3.8.8 \(\int \frac {A+B x}{x^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=240 \[ \frac {231 \sqrt {b} (13 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5} \]

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Rubi [A]  time = 0.12, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {27, 78, 51, 63, 205} \begin {gather*} \frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}+\frac {231 \sqrt {b} (13 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-77*(13*A*b - 3*a*B))/(128*a^6*b*x^(3/2)) + (231*(13*A*b - 3*a*B))/(128*a^7*Sqrt[x]) + (A*b - a*B)/(5*a*b*x^(
3/2)*(a + b*x)^5) + (13*A*b - 3*a*B)/(40*a^2*b*x^(3/2)*(a + b*x)^4) + (11*(13*A*b - 3*a*B))/(240*a^3*b*x^(3/2)
*(a + b*x)^3) + (33*(13*A*b - 3*a*B))/(320*a^4*b*x^(3/2)*(a + b*x)^2) + (231*(13*A*b - 3*a*B))/(640*a^5*b*x^(3
/2)*(a + b*x)) + (231*Sqrt[b]*(13*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(15/2))

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin {align*} \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {A+B x}{x^{5/2} (a+b x)^6} \, dx\\ &=\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}-\frac {\left (-\frac {13 A b}{2}+\frac {3 a B}{2}\right ) \int \frac {1}{x^{5/2} (a+b x)^5} \, dx}{5 a b}\\ &=\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {(11 (13 A b-3 a B)) \int \frac {1}{x^{5/2} (a+b x)^4} \, dx}{80 a^2 b}\\ &=\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {(33 (13 A b-3 a B)) \int \frac {1}{x^{5/2} (a+b x)^3} \, dx}{160 a^3 b}\\ &=\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {(231 (13 A b-3 a B)) \int \frac {1}{x^{5/2} (a+b x)^2} \, dx}{640 a^4 b}\\ &=\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {(231 (13 A b-3 a B)) \int \frac {1}{x^{5/2} (a+b x)} \, dx}{256 a^5 b}\\ &=-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}-\frac {(231 (13 A b-3 a B)) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{256 a^6}\\ &=-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {(231 b (13 A b-3 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 a^7}\\ &=-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {(231 b (13 A b-3 a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 a^7}\\ &=-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {231 \sqrt {b} (13 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 61, normalized size = 0.25 \begin {gather*} \frac {\frac {3 a^5 (A b-a B)}{(a+b x)^5}+(3 a B-13 A b) \, _2F_1\left (-\frac {3}{2},5;-\frac {1}{2};-\frac {b x}{a}\right )}{15 a^6 b x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((3*a^5*(A*b - a*B))/(a + b*x)^5 + (-13*A*b + 3*a*B)*Hypergeometric2F1[-3/2, 5, -1/2, -((b*x)/a)])/(15*a^6*b*x
^(3/2))

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IntegrateAlgebraic [A]  time = 0.43, size = 197, normalized size = 0.82 \begin {gather*} \frac {-1280 a^6 A-3840 a^6 B x+16640 a^5 A b x-31845 a^5 b B x^2+137995 a^4 A b^2 x^2-78210 a^4 b^2 B x^3+338910 a^3 A b^3 x^3-88704 a^3 b^3 B x^4+384384 a^2 A b^4 x^4-48510 a^2 b^4 B x^5+210210 a A b^5 x^5-10395 a b^5 B x^6+45045 A b^6 x^6}{1920 a^7 x^{3/2} (a+b x)^5}-\frac {231 \left (3 a \sqrt {b} B-13 A b^{3/2}\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(x^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-1280*a^6*A + 16640*a^5*A*b*x - 3840*a^6*B*x + 137995*a^4*A*b^2*x^2 - 31845*a^5*b*B*x^2 + 338910*a^3*A*b^3*x^
3 - 78210*a^4*b^2*B*x^3 + 384384*a^2*A*b^4*x^4 - 88704*a^3*b^3*B*x^4 + 210210*a*A*b^5*x^5 - 48510*a^2*b^4*B*x^
5 + 45045*A*b^6*x^6 - 10395*a*b^5*B*x^6)/(1920*a^7*x^(3/2)*(a + b*x)^5) - (231*(-13*A*b^(3/2) + 3*a*Sqrt[b]*B)
*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(15/2))

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fricas [A]  time = 0.46, size = 734, normalized size = 3.06 \begin {gather*} \left [-\frac {3465 \, {\left ({\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{7} + 5 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{6} + 10 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{5} + 10 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + 5 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{3} + {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{7} b^{5} x^{7} + 5 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{5} + 10 \, a^{10} b^{2} x^{4} + 5 \, a^{11} b x^{3} + a^{12} x^{2}\right )}}, \frac {3465 \, {\left ({\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{7} + 5 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{6} + 10 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{5} + 10 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + 5 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{3} + {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{7} b^{5} x^{7} + 5 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{5} + 10 \, a^{10} b^{2} x^{4} + 5 \, a^{11} b x^{3} + a^{12} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(3465*((3*B*a*b^5 - 13*A*b^6)*x^7 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*x^6 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4
)*x^5 + 10*(3*B*a^4*b^2 - 13*A*a^3*b^3)*x^4 + 5*(3*B*a^5*b - 13*A*a^4*b^2)*x^3 + (3*B*a^6 - 13*A*a^5*b)*x^2)*s
qrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(1280*A*a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6
+ 16170*(3*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 29568*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 26070*(3*B*a^4*b^2 - 13*A*a^
3*b^3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280*(3*B*a^6 - 13*A*a^5*b)*x)*sqrt(x))/(a^7*b^5*x^7 + 5*a
^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10*a^10*b^2*x^4 + 5*a^11*b*x^3 + a^12*x^2), 1/1920*(3465*((3*B*a*b^5 - 13*A*b^6)
*x^7 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*x^6 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^5 + 10*(3*B*a^4*b^2 - 13*A*a^3*b^3
)*x^4 + 5*(3*B*a^5*b - 13*A*a^4*b^2)*x^3 + (3*B*a^6 - 13*A*a^5*b)*x^2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x)
)) - (1280*A*a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6 + 16170*(3*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 29568*(3*B*a^3*b^3
- 13*A*a^2*b^4)*x^4 + 26070*(3*B*a^4*b^2 - 13*A*a^3*b^3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280*(3
*B*a^6 - 13*A*a^5*b)*x)*sqrt(x))/(a^7*b^5*x^7 + 5*a^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10*a^10*b^2*x^4 + 5*a^11*b*x^
3 + a^12*x^2)]

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giac [A]  time = 0.18, size = 180, normalized size = 0.75 \begin {gather*} -\frac {231 \, {\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{7}} - \frac {2 \, {\left (3 \, B a x - 18 \, A b x + A a\right )}}{3 \, a^{7} x^{\frac {3}{2}}} - \frac {6555 \, B a b^{5} x^{\frac {9}{2}} - 22005 \, A b^{6} x^{\frac {9}{2}} + 29310 \, B a^{2} b^{4} x^{\frac {7}{2}} - 96290 \, A a b^{5} x^{\frac {7}{2}} + 50304 \, B a^{3} b^{3} x^{\frac {5}{2}} - 160384 \, A a^{2} b^{4} x^{\frac {5}{2}} + 39810 \, B a^{4} b^{2} x^{\frac {3}{2}} - 121310 \, A a^{3} b^{3} x^{\frac {3}{2}} + 12645 \, B a^{5} b \sqrt {x} - 35595 \, A a^{4} b^{2} \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-231/128*(3*B*a*b - 13*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7) - 2/3*(3*B*a*x - 18*A*b*x + A*a)/(a^
7*x^(3/2)) - 1/1920*(6555*B*a*b^5*x^(9/2) - 22005*A*b^6*x^(9/2) + 29310*B*a^2*b^4*x^(7/2) - 96290*A*a*b^5*x^(7
/2) + 50304*B*a^3*b^3*x^(5/2) - 160384*A*a^2*b^4*x^(5/2) + 39810*B*a^4*b^2*x^(3/2) - 121310*A*a^3*b^3*x^(3/2)
+ 12645*B*a^5*b*sqrt(x) - 35595*A*a^4*b^2*sqrt(x))/((b*x + a)^5*a^7)

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maple [A]  time = 0.10, size = 266, normalized size = 1.11 \begin {gather*} \frac {1467 A \,b^{6} x^{\frac {9}{2}}}{128 \left (b x +a \right )^{5} a^{7}}-\frac {437 B \,b^{5} x^{\frac {9}{2}}}{128 \left (b x +a \right )^{5} a^{6}}+\frac {9629 A \,b^{5} x^{\frac {7}{2}}}{192 \left (b x +a \right )^{5} a^{6}}-\frac {977 B \,b^{4} x^{\frac {7}{2}}}{64 \left (b x +a \right )^{5} a^{5}}+\frac {1253 A \,b^{4} x^{\frac {5}{2}}}{15 \left (b x +a \right )^{5} a^{5}}-\frac {131 B \,b^{3} x^{\frac {5}{2}}}{5 \left (b x +a \right )^{5} a^{4}}+\frac {12131 A \,b^{3} x^{\frac {3}{2}}}{192 \left (b x +a \right )^{5} a^{4}}-\frac {1327 B \,b^{2} x^{\frac {3}{2}}}{64 \left (b x +a \right )^{5} a^{3}}+\frac {2373 A \,b^{2} \sqrt {x}}{128 \left (b x +a \right )^{5} a^{3}}-\frac {843 B b \sqrt {x}}{128 \left (b x +a \right )^{5} a^{2}}+\frac {3003 A \,b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, a^{7}}-\frac {693 B b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, a^{6}}+\frac {12 A b}{a^{7} \sqrt {x}}-\frac {2 B}{a^{6} \sqrt {x}}-\frac {2 A}{3 a^{6} x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1467/128/a^7*b^6/(b*x+a)^5*x^(9/2)*A-437/128/a^6*b^5/(b*x+a)^5*x^(9/2)*B+9629/192/a^6*b^5/(b*x+a)^5*A*x^(7/2)-
977/64/a^5*b^4/(b*x+a)^5*B*x^(7/2)+1253/15/a^5*b^4/(b*x+a)^5*x^(5/2)*A-131/5/a^4*b^3/(b*x+a)^5*x^(5/2)*B+12131
/192/a^4*b^3/(b*x+a)^5*x^(3/2)*A-1327/64/a^3*b^2/(b*x+a)^5*x^(3/2)*B+2373/128/a^3*b^2/(b*x+a)^5*x^(1/2)*A-843/
128/a^2*b/(b*x+a)^5*x^(1/2)*B+3003/128/a^7*b^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))*A-693/128/a^6*b/(a*
b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))*B-2/3*A/a^6/x^(3/2)+12/a^7/x^(1/2)*A*b-2/a^6/x^(1/2)*B

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maxima [A]  time = 1.44, size = 233, normalized size = 0.97 \begin {gather*} -\frac {1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x}{1920 \, {\left (a^{7} b^{5} x^{\frac {13}{2}} + 5 \, a^{8} b^{4} x^{\frac {11}{2}} + 10 \, a^{9} b^{3} x^{\frac {9}{2}} + 10 \, a^{10} b^{2} x^{\frac {7}{2}} + 5 \, a^{11} b x^{\frac {5}{2}} + a^{12} x^{\frac {3}{2}}\right )}} - \frac {231 \, {\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/1920*(1280*A*a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6 + 16170*(3*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 29568*(3*B*a^3*
b^3 - 13*A*a^2*b^4)*x^4 + 26070*(3*B*a^4*b^2 - 13*A*a^3*b^3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280
*(3*B*a^6 - 13*A*a^5*b)*x)/(a^7*b^5*x^(13/2) + 5*a^8*b^4*x^(11/2) + 10*a^9*b^3*x^(9/2) + 10*a^10*b^2*x^(7/2) +
5*a^11*b*x^(5/2) + a^12*x^(3/2)) - 231/128*(3*B*a*b - 13*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7)

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mupad [B]  time = 1.33, size = 207, normalized size = 0.86 \begin {gather*} \frac {\frac {2\,x\,\left (13\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {2\,A}{3\,a}+\frac {869\,b^2\,x^3\,\left (13\,A\,b-3\,B\,a\right )}{64\,a^4}+\frac {77\,b^3\,x^4\,\left (13\,A\,b-3\,B\,a\right )}{5\,a^5}+\frac {539\,b^4\,x^5\,\left (13\,A\,b-3\,B\,a\right )}{64\,a^6}+\frac {231\,b^5\,x^6\,\left (13\,A\,b-3\,B\,a\right )}{128\,a^7}+\frac {2123\,b\,x^2\,\left (13\,A\,b-3\,B\,a\right )}{384\,a^3}}{a^5\,x^{3/2}+b^5\,x^{13/2}+5\,a^4\,b\,x^{5/2}+5\,a\,b^4\,x^{11/2}+10\,a^3\,b^2\,x^{7/2}+10\,a^2\,b^3\,x^{9/2}}+\frac {231\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (13\,A\,b-3\,B\,a\right )}{128\,a^{15/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

((2*x*(13*A*b - 3*B*a))/(3*a^2) - (2*A)/(3*a) + (869*b^2*x^3*(13*A*b - 3*B*a))/(64*a^4) + (77*b^3*x^4*(13*A*b
- 3*B*a))/(5*a^5) + (539*b^4*x^5*(13*A*b - 3*B*a))/(64*a^6) + (231*b^5*x^6*(13*A*b - 3*B*a))/(128*a^7) + (2123
*b*x^2*(13*A*b - 3*B*a))/(384*a^3))/(a^5*x^(3/2) + b^5*x^(13/2) + 5*a^4*b*x^(5/2) + 5*a*b^4*x^(11/2) + 10*a^3*
b^2*x^(7/2) + 10*a^2*b^3*x^(9/2)) + (231*b^(1/2)*atan((b^(1/2)*x^(1/2))/a^(1/2))*(13*A*b - 3*B*a))/(128*a^(15/
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)/x**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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