3.1823 \(\int \frac{A+B x}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=291 \[ \frac{35 e^2 (a B e-3 A b e+2 b B d)}{8 \sqrt{d+e x} (b d-a e)^5}+\frac{35 e^2 (a B e-3 A b e+2 b B d)}{24 b (d+e x)^{3/2} (b d-a e)^4}-\frac{35 \sqrt{b} e^2 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}}+\frac{7 e (a B e-3 A b e+2 b B d)}{8 b (a+b x) (d+e x)^{3/2} (b d-a e)^3}-\frac{a B e-3 A b e+2 b B d}{4 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)} \]

[Out]

(35*e^2*(2*b*B*d - 3*A*b*e + a*B*e))/(24*b*(b*d - a*e)^4*(d + e*x)^(3/2)) - (A*b - a*B)/(3*b*(b*d - a*e)*(a +
b*x)^3*(d + e*x)^(3/2)) - (2*b*B*d - 3*A*b*e + a*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)) + (7*e*(
2*b*B*d - 3*A*b*e + a*B*e))/(8*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)) + (35*e^2*(2*b*B*d - 3*A*b*e + a*B*e
))/(8*(b*d - a*e)^5*Sqrt[d + e*x]) - (35*Sqrt[b]*e^2*(2*b*B*d - 3*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x
])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(11/2))

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Rubi [A]  time = 0.30339, antiderivative size = 291, 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} \[ \frac{35 e^2 (a B e-3 A b e+2 b B d)}{8 \sqrt{d+e x} (b d-a e)^5}+\frac{35 e^2 (a B e-3 A b e+2 b B d)}{24 b (d+e x)^{3/2} (b d-a e)^4}-\frac{35 \sqrt{b} e^2 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}}+\frac{7 e (a B e-3 A b e+2 b B d)}{8 b (a+b x) (d+e x)^{3/2} (b d-a e)^3}-\frac{a B e-3 A b e+2 b B d}{4 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*e^2*(2*b*B*d - 3*A*b*e + a*B*e))/(24*b*(b*d - a*e)^4*(d + e*x)^(3/2)) - (A*b - a*B)/(3*b*(b*d - a*e)*(a +
b*x)^3*(d + e*x)^(3/2)) - (2*b*B*d - 3*A*b*e + a*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)) + (7*e*(
2*b*B*d - 3*A*b*e + a*B*e))/(8*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)) + (35*e^2*(2*b*B*d - 3*A*b*e + a*B*e
))/(8*(b*d - a*e)^5*Sqrt[d + e*x]) - (35*Sqrt[b]*e^2*(2*b*B*d - 3*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x
])/Sqrt[b*d - a*e]])/(8*(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}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{A+B x}{(a+b x)^4 (d+e x)^{5/2}} \, dx\\ &=-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac{(2 b B d-3 A b e+a B e) \int \frac{1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{2 b (b d-a e)}\\ &=-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac{2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac{(7 e (2 b B d-3 A b e+a B e)) \int \frac{1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac{2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac{7 e (2 b B d-3 A b e+a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac{\left (35 e^2 (2 b B d-3 A b e+a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 b (b d-a e)^3}\\ &=\frac{35 e^2 (2 b B d-3 A b e+a B e)}{24 b (b d-a e)^4 (d+e x)^{3/2}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac{2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac{7 e (2 b B d-3 A b e+a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac{\left (35 e^2 (2 b B d-3 A b e+a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^4}\\ &=\frac{35 e^2 (2 b B d-3 A b e+a B e)}{24 b (b d-a e)^4 (d+e x)^{3/2}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac{2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac{7 e (2 b B d-3 A b e+a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac{35 e^2 (2 b B d-3 A b e+a B e)}{8 (b d-a e)^5 \sqrt{d+e x}}+\frac{\left (35 b e^2 (2 b B d-3 A b e+a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 (b d-a e)^5}\\ &=\frac{35 e^2 (2 b B d-3 A b e+a B e)}{24 b (b d-a e)^4 (d+e x)^{3/2}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac{2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac{7 e (2 b B d-3 A b e+a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac{35 e^2 (2 b B d-3 A b e+a B e)}{8 (b d-a e)^5 \sqrt{d+e x}}+\frac{(35 b e (2 b B d-3 A b e+a B e)) \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 d-a e)^5}\\ &=\frac{35 e^2 (2 b B d-3 A b e+a B e)}{24 b (b d-a e)^4 (d+e x)^{3/2}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac{2 b B d-3 A b e+a B e}{4 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac{7 e (2 b B d-3 A b e+a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac{35 e^2 (2 b B d-3 A b e+a B e)}{8 (b d-a e)^5 \sqrt{d+e x}}-\frac{35 \sqrt{b} e^2 (2 b B d-3 A b e+a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.078402, size = 100, normalized size = 0.34 \[ \frac{\frac{3 (a B-A b)}{(a+b x)^3}-\frac{3 e^2 (-a B e+3 A b e-2 b B d) \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}}{9 b (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.032, size = 853, normalized size = 2.9 \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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

-2/3*e^3/(a*e-b*d)^4/(e*x+d)^(3/2)*A+2/3*e^2/(a*e-b*d)^4/(e*x+d)^(3/2)*B*d+8*e^3/(a*e-b*d)^5/(e*x+d)^(1/2)*A*b
-2*e^3/(a*e-b*d)^5/(e*x+d)^(1/2)*a*B-6*e^2/(a*e-b*d)^5/(e*x+d)^(1/2)*B*b*d+41/8*e^3/(a*e-b*d)^5*b^4/(b*e*x+a*e
)^3*(e*x+d)^(5/2)*A-19/8*e^3/(a*e-b*d)^5*b^3/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a-11/4*e^2/(a*e-b*d)^5*b^4/(b*e*x+a
*e)^3*(e*x+d)^(5/2)*B*d+35/3*e^4/(a*e-b*d)^5*b^3/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a-35/3*e^3/(a*e-b*d)^5*b^4/(b*e
*x+a*e)^3*A*(e*x+d)^(3/2)*d-17/3*e^4/(a*e-b*d)^5*b^2/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^2-1/3*e^3/(a*e-b*d)^5*b^3
/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a*d+6*e^2/(a*e-b*d)^5*b^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*d^2+55/8*e^5/(a*e-b*d)^
5*b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^2-55/4*e^4/(a*e-b*d)^5*b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a*d+55/8*e^3/(a
*e-b*d)^5*b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*d^2-29/8*e^5/(a*e-b*d)^5*b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^3+23/8*
e^3/(a*e-b*d)^5*b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a*d^2-13/4*e^2/(a*e-b*d)^5*b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B
*d^3+4*e^4/(a*e-b*d)^5*b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^2*d+105/8*e^3/(a*e-b*d)^5*b^2/((a*e-b*d)*b)^(1/2)*a
rctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A-35/8*e^3/(a*e-b*d)^5*b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b
/((a*e-b*d)*b)^(1/2))*a*B-35/4*e^2/(a*e-b*d)^5*b^2/((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)^(5/2)/(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 = 1.68774, size = 5436, normalized size = 18.68 \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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

[1/48*(105*(2*B*a^3*b*d^3*e^2 + (B*a^4 - 3*A*a^3*b)*d^2*e^3 + (2*B*b^4*d*e^4 + (B*a*b^3 - 3*A*b^4)*e^5)*x^5 +
(4*B*b^4*d^2*e^3 + 2*(4*B*a*b^3 - 3*A*b^4)*d*e^4 + 3*(B*a^2*b^2 - 3*A*a*b^3)*e^5)*x^4 + (2*B*b^4*d^3*e^2 + (13
*B*a*b^3 - 3*A*b^4)*d^2*e^3 + 6*(2*B*a^2*b^2 - 3*A*a*b^3)*d*e^4 + 3*(B*a^3*b - 3*A*a^2*b^2)*e^5)*x^3 + (6*B*a*
b^3*d^3*e^2 + 3*(5*B*a^2*b^2 - 3*A*a*b^3)*d^2*e^3 + 2*(4*B*a^3*b - 9*A*a^2*b^2)*d*e^4 + (B*a^4 - 3*A*a^3*b)*e^
5)*x^2 + (6*B*a^2*b^2*d^3*e^2 + (7*B*a^3*b - 9*A*a^2*b^2)*d^2*e^3 + 2*(B*a^4 - 3*A*a^3*b)*d*e^4)*x)*sqrt(b/(b*
d - a*e))*log((b*e*x + 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 2*(16*A*a^4
*e^4 - 4*(B*a*b^3 + 2*A*b^4)*d^4 + 10*(4*B*a^2*b^2 + 5*A*a*b^3)*d^3*e + (247*B*a^3*b - 165*A*a^2*b^2)*d^2*e^2
+ 16*(2*B*a^4 - 13*A*a^3*b)*d*e^3 + 105*(2*B*b^4*d*e^3 + (B*a*b^3 - 3*A*b^4)*e^4)*x^4 + 140*(2*B*b^4*d^2*e^2 +
 (5*B*a*b^3 - 3*A*b^4)*d*e^3 + 2*(B*a^2*b^2 - 3*A*a*b^3)*e^4)*x^3 + 21*(2*B*b^4*d^3*e + (37*B*a*b^3 - 3*A*b^4)
*d^2*e^2 + 2*(20*B*a^2*b^2 - 27*A*a*b^3)*d*e^3 + 11*(B*a^3*b - 3*A*a^2*b^2)*e^4)*x^2 - 6*(2*B*b^4*d^4 - (19*B*
a*b^3 + 3*A*b^4)*d^3*e - 2*(58*B*a^2*b^2 - 15*A*a*b^3)*d^2*e^2 - 3*(23*B*a^3*b - 53*A*a^2*b^2)*d*e^3 - 8*(B*a^
4 - 3*A*a^3*b)*e^4)*x)*sqrt(e*x + d))/(a^3*b^5*d^7 - 5*a^4*b^4*d^6*e + 10*a^5*b^3*d^5*e^2 - 10*a^6*b^2*d^4*e^3
 + 5*a^7*b*d^3*e^4 - a^8*d^2*e^5 + (b^8*d^5*e^2 - 5*a*b^7*d^4*e^3 + 10*a^2*b^6*d^3*e^4 - 10*a^3*b^5*d^2*e^5 +
5*a^4*b^4*d*e^6 - a^5*b^3*e^7)*x^5 + (2*b^8*d^6*e - 7*a*b^7*d^5*e^2 + 5*a^2*b^6*d^4*e^3 + 10*a^3*b^5*d^3*e^4 -
 20*a^4*b^4*d^2*e^5 + 13*a^5*b^3*d*e^6 - 3*a^6*b^2*e^7)*x^4 + (b^8*d^7 + a*b^7*d^6*e - 17*a^2*b^6*d^5*e^2 + 35
*a^3*b^5*d^4*e^3 - 25*a^4*b^4*d^3*e^4 - a^5*b^3*d^2*e^5 + 9*a^6*b^2*d*e^6 - 3*a^7*b*e^7)*x^3 + (3*a*b^7*d^7 -
9*a^2*b^6*d^6*e + a^3*b^5*d^5*e^2 + 25*a^4*b^4*d^4*e^3 - 35*a^5*b^3*d^3*e^4 + 17*a^6*b^2*d^2*e^5 - a^7*b*d*e^6
 - a^8*e^7)*x^2 + (3*a^2*b^6*d^7 - 13*a^3*b^5*d^6*e + 20*a^4*b^4*d^5*e^2 - 10*a^5*b^3*d^4*e^3 - 5*a^6*b^2*d^3*
e^4 + 7*a^7*b*d^2*e^5 - 2*a^8*d*e^6)*x), -1/24*(105*(2*B*a^3*b*d^3*e^2 + (B*a^4 - 3*A*a^3*b)*d^2*e^3 + (2*B*b^
4*d*e^4 + (B*a*b^3 - 3*A*b^4)*e^5)*x^5 + (4*B*b^4*d^2*e^3 + 2*(4*B*a*b^3 - 3*A*b^4)*d*e^4 + 3*(B*a^2*b^2 - 3*A
*a*b^3)*e^5)*x^4 + (2*B*b^4*d^3*e^2 + (13*B*a*b^3 - 3*A*b^4)*d^2*e^3 + 6*(2*B*a^2*b^2 - 3*A*a*b^3)*d*e^4 + 3*(
B*a^3*b - 3*A*a^2*b^2)*e^5)*x^3 + (6*B*a*b^3*d^3*e^2 + 3*(5*B*a^2*b^2 - 3*A*a*b^3)*d^2*e^3 + 2*(4*B*a^3*b - 9*
A*a^2*b^2)*d*e^4 + (B*a^4 - 3*A*a^3*b)*e^5)*x^2 + (6*B*a^2*b^2*d^3*e^2 + (7*B*a^3*b - 9*A*a^2*b^2)*d^2*e^3 + 2
*(B*a^4 - 3*A*a^3*b)*d*e^4)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*
e*x + b*d)) - (16*A*a^4*e^4 - 4*(B*a*b^3 + 2*A*b^4)*d^4 + 10*(4*B*a^2*b^2 + 5*A*a*b^3)*d^3*e + (247*B*a^3*b -
165*A*a^2*b^2)*d^2*e^2 + 16*(2*B*a^4 - 13*A*a^3*b)*d*e^3 + 105*(2*B*b^4*d*e^3 + (B*a*b^3 - 3*A*b^4)*e^4)*x^4 +
 140*(2*B*b^4*d^2*e^2 + (5*B*a*b^3 - 3*A*b^4)*d*e^3 + 2*(B*a^2*b^2 - 3*A*a*b^3)*e^4)*x^3 + 21*(2*B*b^4*d^3*e +
 (37*B*a*b^3 - 3*A*b^4)*d^2*e^2 + 2*(20*B*a^2*b^2 - 27*A*a*b^3)*d*e^3 + 11*(B*a^3*b - 3*A*a^2*b^2)*e^4)*x^2 -
6*(2*B*b^4*d^4 - (19*B*a*b^3 + 3*A*b^4)*d^3*e - 2*(58*B*a^2*b^2 - 15*A*a*b^3)*d^2*e^2 - 3*(23*B*a^3*b - 53*A*a
^2*b^2)*d*e^3 - 8*(B*a^4 - 3*A*a^3*b)*e^4)*x)*sqrt(e*x + d))/(a^3*b^5*d^7 - 5*a^4*b^4*d^6*e + 10*a^5*b^3*d^5*e
^2 - 10*a^6*b^2*d^4*e^3 + 5*a^7*b*d^3*e^4 - a^8*d^2*e^5 + (b^8*d^5*e^2 - 5*a*b^7*d^4*e^3 + 10*a^2*b^6*d^3*e^4
- 10*a^3*b^5*d^2*e^5 + 5*a^4*b^4*d*e^6 - a^5*b^3*e^7)*x^5 + (2*b^8*d^6*e - 7*a*b^7*d^5*e^2 + 5*a^2*b^6*d^4*e^3
 + 10*a^3*b^5*d^3*e^4 - 20*a^4*b^4*d^2*e^5 + 13*a^5*b^3*d*e^6 - 3*a^6*b^2*e^7)*x^4 + (b^8*d^7 + a*b^7*d^6*e -
17*a^2*b^6*d^5*e^2 + 35*a^3*b^5*d^4*e^3 - 25*a^4*b^4*d^3*e^4 - a^5*b^3*d^2*e^5 + 9*a^6*b^2*d*e^6 - 3*a^7*b*e^7
)*x^3 + (3*a*b^7*d^7 - 9*a^2*b^6*d^6*e + a^3*b^5*d^5*e^2 + 25*a^4*b^4*d^4*e^3 - 35*a^5*b^3*d^3*e^4 + 17*a^6*b^
2*d^2*e^5 - a^7*b*d*e^6 - a^8*e^7)*x^2 + (3*a^2*b^6*d^7 - 13*a^3*b^5*d^6*e + 20*a^4*b^4*d^5*e^2 - 10*a^5*b^3*d
^4*e^3 - 5*a^6*b^2*d^3*e^4 + 7*a^7*b*d^2*e^5 - 2*a^8*d*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)/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.19449, size = 1013, normalized size = 3.48 \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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

35/8*(2*B*b^2*d*e^2 + B*a*b*e^3 - 3*A*b^2*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^5 - 5*a*b^
4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(-b^2*d + a*b*e)) + 1/24*(210
*(x*e + d)^4*B*b^4*d*e^2 - 560*(x*e + d)^3*B*b^4*d^2*e^2 + 462*(x*e + d)^2*B*b^4*d^3*e^2 - 96*(x*e + d)*B*b^4*
d^4*e^2 - 16*B*b^4*d^5*e^2 + 105*(x*e + d)^4*B*a*b^3*e^3 - 315*(x*e + d)^4*A*b^4*e^3 + 280*(x*e + d)^3*B*a*b^3
*d*e^3 + 840*(x*e + d)^3*A*b^4*d*e^3 - 693*(x*e + d)^2*B*a*b^3*d^2*e^3 - 693*(x*e + d)^2*A*b^4*d^2*e^3 + 240*(
x*e + d)*B*a*b^3*d^3*e^3 + 144*(x*e + d)*A*b^4*d^3*e^3 + 64*B*a*b^3*d^4*e^3 + 16*A*b^4*d^4*e^3 + 280*(x*e + d)
^3*B*a^2*b^2*e^4 - 840*(x*e + d)^3*A*a*b^3*e^4 + 1386*(x*e + d)^2*A*a*b^3*d*e^4 - 144*(x*e + d)*B*a^2*b^2*d^2*
e^4 - 432*(x*e + d)*A*a*b^3*d^2*e^4 - 96*B*a^2*b^2*d^3*e^4 - 64*A*a*b^3*d^3*e^4 + 231*(x*e + d)^2*B*a^3*b*e^5
- 693*(x*e + d)^2*A*a^2*b^2*e^5 - 48*(x*e + d)*B*a^3*b*d*e^5 + 432*(x*e + d)*A*a^2*b^2*d*e^5 + 64*B*a^3*b*d^2*
e^5 + 96*A*a^2*b^2*d^2*e^5 + 48*(x*e + d)*B*a^4*e^6 - 144*(x*e + d)*A*a^3*b*e^6 - 16*B*a^4*d*e^6 - 64*A*a^3*b*
d*e^6 + 16*A*a^4*e^7)/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^
5*e^5)*((x*e + d)^(3/2)*b - sqrt(x*e + d)*b*d + sqrt(x*e + d)*a*e)^3)