\(\int \frac {1}{\sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [1672]

Optimal result

Integrand size = 28, antiderivative size = 213 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac {21 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac {63 e^4 \sqrt {d+e x}}{128 (b d-a e)^5 (a+b x)}+\frac {63 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 \sqrt {b} (b d-a e)^{11/2}} \]

[Out]

63/128*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(11/2)/b^(1/2)-1/5*(e*x+d)^(1/2)/(-a*e+b
*d)/(b*x+a)^5+9/40*e*(e*x+d)^(1/2)/(-a*e+b*d)^2/(b*x+a)^4-21/80*e^2*(e*x+d)^(1/2)/(-a*e+b*d)^3/(b*x+a)^3+21/64
*e^3*(e*x+d)^(1/2)/(-a*e+b*d)^4/(b*x+a)^2-63/128*e^4*(e*x+d)^(1/2)/(-a*e+b*d)^5/(b*x+a)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 44, 65, 214} \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {63 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 \sqrt {b} (b d-a e)^{11/2}}-\frac {63 e^4 \sqrt {d+e x}}{128 (a+b x) (b d-a e)^5}+\frac {21 e^3 \sqrt {d+e x}}{64 (a+b x)^2 (b d-a e)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (a+b x)^3 (b d-a e)^3}+\frac {9 e \sqrt {d+e x}}{40 (a+b x)^4 (b d-a e)^2}-\frac {\sqrt {d+e x}}{5 (a+b x)^5 (b d-a e)} \]

[In]

Int[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

-1/5*Sqrt[d + e*x]/((b*d - a*e)*(a + b*x)^5) + (9*e*Sqrt[d + e*x])/(40*(b*d - a*e)^2*(a + b*x)^4) - (21*e^2*Sq
rt[d + e*x])/(80*(b*d - a*e)^3*(a + b*x)^3) + (21*e^3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*(a + b*x)^2) - (63*e^4*
Sqrt[d + e*x])/(128*(b*d - a*e)^5*(a + b*x)) + (63*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*
Sqrt[b]*(b*d - a*e)^(11/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 44

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^6 \sqrt {d+e x}} \, dx \\ & = -\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}-\frac {(9 e) \int \frac {1}{(a+b x)^5 \sqrt {d+e x}} \, dx}{10 (b d-a e)} \\ & = -\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}+\frac {\left (63 e^2\right ) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{80 (b d-a e)^2} \\ & = -\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}-\frac {\left (21 e^3\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{32 (b d-a e)^3} \\ & = -\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac {21 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)^2}+\frac {\left (63 e^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 (b d-a e)^4} \\ & = -\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac {21 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac {63 e^4 \sqrt {d+e x}}{128 (b d-a e)^5 (a+b x)}-\frac {\left (63 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^5} \\ & = -\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac {21 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac {63 e^4 \sqrt {d+e x}}{128 (b d-a e)^5 (a+b x)}-\frac {\left (63 e^4\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 (b d-a e)^5} \\ & = -\frac {\sqrt {d+e x}}{5 (b d-a e) (a+b x)^5}+\frac {9 e \sqrt {d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac {21 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac {63 e^4 \sqrt {d+e x}}{128 (b d-a e)^5 (a+b x)}+\frac {63 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 \sqrt {b} (b d-a e)^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1}{640} \left (\frac {\sqrt {d+e x} \left (965 a^4 e^4+10 a^3 b e^3 (-149 d+237 e x)+6 a^2 b^2 e^2 \left (228 d^2-289 d e x+448 e^2 x^2\right )+2 a b^3 e \left (-328 d^3+384 d^2 e x-483 d e^2 x^2+735 e^3 x^3\right )+b^4 \left (128 d^4-144 d^3 e x+168 d^2 e^2 x^2-210 d e^3 x^3+315 e^4 x^4\right )\right )}{(-b d+a e)^5 (a+b x)^5}+\frac {315 e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} (-b d+a e)^{11/2}}\right ) \]

[In]

Integrate[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((Sqrt[d + e*x]*(965*a^4*e^4 + 10*a^3*b*e^3*(-149*d + 237*e*x) + 6*a^2*b^2*e^2*(228*d^2 - 289*d*e*x + 448*e^2*
x^2) + 2*a*b^3*e*(-328*d^3 + 384*d^2*e*x - 483*d*e^2*x^2 + 735*e^3*x^3) + b^4*(128*d^4 - 144*d^3*e*x + 168*d^2
*e^2*x^2 - 210*d*e^3*x^3 + 315*e^4*x^4)))/((-(b*d) + a*e)^5*(a + b*x)^5) + (315*e^5*ArcTan[(Sqrt[b]*Sqrt[d + e
*x])/Sqrt[-(b*d) + a*e]])/(Sqrt[b]*(-(b*d) + a*e)^(11/2)))/640

Maple [A] (verified)

Time = 3.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(\frac {\frac {63 e^{5} \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128}+\frac {193 \left (\frac {\left (63 e^{4} x^{4}-42 d \,e^{3} x^{3}+\frac {168}{5} d^{2} e^{2} x^{2}-\frac {144}{5} d^{3} e x +\frac {128}{5} d^{4}\right ) b^{4}}{193}-\frac {656 \left (-\frac {735}{328} e^{3} x^{3}+\frac {483}{328} d \,e^{2} x^{2}-\frac {48}{41} d^{2} e x +d^{3}\right ) e a \,b^{3}}{965}+\frac {1368 \left (\frac {112}{57} x^{2} e^{2}-\frac {289}{228} d e x +d^{2}\right ) e^{2} a^{2} b^{2}}{965}-\frac {298 \left (-\frac {237 e x}{149}+d \right ) e^{3} a^{3} b}{193}+e^{4} a^{4}\right ) \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}}{128}}{\left (a e -b d \right )^{5} \left (b x +a \right )^{5} \sqrt {\left (a e -b d \right ) b}}\)	\(219\)
derivativedivides	\(2 e^{5} \left (\frac {\sqrt {e x +d}}{10 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {\frac {9 \sqrt {e x +d}}{80 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {9 \left (\frac {7 \sqrt {e x +d}}{48 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}\right )}{6 \left (a e -b d \right )}\right )}{8 \left (a e -b d \right )}\right )}{10 \left (a e -b d \right )}}{a e -b d}\right )\)	\(285\)
default	\(2 e^{5} \left (\frac {\sqrt {e x +d}}{10 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {\frac {9 \sqrt {e x +d}}{80 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {9 \left (\frac {7 \sqrt {e x +d}}{48 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}\right )}{6 \left (a e -b d \right )}\right )}{8 \left (a e -b d \right )}\right )}{10 \left (a e -b d \right )}}{a e -b d}\right )\)	\(285\)

[In]

int(1/(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

193/128*(63/193*e^5*(b*x+a)^5*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))+(1/193*(63*e^4*x^4-42*d*e^3*x^3+168/
5*d^2*e^2*x^2-144/5*d^3*e*x+128/5*d^4)*b^4-656/965*(-735/328*e^3*x^3+483/328*d*e^2*x^2-48/41*d^2*e*x+d^3)*e*a*
b^3+1368/965*(112/57*x^2*e^2-289/228*d*e*x+d^2)*e^2*a^2*b^2-298/193*(-237/149*e*x+d)*e^3*a^3*b+e^4*a^4)*(e*x+d
)^(1/2)*((a*e-b*d)*b)^(1/2))/((a*e-b*d)*b)^(1/2)/(a*e-b*d)^5/(b*x+a)^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 905 vs. \(2 (181) = 362\).

Time = 0.48 (sec) , antiderivative size = 1824, normalized size of antiderivative = 8.56 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/1280*(315*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e
^5)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) + 2*(128*b^
6*d^5 - 784*a*b^5*d^4*e + 2024*a^2*b^4*d^3*e^2 - 2858*a^3*b^3*d^2*e^3 + 2455*a^4*b^2*d*e^4 - 965*a^5*b*e^5 + 3
15*(b^6*d*e^4 - a*b^5*e^5)*x^4 - 210*(b^6*d^2*e^3 - 8*a*b^5*d*e^4 + 7*a^2*b^4*e^5)*x^3 + 42*(4*b^6*d^3*e^2 - 2
7*a*b^5*d^2*e^3 + 87*a^2*b^4*d*e^4 - 64*a^3*b^3*e^5)*x^2 - 6*(24*b^6*d^4*e - 152*a*b^5*d^3*e^2 + 417*a^2*b^4*d
^2*e^3 - 684*a^3*b^3*d*e^4 + 395*a^4*b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d^6 - 6*a^6*b^6*d^5*e + 15*a^7*b^5*d^
4*e^2 - 20*a^8*b^4*d^3*e^3 + 15*a^9*b^3*d^2*e^4 - 6*a^10*b^2*d*e^5 + a^11*b*e^6 + (b^12*d^6 - 6*a*b^11*d^5*e +
15*a^2*b^10*d^4*e^2 - 20*a^3*b^9*d^3*e^3 + 15*a^4*b^8*d^2*e^4 - 6*a^5*b^7*d*e^5 + a^6*b^6*e^6)*x^5 + 5*(a*b^1
1*d^6 - 6*a^2*b^10*d^5*e + 15*a^3*b^9*d^4*e^2 - 20*a^4*b^8*d^3*e^3 + 15*a^5*b^7*d^2*e^4 - 6*a^6*b^6*d*e^5 + a^
7*b^5*e^6)*x^4 + 10*(a^2*b^10*d^6 - 6*a^3*b^9*d^5*e + 15*a^4*b^8*d^4*e^2 - 20*a^5*b^7*d^3*e^3 + 15*a^6*b^6*d^2
*e^4 - 6*a^7*b^5*d*e^5 + a^8*b^4*e^6)*x^3 + 10*(a^3*b^9*d^6 - 6*a^4*b^8*d^5*e + 15*a^5*b^7*d^4*e^2 - 20*a^6*b^
6*d^3*e^3 + 15*a^7*b^5*d^2*e^4 - 6*a^8*b^4*d*e^5 + a^9*b^3*e^6)*x^2 + 5*(a^4*b^8*d^6 - 6*a^5*b^7*d^5*e + 15*a^
6*b^6*d^4*e^2 - 20*a^7*b^5*d^3*e^3 + 15*a^8*b^4*d^2*e^4 - 6*a^9*b^3*d*e^5 + a^10*b^2*e^6)*x), -1/640*(315*(b^5
*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*sqrt(-b^2*d +
a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (128*b^6*d^5 - 784*a*b^5*d^4*e + 2024*a^2*b^
4*d^3*e^2 - 2858*a^3*b^3*d^2*e^3 + 2455*a^4*b^2*d*e^4 - 965*a^5*b*e^5 + 315*(b^6*d*e^4 - a*b^5*e^5)*x^4 - 210*
(b^6*d^2*e^3 - 8*a*b^5*d*e^4 + 7*a^2*b^4*e^5)*x^3 + 42*(4*b^6*d^3*e^2 - 27*a*b^5*d^2*e^3 + 87*a^2*b^4*d*e^4 -
64*a^3*b^3*e^5)*x^2 - 6*(24*b^6*d^4*e - 152*a*b^5*d^3*e^2 + 417*a^2*b^4*d^2*e^3 - 684*a^3*b^3*d*e^4 + 395*a^4*
b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d^6 - 6*a^6*b^6*d^5*e + 15*a^7*b^5*d^4*e^2 - 20*a^8*b^4*d^3*e^3 + 15*a^9*b
^3*d^2*e^4 - 6*a^10*b^2*d*e^5 + a^11*b*e^6 + (b^12*d^6 - 6*a*b^11*d^5*e + 15*a^2*b^10*d^4*e^2 - 20*a^3*b^9*d^3
*e^3 + 15*a^4*b^8*d^2*e^4 - 6*a^5*b^7*d*e^5 + a^6*b^6*e^6)*x^5 + 5*(a*b^11*d^6 - 6*a^2*b^10*d^5*e + 15*a^3*b^9
*d^4*e^2 - 20*a^4*b^8*d^3*e^3 + 15*a^5*b^7*d^2*e^4 - 6*a^6*b^6*d*e^5 + a^7*b^5*e^6)*x^4 + 10*(a^2*b^10*d^6 - 6
*a^3*b^9*d^5*e + 15*a^4*b^8*d^4*e^2 - 20*a^5*b^7*d^3*e^3 + 15*a^6*b^6*d^2*e^4 - 6*a^7*b^5*d*e^5 + a^8*b^4*e^6)
*x^3 + 10*(a^3*b^9*d^6 - 6*a^4*b^8*d^5*e + 15*a^5*b^7*d^4*e^2 - 20*a^6*b^6*d^3*e^3 + 15*a^7*b^5*d^2*e^4 - 6*a^
8*b^4*d*e^5 + a^9*b^3*e^6)*x^2 + 5*(a^4*b^8*d^6 - 6*a^5*b^7*d^5*e + 15*a^6*b^6*d^4*e^2 - 20*a^7*b^5*d^3*e^3 +
15*a^8*b^4*d^2*e^4 - 6*a^9*b^3*d*e^5 + a^10*b^2*e^6)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,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-b*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (181) = 362\).

Time = 0.28 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.14 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {63 \, e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e}} - \frac {315 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 1470 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} d e^{5} + 2688 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} - 2370 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} + 965 \, \sqrt {e x + d} b^{4} d^{4} e^{5} + 1470 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{3} e^{6} - 5376 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} + 7110 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} - 3860 \, \sqrt {e x + d} a b^{3} d^{3} e^{6} + 2688 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} - 7110 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} + 5790 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{7} + 2370 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{8} - 3860 \, \sqrt {e x + d} a^{3} b d e^{8} + 965 \, \sqrt {e x + d} a^{4} e^{9}}{640 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]

[In]

integrate(1/(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-63/128*e^5*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a
^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(-b^2*d + a*b*e)) - 1/640*(315*(e*x + d)^(9/2)*b^4*e^5 - 1470*(e
*x + d)^(7/2)*b^4*d*e^5 + 2688*(e*x + d)^(5/2)*b^4*d^2*e^5 - 2370*(e*x + d)^(3/2)*b^4*d^3*e^5 + 965*sqrt(e*x +
d)*b^4*d^4*e^5 + 1470*(e*x + d)^(7/2)*a*b^3*e^6 - 5376*(e*x + d)^(5/2)*a*b^3*d*e^6 + 7110*(e*x + d)^(3/2)*a*b
^3*d^2*e^6 - 3860*sqrt(e*x + d)*a*b^3*d^3*e^6 + 2688*(e*x + d)^(5/2)*a^2*b^2*e^7 - 7110*(e*x + d)^(3/2)*a^2*b^
2*d*e^7 + 5790*sqrt(e*x + d)*a^2*b^2*d^2*e^7 + 2370*(e*x + d)^(3/2)*a^3*b*e^8 - 3860*sqrt(e*x + d)*a^3*b*d*e^8
+ 965*sqrt(e*x + d)*a^4*e^9)/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*
e^4 - a^5*e^5)*((e*x + d)*b - b*d + a*e)^5)

Mupad [B] (verification not implemented)

Time = 9.60 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {965\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{9/2}\,\sqrt {d+e\,x}+2370\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{7/2}\,{\left (d+e\,x\right )}^{3/2}+2688\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{5/2}\,{\left (d+e\,x\right )}^{5/2}+1470\,b^{7/2}\,{\left (a\,e-b\,d\right )}^{3/2}\,{\left (d+e\,x\right )}^{7/2}+315\,b^{9/2}\,\sqrt {a\,e-b\,d}\,{\left (d+e\,x\right )}^{9/2}+315\,b^5\,e^5\,x^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{640\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{11/2}}-\frac {63\,b^{9/2}\,e^5\,x^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,{\left (a\,e-b\,d\right )}^{11/2}}}{{\left (a+b\,x\right )}^5}+\frac {63\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{11/2}} \]

[In]

int(1/((d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

((965*b^(1/2)*(a*e - b*d)^(9/2)*(d + e*x)^(1/2) + 2370*b^(3/2)*(a*e - b*d)^(7/2)*(d + e*x)^(3/2) + 2688*b^(5/2
)*(a*e - b*d)^(5/2)*(d + e*x)^(5/2) + 1470*b^(7/2)*(a*e - b*d)^(3/2)*(d + e*x)^(7/2) + 315*b^(9/2)*(a*e - b*d)
^(1/2)*(d + e*x)^(9/2) + 315*b^5*e^5*x^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(640*b^(1/2)*(a*e
- b*d)^(11/2)) - (63*b^(9/2)*e^5*x^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(128*(a*e - b*d)^(11/2
)))/(a + b*x)^5 + (63*e^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(128*b^(1/2)*(a*e - b*d)^(11/2))