720*a^4*c^6*d^7*e^8*(d +
e*x)^6 - 3820544*a^5*c^5*d^5*e^10*(d + e*x)^6 - 7383040*c^10*d^14*(d + e*x)^7 + 29532160*a*c^9*d^12*e^2*(d +
e*x)^7 - 44298240*a^2*c^8*d^10*e^4*(d + e*x)^7 + 29532160*a^3*c^7*d^8*e^6*(d + e*x)^7 - 7383040*a^4*c^6*d^6*e^
8*(d + e*x)^7 + 9043968*c^10*d^13*(d + e*x)^8 - 27131904*a*c^9*d^11*e^2*(d + e*x)^8 + 27131904*a^2*c^8*d^9*e^4
*(d + e*x)^8 - 9043968*a^3*c^7*d^7*e^6*(d + e*x)^8 - 6815744*c^10*d^12*(d + e*x)^9 + 13631488*a*c^9*d^10*e^2*(
d + e*x)^9 - 6815744*a^2*c^8*d^8*e^4*(d + e*x)^9 + 2883584*c^10*d^11*(d + e*x)^10 - 2883584*a*c^9*d^9*e^2*(d +
e*x)^10 - 524288*c^10*d^10*(d + e*x)^11))/(4*e*Sqrt[c*d*e]*(-(c*d^2) + a*e^2)^2*(d + e*x)^(5/2)*Sqrt[-((c*d^2
*(d + e*x))/e) + a*e*(d + e*x) + (c*d*(d + e*x)^2)/e]*(20*c^9*d^18 - 180*a*c^8*d^16*e^2 + 720*a^2*c^7*d^14*e^4
- 1680*a^3*c^6*d^12*e^6 + 2520*a^4*c^5*d^10*e^8 - 2520*a^5*c^4*d^8*e^10 + 1680*a^6*c^3*d^6*e^12 - 720*a^7*c^2
*d^4*e^14 + 180*a^8*c*d^2*e^16 - 20*a^9*e^18 - 1320*c^9*d^17*(d + e*x) + 10560*a*c^8*d^15*e^2*(d + e*x) - 3696
0*a^2*c^7*d^13*e^4*(d + e*x) + 73920*a^3*c^6*d^11*e^6*(d + e*x) - 92400*a^4*c^5*d^9*e^8*(d + e*x) + 73920*a^5*
c^4*d^7*e^10*(d + e*x) - 36960*a^6*c^3*d^5*e^12*(d + e*x) + 10560*a^7*c^2*d^3*e^14*(d + e*x) - 1320*a^8*c*d*e^
16*(d + e*x) + 25344*c^9*d^16*(d + e*x)^2 - 177408*a*c^8*d^14*e^2*(d + e*x)^2 + 532224*a^2*c^7*d^12*e^4*(d + e
*x)^2 - 887040*a^3*c^6*d^10*e^6*(d + e*x)^2 + 887040*a^4*c^5*d^8*e^8*(d + e*x)^2 - 532224*a^5*c^4*d^6*e^10*(d
+ e*x)^2 + 177408*a^6*c^3*d^4*e^12*(d + e*x)^2 - 25344*a^7*c^2*d^2*e^14*(d + e*x)^2 - 219648*c^9*d^15*(d + e*x
)^3 + 1317888*a*c^8*d^13*e^2*(d + e*x)^3 - 3294720*a^2*c^7*d^11*e^4*(d + e*x)^3 + 4392960*a^3*c^6*d^9*e^6*(d +
e*x)^3 - 3294720*a^4*c^5*d^7*e^8*(d + e*x)^3 + 1317888*a^5*c^4*d^5*e^10*(d + e*x)^3 - 219648*a^6*c^3*d^3*e^12
*(d + e*x)^3 + 1025024*c^9*d^14*(d + e*x)^4 - 5125120*a*c^8*d^12*e^2*(d + e*x)^4 + 10250240*a^2*c^7*d^10*e^4*(
d + e*x)^4 - 10250240*a^3*c^6*d^8*e^6*(d + e*x)^4 + 5125120*a^4*c^5*d^6*e^8*(d + e*x)^4 - 1025024*a^5*c^4*d^4*
e^10*(d + e*x)^4 - 2795520*c^9*d^13*(d + e*x)^5 + 11182080*a*c^8*d^11*e^2*(d + e*x)^5 - 16773120*a^2*c^7*d^9*e
^4*(d + e*x)^5 + 11182080*a^3*c^6*d^7*e^6*(d + e*x)^5 - 2795520*a^4*c^5*d^5*e^8*(d + e*x)^5 + 4587520*c^9*d^12
*(d + e*x)^6 - 13762560*a*c^8*d^10*e^2*(d + e*x)^6 + 13762560*a^2*c^7*d^8*e^4*(d + e*x)^6 - 4587520*a^3*c^6*d^
6*e^6*(d + e*x)^6 - 4456448*c^9*d^11*(d + e*x)^7 + 8912896*a*c^8*d^9*e^2*(d + e*x)^7 - 4456448*a^2*c^7*d^7*e^4
*(d + e*x)^7 + 2359296*c^9*d^10*(d + e*x)^8 - 2359296*a*c^8*d^8*e^2*(d + e*x)^8 - 524288*c^9*d^9*(d + e*x)^9)
+ 4*e*(-(c*d^2) + a*e^2)^2*(d + e*x)^(5/2)*(c^10*d^20 - 10*a*c^9*d^18*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*
d^14*e^6 + 210*a^4*c^6*d^12*e^8 - 252*a^5*c^5*d^10*e^10 + 210*a^6*c^4*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8
*c^2*d^4*e^16 - 10*a^9*c*d^2*e^18 + a^10*e^20 - 200*c^10*d^19*(d + e*x) + 1800*a*c^9*d^17*e^2*(d + e*x) - 7200
*a^2*c^8*d^15*e^4*(d + e*x) + 16800*a^3*c^7*d^13*e^6*(d + e*x) - 25200*a^4*c^6*d^11*e^8*(d + e*x) + 25200*a^5*
c^5*d^9*e^10*(d + e*x) - 16800*a^6*c^4*d^7*e^12*(d + e*x) + 7200*a^7*c^3*d^5*e^14*(d + e*x) - 1800*a^8*c^2*d^3
*e^16*(d + e*x) + 200*a^9*c*d*e^18*(d + e*x) + 6600*c^10*d^18*(d + e*x)^2 - 52800*a*c^9*d^16*e^2*(d + e*x)^2 +
184800*a^2*c^8*d^14*e^4*(d + e*x)^2 - 369600*a^3*c^7*d^12*e^6*(d + e*x)^2 + 462000*a^4*c^6*d^10*e^8*(d + e*x)
^2 - 369600*a^5*c^5*d^8*e^10*(d + e*x)^2 + 184800*a^6*c^4*d^6*e^12*(d + e*x)^2 - 52800*a^7*c^3*d^4*e^14*(d + e
*x)^2 + 6600*a^8*c^2*d^2*e^16*(d + e*x)^2 - 84480*c^10*d^17*(d + e*x)^3 + 591360*a*c^9*d^15*e^2*(d + e*x)^3 -
1774080*a^2*c^8*d^13*e^4*(d + e*x)^3 + 2956800*a^3*c^7*d^11*e^6*(d + e*x)^3 - 2956800*a^4*c^6*d^9*e^8*(d + e*x
)^3 + 1774080*a^5*c^5*d^7*e^10*(d + e*x)^3 - 591360*a^6*c^4*d^5*e^12*(d + e*x)^3 + 84480*a^7*c^3*d^3*e^14*(d +
e*x)^3 + 549120*c^10*d^16*(d + e*x)^4 - 3294720*a*c^9*d^14*e^2*(d + e*x)^4 + 8236800*a^2*c^8*d^12*e^4*(d + e*
x)^4 - 10982400*a^3*c^7*d^10*e^6*(d + e*x)^4 + 8236800*a^4*c^6*d^8*e^8*(d + e*x)^4 - 3294720*a^5*c^5*d^6*e^10*
(d + e*x)^4 + 549120*a^6*c^4*d^4*e^12*(d + e*x)^4 - 2050048*c^10*d^15*(d + e*x)^5 + 10250240*a*c^9*d^13*e^2*(d
+ e*x)^5 - 20500480*a^2*c^8*d^11*e^4*(d + e*x)^5 + 20500480*a^3*c^7*d^9*e^6*(d + e*x)^5 - 10250240*a^4*c^6*d^
7*e^8*(d + e*x)^5 + 2050048*a^5*c^5*d^5*e^10*(d + e*x)^5 + 4659200*c^10*d^14*(d + e*x)^6 - 18636800*a*c^9*d^12
*e^2*(d + e*x)^6 + 27955200*a^2*c^8*d^10*e^4*(d + e*x)^6 - 18636800*a^3*c^7*d^8*e^6*(d + e*x)^6 + 4659200*a^4*
c^6*d^6*e^8*(d + e*x)^6 - 6553600*c^10*d^13*(d + e*x)^7 + 19660800*a*c^9*d^11*e^2*(d + e*x)^7 - 19660800*a^2*c
^8*d^9*e^4*(d + e*x)^7 + 6553600*a^3*c^7*d^7*e^6*(d + e*x)^7 + 5570560*c^10*d^12*(d + e*x)^8 - 11141120*a*c^9*
d^10*e^2*(d + e*x)^8 + 5570560*a^2*c^8*d^8*e^4*(d + e*x)^8 - 2621440*c^10*d^11*(d + e*x)^9 + 2621440*a*c^9*d^9
*e^2*(d + e*x)^9 + 524288*c^10*d^10*(d + e*x)^10)) - (3*c^2*d^2*ArcTan[(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*Sqrt[d + e
*x])/(-(Sqrt[c*d*e]*(d + e*x)) + e*Sqrt[-((c*d^2*(d + e*x))/e) + a*e*(d + e*x) + (c*d*(d + e*x)^2)/e])])/(2*Sq
rt[e]*(c*d^2 - a*e^2)^(5/2))
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fricas [B] time = 0.44, size = 870, normalized size = 4.20 \begin {gather*} \left [-\frac {3 \, {\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (-\frac {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 + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d^{2} e + a e^{3}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (5 \, c^{2} d^{4} e - 7 \, a c d^{2} e^{3} + 2 \, a^{2} e^{5} + 3 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{8 \, {\left (c^{3} d^{9} e - 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} - a^{3} d^{3} e^{7} + {\left (c^{3} d^{6} e^{4} - 3 \, a c^{2} d^{4} e^{6} + 3 \, a^{2} c d^{2} e^{8} - a^{3} e^{10}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{3} - 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} - a^{3} d e^{9}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e^{2} - 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} - a^{3} d^{2} e^{8}\right )} x\right )}}, -\frac {3 \, {\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d^{2} e - a e^{3}} \sqrt {e x + d}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) - {\left (5 \, c^{2} d^{4} e - 7 \, a c d^{2} e^{3} + 2 \, a^{2} e^{5} + 3 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{4 \, {\left (c^{3} d^{9} e - 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} - a^{3} d^{3} e^{7} + {\left (c^{3} d^{6} e^{4} - 3 \, a c^{2} d^{4} e^{6} + 3 \, a^{2} c d^{2} e^{8} - a^{3} e^{10}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{3} - 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} - a^{3} d e^{9}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e^{2} - 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} - a^{3} d^{2} e^{8}\right )} x\right )}}\right ] \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)^(1/2),x, algorithm="fricas")
[Out]
[-1/8*(3*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x^2 + 3*c^2*d^4*e*x + c^2*d^5)*sqrt(-c*d^2*e + a*e^3)*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)*sqrt(-c*d^2*e + a*e^3)*sqr
t(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(5*c^2*d^4*e - 7*a*c*d^2*e^3 + 2*a^2*e^5 + 3*(c^2*d^3*e^2 - a*c*d*e
^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^9*e - 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e
^5 - a^3*d^3*e^7 + (c^3*d^6*e^4 - 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 - a^3*e^10)*x^3 + 3*(c^3*d^7*e^3 - 3*a*c^2
*d^5*e^5 + 3*a^2*c*d^3*e^7 - a^3*d*e^9)*x^2 + 3*(c^3*d^8*e^2 - 3*a*c^2*d^6*e^4 + 3*a^2*c*d^4*e^6 - a^3*d^2*e^8
)*x), -1/4*(3*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x^2 + 3*c^2*d^4*e*x + c^2*d^5)*sqrt(c*d^2*e - a*e^3)*arctan(sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)*sqrt(e*x + d)/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e
+ a*e^3)*x)) - (5*c^2*d^4*e - 7*a*c*d^2*e^3 + 2*a^2*e^5 + 3*(c^2*d^3*e^2 - a*c*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d
*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^9*e - 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 - a^3*d^3*e^7 + (c^3*d^6
*e^4 - 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 - a^3*e^10)*x^3 + 3*(c^3*d^7*e^3 - 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7
- a^3*d*e^9)*x^2 + 3*(c^3*d^8*e^2 - 3*a*c^2*d^6*e^4 + 3*a^2*c*d^4*e^6 - a^3*d^2*e^8)*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)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")
[Out]
sage0*x
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maple [A] time = 0.08, size = 292, normalized size = 1.41 \begin {gather*} -\frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (3 c^{2} d^{2} e^{2} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+6 c^{2} d^{3} e x \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+3 c^{2} d^{4} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )-3 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c d e x +2 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,e^{2}-5 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c \,d^{2}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \sqrt {c d x +a e}\, \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^(5/2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2),x)
[Out]
-1/4*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(3*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*x^2*c^2*d
^2*e^2+6*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*x*c^2*d^3*e+3*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-
c*d^2)*e)^(1/2)*e)*c^2*d^4-3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*c*d*e*x+2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d
^2)*e)^(1/2)*a*e^2-5*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*c*d^2)/(e*x+d)^(5/2)/(c*d*x+a*e)^(1/2)/(a*e^2-c
*d^2)^2/((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*} \int \frac {1}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \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)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(5/2)), x)
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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}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,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)^(1/2)),x)
[Out]
int(1/((d + e*x)^(5/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \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)**(1/2),x)
[Out]
Integral(1/(sqrt((d + e*x)*(a*e + c*d*x))*(d + e*x)**(5/2)), x)
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