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

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

________________________________________________________________________________________

Rubi [A]  time = 0.30, antiderivative size = 291, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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 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}{(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 = 0.06, size = 100, normalized size = 0.34 \begin {gather*} \frac {\frac {3 a B-3 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)} \end {gather*}

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 + 3*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))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 2.06, size = 638, normalized size = 2.19 \begin {gather*} -\frac {e^2 \left (16 a^4 A e^5+48 a^4 B e^4 (d+e x)-16 a^4 B d e^4-144 a^3 A b e^4 (d+e x)-64 a^3 A b d e^4+64 a^3 b B d^2 e^3-48 a^3 b B d e^3 (d+e x)+231 a^3 b B e^3 (d+e x)^2+96 a^2 A b^2 d^2 e^3+432 a^2 A b^2 d e^3 (d+e x)-693 a^2 A b^2 e^3 (d+e x)^2-96 a^2 b^2 B d^3 e^2-144 a^2 b^2 B d^2 e^2 (d+e x)+280 a^2 b^2 B e^2 (d+e x)^3-64 a A b^3 d^3 e^2-432 a A b^3 d^2 e^2 (d+e x)+1386 a A b^3 d e^2 (d+e x)^2-840 a A b^3 e^2 (d+e x)^3+64 a b^3 B d^4 e+240 a b^3 B d^3 e (d+e x)-693 a b^3 B d^2 e (d+e x)^2+280 a b^3 B d e (d+e x)^3+105 a b^3 B e (d+e x)^4+16 A b^4 d^4 e+144 A b^4 d^3 e (d+e x)-693 A b^4 d^2 e (d+e x)^2+840 A b^4 d e (d+e x)^3-315 A b^4 e (d+e x)^4-16 b^4 B d^5-96 b^4 B d^4 (d+e x)+462 b^4 B d^3 (d+e x)^2-560 b^4 B d^2 (d+e x)^3+210 b^4 B d (d+e x)^4\right )}{24 (d+e x)^{3/2} (b d-a e)^5 (-a e-b (d+e x)+b d)^3}-\frac {35 \left (a \sqrt {b} B e^3-3 A b^{3/2} e^3+2 b^{3/2} B d e^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{8 (b d-a e)^5 \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.50, size = 2648, normalized size = 9.10

result too large to display

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)]

________________________________________________________________________________________

giac [B]  time = 0.29, size = 750, normalized size = 2.58 \begin {gather*} \frac {35 \, {\left (2 \, B b^{2} d e^{2} + B a b e^{3} - 3 \, A b^{2} e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (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}\right )} \sqrt {-b^{2} d + a b e}} + \frac {210 \, {\left (x e + d\right )}^{4} B b^{4} d e^{2} - 560 \, {\left (x e + d\right )}^{3} B b^{4} d^{2} e^{2} + 462 \, {\left (x e + d\right )}^{2} B b^{4} d^{3} e^{2} - 96 \, {\left (x e + d\right )} B b^{4} d^{4} e^{2} - 16 \, B b^{4} d^{5} e^{2} + 105 \, {\left (x e + d\right )}^{4} B a b^{3} e^{3} - 315 \, {\left (x e + d\right )}^{4} A b^{4} e^{3} + 280 \, {\left (x e + d\right )}^{3} B a b^{3} d e^{3} + 840 \, {\left (x e + d\right )}^{3} A b^{4} d e^{3} - 693 \, {\left (x e + d\right )}^{2} B a b^{3} d^{2} e^{3} - 693 \, {\left (x e + d\right )}^{2} A b^{4} d^{2} e^{3} + 240 \, {\left (x e + d\right )} B a b^{3} d^{3} e^{3} + 144 \, {\left (x e + d\right )} 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 \, {\left (x e + d\right )}^{3} B a^{2} b^{2} e^{4} - 840 \, {\left (x e + d\right )}^{3} A a b^{3} e^{4} + 1386 \, {\left (x e + d\right )}^{2} A a b^{3} d e^{4} - 144 \, {\left (x e + d\right )} B a^{2} b^{2} d^{2} e^{4} - 432 \, {\left (x e + d\right )} 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 \, {\left (x e + d\right )}^{2} B a^{3} b e^{5} - 693 \, {\left (x e + d\right )}^{2} A a^{2} b^{2} e^{5} - 48 \, {\left (x e + d\right )} B a^{3} b d e^{5} + 432 \, {\left (x e + d\right )} 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 \, {\left (x e + d\right )} B a^{4} e^{6} - 144 \, {\left (x e + d\right )} 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}}{24 \, {\left (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}\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} b - \sqrt {x e + d} b d + \sqrt {x e + d} a e\right )}^{3}} \end {gather*}

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)

________________________________________________________________________________________

maple [B]  time = 0.08, size = 853, normalized size = 2.93 \begin {gather*} \frac {55 \sqrt {e x +d}\, A \,a^{2} b^{2} e^{5}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {55 \sqrt {e x +d}\, A a \,b^{3} d \,e^{4}}{4 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {55 \sqrt {e x +d}\, A \,b^{4} d^{2} e^{3}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {29 \sqrt {e x +d}\, B \,a^{3} b \,e^{5}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {4 \sqrt {e x +d}\, B \,a^{2} b^{2} d \,e^{4}}{\left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {23 \sqrt {e x +d}\, B a \,b^{3} d^{2} e^{3}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {13 \sqrt {e x +d}\, B \,b^{4} d^{3} e^{2}}{4 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {35 \left (e x +d \right )^{\frac {3}{2}} A a \,b^{3} e^{4}}{3 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {35 \left (e x +d \right )^{\frac {3}{2}} A \,b^{4} d \,e^{3}}{3 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {17 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} b^{2} e^{4}}{3 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {\left (e x +d \right )^{\frac {3}{2}} B a \,b^{3} d \,e^{3}}{3 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {6 \left (e x +d \right )^{\frac {3}{2}} B \,b^{4} d^{2} e^{2}}{\left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {41 \left (e x +d \right )^{\frac {5}{2}} A \,b^{4} e^{3}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {19 \left (e x +d \right )^{\frac {5}{2}} B a \,b^{3} e^{3}}{8 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}-\frac {11 \left (e x +d \right )^{\frac {5}{2}} B \,b^{4} d \,e^{2}}{4 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{3}}+\frac {105 A \,b^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}-\frac {35 B a b \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}-\frac {35 B \,b^{2} d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}+\frac {8 A b \,e^{3}}{\left (a e -b d \right )^{5} \sqrt {e x +d}}-\frac {2 B a \,e^{3}}{\left (a e -b d \right )^{5} \sqrt {e x +d}}-\frac {6 B b d \,e^{2}}{\left (a e -b d \right )^{5} \sqrt {e x +d}}-\frac {2 A \,e^{3}}{3 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 B d \,e^{2}}{3 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {3}{2}}} \end {gather*}

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]

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)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A-35/8*e^3/(a*e-b*d)^5*b/((a*e-b
*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a*B-35/4*e^2/(a*e-b*d)^5*b^2/((a*e-b*d)*b)^(1/2)*arct
an((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*d-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

________________________________________________________________________________________

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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^2,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?

________________________________________________________________________________________

mupad [B]  time = 2.51, size = 483, normalized size = 1.66 \begin {gather*} -\frac {\frac {2\,\left (A\,e^3-B\,d\,e^2\right )}{3\,\left (a\,e-b\,d\right )}+\frac {77\,{\left (d+e\,x\right )}^2\,\left (-3\,A\,b^2\,e^3+2\,B\,d\,b^2\,e^2+B\,a\,b\,e^3\right )}{8\,{\left (a\,e-b\,d\right )}^3}+\frac {35\,{\left (d+e\,x\right )}^3\,\left (-3\,A\,b^3\,e^3+2\,B\,d\,b^3\,e^2+B\,a\,b^2\,e^3\right )}{3\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,\left (d+e\,x\right )\,\left (B\,a\,e^3-3\,A\,b\,e^3+2\,B\,b\,d\,e^2\right )}{{\left (a\,e-b\,d\right )}^2}+\frac {35\,b^3\,{\left (d+e\,x\right )}^4\,\left (B\,a\,e^3-3\,A\,b\,e^3+2\,B\,b\,d\,e^2\right )}{8\,{\left (a\,e-b\,d\right )}^5}}{{\left (d+e\,x\right )}^{3/2}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )+b^3\,{\left (d+e\,x\right )}^{9/2}-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{7/2}+{\left (d+e\,x\right )}^{5/2}\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}-\frac {35\,\sqrt {b}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^2\,\sqrt {d+e\,x}\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )\,\left (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\right )}{{\left (a\,e-b\,d\right )}^{11/2}\,\left (B\,a\,e^3-3\,A\,b\,e^3+2\,B\,b\,d\,e^2\right )}\right )\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{8\,{\left (a\,e-b\,d\right )}^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*(A*e^3 - B*d*e^2))/(3*(a*e - b*d)) + (77*(d + e*x)^2*(B*a*b*e^3 - 3*A*b^2*e^3 + 2*B*b^2*d*e^2))/(8*(a*e
- b*d)^3) + (35*(d + e*x)^3*(B*a*b^2*e^3 - 3*A*b^3*e^3 + 2*B*b^3*d*e^2))/(3*(a*e - b*d)^4) + (2*(d + e*x)*(B*a
*e^3 - 3*A*b*e^3 + 2*B*b*d*e^2))/(a*e - b*d)^2 + (35*b^3*(d + e*x)^4*(B*a*e^3 - 3*A*b*e^3 + 2*B*b*d*e^2))/(8*(
a*e - b*d)^5))/((d + e*x)^(3/2)*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2) + b^3*(d + e*x)^(9/2) - (3
*b^3*d - 3*a*b^2*e)*(d + e*x)^(7/2) + (d + e*x)^(5/2)*(3*b^3*d^2 + 3*a^2*b*e^2 - 6*a*b^2*d*e)) - (35*b^(1/2)*e
^2*atan((b^(1/2)*e^2*(d + e*x)^(1/2)*(B*a*e - 3*A*b*e + 2*B*b*d)*(a^5*e^5 - b^5*d^5 - 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))/((a*e - b*d)^(11/2)*(B*a*e^3 - 3*A*b*e^3 + 2*B*b*d*e^2)))*(B
*a*e - 3*A*b*e + 2*B*b*d))/(8*(a*e - b*d)^(11/2))

________________________________________________________________________________________

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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

________________________________________________________________________________________