3.1879 \(\int \frac{(A+B x) \sqrt{d+e x}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=359 \[ \frac{e^2 \sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{64 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{e^3 (a+b x) (-3 a B e-5 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}-\frac{e \sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{96 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{(d+e x)^{3/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(e^2*(8*b*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(64*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b
*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*(
8*b*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(96*b^2*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
(A*b - a*B)*(d + e*x)^(3/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^3*(8*b*B*d - 5*A
*b*e - 3*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(5/2)*(b*d - a*e)^(7/2)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.350075, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {770, 78, 47, 51, 63, 208} \[ \frac{e^2 \sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{64 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{e^3 (a+b x) (-3 a B e-5 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}-\frac{e \sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{96 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{(d+e x)^{3/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*(8*b*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(64*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b
*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*(
8*b*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(96*b^2*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
(A*b - a*B)*(d + e*x)^(3/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^3*(8*b*B*d - 5*A
*b*e - 3*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(5/2)*(b*d - a*e)^(7/2)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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{(A+B x) \sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{(A+B x) \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{(A b-a B) (d+e x)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b^2 (8 b B d-5 A b e-3 a B e) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{\left (a b+b^2 x\right )^4} \, dx}{8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (e (8 b B d-5 A b e-3 a B e) \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{(8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{96 b^2 (b d-a e)^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (e^2 (8 b B d-5 A b e-3 a B e) \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 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{e^2 (8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{64 b^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{96 b^2 (b d-a e)^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (e^3 (8 b B d-5 A b e-3 a B e) \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^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{e^2 (8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{64 b^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{96 b^2 (b d-a e)^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (e^2 (8 b B d-5 A b e-3 a B e) \left (a b+b^2 x\right )\right ) \operatorname{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^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{e^2 (8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{64 b^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (8 b B d-5 A b e-3 a B e) \sqrt{d+e x}}{96 b^2 (b d-a e)^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^3 (8 b B d-5 A b e-3 a B e) (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{5/2} (b d-a e)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.124051, size = 116, normalized size = 0.32 \[ \frac{(d+e x)^{3/2} \left (-\frac{e^3 (a+b x)^4 (3 a B e+5 A b e-8 b B d) \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+3 a B-3 A b\right )}{12 b (a+b x)^3 \sqrt{(a+b x)^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(3/2)*(-3*A*b + 3*a*B - (e^3*(-8*b*B*d + 5*A*b*e + 3*a*B*e)*(a + b*x)^4*Hypergeometric2F1[3/2, 4, 5
/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^4))/(12*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.023, size = 1296, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(b*x+a)/e*(-15*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b*e^4+9*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a*b^3
*e+55*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^3*e^2-55*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*b^4*d*e+9*B*arctan(
(e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^4*a*b^4*e^5-24*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^4*b^5*d*
e^4+60*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^3*a*b^4*e^5+36*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(
1/2))*x^3*a^2*b^3*e^5+36*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^4*b*e^5+73*A*((a*e-b*d)*b)^(1/2)*(e
*x+d)^(3/2)*a^2*b^2*e^3+73*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^4*d^2*e-33*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2
)*a^3*b*e^3-24*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^4*b*d*e^4+33*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2
)*a^2*b^2*e^2+60*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^3*b^2*e^5+15*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^
(1/2)*b^4*d^3*e+90*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a^2*b^3*e^5+54*B*arctan((e*x+d)^(1/2)*b/(
(a*e-b*d)*b)^(1/2))*x^2*a^3*b^2*e^5+9*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^5*e^5-24*B*((a*e-b*d)*b)
^(1/2)*(e*x+d)^(7/2)*b^4*d+88*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*b^4*d^2+15*A*arctan((e*x+d)^(1/2)*b/((a*e-b*
d)*b)^(1/2))*x^4*b^5*e^5+15*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^4*e-9*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^
4*e^4-24*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*b^4*d^4+15*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^4*b*e^
5-40*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^4*d^3-146*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^3*d*e^2+26*B*((a*
e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^2*d*e^2+51*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b*d*e^3-144*B*arctan((e
*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a^2*b^3*d*e^4-121*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^3*d*e-96*B*ar
ctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^3*a*b^4*d*e^4-99*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d^2*e
^2+81*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e+47*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^3*d^2*e+45*A*
((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d*e^3-45*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^3*d^2*e^2-96*B*arct
an((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^3*b^2*d*e^4)/((a*e-b*d)*b)^(1/2)/b^2/(a*e-b*d)/(a^2*e^2-2*a*b*d*e+
b^2*d^2)/((b*x+a)^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )} \sqrt{e x + d}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.81524, size = 3784, normalized size = 10.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/384*(3*(8*B*a^4*b*d*e^3 - (3*B*a^5 + 5*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (3*B*a*b^4 + 5*A*b^5)*e^4)*x^4 + 4*(
8*B*a*b^4*d*e^3 - (3*B*a^2*b^3 + 5*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (3*B*a^3*b^2 + 5*A*a^2*b^3)*e^4)
*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (3*B*a^4*b + 5*A*a^3*b^2)*e^4)*x)*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*(16*(B*a*b^5 + 3*A*b^6)*d^4 - 8*(7*B*a^2*b^4 + 23*A*a*b^5
)*d^3*e + 2*(29*B*a^3*b^3 + 127*A*a^2*b^4)*d^2*e^2 - (27*B*a^4*b^2 + 133*A*a^3*b^3)*d*e^3 + 3*(3*B*a^5*b + 5*A
*a^4*b^2)*e^4 - 3*(8*B*b^6*d^2*e^2 - (11*B*a*b^5 + 5*A*b^6)*d*e^3 + (3*B*a^2*b^4 + 5*A*a*b^5)*e^4)*x^3 + (16*B
*b^6*d^3*e - 10*(11*B*a*b^5 + A*b^6)*d^2*e^2 + (127*B*a^2*b^4 + 65*A*a*b^5)*d*e^3 - 11*(3*B*a^3*b^3 + 5*A*a^2*
b^4)*e^4)*x^2 + (64*B*b^6*d^4 - 8*(29*B*a*b^5 - A*b^6)*d^3*e + 4*(65*B*a^2*b^4 - 11*A*a*b^5)*d^2*e^2 - (125*B*
a^3*b^3 - 109*A*a^2*b^4)*d*e^3 + (33*B*a^4*b^2 - 73*A*a^3*b^3)*e^4)*x)*sqrt(e*x + d))/(a^4*b^7*d^4 - 4*a^5*b^6
*d^3*e + 6*a^6*b^5*d^2*e^2 - 4*a^7*b^4*d*e^3 + a^8*b^3*e^4 + (b^11*d^4 - 4*a*b^10*d^3*e + 6*a^2*b^9*d^2*e^2 -
4*a^3*b^8*d*e^3 + a^4*b^7*e^4)*x^4 + 4*(a*b^10*d^4 - 4*a^2*b^9*d^3*e + 6*a^3*b^8*d^2*e^2 - 4*a^4*b^7*d*e^3 + a
^5*b^6*e^4)*x^3 + 6*(a^2*b^9*d^4 - 4*a^3*b^8*d^3*e + 6*a^4*b^7*d^2*e^2 - 4*a^5*b^6*d*e^3 + a^6*b^5*e^4)*x^2 +
4*(a^3*b^8*d^4 - 4*a^4*b^7*d^3*e + 6*a^5*b^6*d^2*e^2 - 4*a^6*b^5*d*e^3 + a^7*b^4*e^4)*x), 1/192*(3*(8*B*a^4*b*
d*e^3 - (3*B*a^5 + 5*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (3*B*a*b^4 + 5*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (3*
B*a^2*b^3 + 5*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (3*B*a^3*b^2 + 5*A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2
*d*e^3 - (3*B*a^4*b + 5*A*a^3*b^2)*e^4)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e
*x + b*d)) - (16*(B*a*b^5 + 3*A*b^6)*d^4 - 8*(7*B*a^2*b^4 + 23*A*a*b^5)*d^3*e + 2*(29*B*a^3*b^3 + 127*A*a^2*b^
4)*d^2*e^2 - (27*B*a^4*b^2 + 133*A*a^3*b^3)*d*e^3 + 3*(3*B*a^5*b + 5*A*a^4*b^2)*e^4 - 3*(8*B*b^6*d^2*e^2 - (11
*B*a*b^5 + 5*A*b^6)*d*e^3 + (3*B*a^2*b^4 + 5*A*a*b^5)*e^4)*x^3 + (16*B*b^6*d^3*e - 10*(11*B*a*b^5 + A*b^6)*d^2
*e^2 + (127*B*a^2*b^4 + 65*A*a*b^5)*d*e^3 - 11*(3*B*a^3*b^3 + 5*A*a^2*b^4)*e^4)*x^2 + (64*B*b^6*d^4 - 8*(29*B*
a*b^5 - A*b^6)*d^3*e + 4*(65*B*a^2*b^4 - 11*A*a*b^5)*d^2*e^2 - (125*B*a^3*b^3 - 109*A*a^2*b^4)*d*e^3 + (33*B*a
^4*b^2 - 73*A*a^3*b^3)*e^4)*x)*sqrt(e*x + d))/(a^4*b^7*d^4 - 4*a^5*b^6*d^3*e + 6*a^6*b^5*d^2*e^2 - 4*a^7*b^4*d
*e^3 + a^8*b^3*e^4 + (b^11*d^4 - 4*a*b^10*d^3*e + 6*a^2*b^9*d^2*e^2 - 4*a^3*b^8*d*e^3 + a^4*b^7*e^4)*x^4 + 4*(
a*b^10*d^4 - 4*a^2*b^9*d^3*e + 6*a^3*b^8*d^2*e^2 - 4*a^4*b^7*d*e^3 + a^5*b^6*e^4)*x^3 + 6*(a^2*b^9*d^4 - 4*a^3
*b^8*d^3*e + 6*a^4*b^7*d^2*e^2 - 4*a^5*b^6*d*e^3 + a^6*b^5*e^4)*x^2 + 4*(a^3*b^8*d^4 - 4*a^4*b^7*d^3*e + 6*a^5
*b^6*d^2*e^2 - 4*a^6*b^5*d*e^3 + a^7*b^4*e^4)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.34771, size = 1062, normalized size = 2.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(8*B*b*d*e^3 - 3*B*a*e^4 - 5*A*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^3*sgn((x*e + d
)*b*e - b*d*e + a*e^2) - 3*a*b^4*d^2*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 3*a^2*b^3*d*e^2*sgn((x*e + d)*b*e
- b*d*e + a*e^2) - a^3*b^2*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) + 1/192*(24*(x*e + d)
^(7/2)*B*b^4*d*e^3 - 88*(x*e + d)^(5/2)*B*b^4*d^2*e^3 + 40*(x*e + d)^(3/2)*B*b^4*d^3*e^3 + 24*sqrt(x*e + d)*B*
b^4*d^4*e^3 - 9*(x*e + d)^(7/2)*B*a*b^3*e^4 - 15*(x*e + d)^(7/2)*A*b^4*e^4 + 121*(x*e + d)^(5/2)*B*a*b^3*d*e^4
+ 55*(x*e + d)^(5/2)*A*b^4*d*e^4 - 47*(x*e + d)^(3/2)*B*a*b^3*d^2*e^4 - 73*(x*e + d)^(3/2)*A*b^4*d^2*e^4 - 81
*sqrt(x*e + d)*B*a*b^3*d^3*e^4 - 15*sqrt(x*e + d)*A*b^4*d^3*e^4 - 33*(x*e + d)^(5/2)*B*a^2*b^2*e^5 - 55*(x*e +
d)^(5/2)*A*a*b^3*e^5 - 26*(x*e + d)^(3/2)*B*a^2*b^2*d*e^5 + 146*(x*e + d)^(3/2)*A*a*b^3*d*e^5 + 99*sqrt(x*e +
d)*B*a^2*b^2*d^2*e^5 + 45*sqrt(x*e + d)*A*a*b^3*d^2*e^5 + 33*(x*e + d)^(3/2)*B*a^3*b*e^6 - 73*(x*e + d)^(3/2)
*A*a^2*b^2*e^6 - 51*sqrt(x*e + d)*B*a^3*b*d*e^6 - 45*sqrt(x*e + d)*A*a^2*b^2*d*e^6 + 9*sqrt(x*e + d)*B*a^4*e^7
+ 15*sqrt(x*e + d)*A*a^3*b*e^7)/((b^5*d^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 3*a*b^4*d^2*e*sgn((x*e + d)*b*
e - b*d*e + a*e^2) + 3*a^2*b^3*d*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - a^3*b^2*e^3*sgn((x*e + d)*b*e - b*d*
e + a*e^2))*((x*e + d)*b - b*d + a*e)^4)