### 3.1659 $$\int \frac{\sqrt{d+e x}}{(a^2+2 a b x+b^2 x^2)^2} \, dx$$

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

[Out]

-Sqrt[d + e*x]/(3*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
))

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Rubi [A]  time = 0.0687249, antiderivative size = 146, normalized size of antiderivative = 1., 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, 47, 51, 63, 208} $-\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{5/2}}+\frac{e^2 \sqrt{d+e x}}{8 b (a+b x) (b d-a e)^2}-\frac{e \sqrt{d+e x}}{12 b (a+b x)^2 (b d-a e)}-\frac{\sqrt{d+e x}}{3 b (a+b x)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sqrt[d + e*x]/(3*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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 208

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

Mathematica [C]  time = 0.0134836, size = 52, normalized size = 0.36 $\frac{2 e^3 (d+e x)^{3/2} \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};-\frac{b (d+e x)}{a e-b d}\right )}{3 (a e-b d)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*e^3*(d + e*x)^(3/2)*Hypergeometric2F1[3/2, 4, 5/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(3*(-(b*d) + a*e)^4)

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Maple [A]  time = 0.204, size = 170, normalized size = 1.2 \begin{align*}{\frac{{e}^{3}b}{8\, \left ( bxe+ae \right ) ^{3} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{3}}{3\, \left ( bxe+ae \right ) ^{3} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{3}}{8\, \left ( bxe+ae \right ) ^{3}b}\sqrt{ex+d}}+{\frac{{e}^{3}}{8\,b \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \end{align*}

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)

[Out]

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

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

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

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Fricas [B]  time = 2.10518, size = 1581, normalized size = 10.83 \begin{align*} \left [\frac{3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\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 (8 \, b^{4} d^{3} - 22 \, a b^{3} d^{2} e + 17 \, a^{2} b^{2} d e^{2} - 3 \, a^{3} b e^{3} - 3 \,{\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \,{\left (b^{4} d^{2} e - 5 \, a b^{3} d e^{2} + 4 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{48 \,{\left (a^{3} b^{5} d^{3} - 3 \, a^{4} b^{4} d^{2} e + 3 \, a^{5} b^{3} d e^{2} - a^{6} b^{2} e^{3} +{\left (b^{8} d^{3} - 3 \, a b^{7} d^{2} e + 3 \, a^{2} b^{6} d e^{2} - a^{3} b^{5} e^{3}\right )} x^{3} + 3 \,{\left (a b^{7} d^{3} - 3 \, a^{2} b^{6} d^{2} e + 3 \, a^{3} b^{5} d e^{2} - a^{4} b^{4} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{6} d^{3} - 3 \, a^{3} b^{5} d^{2} e + 3 \, a^{4} b^{4} d e^{2} - a^{5} b^{3} e^{3}\right )} x\right )}}, \frac{3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\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 (8 \, b^{4} d^{3} - 22 \, a b^{3} d^{2} e + 17 \, a^{2} b^{2} d e^{2} - 3 \, a^{3} b e^{3} - 3 \,{\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \,{\left (b^{4} d^{2} e - 5 \, a b^{3} d e^{2} + 4 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (a^{3} b^{5} d^{3} - 3 \, a^{4} b^{4} d^{2} e + 3 \, a^{5} b^{3} d e^{2} - a^{6} b^{2} e^{3} +{\left (b^{8} d^{3} - 3 \, a b^{7} d^{2} e + 3 \, a^{2} b^{6} d e^{2} - a^{3} b^{5} e^{3}\right )} x^{3} + 3 \,{\left (a b^{7} d^{3} - 3 \, a^{2} b^{6} d^{2} e + 3 \, a^{3} b^{5} d e^{2} - a^{4} b^{4} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{6} d^{3} - 3 \, a^{3} b^{5} d^{2} e + 3 \, a^{4} b^{4} d e^{2} - a^{5} b^{3} e^{3}\right )} x\right )}}\right ] \end{align*}

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*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*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*(8*b^4*d^3 - 22*a*b^3*d^2*e + 17*a^2*b^2*d*e^2 - 3*a^3
*b*e^3 - 3*(b^4*d*e^2 - a*b^3*e^3)*x^2 + 2*(b^4*d^2*e - 5*a*b^3*d*e^2 + 4*a^2*b^2*e^3)*x)*sqrt(e*x + d))/(a^3*
b^5*d^3 - 3*a^4*b^4*d^2*e + 3*a^5*b^3*d*e^2 - a^6*b^2*e^3 + (b^8*d^3 - 3*a*b^7*d^2*e + 3*a^2*b^6*d*e^2 - a^3*b
^5*e^3)*x^3 + 3*(a*b^7*d^3 - 3*a^2*b^6*d^2*e + 3*a^3*b^5*d*e^2 - a^4*b^4*e^3)*x^2 + 3*(a^2*b^6*d^3 - 3*a^3*b^5
*d^2*e + 3*a^4*b^4*d*e^2 - a^5*b^3*e^3)*x), 1/24*(3*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*
sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (8*b^4*d^3 - 22*a*b^3*d^2*e +
17*a^2*b^2*d*e^2 - 3*a^3*b*e^3 - 3*(b^4*d*e^2 - a*b^3*e^3)*x^2 + 2*(b^4*d^2*e - 5*a*b^3*d*e^2 + 4*a^2*b^2*e^3)
*x)*sqrt(e*x + d))/(a^3*b^5*d^3 - 3*a^4*b^4*d^2*e + 3*a^5*b^3*d*e^2 - a^6*b^2*e^3 + (b^8*d^3 - 3*a*b^7*d^2*e +
3*a^2*b^6*d*e^2 - a^3*b^5*e^3)*x^3 + 3*(a*b^7*d^3 - 3*a^2*b^6*d^2*e + 3*a^3*b^5*d*e^2 - a^4*b^4*e^3)*x^2 + 3*
(a^2*b^6*d^3 - 3*a^3*b^5*d^2*e + 3*a^4*b^4*d*e^2 - a^5*b^3*e^3)*x)]

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Sympy [B]  time = 108.858, size = 4592, normalized size = 31.45 \begin{align*} \text{result too large to display} \end{align*}

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**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*b**2*d**2*e**3*(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*b**2*d*e**3*(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 - 4
8*b**6*d**3*(d + e*x)**3) - 5*d*e**3*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 + 5*d*e**3*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*s
qrt(-1/(b*(a*e - b*d)**7)) + sqrt(d + e*x))/16 - 10*d*e**3*sqrt(d + e*x)/(8*a**4*e**4 - 16*a**3*b*d*e**3 + 16*
a**3*b*e**4*x - 48*a**2*b**2*d*e**3*x + 8*a**2*b**2*e**2*(d + e*x)**2 + 16*a*b**3*d**3*e + 48*a*b**3*d**2*e**2
*x - 16*a*b**3*d*e*(d + e*x)**2 - 8*b**4*d**4 - 16*b**4*d**3*e*x + 8*b**4*d**2*(d + e*x)**2) + 6*e**3*(d + e*x
)**(3/2)/(8*a**4*e**4 - 16*a**3*b*d*e**3 + 16*a**3*b*e**4*x - 48*a**2*b**2*d*e**3*x + 8*a**2*b**2*e**2*(d + e*
x)**2 + 16*a*b**3*d**3*e + 48*a*b**3*d**2*e**2*x - 16*a*b**3*d*e*(d + e*x)**2 - 8*b**4*d**4 - 16*b**4*d**3*e*x
+ 8*b**4*d**2*(d + e*x)**2) - 3*e**3*sqrt(-1/(b*(a*e - b*d)**5))*log(-a**3*e**3*sqrt(-1/(b*(a*e - b*d)**5)) +
3*a**2*b*d*e**2*sqrt(-1/(b*(a*e - b*d)**5)) - 3*a*b**2*d**2*e*sqrt(-1/(b*(a*e - b*d)**5)) + b**3*d**3*sqrt(-1
/(b*(a*e - b*d)**5)) + sqrt(d + e*x))/(8*b) + 3*e**3*sqrt(-1/(b*(a*e - b*d)**5))*log(a**3*e**3*sqrt(-1/(b*(a*e
- b*d)**5)) - 3*a**2*b*d*e**2*sqrt(-1/(b*(a*e - b*d)**5)) + 3*a*b**2*d**2*e*sqrt(-1/(b*(a*e - b*d)**5)) - b**
3*d**3*sqrt(-1/(b*(a*e - b*d)**5)) + sqrt(d + e*x))/(8*b)

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Giac [A]  time = 1.25656, size = 285, normalized size = 1.95 \begin{align*} \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{align*}

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)