\(\int \frac {\sqrt {d+e x}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1671]

Optimal result

Integrand size = 28, antiderivative size = 218 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}-\frac {7 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}} \]

[Out]

-7/128*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(3/2)/(-a*e+b*d)^(9/2)-1/5*(e*x+d)^(1/2)/b/(b*x+a
)^5-1/40*e*(e*x+d)^(1/2)/b/(-a*e+b*d)/(b*x+a)^4+7/240*e^2*(e*x+d)^(1/2)/b/(-a*e+b*d)^2/(b*x+a)^3-7/192*e^3*(e*
x+d)^(1/2)/b/(-a*e+b*d)^3/(b*x+a)^2+7/128*e^4*(e*x+d)^(1/2)/b/(-a*e+b*d)^4/(b*x+a)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 44, 65, 214} \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {7 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}+\frac {7 e^4 \sqrt {d+e x}}{128 b (a+b x) (b d-a e)^4}-\frac {7 e^3 \sqrt {d+e x}}{192 b (a+b x)^2 (b d-a e)^3}+\frac {7 e^2 \sqrt {d+e x}}{240 b (a+b x)^3 (b d-a e)^2}-\frac {e \sqrt {d+e x}}{40 b (a+b x)^4 (b d-a e)}-\frac {\sqrt {d+e x}}{5 b (a+b x)^5} \]

[In]

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

[Out]

-1/5*Sqrt[d + e*x]/(b*(a + b*x)^5) - (e*Sqrt[d + e*x])/(40*b*(b*d - a*e)*(a + b*x)^4) + (7*e^2*Sqrt[d + e*x])/
(240*b*(b*d - a*e)^2*(a + b*x)^3) - (7*e^3*Sqrt[d + e*x])/(192*b*(b*d - a*e)^3*(a + b*x)^2) + (7*e^4*Sqrt[d +
e*x])/(128*b*(b*d - a*e)^4*(a + b*x)) - (7*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(3/2)*
(b*d - a*e)^(9/2))

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 43

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

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 {\sqrt {d+e x}}{(a+b x)^6} \, dx \\ & = -\frac {\sqrt {d+e x}}{5 b (a+b x)^5}+\frac {e \int \frac {1}{(a+b x)^5 \sqrt {d+e x}} \, dx}{10 b} \\ & = -\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}-\frac {\left (7 e^2\right ) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{80 b (b d-a e)} \\ & = -\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}+\frac {\left (7 e^3\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{96 b (b d-a e)^2} \\ & = -\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}-\frac {\left (7 e^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b (b d-a e)^3} \\ & = -\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}+\frac {\left (7 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b (b d-a e)^4} \\ & = -\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}+\frac {\left (7 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 (b d-a e)^4} \\ & = -\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}-\frac {7 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {d+e x} \left (-105 a^4 e^4+10 a^3 b e^3 (121 d+79 e x)+2 a^2 b^2 e^2 \left (-1052 d^2-289 d e x+448 e^2 x^2\right )+2 a b^3 e \left (744 d^3+128 d^2 e x-161 d e^2 x^2+245 e^3 x^3\right )+b^4 \left (-384 d^4-48 d^3 e x+56 d^2 e^2 x^2-70 d e^3 x^3+105 e^4 x^4\right )\right )}{1920 b (b d-a e)^4 (a+b x)^5}+\frac {7 e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{3/2} (-b d+a e)^{9/2}} \]

[In]

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

[Out]

(Sqrt[d + e*x]*(-105*a^4*e^4 + 10*a^3*b*e^3*(121*d + 79*e*x) + 2*a^2*b^2*e^2*(-1052*d^2 - 289*d*e*x + 448*e^2*
x^2) + 2*a*b^3*e*(744*d^3 + 128*d^2*e*x - 161*d*e^2*x^2 + 245*e^3*x^3) + b^4*(-384*d^4 - 48*d^3*e*x + 56*d^2*e
^2*x^2 - 70*d*e^3*x^3 + 105*e^4*x^4)))/(1920*b*(b*d - a*e)^4*(a + b*x)^5) + (7*e^5*ArcTan[(Sqrt[b]*Sqrt[d + e*
x])/Sqrt[-(b*d) + a*e]])/(128*b^(3/2)*(-(b*d) + a*e)^(9/2))

Maple [A] (verified)

Time = 2.53 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(\frac {-\frac {7 \sqrt {\left (a e -b d \right ) b}\, \left (\left (-b^{4} x^{4}-\frac {14}{3} a \,b^{3} x^{3}-\frac {128}{15} a^{2} b^{2} x^{2}-\frac {158}{21} a^{3} b x +a^{4}\right ) e^{4}-\frac {242 b d \left (-\frac {7}{121} b^{3} x^{3}-\frac {161}{605} a \,b^{2} x^{2}-\frac {289}{605} a^{2} b x +a^{3}\right ) e^{3}}{21}+\frac {2104 b^{2} \left (-\frac {7}{263} b^{2} x^{2}-\frac {32}{263} a b x +a^{2}\right ) d^{2} e^{2}}{105}-\frac {496 \left (-\frac {b x}{31}+a \right ) b^{3} d^{3} e}{35}+\frac {128 b^{4} d^{4}}{35}\right ) \sqrt {e x +d}}{128}+\frac {7 e^{5} \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128}}{\sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{5} b \left (a e -b d \right )^{4}}\)	\(220\)
derivativedivides	\(2 e^{5} \left (\frac {\frac {7 b^{3} \left (e x +d \right )^{\frac {9}{2}}}{256 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {49 b^{2} \left (e x +d \right )^{\frac {7}{2}}}{384 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 b \left (e x +d \right )^{\frac {5}{2}}}{30 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {79 \left (e x +d \right )^{\frac {3}{2}}}{384 \left (a e -b d \right )}-\frac {7 \sqrt {e x +d}}{256 b}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\)	\(293\)
default	\(2 e^{5} \left (\frac {\frac {7 b^{3} \left (e x +d \right )^{\frac {9}{2}}}{256 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {49 b^{2} \left (e x +d \right )^{\frac {7}{2}}}{384 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 b \left (e x +d \right )^{\frac {5}{2}}}{30 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {79 \left (e x +d \right )^{\frac {3}{2}}}{384 \left (a e -b d \right )}-\frac {7 \sqrt {e x +d}}{256 b}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\)	\(293\)

[In]

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

[Out]

7/128/((a*e-b*d)*b)^(1/2)*(-((a*e-b*d)*b)^(1/2)*((-b^4*x^4-14/3*a*b^3*x^3-128/15*a^2*b^2*x^2-158/21*a^3*b*x+a^
4)*e^4-242/21*b*d*(-7/121*b^3*x^3-161/605*a*b^2*x^2-289/605*a^2*b*x+a^3)*e^3+2104/105*b^2*(-7/263*b^2*x^2-32/2
63*a*b*x+a^2)*d^2*e^2-496/35*(-1/31*b*x+a)*b^3*d^3*e+128/35*b^4*d^4)*(e*x+d)^(1/2)+e^5*(b*x+a)^5*arctan(b*(e*x
+d)^(1/2)/((a*e-b*d)*b)^(1/2)))/(b*x+a)^5/b/(a*e-b*d)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 830 vs. \(2 (186) = 372\).

Time = 0.45 (sec) , antiderivative size = 1673, normalized size of antiderivative = 7.67 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/3840*(105*(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*(384*b^6
*d^5 - 1872*a*b^5*d^4*e + 3592*a^2*b^4*d^3*e^2 - 3314*a^3*b^3*d^2*e^3 + 1315*a^4*b^2*d*e^4 - 105*a^5*b*e^5 - 1
05*(b^6*d*e^4 - a*b^5*e^5)*x^4 + 70*(b^6*d^2*e^3 - 8*a*b^5*d*e^4 + 7*a^2*b^4*e^5)*x^3 - 14*(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 + 2*(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^5 - 5*a^6*b^6*d^4*e + 10*a^7*b^5*d^3
*e^2 - 10*a^8*b^4*d^2*e^3 + 5*a^9*b^3*d*e^4 - a^10*b^2*e^5 + (b^12*d^5 - 5*a*b^11*d^4*e + 10*a^2*b^10*d^3*e^2
- 10*a^3*b^9*d^2*e^3 + 5*a^4*b^8*d*e^4 - a^5*b^7*e^5)*x^5 + 5*(a*b^11*d^5 - 5*a^2*b^10*d^4*e + 10*a^3*b^9*d^3*
e^2 - 10*a^4*b^8*d^2*e^3 + 5*a^5*b^7*d*e^4 - a^6*b^6*e^5)*x^4 + 10*(a^2*b^10*d^5 - 5*a^3*b^9*d^4*e + 10*a^4*b^
8*d^3*e^2 - 10*a^5*b^7*d^2*e^3 + 5*a^6*b^6*d*e^4 - a^7*b^5*e^5)*x^3 + 10*(a^3*b^9*d^5 - 5*a^4*b^8*d^4*e + 10*a
^5*b^7*d^3*e^2 - 10*a^6*b^6*d^2*e^3 + 5*a^7*b^5*d*e^4 - a^8*b^4*e^5)*x^2 + 5*(a^4*b^8*d^5 - 5*a^5*b^7*d^4*e +
10*a^6*b^6*d^3*e^2 - 10*a^7*b^5*d^2*e^3 + 5*a^8*b^4*d*e^4 - a^9*b^3*e^5)*x), 1/1920*(105*(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(sqr
t(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (384*b^6*d^5 - 1872*a*b^5*d^4*e + 3592*a^2*b^4*d^3*e^2 - 3314
*a^3*b^3*d^2*e^3 + 1315*a^4*b^2*d*e^4 - 105*a^5*b*e^5 - 105*(b^6*d*e^4 - a*b^5*e^5)*x^4 + 70*(b^6*d^2*e^3 - 8*
a*b^5*d*e^4 + 7*a^2*b^4*e^5)*x^3 - 14*(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 + 2*(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^5 - 5*a^6*b^6*d^4*e + 10*a^7*b^5*d^3*e^2 - 10*a^8*b^4*d^2*e^3 + 5*a^9*b^3*d*e^4 - a^10*b^
2*e^5 + (b^12*d^5 - 5*a*b^11*d^4*e + 10*a^2*b^10*d^3*e^2 - 10*a^3*b^9*d^2*e^3 + 5*a^4*b^8*d*e^4 - a^5*b^7*e^5)
*x^5 + 5*(a*b^11*d^5 - 5*a^2*b^10*d^4*e + 10*a^3*b^9*d^3*e^2 - 10*a^4*b^8*d^2*e^3 + 5*a^5*b^7*d*e^4 - a^6*b^6*
e^5)*x^4 + 10*(a^2*b^10*d^5 - 5*a^3*b^9*d^4*e + 10*a^4*b^8*d^3*e^2 - 10*a^5*b^7*d^2*e^3 + 5*a^6*b^6*d*e^4 - a^
7*b^5*e^5)*x^3 + 10*(a^3*b^9*d^5 - 5*a^4*b^8*d^4*e + 10*a^5*b^7*d^3*e^2 - 10*a^6*b^6*d^2*e^3 + 5*a^7*b^5*d*e^4
- a^8*b^4*e^5)*x^2 + 5*(a^4*b^8*d^5 - 5*a^5*b^7*d^4*e + 10*a^6*b^6*d^3*e^2 - 10*a^7*b^5*d^2*e^3 + 5*a^8*b^4*d
*e^4 - a^9*b^3*e^5)*x)]

Sympy [F(-1)]

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

[In]

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

[In]

integrate((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. 432 vs. \(2 (186) = 372\).

Time = 0.32 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {7 \, e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {105 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 490 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} d e^{5} + 896 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} - 790 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} - 105 \, \sqrt {e x + d} b^{4} d^{4} e^{5} + 490 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{3} e^{6} - 1792 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} + 2370 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} + 420 \, \sqrt {e x + d} a b^{3} d^{3} e^{6} + 896 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} - 2370 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} - 630 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{7} + 790 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{8} + 420 \, \sqrt {e x + d} a^{3} b d e^{8} - 105 \, \sqrt {e x + d} a^{4} e^{9}}{1920 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]

[In]

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

[Out]

7/128*e^5*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b
^2*d*e^3 + a^4*b*e^4)*sqrt(-b^2*d + a*b*e)) + 1/1920*(105*(e*x + d)^(9/2)*b^4*e^5 - 490*(e*x + d)^(7/2)*b^4*d*
e^5 + 896*(e*x + d)^(5/2)*b^4*d^2*e^5 - 790*(e*x + d)^(3/2)*b^4*d^3*e^5 - 105*sqrt(e*x + d)*b^4*d^4*e^5 + 490*
(e*x + d)^(7/2)*a*b^3*e^6 - 1792*(e*x + d)^(5/2)*a*b^3*d*e^6 + 2370*(e*x + d)^(3/2)*a*b^3*d^2*e^6 + 420*sqrt(e
*x + d)*a*b^3*d^3*e^6 + 896*(e*x + d)^(5/2)*a^2*b^2*e^7 - 2370*(e*x + d)^(3/2)*a^2*b^2*d*e^7 - 630*sqrt(e*x +
d)*a^2*b^2*d^2*e^7 + 790*(e*x + d)^(3/2)*a^3*b*e^8 + 420*sqrt(e*x + d)*a^3*b*d*e^8 - 105*sqrt(e*x + d)*a^4*e^9
)/((b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*((e*x + d)*b - b*d + a*e)^5)

Mupad [B] (verification not implemented)

Time = 9.54 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.84 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {79\,e^5\,{\left (d+e\,x\right )}^{3/2}}{192\,\left (a\,e-b\,d\right )}-\frac {7\,e^5\,\sqrt {d+e\,x}}{128\,b}+\frac {49\,b^2\,e^5\,{\left (d+e\,x\right )}^{7/2}}{192\,{\left (a\,e-b\,d\right )}^3}+\frac {7\,b^3\,e^5\,{\left (d+e\,x\right )}^{9/2}}{128\,{\left (a\,e-b\,d\right )}^4}+\frac {7\,b\,e^5\,{\left (d+e\,x\right )}^{5/2}}{15\,{\left (a\,e-b\,d\right )}^2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4}+\frac {7\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{9/2}} \]

[In]

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

[Out]

((79*e^5*(d + e*x)^(3/2))/(192*(a*e - b*d)) - (7*e^5*(d + e*x)^(1/2))/(128*b) + (49*b^2*e^5*(d + e*x)^(7/2))/(
192*(a*e - b*d)^3) + (7*b^3*e^5*(d + e*x)^(9/2))/(128*(a*e - b*d)^4) + (7*b*e^5*(d + e*x)^(5/2))/(15*(a*e - b*
d)^2))/((d + e*x)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3*e) - (d + e*
x)^2*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b^4*d^2*e) + b^5*(d + e*x)^5 - (5*b^5*d - 5*a*b^4*
e)*(d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e) - 10*a^2*b^3*d^3
*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) + (7*e^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)
^(1/2)))/(128*b^(3/2)*(a*e - b*d)^(9/2))