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

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

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Rubi [A]  time = 0.29, antiderivative size = 313, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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)} \end {gather*}

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 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 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 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*}

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Mathematica [C]  time = 0.06, size = 99, normalized size = 0.32 \begin {gather*} \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-3 A b}{(a+b x)^5}\right )}{15 b (b d-a e)} \end {gather*}

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 + 3*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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IntegrateAlgebraic [B]  time = 2.72, size = 676, normalized size = 2.16 \begin {gather*} \frac {e^4 \sqrt {d+e x} \left (45 a^5 B e^5+105 a^4 A b e^5+210 a^4 b B e^4 (d+e x)-330 a^4 b B d e^4-790 a^3 A b^2 e^4 (d+e x)-420 a^3 A b^2 d e^4+870 a^3 b^2 B d^2 e^3-384 a^3 b^2 B e^3 (d+e x)^2-50 a^3 b^2 B d e^3 (d+e x)+630 a^2 A b^3 d^2 e^3-896 a^2 A b^3 e^3 (d+e x)^2+2370 a^2 A b^3 d e^3 (d+e x)-1080 a^2 b^3 B d^3 e^2-1110 a^2 b^3 B d^2 e^2 (d+e x)-210 a^2 b^3 B e^2 (d+e x)^3+2048 a^2 b^3 B d e^2 (d+e x)^2-420 a A b^4 d^3 e^2-2370 a A b^4 d^2 e^2 (d+e x)-490 a A b^4 e^2 (d+e x)^3+1792 a A b^4 d e^2 (d+e x)^2+645 a b^4 B d^4 e+1530 a b^4 B d^3 e (d+e x)-2944 a b^4 B d^2 e (d+e x)^2-45 a b^4 B e (d+e x)^4+910 a b^4 B d e (d+e x)^3+105 A b^5 d^4 e+790 A b^5 d^3 e (d+e x)-896 A b^5 d^2 e (d+e x)^2-105 A b^5 e (d+e x)^4+490 A b^5 d e (d+e x)^3-150 b^5 B d^5-580 b^5 B d^4 (d+e x)+1280 b^5 B d^3 (d+e x)^2-700 b^5 B d^2 (d+e x)^3+150 b^5 B d (d+e x)^4\right )}{1920 b^2 (b d-a e)^4 (-a e-b (d+e x)+b d)^5}+\frac {\left (-3 a B e^5-7 A b e^5+10 b B d e^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{5/2} (b d-a e)^4 \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.50, size = 2580, normalized size = 8.24

result too large to display

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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giac [B]  time = 0.27, size = 857, normalized size = 2.74 \begin {gather*} -\frac {{\left (10 \, B b d e^{4} - 3 \, B a e^{5} - 7 \, A b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (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}\right )} \sqrt {-b^{2} d + a b e}} - \frac {150 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 700 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 1280 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 580 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} - 150 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 45 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 105 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 910 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 490 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 2944 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 1530 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 790 \, {\left (x e + d\right )}^{\frac {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 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 490 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 2048 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 1792 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 1110 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 2370 \, {\left (x e + d\right )}^{\frac {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 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} - 50 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 2370 \, {\left (x e + d\right )}^{\frac {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 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 790 \, {\left (x e + d\right )}^{\frac {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} + 105 \, \sqrt {x e + d} A a^{4} b e^{9}}{1920 \, {\left (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}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end {gather*}

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)

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maple [B]  time = 0.09, size = 1037, normalized size = 3.31 \begin {gather*} \frac {7 \left (e x +d \right )^{\frac {9}{2}} A \,b^{3} e^{5}}{128 \left (b e x +a e \right )^{5} \left (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}\right )}+\frac {3 \left (e x +d \right )^{\frac {9}{2}} B a \,b^{2} e^{5}}{128 \left (b e x +a e \right )^{5} \left (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}\right )}-\frac {5 \left (e x +d \right )^{\frac {9}{2}} B \,b^{3} d \,e^{4}}{64 \left (b e x +a e \right )^{5} \left (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}\right )}+\frac {49 \left (e x +d \right )^{\frac {7}{2}} A \,b^{2} e^{5}}{192 \left (b e x +a e \right )^{5} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 \left (e x +d \right )^{\frac {7}{2}} B a b \,e^{5}}{64 \left (b e x +a e \right )^{5} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {35 \left (e x +d \right )^{\frac {7}{2}} B \,b^{2} d \,e^{4}}{96 \left (b e x +a e \right )^{5} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 \left (e x +d \right )^{\frac {5}{2}} A b \,e^{5}}{15 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {\left (e x +d \right )^{\frac {5}{2}} B a \,e^{5}}{5 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {2 \left (e x +d \right )^{\frac {5}{2}} B b d \,e^{4}}{3 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {7 A \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (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}\right ) \sqrt {\left (a e -b d \right ) b}\, b}+\frac {79 \left (e x +d \right )^{\frac {3}{2}} A \,e^{5}}{192 \left (b e x +a e \right )^{5} \left (a e -b d \right )}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} B a \,e^{5}}{64 \left (b e x +a e \right )^{5} \left (a e -b d \right ) b}+\frac {3 B a \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (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}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{2}}-\frac {5 B d \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \left (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}\right ) \sqrt {\left (a e -b d \right ) b}\, b}-\frac {29 \left (e x +d \right )^{\frac {3}{2}} B d \,e^{4}}{96 \left (b e x +a e \right )^{5} \left (a e -b d \right )}-\frac {7 \sqrt {e x +d}\, A \,e^{5}}{128 \left (b e x +a e \right )^{5} b}-\frac {3 \sqrt {e x +d}\, B a \,e^{5}}{128 \left (b e x +a e \right )^{5} b^{2}}+\frac {5 \sqrt {e x +d}\, B d \,e^{4}}{64 \left (b e x +a e \right )^{5} b} \end {gather*}

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)/
((a*e-b*d)*b)^(1/2)*b)*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)/((a*e-b*d)*b)^(1/2)*b)*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)/((a*e-b*d)*b)^(1/2)*b)*B*d

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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((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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d positive or negative?

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mupad [B]  time = 2.24, size = 564, normalized size = 1.80 \begin {gather*} \frac {\frac {7\,{\left (d+e\,x\right )}^{7/2}\,\left (7\,A\,b^2\,e^5-10\,B\,d\,b^2\,e^4+3\,B\,a\,b\,e^5\right )}{192\,{\left (a\,e-b\,d\right )}^3}-\frac {\sqrt {d+e\,x}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,b^2}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,{\left (a\,e-b\,d\right )}^2}+\frac {b^2\,{\left (d+e\,x\right )}^{9/2}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^4}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (21\,B\,a\,e^5-79\,A\,b\,e^5+58\,B\,b\,d\,e^4\right )}{192\,b\,\left (a\,e-b\,d\right )}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4}+\frac {e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (7\,A\,b\,e+3\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (7\,A\,b\,e+3\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(1/2))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((7*(d + e*x)^(7/2)*(7*A*b^2*e^5 + 3*B*a*b*e^5 - 10*B*b^2*d*e^4))/(192*(a*e - b*d)^3) - ((d + e*x)^(1/2)*(7*A*
b*e^5 + 3*B*a*e^5 - 10*B*b*d*e^4))/(128*b^2) + ((d + e*x)^(5/2)*(7*A*b*e^5 + 3*B*a*e^5 - 10*B*b*d*e^4))/(15*(a
*e - b*d)^2) + (b^2*(d + e*x)^(9/2)*(7*A*b*e^5 + 3*B*a*e^5 - 10*B*b*d*e^4))/(128*(a*e - b*d)^4) - ((d + e*x)^(
3/2)*(21*B*a*e^5 - 79*A*b*e^5 + 58*B*b*d*e^4))/(192*b*(a*e - b*d)))/((d + e*x)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a
^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3*e) - (d + e*x)^2*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*
e^2 - 30*a*b^4*d^2*e) + b^5*(d + e*x)^5 - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*
(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e) - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4
*b*d*e^4) + (e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(7*A*b*e + 3*B*a*e - 10*B*b*d))/((a*e - b*d)^(1/2)*(7*A*b*e
^5 + 3*B*a*e^5 - 10*B*b*d*e^4)))*(7*A*b*e + 3*B*a*e - 10*B*b*d))/(128*b^(5/2)*(a*e - b*d)^(9/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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