\(\int \frac {\sqrt {d+e x}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1726]

Optimal result

Integrand size = 30, antiderivative size = 280 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {5 e^3 \sqrt {d+e x}}{64 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^2 \sqrt {d+e x}}{96 b (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{3/2} (b d-a e)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

5/64*e^4*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(3/2)/(-a*e+b*d)^(7/2)/((b*x+a)^2)^(1/2)-5/
64*e^3*(e*x+d)^(1/2)/b/(-a*e+b*d)^3/((b*x+a)^2)^(1/2)-1/4*(e*x+d)^(1/2)/b/(b*x+a)^3/((b*x+a)^2)^(1/2)-1/24*e*(
e*x+d)^(1/2)/b/(-a*e+b*d)/(b*x+a)^2/((b*x+a)^2)^(1/2)+5/96*e^2*(e*x+d)^(1/2)/b/(-a*e+b*d)^2/(b*x+a)/((b*x+a)^2
)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 44, 65, 214} \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {5 e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}-\frac {5 e^3 \sqrt {d+e x}}{64 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac {5 e^2 \sqrt {d+e x}}{96 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {e \sqrt {d+e x}}{24 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]

[In]

Int[Sqrt[d + e*x]/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-5*e^3*Sqrt[d + e*x])/(64*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d + e*x]/(4*b*(a + b*x)^3*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]) - (e*Sqrt[d + e*x])/(24*b*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(5*e^2*Sqrt[d + e*x])/(96*b*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^4*(a + b*x)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(3/2)*(b*d - a*e)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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]

Rule 660

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 \sqrt {d+e x}} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 b e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 \sqrt {d+e x}} \, dx}{48 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^2 \sqrt {d+e x}}{96 b (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{64 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {5 e^3 \sqrt {d+e x}}{64 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^2 \sqrt {d+e x}}{96 b (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {5 e^3 \sqrt {d+e x}}{64 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^2 \sqrt {d+e x}}{96 b (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 e^3 \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {5 e^3 \sqrt {d+e x}}{64 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^2 \sqrt {d+e x}}{96 b (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{3/2} (b d-a e)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x)^5 \left (\frac {\sqrt {b} \sqrt {d+e x} \left (-15 a^3 e^3+a^2 b e^2 (118 d+73 e x)+a b^2 e \left (-136 d^2-36 d e x+55 e^2 x^2\right )+b^3 \left (48 d^3+8 d^2 e x-10 d e^2 x^2+15 e^3 x^3\right )\right )}{e^4 (-b d+a e)^3 (a+b x)^4}+\frac {15 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{7/2}}\right )}{192 b^{3/2} \left ((a+b x)^2\right )^{5/2}} \]

[In]

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

[Out]

(e^4*(a + b*x)^5*((Sqrt[b]*Sqrt[d + e*x]*(-15*a^3*e^3 + a^2*b*e^2*(118*d + 73*e*x) + a*b^2*e*(-136*d^2 - 36*d*
e*x + 55*e^2*x^2) + b^3*(48*d^3 + 8*d^2*e*x - 10*d*e^2*x^2 + 15*e^3*x^3)))/(e^4*(-(b*d) + a*e)^3*(a + b*x)^4)
+ (15*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(7/2)))/(192*b^(3/2)*((a + b*x)^2)^(5
/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(499\) vs. \(2(197)=394\).

Time = 2.29 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.79

method	result	size
default	\(\frac {\left (b x +a \right ) \left (15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{4} e^{4} x^{4}+60 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{3} e^{4} x^{3}+15 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {7}{2}} b^{3}+90 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{2} e^{4} x^{2}+55 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e -55 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{3} d +60 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b \,e^{4} x +73 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a^{2} b \,e^{2}-146 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} d e +73 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} d^{2}+15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} e^{4}-15 \sqrt {e x +d}\, a^{3} e^{3} \sqrt {\left (a e -b d \right ) b}+45 \sqrt {e x +d}\, a^{2} d \,e^{2} b \sqrt {\left (a e -b d \right ) b}-45 \sqrt {e x +d}\, a \,d^{2} e \,b^{2} \sqrt {\left (a e -b d \right ) b}+15 \sqrt {e x +d}\, d^{3} b^{3} \sqrt {\left (a e -b d \right ) b}\right )}{192 \sqrt {\left (a e -b d \right ) b}\, b \left (a e -b d \right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(500\)

[In]

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

[Out]

1/192*(b*x+a)*(15*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*b^4*e^4*x^4+60*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)
*b)^(1/2))*a*b^3*e^4*x^3+15*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^3+90*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2
))*a^2*b^2*e^4*x^2+55*(e*x+d)^(5/2)*((a*e-b*d)*b)^(1/2)*a*b^2*e-55*(e*x+d)^(5/2)*((a*e-b*d)*b)^(1/2)*b^3*d+60*
arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^3*b*e^4*x+73*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2-146*((a
*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^2*d*e+73*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^3*d^2+15*arctan(b*(e*x+d)^(1/2
)/((a*e-b*d)*b)^(1/2))*a^4*e^4-15*(e*x+d)^(1/2)*a^3*e^3*((a*e-b*d)*b)^(1/2)+45*(e*x+d)^(1/2)*a^2*d*e^2*b*((a*e
-b*d)*b)^(1/2)-45*(e*x+d)^(1/2)*a*d^2*e*b^2*((a*e-b*d)*b)^(1/2)+15*(e*x+d)^(1/2)*d^3*b^3*((a*e-b*d)*b)^(1/2))/
((a*e-b*d)*b)^(1/2)/b/(a*e-b*d)/(a^2*e^2-2*a*b*d*e+b^2*d^2)/((b*x+a)^2)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (197) = 394\).

Time = 0.34 (sec) , antiderivative size = 1176, normalized size of antiderivative = 4.20 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [-\frac {15 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) + 2 \, {\left (48 \, b^{5} d^{4} - 184 \, a b^{4} d^{3} e + 254 \, a^{2} b^{3} d^{2} e^{2} - 133 \, a^{3} b^{2} d e^{3} + 15 \, a^{4} b e^{4} + 15 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} - 5 \, {\left (2 \, b^{5} d^{2} e^{2} - 13 \, a b^{4} d e^{3} + 11 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (8 \, b^{5} d^{3} e - 44 \, a b^{4} d^{2} e^{2} + 109 \, a^{2} b^{3} d e^{3} - 73 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{384 \, {\left (a^{4} b^{6} d^{4} - 4 \, a^{5} b^{5} d^{3} e + 6 \, a^{6} b^{4} d^{2} e^{2} - 4 \, a^{7} b^{3} d e^{3} + a^{8} b^{2} e^{4} + {\left (b^{10} d^{4} - 4 \, a b^{9} d^{3} e + 6 \, a^{2} b^{8} d^{2} e^{2} - 4 \, a^{3} b^{7} d e^{3} + a^{4} b^{6} e^{4}\right )} x^{4} + 4 \, {\left (a b^{9} d^{4} - 4 \, a^{2} b^{8} d^{3} e + 6 \, a^{3} b^{7} d^{2} e^{2} - 4 \, a^{4} b^{6} d e^{3} + a^{5} b^{5} e^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{4} - 4 \, a^{3} b^{7} d^{3} e + 6 \, a^{4} b^{6} d^{2} e^{2} - 4 \, a^{5} b^{5} d e^{3} + a^{6} b^{4} e^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{4} - 4 \, a^{4} b^{6} d^{3} e + 6 \, a^{5} b^{5} d^{2} e^{2} - 4 \, a^{6} b^{4} d e^{3} + a^{7} b^{3} e^{4}\right )} x\right )}}, -\frac {15 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (48 \, b^{5} d^{4} - 184 \, a b^{4} d^{3} e + 254 \, a^{2} b^{3} d^{2} e^{2} - 133 \, a^{3} b^{2} d e^{3} + 15 \, a^{4} b e^{4} + 15 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} - 5 \, {\left (2 \, b^{5} d^{2} e^{2} - 13 \, a b^{4} d e^{3} + 11 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (8 \, b^{5} d^{3} e - 44 \, a b^{4} d^{2} e^{2} + 109 \, a^{2} b^{3} d e^{3} - 73 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{192 \, {\left (a^{4} b^{6} d^{4} - 4 \, a^{5} b^{5} d^{3} e + 6 \, a^{6} b^{4} d^{2} e^{2} - 4 \, a^{7} b^{3} d e^{3} + a^{8} b^{2} e^{4} + {\left (b^{10} d^{4} - 4 \, a b^{9} d^{3} e + 6 \, a^{2} b^{8} d^{2} e^{2} - 4 \, a^{3} b^{7} d e^{3} + a^{4} b^{6} e^{4}\right )} x^{4} + 4 \, {\left (a b^{9} d^{4} - 4 \, a^{2} b^{8} d^{3} e + 6 \, a^{3} b^{7} d^{2} e^{2} - 4 \, a^{4} b^{6} d e^{3} + a^{5} b^{5} e^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{4} - 4 \, a^{3} b^{7} d^{3} e + 6 \, a^{4} b^{6} d^{2} e^{2} - 4 \, a^{5} b^{5} d e^{3} + a^{6} b^{4} e^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{4} - 4 \, a^{4} b^{6} d^{3} e + 6 \, a^{5} b^{5} d^{2} e^{2} - 4 \, a^{6} b^{4} d e^{3} + a^{7} b^{3} e^{4}\right )} x\right )}}\right ] \]

[In]

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

[Out]

[-1/384*(15*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*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*(48*b^5*d^4 - 184*a*b^4*d^3*e +
254*a^2*b^3*d^2*e^2 - 133*a^3*b^2*d*e^3 + 15*a^4*b*e^4 + 15*(b^5*d*e^3 - a*b^4*e^4)*x^3 - 5*(2*b^5*d^2*e^2 -
13*a*b^4*d*e^3 + 11*a^2*b^3*e^4)*x^2 + (8*b^5*d^3*e - 44*a*b^4*d^2*e^2 + 109*a^2*b^3*d*e^3 - 73*a^3*b^2*e^4)*x
)*sqrt(e*x + d))/(a^4*b^6*d^4 - 4*a^5*b^5*d^3*e + 6*a^6*b^4*d^2*e^2 - 4*a^7*b^3*d*e^3 + a^8*b^2*e^4 + (b^10*d^
4 - 4*a*b^9*d^3*e + 6*a^2*b^8*d^2*e^2 - 4*a^3*b^7*d*e^3 + a^4*b^6*e^4)*x^4 + 4*(a*b^9*d^4 - 4*a^2*b^8*d^3*e +
6*a^3*b^7*d^2*e^2 - 4*a^4*b^6*d*e^3 + a^5*b^5*e^4)*x^3 + 6*(a^2*b^8*d^4 - 4*a^3*b^7*d^3*e + 6*a^4*b^6*d^2*e^2
- 4*a^5*b^5*d*e^3 + a^6*b^4*e^4)*x^2 + 4*(a^3*b^7*d^4 - 4*a^4*b^6*d^3*e + 6*a^5*b^5*d^2*e^2 - 4*a^6*b^4*d*e^3
+ a^7*b^3*e^4)*x), -1/192*(15*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*sq
rt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (48*b^5*d^4 - 184*a*b^4*d^3*e +
254*a^2*b^3*d^2*e^2 - 133*a^3*b^2*d*e^3 + 15*a^4*b*e^4 + 15*(b^5*d*e^3 - a*b^4*e^4)*x^3 - 5*(2*b^5*d^2*e^2 - 1
3*a*b^4*d*e^3 + 11*a^2*b^3*e^4)*x^2 + (8*b^5*d^3*e - 44*a*b^4*d^2*e^2 + 109*a^2*b^3*d*e^3 - 73*a^3*b^2*e^4)*x)
*sqrt(e*x + d))/(a^4*b^6*d^4 - 4*a^5*b^5*d^3*e + 6*a^6*b^4*d^2*e^2 - 4*a^7*b^3*d*e^3 + a^8*b^2*e^4 + (b^10*d^4
- 4*a*b^9*d^3*e + 6*a^2*b^8*d^2*e^2 - 4*a^3*b^7*d*e^3 + a^4*b^6*e^4)*x^4 + 4*(a*b^9*d^4 - 4*a^2*b^8*d^3*e + 6
*a^3*b^7*d^2*e^2 - 4*a^4*b^6*d*e^3 + a^5*b^5*e^4)*x^3 + 6*(a^2*b^8*d^4 - 4*a^3*b^7*d^3*e + 6*a^4*b^6*d^2*e^2 -
4*a^5*b^5*d*e^3 + a^6*b^4*e^4)*x^2 + 4*(a^3*b^7*d^4 - 4*a^4*b^6*d^3*e + 6*a^5*b^5*d^2*e^2 - 4*a^6*b^4*d*e^3 +
a^7*b^3*e^4)*x)]

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(d + e*x)/((a + b*x)**2)**(5/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {5 \, e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 55 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} d e^{4} + 73 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} + 15 \, \sqrt {e x + d} b^{3} d^{3} e^{4} + 55 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{2} e^{5} - 146 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} - 45 \, \sqrt {e x + d} a b^{2} d^{2} e^{5} + 73 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b e^{6} + 45 \, \sqrt {e x + d} a^{2} b d e^{6} - 15 \, \sqrt {e x + d} a^{3} e^{7}}{192 \, {\left (b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]

[In]

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

[Out]

-5/64*e^4*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d^3*sgn(b*x + a) - 3*a*b^3*d^2*e*sgn(b*x + a) + 3
*a^2*b^2*d*e^2*sgn(b*x + a) - a^3*b*e^3*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) - 1/192*(15*(e*x + d)^(7/2)*b^3*e^
4 - 55*(e*x + d)^(5/2)*b^3*d*e^4 + 73*(e*x + d)^(3/2)*b^3*d^2*e^4 + 15*sqrt(e*x + d)*b^3*d^3*e^4 + 55*(e*x + d
)^(5/2)*a*b^2*e^5 - 146*(e*x + d)^(3/2)*a*b^2*d*e^5 - 45*sqrt(e*x + d)*a*b^2*d^2*e^5 + 73*(e*x + d)^(3/2)*a^2*
b*e^6 + 45*sqrt(e*x + d)*a^2*b*d*e^6 - 15*sqrt(e*x + d)*a^3*e^7)/((b^4*d^3*sgn(b*x + a) - 3*a*b^3*d^2*e*sgn(b*
x + a) + 3*a^2*b^2*d*e^2*sgn(b*x + a) - a^3*b*e^3*sgn(b*x + a))*((e*x + d)*b - b*d + a*e)^4)

Mupad [F(-1)]

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

[In]

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

[Out]

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