3.17.59 \(\int \frac {\sqrt {d+e x}}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [1659]

Optimal. Leaf size=146 \[ -\frac {\sqrt {d+e x}}{3 b (a+b x)^3}-\frac {e \sqrt {d+e x}}{12 b (b d-a e) (a+b x)^2}+\frac {e^2 \sqrt {d+e x}}{8 b (b d-a e)^2 (a+b x)}-\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2}} \]

[Out]

-1/8*e^3*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(3/2)/(-a*e+b*d)^(5/2)-1/3*(e*x+d)^(1/2)/b/(b*x+a)^
3-1/12*e*(e*x+d)^(1/2)/b/(-a*e+b*d)/(b*x+a)^2+1/8*e^2*(e*x+d)^(1/2)/b/(-a*e+b*d)^2/(b*x+a)

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Rubi [A]
time = 0.05, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 44, 65, 214} \begin {gather*} -\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2}}+\frac {e^2 \sqrt {d+e x}}{8 b (a+b x) (b d-a e)^2}-\frac {e \sqrt {d+e x}}{12 b (a+b x)^2 (b d-a e)}-\frac {\sqrt {d+e x}}{3 b (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Sqrt[d + e*x]/(b*(a + b*x)^3) - (e*Sqrt[d + e*x])/(12*b*(b*d - a*e)*(a + b*x)^2) + (e^2*Sqrt[d + e*x])/(8
*b*(b*d - a*e)^2*(a + b*x)) - (e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(3/2)*(b*d - a*e)^(5
/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*} \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {\sqrt {d+e x}}{(a+b x)^4} \, dx\\ &=-\frac {\sqrt {d+e x}}{3 b (a+b x)^3}+\frac {e \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{6 b}\\ &=-\frac {\sqrt {d+e x}}{3 b (a+b x)^3}-\frac {e \sqrt {d+e x}}{12 b (b d-a e) (a+b x)^2}-\frac {e^2 \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{8 b (b d-a e)}\\ &=-\frac {\sqrt {d+e x}}{3 b (a+b x)^3}-\frac {e \sqrt {d+e x}}{12 b (b d-a e) (a+b x)^2}+\frac {e^2 \sqrt {d+e x}}{8 b (b d-a e)^2 (a+b x)}+\frac {e^3 \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b (b d-a e)^2}\\ &=-\frac {\sqrt {d+e x}}{3 b (a+b x)^3}-\frac {e \sqrt {d+e x}}{12 b (b d-a e) (a+b x)^2}+\frac {e^2 \sqrt {d+e x}}{8 b (b d-a e)^2 (a+b x)}+\frac {e^2 \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b (b d-a e)^2}\\ &=-\frac {\sqrt {d+e x}}{3 b (a+b x)^3}-\frac {e \sqrt {d+e x}}{12 b (b d-a e) (a+b x)^2}+\frac {e^2 \sqrt {d+e x}}{8 b (b d-a e)^2 (a+b x)}-\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 131, normalized size = 0.90 \begin {gather*} \frac {\sqrt {d+e x} \left (-3 a^2 e^2+2 a b e (7 d+4 e x)+b^2 \left (-8 d^2-2 d e x+3 e^2 x^2\right )\right )}{24 b (b d-a e)^2 (a+b x)^3}+\frac {e^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 b^{3/2} (-b d+a e)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(-3*a^2*e^2 + 2*a*b*e*(7*d + 4*e*x) + b^2*(-8*d^2 - 2*d*e*x + 3*e^2*x^2)))/(24*b*(b*d - a*e)^2*
(a + b*x)^3) + (e^3*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(8*b^(3/2)*(-(b*d) + a*e)^(5/2))

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Maple [A]
time = 0.69, size = 152, normalized size = 1.04

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {\frac {b \left (e x +d \right )^{\frac {5}{2}}}{16 a^{2} e^{2}-32 a b d e +16 b^{2} d^{2}}+\frac {\left (e x +d \right )^{\frac {3}{2}}}{6 a e -6 b d}-\frac {\sqrt {e x +d}}{16 b}}{\left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {b \left (a e -b d \right )}}\right )\)	\(152\)
default	\(2 e^{3} \left (\frac {\frac {b \left (e x +d \right )^{\frac {5}{2}}}{16 a^{2} e^{2}-32 a b d e +16 b^{2} d^{2}}+\frac {\left (e x +d \right )^{\frac {3}{2}}}{6 a e -6 b d}-\frac {\sqrt {e x +d}}{16 b}}{\left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {b \left (a e -b d \right )}}\right )\)	\(152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^3*((1/16*b/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(5/2)+1/6/(a*e-b*d)*(e*x+d)^(3/2)-1/16/b*(e*x+d)^(1/2))/((e
*x+d)*b+a*e-b*d)^3+1/16/b/(a^2*e^2-2*a*b*d*e+b^2*d^2)/(b*(a*e-b*d))^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))
^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^2,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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (129) = 258\).
time = 3.01, size = 754, normalized size = 5.16 \begin {gather*} \left [\frac {3 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt {b^{2} d - a b e} e^{3} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) - 2 \, {\left (8 \, b^{4} d^{3} + {\left (3 \, a b^{3} x^{2} + 8 \, a^{2} b^{2} x - 3 \, a^{3} b\right )} e^{3} - {\left (3 \, b^{4} d x^{2} + 10 \, a b^{3} d x - 17 \, a^{2} b^{2} d\right )} e^{2} + 2 \, {\left (b^{4} d^{2} x - 11 \, a b^{3} d^{2}\right )} e\right )} \sqrt {x e + d}}{48 \, {\left (b^{8} d^{3} x^{3} + 3 \, a b^{7} d^{3} x^{2} + 3 \, a^{2} b^{6} d^{3} x + a^{3} b^{5} d^{3} - {\left (a^{3} b^{5} x^{3} + 3 \, a^{4} b^{4} x^{2} + 3 \, a^{5} b^{3} x + a^{6} b^{2}\right )} e^{3} + 3 \, {\left (a^{2} b^{6} d x^{3} + 3 \, a^{3} b^{5} d x^{2} + 3 \, a^{4} b^{4} d x + a^{5} b^{3} d\right )} e^{2} - 3 \, {\left (a b^{7} d^{2} x^{3} + 3 \, a^{2} b^{6} d^{2} x^{2} + 3 \, a^{3} b^{5} d^{2} x + a^{4} b^{4} d^{2}\right )} e\right )}}, \frac {3 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) e^{3} - {\left (8 \, b^{4} d^{3} + {\left (3 \, a b^{3} x^{2} + 8 \, a^{2} b^{2} x - 3 \, a^{3} b\right )} e^{3} - {\left (3 \, b^{4} d x^{2} + 10 \, a b^{3} d x - 17 \, a^{2} b^{2} d\right )} e^{2} + 2 \, {\left (b^{4} d^{2} x - 11 \, a b^{3} d^{2}\right )} e\right )} \sqrt {x e + d}}{24 \, {\left (b^{8} d^{3} x^{3} + 3 \, a b^{7} d^{3} x^{2} + 3 \, a^{2} b^{6} d^{3} x + a^{3} b^{5} d^{3} - {\left (a^{3} b^{5} x^{3} + 3 \, a^{4} b^{4} x^{2} + 3 \, a^{5} b^{3} x + a^{6} b^{2}\right )} e^{3} + 3 \, {\left (a^{2} b^{6} d x^{3} + 3 \, a^{3} b^{5} d x^{2} + 3 \, a^{4} b^{4} d x + a^{5} b^{3} d\right )} e^{2} - 3 \, {\left (a b^{7} d^{2} x^{3} + 3 \, a^{2} b^{6} d^{2} x^{2} + 3 \, a^{3} b^{5} d^{2} x + a^{4} b^{4} d^{2}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(3*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(b^2*d - a*b*e)*e^3*log((2*b*d + (b*x - a)*e - 2*sqrt(b
^2*d - a*b*e)*sqrt(x*e + d))/(b*x + a)) - 2*(8*b^4*d^3 + (3*a*b^3*x^2 + 8*a^2*b^2*x - 3*a^3*b)*e^3 - (3*b^4*d*
x^2 + 10*a*b^3*d*x - 17*a^2*b^2*d)*e^2 + 2*(b^4*d^2*x - 11*a*b^3*d^2)*e)*sqrt(x*e + d))/(b^8*d^3*x^3 + 3*a*b^7
*d^3*x^2 + 3*a^2*b^6*d^3*x + a^3*b^5*d^3 - (a^3*b^5*x^3 + 3*a^4*b^4*x^2 + 3*a^5*b^3*x + a^6*b^2)*e^3 + 3*(a^2*
b^6*d*x^3 + 3*a^3*b^5*d*x^2 + 3*a^4*b^4*d*x + a^5*b^3*d)*e^2 - 3*(a*b^7*d^2*x^3 + 3*a^2*b^6*d^2*x^2 + 3*a^3*b^
5*d^2*x + a^4*b^4*d^2)*e), 1/24*(3*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(
-b^2*d + a*b*e)*sqrt(x*e + d)/(b*x*e + b*d))*e^3 - (8*b^4*d^3 + (3*a*b^3*x^2 + 8*a^2*b^2*x - 3*a^3*b)*e^3 - (3
*b^4*d*x^2 + 10*a*b^3*d*x - 17*a^2*b^2*d)*e^2 + 2*(b^4*d^2*x - 11*a*b^3*d^2)*e)*sqrt(x*e + d))/(b^8*d^3*x^3 +
3*a*b^7*d^3*x^2 + 3*a^2*b^6*d^3*x + a^3*b^5*d^3 - (a^3*b^5*x^3 + 3*a^4*b^4*x^2 + 3*a^5*b^3*x + a^6*b^2)*e^3 +
3*(a^2*b^6*d*x^3 + 3*a^3*b^5*d*x^2 + 3*a^4*b^4*d*x + a^5*b^3*d)*e^2 - 3*(a*b^7*d^2*x^3 + 3*a^2*b^6*d^2*x^2 + 3
*a^3*b^5*d^2*x + a^4*b^4*d^2)*e)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 4592 vs. \(2 (119) = 238\).
time = 64.22, size = 4592, normalized size = 31.45 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-66*a**3*e**6*sqrt(d + e*x)/(48*a**6*b*e**6 - 144*a**5*b**2*d*e**5 + 144*a**5*b**2*e**6*x - 720*a**4*b**3*d*e*
*5*x + 144*a**4*b**3*e**4*(d + e*x)**2 + 480*a**3*b**4*d**3*e**3 + 1440*a**3*b**4*d**2*e**4*x - 576*a**3*b**4*
d*e**3*(d + e*x)**2 + 48*a**3*b**4*e**3*(d + e*x)**3 - 720*a**2*b**5*d**4*e**2 - 1440*a**2*b**5*d**3*e**3*x +
864*a**2*b**5*d**2*e**2*(d + e*x)**2 - 144*a**2*b**5*d*e**2*(d + e*x)**3 + 432*a*b**6*d**5*e + 720*a*b**6*d**4
*e**2*x - 576*a*b**6*d**3*e*(d + e*x)**2 + 144*a*b**6*d**2*e*(d + e*x)**3 - 96*b**7*d**6 - 144*b**7*d**5*e*x +
144*b**7*d**4*(d + e*x)**2 - 48*b**7*d**3*(d + e*x)**3) + 198*a**2*d*e**5*sqrt(d + e*x)/(48*a**6*e**6 - 144*a
**5*b*d*e**5 + 144*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x + 144*a**4*b**2*e**4*(d + e*x)**2 + 480*a**3*b**3*d*
*3*e**3 + 1440*a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e**3*(d + e*x)**2 + 48*a**3*b**3*e**3*(d + e*x)**3 - 72
0*a**2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**3*x + 864*a**2*b**4*d**2*e**2*(d + e*x)**2 - 144*a**2*b**4*d*e*
*2*(d + e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*d**4*e**2*x - 576*a*b**5*d**3*e*(d + e*x)**2 + 144*a*b**5*d**
2*e*(d + e*x)**3 - 96*b**6*d**6 - 144*b**6*d**5*e*x + 144*b**6*d**4*(d + e*x)**2 - 48*b**6*d**3*(d + e*x)**3)
- 80*a**2*e**5*(d + e*x)**(3/2)/(48*a**6*e**6 - 144*a**5*b*d*e**5 + 144*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x
+ 144*a**4*b**2*e**4*(d + e*x)**2 + 480*a**3*b**3*d**3*e**3 + 1440*a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e*
*3*(d + e*x)**2 + 48*a**3*b**3*e**3*(d + e*x)**3 - 720*a**2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**3*x + 864*
a**2*b**4*d**2*e**2*(d + e*x)**2 - 144*a**2*b**4*d*e**2*(d + e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*d**4*e**
2*x - 576*a*b**5*d**3*e*(d + e*x)**2 + 144*a*b**5*d**2*e*(d + e*x)**3 - 96*b**6*d**6 - 144*b**6*d**5*e*x + 144
*b**6*d**4*(d + e*x)**2 - 48*b**6*d**3*(d + e*x)**3) - 198*a*b*d**2*e**4*sqrt(d + e*x)/(48*a**6*e**6 - 144*a**
5*b*d*e**5 + 144*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x + 144*a**4*b**2*e**4*(d + e*x)**2 + 480*a**3*b**3*d**3
*e**3 + 1440*a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e**3*(d + e*x)**2 + 48*a**3*b**3*e**3*(d + e*x)**3 - 720*
a**2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**3*x + 864*a**2*b**4*d**2*e**2*(d + e*x)**2 - 144*a**2*b**4*d*e**2
*(d + e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*d**4*e**2*x - 576*a*b**5*d**3*e*(d + e*x)**2 + 144*a*b**5*d**2*
e*(d + e*x)**3 - 96*b**6*d**6 - 144*b**6*d**5*e*x + 144*b**6*d**4*(d + e*x)**2 - 48*b**6*d**3*(d + e*x)**3) +
160*a*b*d*e**4*(d + e*x)**(3/2)/(48*a**6*e**6 - 144*a**5*b*d*e**5 + 144*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x
+ 144*a**4*b**2*e**4*(d + e*x)**2 + 480*a**3*b**3*d**3*e**3 + 1440*a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e*
*3*(d + e*x)**2 + 48*a**3*b**3*e**3*(d + e*x)**3 - 720*a**2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**3*x + 864*
a**2*b**4*d**2*e**2*(d + e*x)**2 - 144*a**2*b**4*d*e**2*(d + e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*d**4*e**
2*x - 576*a*b**5*d**3*e*(d + e*x)**2 + 144*a*b**5*d**2*e*(d + e*x)**3 - 96*b**6*d**6 - 144*b**6*d**5*e*x + 144
*b**6*d**4*(d + e*x)**2 - 48*b**6*d**3*(d + e*x)**3) - 30*a*b*e**4*(d + e*x)**(5/2)/(48*a**6*e**6 - 144*a**5*b
*d*e**5 + 144*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x + 144*a**4*b**2*e**4*(d + e*x)**2 + 480*a**3*b**3*d**3*e*
*3 + 1440*a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e**3*(d + e*x)**2 + 48*a**3*b**3*e**3*(d + e*x)**3 - 720*a**
2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**3*x + 864*a**2*b**4*d**2*e**2*(d + e*x)**2 - 144*a**2*b**4*d*e**2*(d
+ e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*d**4*e**2*x - 576*a*b**5*d**3*e*(d + e*x)**2 + 144*a*b**5*d**2*e*(
d + e*x)**3 - 96*b**6*d**6 - 144*b**6*d**5*e*x + 144*b**6*d**4*(d + e*x)**2 - 48*b**6*d**3*(d + e*x)**3) + 10*
a*e**4*sqrt(d + e*x)/(8*a**4*b*e**4 - 16*a**3*b**2*d*e**3 + 16*a**3*b**2*e**4*x - 48*a**2*b**3*d*e**3*x + 8*a*
*2*b**3*e**2*(d + e*x)**2 + 16*a*b**4*d**3*e + 48*a*b**4*d**2*e**2*x - 16*a*b**4*d*e*(d + e*x)**2 - 8*b**5*d**
4 - 16*b**5*d**3*e*x + 8*b**5*d**2*(d + e*x)**2) + 5*a*e**4*sqrt(-1/(b*(a*e - b*d)**7))*log(-a**4*e**4*sqrt(-1
/(b*(a*e - b*d)**7)) + 4*a**3*b*d*e**3*sqrt(-1/(b*(a*e - b*d)**7)) - 6*a**2*b**2*d**2*e**2*sqrt(-1/(b*(a*e - b
*d)**7)) + 4*a*b**3*d**3*e*sqrt(-1/(b*(a*e - b*d)**7)) - b**4*d**4*sqrt(-1/(b*(a*e - b*d)**7)) + sqrt(d + e*x)
)/(16*b) - 5*a*e**4*sqrt(-1/(b*(a*e - b*d)**7))*log(a**4*e**4*sqrt(-1/(b*(a*e - b*d)**7)) - 4*a**3*b*d*e**3*sq
rt(-1/(b*(a*e - b*d)**7)) + 6*a**2*b**2*d**2*e**2*sqrt(-1/(b*(a*e - b*d)**7)) - 4*a*b**3*d**3*e*sqrt(-1/(b*(a*
e - b*d)**7)) + b**4*d**4*sqrt(-1/(b*(a*e - b*d)**7)) + sqrt(d + e*x))/(16*b) + 66*b**2*d**3*e**3*sqrt(d + e*x
)/(48*a**6*e**6 - 144*a**5*b*d*e**5 + 144*a**5*b*e**6*x - 720*a**4*b**2*d*e**5*x + 144*a**4*b**2*e**4*(d + e*x
)**2 + 480*a**3*b**3*d**3*e**3 + 1440*a**3*b**3*d**2*e**4*x - 576*a**3*b**3*d*e**3*(d + e*x)**2 + 48*a**3*b**3
*e**3*(d + e*x)**3 - 720*a**2*b**4*d**4*e**2 - 1440*a**2*b**4*d**3*e**3*x + 864*a**2*b**4*d**2*e**2*(d + e*x)*
*2 - 144*a**2*b**4*d*e**2*(d + e*x)**3 + 432*a*b**5*d**5*e + 720*a*b**5*d**4*e**2*x - 576*a*b**5*d**3*e*(d + e
*x)**2 + 144*a*b**5*d**2*e*(d + e*x)**3 - 96*b*...

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Giac [A]
time = 0.71, size = 211, normalized size = 1.45 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{3}}{8 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} \sqrt {-b^{2} d + a b e}} + \frac {3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} e^{3} - 8 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} d e^{3} - 3 \, \sqrt {x e + d} b^{2} d^{2} e^{3} + 8 \, {\left (x e + d\right )}^{\frac {3}{2}} a b e^{4} + 6 \, \sqrt {x e + d} a b d e^{4} - 3 \, \sqrt {x e + d} a^{2} e^{5}}{24 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*sqrt(-b^2*d + a*b*e)
) + 1/24*(3*(x*e + d)^(5/2)*b^2*e^3 - 8*(x*e + d)^(3/2)*b^2*d*e^3 - 3*sqrt(x*e + d)*b^2*d^2*e^3 + 8*(x*e + d)^
(3/2)*a*b*e^4 + 6*sqrt(x*e + d)*a*b*d*e^4 - 3*sqrt(x*e + d)*a^2*e^5)/((b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*((x*
e + d)*b - b*d + a*e)^3)

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Mupad [B]
time = 0.65, size = 207, normalized size = 1.42 \begin {gather*} \frac {\frac {e^3\,{\left (d+e\,x\right )}^{3/2}}{3\,\left (a\,e-b\,d\right )}-\frac {e^3\,\sqrt {d+e\,x}}{8\,b}+\frac {b\,e^3\,{\left (d+e\,x\right )}^{5/2}}{8\,{\left (a\,e-b\,d\right )}^2}}{\left (d+e\,x\right )\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )+b^3\,{\left (d+e\,x\right )}^3-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^2+a^3\,e^3-b^3\,d^3+3\,a\,b^2\,d^2\,e-3\,a^2\,b\,d\,e^2}+\frac {e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{8\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((e^3*(d + e*x)^(3/2))/(3*(a*e - b*d)) - (e^3*(d + e*x)^(1/2))/(8*b) + (b*e^3*(d + e*x)^(5/2))/(8*(a*e - b*d)^
2))/((d + e*x)*(3*b^3*d^2 + 3*a^2*b*e^2 - 6*a*b^2*d*e) + b^3*(d + e*x)^3 - (3*b^3*d - 3*a*b^2*e)*(d + e*x)^2 +
a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2) + (e^3*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/
(8*b^(3/2)*(a*e - b*d)^(5/2))

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