3.1831 \(\int \frac{A+B x}{\sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=313 \[ -\frac{7 e^4 (-a B e-9 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{11/2}}+\frac{7 e^3 \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{128 b (a+b x) (b d-a e)^5}-\frac{7 e^2 \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{192 b (a+b x)^2 (b d-a e)^4}+\frac{7 e \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{240 b (a+b x)^3 (b d-a e)^3}-\frac{\sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{40 b (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

[Out]

-((A*b - a*B)*Sqrt[d + e*x])/(5*b*(b*d - a*e)*(a + b*x)^5) - ((10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(40*
b*(b*d - a*e)^2*(a + b*x)^4) + (7*e*(10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(240*b*(b*d - a*e)^3*(a + b*x)
^3) - (7*e^2*(10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(192*b*(b*d - a*e)^4*(a + b*x)^2) + (7*e^3*(10*b*B*d
- 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(128*b*(b*d - a*e)^5*(a + b*x)) - (7*e^4*(10*b*B*d - 9*A*b*e - a*B*e)*ArcTan
h[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(3/2)*(b*d - a*e)^(11/2))

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Rubi [A]  time = 0.314548, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {27, 78, 51, 63, 208} \[ -\frac{7 e^4 (-a B e-9 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{11/2}}+\frac{7 e^3 \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{128 b (a+b x) (b d-a e)^5}-\frac{7 e^2 \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{192 b (a+b x)^2 (b d-a e)^4}+\frac{7 e \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{240 b (a+b x)^3 (b d-a e)^3}-\frac{\sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{40 b (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

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]

-((A*b - a*B)*Sqrt[d + e*x])/(5*b*(b*d - a*e)*(a + b*x)^5) - ((10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(40*
b*(b*d - a*e)^2*(a + b*x)^4) + (7*e*(10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(240*b*(b*d - a*e)^3*(a + b*x)
^3) - (7*e^2*(10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(192*b*(b*d - a*e)^4*(a + b*x)^2) + (7*e^3*(10*b*B*d
- 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(128*b*(b*d - a*e)^5*(a + b*x)) - (7*e^4*(10*b*B*d - 9*A*b*e - a*B*e)*ArcTan
h[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(3/2)*(b*d - a*e)^(11/2))

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 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}{(a+b x)^6 \sqrt{d+e x}} \, dx\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{5 b (b d-a e) (a+b x)^5}+\frac{(10 b B d-9 A b e-a B e) \int \frac{1}{(a+b x)^5 \sqrt{d+e x}} \, dx}{10 b (b d-a e)}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac{(10 b B d-9 A b e-a B e) \sqrt{d+e x}}{40 b (b d-a e)^2 (a+b x)^4}-\frac{(7 e (10 b B d-9 A b e-a B e)) \int \frac{1}{(a+b x)^4 \sqrt{d+e x}} \, dx}{80 b (b d-a e)^2}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac{(10 b B d-9 A b e-a B e) \sqrt{d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac{7 e (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{240 b (b d-a e)^3 (a+b x)^3}+\frac{\left (7 e^2 (10 b B d-9 A b e-a B e)\right ) \int \frac{1}{(a+b x)^3 \sqrt{d+e x}} \, dx}{96 b (b d-a e)^3}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac{(10 b B d-9 A b e-a B e) \sqrt{d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac{7 e (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{240 b (b d-a e)^3 (a+b x)^3}-\frac{7 e^2 (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{192 b (b d-a e)^4 (a+b x)^2}-\frac{\left (7 e^3 (10 b B d-9 A b e-a B e)\right ) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{128 b (b d-a e)^4}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac{(10 b B d-9 A b e-a B e) \sqrt{d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac{7 e (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{240 b (b d-a e)^3 (a+b x)^3}-\frac{7 e^2 (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{192 b (b d-a e)^4 (a+b x)^2}+\frac{7 e^3 (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{128 b (b d-a e)^5 (a+b x)}+\frac{\left (7 e^4 (10 b B d-9 A b e-a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b (b d-a e)^5}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac{(10 b B d-9 A b e-a B e) \sqrt{d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac{7 e (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{240 b (b d-a e)^3 (a+b x)^3}-\frac{7 e^2 (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{192 b (b d-a e)^4 (a+b x)^2}+\frac{7 e^3 (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{128 b (b d-a e)^5 (a+b x)}+\frac{\left (7 e^3 (10 b B d-9 A b e-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 (b d-a e)^5}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac{(10 b B d-9 A b e-a B e) \sqrt{d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac{7 e (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{240 b (b d-a e)^3 (a+b x)^3}-\frac{7 e^2 (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{192 b (b d-a e)^4 (a+b x)^2}+\frac{7 e^3 (10 b B d-9 A b e-a B e) \sqrt{d+e x}}{128 b (b d-a e)^5 (a+b x)}-\frac{7 e^4 (10 b B d-9 A b e-a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.0631944, size = 97, normalized size = 0.31 \[ \frac{\sqrt{d+e x} \left (\frac{e^4 (a B e+9 A b e-10 b B d) \, _2F_1\left (\frac{1}{2},5;\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac{a B-A b}{(a+b x)^5}\right )}{5 b (b d-a e)} \]

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]

(Sqrt[d + e*x]*((-(A*b) + a*B)/(a + b*x)^5 + (e^4*(-10*b*B*d + 9*A*b*e + a*B*e)*Hypergeometric2F1[1/2, 5, 3/2,
(b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^5))/(5*b*(b*d - a*e))

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Maple [B]  time = 0.021, size = 1274, normalized size = 4.1 \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)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x)

[Out]

63/128*e^5/(b*e*x+a*e)^5*b^4/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^
5)*(e*x+d)^(9/2)*A+7/128*e^5/(b*e*x+a*e)^5*b^3/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*
a*b^4*d^4*e-b^5*d^5)*(e*x+d)^(9/2)*a*B-35/64*e^4/(b*e*x+a*e)^5*b^4/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-1
0*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)*(e*x+d)^(9/2)*B*d+147/64*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)^(7/2)*A+49/192*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)^(7/2)*a*B-245/96*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)^(7/2)*B*d+21/5*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)^(5/2)*A+7/15*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)^(5/2)*a*B-14/3*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)^(5/2)*B*d+2
37/64*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(3/2)*A*b+79/192*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*
d*e+b^2*d^2)*(e*x+d)^(3/2)*a*B-395/96*e^4/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(3/2)*B*b*d+193/12
8*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(1/2)*A-7/128*e^5/(b*e*x+a*e)^5/b/(a*e-b*d)*(e*x+d)^(1/2)*a*B-93/64*e^4/
(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(1/2)*B*d+63/128*e^5/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*
e^2+5*a*b^4*d^4*e-b^5*d^5)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A+7/128*e^5/b/(a^5*
e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)/((a*e-b*d)*b)^(1/2)*arctan((e*x
+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a*B-35/64*e^4/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*
a*b^4*d^4*e-b^5*d^5)/((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)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.82832, size = 5731, normalized size = 18.31 \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)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[1/3840*(105*(10*B*a^5*b*d*e^4 - (B*a^6 + 9*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (B*a*b^5 + 9*A*b^6)*e^5)*x^5 + 5*
(10*B*a*b^5*d*e^4 - (B*a^2*b^4 + 9*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (B*a^3*b^3 + 9*A*a^2*b^4)*e^5)
*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (B*a^4*b^2 + 9*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (B*a^5*b + 9*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*(38*B*a^2*b^5 + 147*A*a*b^6)*d^4*e + 12*(139*B*a^3*b^4 + 506*A*a^2*b^5)*d
^3*e^2 - 6*(456*B*a^4*b^3 + 1429*A*a^3*b^4)*d^2*e^3 + 5*(295*B*a^5*b^2 + 1473*A*a^4*b^3)*d*e^4 + 15*(7*B*a^6*b
- 193*A*a^5*b^2)*e^5 - 105*(10*B*b^7*d^2*e^3 - (11*B*a*b^6 + 9*A*b^7)*d*e^4 + (B*a^2*b^5 + 9*A*a*b^6)*e^5)*x^
4 + 70*(10*B*b^7*d^3*e^2 - 9*(9*B*a*b^6 + A*b^7)*d^2*e^3 + 6*(13*B*a^2*b^5 + 12*A*a*b^6)*d*e^4 - 7*(B*a^3*b^4
+ 9*A*a^2*b^5)*e^5)*x^3 - 14*(40*B*b^7*d^4*e - 2*(137*B*a*b^6 + 18*A*b^7)*d^3*e^2 + 3*(299*B*a^2*b^5 + 81*A*a*
b^6)*d^2*e^3 - (727*B*a^3*b^4 + 783*A*a^2*b^5)*d*e^4 + 64*(B*a^4*b^3 + 9*A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^
5 - 8*(193*B*a*b^6 + 27*A*b^7)*d^4*e + 2*(2161*B*a^2*b^5 + 684*A*a*b^6)*d^3*e^2 - 3*(2419*B*a^3*b^4 + 1251*A*a
^2*b^5)*d^2*e^3 + 2*(2317*B*a^4*b^3 + 3078*A*a^3*b^4)*d*e^4 - 395*(B*a^5*b^2 + 9*A*a^4*b^3)*e^5)*x)*sqrt(e*x +
d))/(a^5*b^8*d^6 - 6*a^6*b^7*d^5*e + 15*a^7*b^6*d^4*e^2 - 20*a^8*b^5*d^3*e^3 + 15*a^9*b^4*d^2*e^4 - 6*a^10*b^
3*d*e^5 + a^11*b^2*e^6 + (b^13*d^6 - 6*a*b^12*d^5*e + 15*a^2*b^11*d^4*e^2 - 20*a^3*b^10*d^3*e^3 + 15*a^4*b^9*d
^2*e^4 - 6*a^5*b^8*d*e^5 + a^6*b^7*e^6)*x^5 + 5*(a*b^12*d^6 - 6*a^2*b^11*d^5*e + 15*a^3*b^10*d^4*e^2 - 20*a^4*
b^9*d^3*e^3 + 15*a^5*b^8*d^2*e^4 - 6*a^6*b^7*d*e^5 + a^7*b^6*e^6)*x^4 + 10*(a^2*b^11*d^6 - 6*a^3*b^10*d^5*e +
15*a^4*b^9*d^4*e^2 - 20*a^5*b^8*d^3*e^3 + 15*a^6*b^7*d^2*e^4 - 6*a^7*b^6*d*e^5 + a^8*b^5*e^6)*x^3 + 10*(a^3*b^
10*d^6 - 6*a^4*b^9*d^5*e + 15*a^5*b^8*d^4*e^2 - 20*a^6*b^7*d^3*e^3 + 15*a^7*b^6*d^2*e^4 - 6*a^8*b^5*d*e^5 + a^
9*b^4*e^6)*x^2 + 5*(a^4*b^9*d^6 - 6*a^5*b^8*d^5*e + 15*a^6*b^7*d^4*e^2 - 20*a^7*b^6*d^3*e^3 + 15*a^8*b^5*d^2*e
^4 - 6*a^9*b^4*d*e^5 + a^10*b^3*e^6)*x), 1/1920*(105*(10*B*a^5*b*d*e^4 - (B*a^6 + 9*A*a^5*b)*e^5 + (10*B*b^6*d
*e^4 - (B*a*b^5 + 9*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (B*a^2*b^4 + 9*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^
4*d*e^4 - (B*a^3*b^3 + 9*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (B*a^4*b^2 + 9*A*a^3*b^3)*e^5)*x^2 + 5
*(10*B*a^4*b^2*d*e^4 - (B*a^5*b + 9*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*(38*B*a^2*b^5 + 147*A*a*b^6)*d^4*e + 12*(139*B*a^3*b
^4 + 506*A*a^2*b^5)*d^3*e^2 - 6*(456*B*a^4*b^3 + 1429*A*a^3*b^4)*d^2*e^3 + 5*(295*B*a^5*b^2 + 1473*A*a^4*b^3)*
d*e^4 + 15*(7*B*a^6*b - 193*A*a^5*b^2)*e^5 - 105*(10*B*b^7*d^2*e^3 - (11*B*a*b^6 + 9*A*b^7)*d*e^4 + (B*a^2*b^5
+ 9*A*a*b^6)*e^5)*x^4 + 70*(10*B*b^7*d^3*e^2 - 9*(9*B*a*b^6 + A*b^7)*d^2*e^3 + 6*(13*B*a^2*b^5 + 12*A*a*b^6)*
d*e^4 - 7*(B*a^3*b^4 + 9*A*a^2*b^5)*e^5)*x^3 - 14*(40*B*b^7*d^4*e - 2*(137*B*a*b^6 + 18*A*b^7)*d^3*e^2 + 3*(29
9*B*a^2*b^5 + 81*A*a*b^6)*d^2*e^3 - (727*B*a^3*b^4 + 783*A*a^2*b^5)*d*e^4 + 64*(B*a^4*b^3 + 9*A*a^3*b^4)*e^5)*
x^2 + 2*(240*B*b^7*d^5 - 8*(193*B*a*b^6 + 27*A*b^7)*d^4*e + 2*(2161*B*a^2*b^5 + 684*A*a*b^6)*d^3*e^2 - 3*(2419
*B*a^3*b^4 + 1251*A*a^2*b^5)*d^2*e^3 + 2*(2317*B*a^4*b^3 + 3078*A*a^3*b^4)*d*e^4 - 395*(B*a^5*b^2 + 9*A*a^4*b^
3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^6 - 6*a^6*b^7*d^5*e + 15*a^7*b^6*d^4*e^2 - 20*a^8*b^5*d^3*e^3 + 15*a^9*b^
4*d^2*e^4 - 6*a^10*b^3*d*e^5 + a^11*b^2*e^6 + (b^13*d^6 - 6*a*b^12*d^5*e + 15*a^2*b^11*d^4*e^2 - 20*a^3*b^10*d
^3*e^3 + 15*a^4*b^9*d^2*e^4 - 6*a^5*b^8*d*e^5 + a^6*b^7*e^6)*x^5 + 5*(a*b^12*d^6 - 6*a^2*b^11*d^5*e + 15*a^3*b
^10*d^4*e^2 - 20*a^4*b^9*d^3*e^3 + 15*a^5*b^8*d^2*e^4 - 6*a^6*b^7*d*e^5 + a^7*b^6*e^6)*x^4 + 10*(a^2*b^11*d^6
- 6*a^3*b^10*d^5*e + 15*a^4*b^9*d^4*e^2 - 20*a^5*b^8*d^3*e^3 + 15*a^6*b^7*d^2*e^4 - 6*a^7*b^6*d*e^5 + a^8*b^5*
e^6)*x^3 + 10*(a^3*b^10*d^6 - 6*a^4*b^9*d^5*e + 15*a^5*b^8*d^4*e^2 - 20*a^6*b^7*d^3*e^3 + 15*a^7*b^6*d^2*e^4 -
6*a^8*b^5*d*e^5 + a^9*b^4*e^6)*x^2 + 5*(a^4*b^9*d^6 - 6*a^5*b^8*d^5*e + 15*a^6*b^7*d^4*e^2 - 20*a^7*b^6*d^3*e
^3 + 15*a^8*b^5*d^2*e^4 - 6*a^9*b^4*d*e^5 + a^10*b^3*e^6)*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)/(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.23958, size = 1189, normalized size = 3.8 \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)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

7/128*(10*B*b*d*e^4 - B*a*e^5 - 9*A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^5 - 5*a*b^5*d^
4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*sqrt(-b^2*d + a*b*e)) + 1/1920*(1
050*(x*e + d)^(9/2)*B*b^5*d*e^4 - 4900*(x*e + d)^(7/2)*B*b^5*d^2*e^4 + 8960*(x*e + d)^(5/2)*B*b^5*d^3*e^4 - 79
00*(x*e + d)^(3/2)*B*b^5*d^4*e^4 + 2790*sqrt(x*e + d)*B*b^5*d^5*e^4 - 105*(x*e + d)^(9/2)*B*a*b^4*e^5 - 945*(x
*e + d)^(9/2)*A*b^5*e^5 + 5390*(x*e + d)^(7/2)*B*a*b^4*d*e^5 + 4410*(x*e + d)^(7/2)*A*b^5*d*e^5 - 18816*(x*e +
d)^(5/2)*B*a*b^4*d^2*e^5 - 8064*(x*e + d)^(5/2)*A*b^5*d^2*e^5 + 24490*(x*e + d)^(3/2)*B*a*b^4*d^3*e^5 + 7110*
(x*e + d)^(3/2)*A*b^5*d^3*e^5 - 11055*sqrt(x*e + d)*B*a*b^4*d^4*e^5 - 2895*sqrt(x*e + d)*A*b^5*d^4*e^5 - 490*(
x*e + d)^(7/2)*B*a^2*b^3*e^6 - 4410*(x*e + d)^(7/2)*A*a*b^4*e^6 + 10752*(x*e + d)^(5/2)*B*a^2*b^3*d*e^6 + 1612
8*(x*e + d)^(5/2)*A*a*b^4*d*e^6 - 26070*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^6 - 21330*(x*e + d)^(3/2)*A*a*b^4*d^2*
e^6 + 16320*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6 + 11580*sqrt(x*e + d)*A*a*b^4*d^3*e^6 - 896*(x*e + d)^(5/2)*B*a^3*
b^2*e^7 - 8064*(x*e + d)^(5/2)*A*a^2*b^3*e^7 + 10270*(x*e + d)^(3/2)*B*a^3*b^2*d*e^7 + 21330*(x*e + d)^(3/2)*A
*a^2*b^3*d*e^7 - 10530*sqrt(x*e + d)*B*a^3*b^2*d^2*e^7 - 17370*sqrt(x*e + d)*A*a^2*b^3*d^2*e^7 - 790*(x*e + d)
^(3/2)*B*a^4*b*e^8 - 7110*(x*e + d)^(3/2)*A*a^3*b^2*e^8 + 2370*sqrt(x*e + d)*B*a^4*b*d*e^8 + 11580*sqrt(x*e +
d)*A*a^3*b^2*d*e^8 + 105*sqrt(x*e + d)*B*a^5*e^9 - 2895*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d^5 - 5*a*b^5*d^4*e +
10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*((x*e + d)*b - b*d + a*e)^5)