3.1830 \(\int \frac{(A+B x) \sqrt{d+e x}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=313 \[ -\frac{e^3 \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{128 b^2 (a+b x) (b d-a e)^4}+\frac{e^2 \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{192 b^2 (a+b x)^2 (b d-a e)^3}+\frac{e^4 (-3 a B e-7 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}-\frac{e \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{240 b^2 (a+b x)^3 (b d-a e)^2}-\frac{\sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{3/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

[Out]

-((10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(40*b^2*(b*d - a*e)*(a + b*x)^4) - (e*(10*b*B*d - 7*A*b*e - 3*
a*B*e)*Sqrt[d + e*x])/(240*b^2*(b*d - a*e)^2*(a + b*x)^3) + (e^2*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])
/(192*b^2*(b*d - a*e)^3*(a + b*x)^2) - (e^3*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(128*b^2*(b*d - a*e)
^4*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(3/2))/(5*b*(b*d - a*e)*(a + b*x)^5) + (e^4*(10*b*B*d - 7*A*b*e - 3*a*B
*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(5/2)*(b*d - a*e)^(9/2))

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Rubi [A]  time = 0.288091, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {27, 78, 47, 51, 63, 208} \[ -\frac{e^3 \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{128 b^2 (a+b x) (b d-a e)^4}+\frac{e^2 \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{192 b^2 (a+b x)^2 (b d-a e)^3}+\frac{e^4 (-3 a B e-7 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}-\frac{e \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{240 b^2 (a+b x)^3 (b d-a e)^2}-\frac{\sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{3/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 0.0662277, size = 99, normalized size = 0.32 \[ \frac{(d+e x)^{3/2} \left (\frac{e^4 (3 a B e+7 A b e-10 b B d) \, _2F_1\left (\frac{3}{2},5;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac{3 (a B-A b)}{(a+b x)^5}\right )}{15 b (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(3/2)*((3*(-(A*b) + a*B))/(a + b*x)^5 + (e^4*(-10*b*B*d + 7*A*b*e + 3*a*B*e)*Hypergeometric2F1[3/2,
5, 5/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^5))/(15*b*(b*d - a*e))

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Maple [B]  time = 0.023, size = 1037, normalized size = 3.3 \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)^3,x)

[Out]

7/128*e^5/(b*e*x+a*e)^5*b^3/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*(e*x+d)^(9/2)*A+3/
128*e^5/(b*e*x+a*e)^5*b^2/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*(e*x+d)^(9/2)*a*B-5/
64*e^4/(b*e*x+a*e)^5*b^3/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*(e*x+d)^(9/2)*B*d+49/
192*e^5/(b*e*x+a*e)^5*b^2/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(7/2)*A+7/64*e^5/(b*e*x+a*e)^5
*b/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(7/2)*a*B-35/96*e^4/(b*e*x+a*e)^5*b^2/(a^3*e^3-3*a^2*
b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(7/2)*B*d+7/15*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(5
/2)*A*b+1/5*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(5/2)*a*B-2/3*e^4/(b*e*x+a*e)^5/(a^2*e^2-2*a
*b*d*e+b^2*d^2)*(e*x+d)^(5/2)*B*b*d+79/192*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(3/2)*A-7/64*e^5/(b*e*x+a*e)^5/
b/(a*e-b*d)*(e*x+d)^(3/2)*a*B-29/96*e^4/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(3/2)*B*d-7/128*e^5/(b*e*x+a*e)^5/b*(e
*x+d)^(1/2)*A-3/128*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*a*B+5/64*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*B*d+7/128*e
^5/b/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*
b/((a*e-b*d)*b)^(1/2))*A+3/128*e^5/b^2/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/((a*e-b
*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a*B-5/64*e^4/b/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e
^2-4*a*b^3*d^3*e+b^4*d^4)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*d

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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((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.57467, size = 5407, normalized size = 17.27 \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)^3,x, algorithm="fricas")

[Out]

[-1/3840*(15*(10*B*a^5*b*d*e^4 - (3*B*a^6 + 7*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (3*B*a*b^5 + 7*A*b^6)*e^5)*x^5
+ 5*(10*B*a*b^5*d*e^4 - (3*B*a^2*b^4 + 7*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (3*B*a^3*b^3 + 7*A*a^2*b
^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (3*B*a^4*b^2 + 7*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (3*B*a^
5*b + 7*A*a^4*b^2)*e^5)*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*(96*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(28*B*a^2*b^5 + 117*A*a*b^6)*d^4*e + 4*(197*B*a^3*b^4 + 898*A
*a^2*b^5)*d^3*e^2 - 2*(278*B*a^4*b^3 + 1657*A*a^3*b^4)*d^2*e^3 + 5*(33*B*a^5*b^2 + 263*A*a^4*b^3)*d*e^4 - 15*(
3*B*a^6*b + 7*A*a^5*b^2)*e^5 + 15*(10*B*b^7*d^2*e^3 - (13*B*a*b^6 + 7*A*b^7)*d*e^4 + (3*B*a^2*b^5 + 7*A*a*b^6)
*e^5)*x^4 - 10*(10*B*b^7*d^3*e^2 - (83*B*a*b^6 + 7*A*b^7)*d^2*e^3 + 2*(47*B*a^2*b^5 + 28*A*a*b^6)*d*e^4 - 7*(3
*B*a^3*b^4 + 7*A*a^2*b^5)*e^5)*x^3 + 2*(40*B*b^7*d^4*e - 2*(141*B*a*b^6 + 14*A*b^7)*d^3*e^2 + 3*(317*B*a^2*b^5
+ 63*A*a*b^6)*d^2*e^3 - (901*B*a^3*b^4 + 609*A*a^2*b^5)*d*e^4 + 64*(3*B*a^4*b^3 + 7*A*a^3*b^4)*e^5)*x^2 + 2*(
240*B*b^7*d^5 - 8*(143*B*a*b^6 - 3*A*b^7)*d^4*e + 2*(1041*B*a^2*b^5 - 76*A*a*b^6)*d^3*e^2 - 3*(529*B*a^3*b^4 -
139*A*a^2*b^5)*d^2*e^3 + 2*(257*B*a^4*b^3 - 342*A*a^3*b^4)*d*e^4 - 5*(21*B*a^5*b^2 - 79*A*a^4*b^3)*e^5)*x)*sq
rt(e*x + d))/(a^5*b^8*d^5 - 5*a^6*b^7*d^4*e + 10*a^7*b^6*d^3*e^2 - 10*a^8*b^5*d^2*e^3 + 5*a^9*b^4*d*e^4 - a^10
*b^3*e^5 + (b^13*d^5 - 5*a*b^12*d^4*e + 10*a^2*b^11*d^3*e^2 - 10*a^3*b^10*d^2*e^3 + 5*a^4*b^9*d*e^4 - a^5*b^8*
e^5)*x^5 + 5*(a*b^12*d^5 - 5*a^2*b^11*d^4*e + 10*a^3*b^10*d^3*e^2 - 10*a^4*b^9*d^2*e^3 + 5*a^5*b^8*d*e^4 - a^6
*b^7*e^5)*x^4 + 10*(a^2*b^11*d^5 - 5*a^3*b^10*d^4*e + 10*a^4*b^9*d^3*e^2 - 10*a^5*b^8*d^2*e^3 + 5*a^6*b^7*d*e^
4 - a^7*b^6*e^5)*x^3 + 10*(a^3*b^10*d^5 - 5*a^4*b^9*d^4*e + 10*a^5*b^8*d^3*e^2 - 10*a^6*b^7*d^2*e^3 + 5*a^7*b^
6*d*e^4 - a^8*b^5*e^5)*x^2 + 5*(a^4*b^9*d^5 - 5*a^5*b^8*d^4*e + 10*a^6*b^7*d^3*e^2 - 10*a^7*b^6*d^2*e^3 + 5*a^
8*b^5*d*e^4 - a^9*b^4*e^5)*x), -1/1920*(15*(10*B*a^5*b*d*e^4 - (3*B*a^6 + 7*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (
3*B*a*b^5 + 7*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (3*B*a^2*b^4 + 7*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*
e^4 - (3*B*a^3*b^3 + 7*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (3*B*a^4*b^2 + 7*A*a^3*b^3)*e^5)*x^2 + 5
*(10*B*a^4*b^2*d*e^4 - (3*B*a^5*b + 7*A*a^4*b^2)*e^5)*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)) + (96*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(28*B*a^2*b^5 + 117*A*a*b^6)*d^4*e + 4*(197*B*a^3*
b^4 + 898*A*a^2*b^5)*d^3*e^2 - 2*(278*B*a^4*b^3 + 1657*A*a^3*b^4)*d^2*e^3 + 5*(33*B*a^5*b^2 + 263*A*a^4*b^3)*d
*e^4 - 15*(3*B*a^6*b + 7*A*a^5*b^2)*e^5 + 15*(10*B*b^7*d^2*e^3 - (13*B*a*b^6 + 7*A*b^7)*d*e^4 + (3*B*a^2*b^5 +
7*A*a*b^6)*e^5)*x^4 - 10*(10*B*b^7*d^3*e^2 - (83*B*a*b^6 + 7*A*b^7)*d^2*e^3 + 2*(47*B*a^2*b^5 + 28*A*a*b^6)*d
*e^4 - 7*(3*B*a^3*b^4 + 7*A*a^2*b^5)*e^5)*x^3 + 2*(40*B*b^7*d^4*e - 2*(141*B*a*b^6 + 14*A*b^7)*d^3*e^2 + 3*(31
7*B*a^2*b^5 + 63*A*a*b^6)*d^2*e^3 - (901*B*a^3*b^4 + 609*A*a^2*b^5)*d*e^4 + 64*(3*B*a^4*b^3 + 7*A*a^3*b^4)*e^5
)*x^2 + 2*(240*B*b^7*d^5 - 8*(143*B*a*b^6 - 3*A*b^7)*d^4*e + 2*(1041*B*a^2*b^5 - 76*A*a*b^6)*d^3*e^2 - 3*(529*
B*a^3*b^4 - 139*A*a^2*b^5)*d^2*e^3 + 2*(257*B*a^4*b^3 - 342*A*a^3*b^4)*d*e^4 - 5*(21*B*a^5*b^2 - 79*A*a^4*b^3)
*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^5 - 5*a^6*b^7*d^4*e + 10*a^7*b^6*d^3*e^2 - 10*a^8*b^5*d^2*e^3 + 5*a^9*b^4*d
*e^4 - a^10*b^3*e^5 + (b^13*d^5 - 5*a*b^12*d^4*e + 10*a^2*b^11*d^3*e^2 - 10*a^3*b^10*d^2*e^3 + 5*a^4*b^9*d*e^4
- a^5*b^8*e^5)*x^5 + 5*(a*b^12*d^5 - 5*a^2*b^11*d^4*e + 10*a^3*b^10*d^3*e^2 - 10*a^4*b^9*d^2*e^3 + 5*a^5*b^8*
d*e^4 - a^6*b^7*e^5)*x^4 + 10*(a^2*b^11*d^5 - 5*a^3*b^10*d^4*e + 10*a^4*b^9*d^3*e^2 - 10*a^5*b^8*d^2*e^3 + 5*a
^6*b^7*d*e^4 - a^7*b^6*e^5)*x^3 + 10*(a^3*b^10*d^5 - 5*a^4*b^9*d^4*e + 10*a^5*b^8*d^3*e^2 - 10*a^6*b^7*d^2*e^3
+ 5*a^7*b^6*d*e^4 - a^8*b^5*e^5)*x^2 + 5*(a^4*b^9*d^5 - 5*a^5*b^8*d^4*e + 10*a^6*b^7*d^3*e^2 - 10*a^7*b^6*d^2
*e^3 + 5*a^8*b^5*d*e^4 - a^9*b^4*e^5)*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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.24455, size = 1157, normalized size = 3.7 \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)^3,x, algorithm="giac")

[Out]

-1/128*(10*B*b*d*e^4 - 3*B*a*e^5 - 7*A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^4 - 4*a*b^5
*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*sqrt(-b^2*d + a*b*e)) - 1/1920*(150*(x*e + d)^(9/2
)*B*b^5*d*e^4 - 700*(x*e + d)^(7/2)*B*b^5*d^2*e^4 + 1280*(x*e + d)^(5/2)*B*b^5*d^3*e^4 - 580*(x*e + d)^(3/2)*B
*b^5*d^4*e^4 - 150*sqrt(x*e + d)*B*b^5*d^5*e^4 - 45*(x*e + d)^(9/2)*B*a*b^4*e^5 - 105*(x*e + d)^(9/2)*A*b^5*e^
5 + 910*(x*e + d)^(7/2)*B*a*b^4*d*e^5 + 490*(x*e + d)^(7/2)*A*b^5*d*e^5 - 2944*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5
- 896*(x*e + d)^(5/2)*A*b^5*d^2*e^5 + 1530*(x*e + d)^(3/2)*B*a*b^4*d^3*e^5 + 790*(x*e + d)^(3/2)*A*b^5*d^3*e^
5 + 645*sqrt(x*e + d)*B*a*b^4*d^4*e^5 + 105*sqrt(x*e + d)*A*b^5*d^4*e^5 - 210*(x*e + d)^(7/2)*B*a^2*b^3*e^6 -
490*(x*e + d)^(7/2)*A*a*b^4*e^6 + 2048*(x*e + d)^(5/2)*B*a^2*b^3*d*e^6 + 1792*(x*e + d)^(5/2)*A*a*b^4*d*e^6 -
1110*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^6 - 2370*(x*e + d)^(3/2)*A*a*b^4*d^2*e^6 - 1080*sqrt(x*e + d)*B*a^2*b^3*d
^3*e^6 - 420*sqrt(x*e + d)*A*a*b^4*d^3*e^6 - 384*(x*e + d)^(5/2)*B*a^3*b^2*e^7 - 896*(x*e + d)^(5/2)*A*a^2*b^3
*e^7 - 50*(x*e + d)^(3/2)*B*a^3*b^2*d*e^7 + 2370*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7 + 870*sqrt(x*e + d)*B*a^3*b^2
*d^2*e^7 + 630*sqrt(x*e + d)*A*a^2*b^3*d^2*e^7 + 210*(x*e + d)^(3/2)*B*a^4*b*e^8 - 790*(x*e + d)^(3/2)*A*a^3*b
^2*e^8 - 330*sqrt(x*e + d)*B*a^4*b*d*e^8 - 420*sqrt(x*e + d)*A*a^3*b^2*d*e^8 + 45*sqrt(x*e + d)*B*a^5*e^9 + 10
5*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*((
x*e + d)*b - b*d + a*e)^5)