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

Optimal. Leaf size=219 \[ -\frac {63 (11 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{13/2} \sqrt {b}}-\frac {63 (11 A b-a B)}{128 a^6 b \sqrt {x}}+\frac {21 (11 A b-a B)}{128 a^5 b \sqrt {x} (a+b x)}+\frac {21 (11 A b-a B)}{320 a^4 b \sqrt {x} (a+b x)^2}+\frac {3 (11 A b-a B)}{80 a^3 b \sqrt {x} (a+b x)^3}+\frac {11 A b-a B}{40 a^2 b \sqrt {x} (a+b x)^4}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {63 (11 A b-a B)}{128 a^6 b \sqrt {x}}+\frac {21 (11 A b-a B)}{128 a^5 b \sqrt {x} (a+b x)}+\frac {21 (11 A b-a B)}{320 a^4 b \sqrt {x} (a+b x)^2}+\frac {3 (11 A b-a B)}{80 a^3 b \sqrt {x} (a+b x)^3}+\frac {11 A b-a B}{40 a^2 b \sqrt {x} (a+b x)^4}-\frac {63 (11 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{13/2} \sqrt {b}}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-63*(11*A*b - a*B))/(128*a^6*b*Sqrt[x]) + (A*b - a*B)/(5*a*b*Sqrt[x]*(a + b*x)^5) + (11*A*b - a*B)/(40*a^2*b*
Sqrt[x]*(a + b*x)^4) + (3*(11*A*b - a*B))/(80*a^3*b*Sqrt[x]*(a + b*x)^3) + (21*(11*A*b - a*B))/(320*a^4*b*Sqrt
[x]*(a + b*x)^2) + (21*(11*A*b - a*B))/(128*a^5*b*Sqrt[x]*(a + b*x)) - (63*(11*A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt
[x])/Sqrt[a]])/(128*a^(13/2)*Sqrt[b])

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^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {A+B x}{x^{3/2} (a+b x)^6} \, dx\\ &=\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}+\frac {(11 A b-a B) \int \frac {1}{x^{3/2} (a+b x)^5} \, dx}{10 a b}\\ &=\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}+\frac {11 A b-a B}{40 a^2 b \sqrt {x} (a+b x)^4}+\frac {(9 (11 A b-a B)) \int \frac {1}{x^{3/2} (a+b x)^4} \, dx}{80 a^2 b}\\ &=\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}+\frac {11 A b-a B}{40 a^2 b \sqrt {x} (a+b x)^4}+\frac {3 (11 A b-a B)}{80 a^3 b \sqrt {x} (a+b x)^3}+\frac {(21 (11 A b-a B)) \int \frac {1}{x^{3/2} (a+b x)^3} \, dx}{160 a^3 b}\\ &=\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}+\frac {11 A b-a B}{40 a^2 b \sqrt {x} (a+b x)^4}+\frac {3 (11 A b-a B)}{80 a^3 b \sqrt {x} (a+b x)^3}+\frac {21 (11 A b-a B)}{320 a^4 b \sqrt {x} (a+b x)^2}+\frac {(21 (11 A b-a B)) \int \frac {1}{x^{3/2} (a+b x)^2} \, dx}{128 a^4 b}\\ &=\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}+\frac {11 A b-a B}{40 a^2 b \sqrt {x} (a+b x)^4}+\frac {3 (11 A b-a B)}{80 a^3 b \sqrt {x} (a+b x)^3}+\frac {21 (11 A b-a B)}{320 a^4 b \sqrt {x} (a+b x)^2}+\frac {21 (11 A b-a B)}{128 a^5 b \sqrt {x} (a+b x)}+\frac {(63 (11 A b-a B)) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{256 a^5 b}\\ &=-\frac {63 (11 A b-a B)}{128 a^6 b \sqrt {x}}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}+\frac {11 A b-a B}{40 a^2 b \sqrt {x} (a+b x)^4}+\frac {3 (11 A b-a B)}{80 a^3 b \sqrt {x} (a+b x)^3}+\frac {21 (11 A b-a B)}{320 a^4 b \sqrt {x} (a+b x)^2}+\frac {21 (11 A b-a B)}{128 a^5 b \sqrt {x} (a+b x)}-\frac {(63 (11 A b-a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 a^6}\\ &=-\frac {63 (11 A b-a B)}{128 a^6 b \sqrt {x}}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}+\frac {11 A b-a B}{40 a^2 b \sqrt {x} (a+b x)^4}+\frac {3 (11 A b-a B)}{80 a^3 b \sqrt {x} (a+b x)^3}+\frac {21 (11 A b-a B)}{320 a^4 b \sqrt {x} (a+b x)^2}+\frac {21 (11 A b-a B)}{128 a^5 b \sqrt {x} (a+b x)}-\frac {(63 (11 A b-a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 a^6}\\ &=-\frac {63 (11 A b-a B)}{128 a^6 b \sqrt {x}}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}+\frac {11 A b-a B}{40 a^2 b \sqrt {x} (a+b x)^4}+\frac {3 (11 A b-a B)}{80 a^3 b \sqrt {x} (a+b x)^3}+\frac {21 (11 A b-a B)}{320 a^4 b \sqrt {x} (a+b x)^2}+\frac {21 (11 A b-a B)}{128 a^5 b \sqrt {x} (a+b x)}-\frac {63 (11 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{13/2} \sqrt {b}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 59, normalized size = 0.27 \begin {gather*} \frac {\frac {a^5 (A b-a B)}{(a+b x)^5}+(a B-11 A b) \, _2F_1\left (-\frac {1}{2},5;\frac {1}{2};-\frac {b x}{a}\right )}{5 a^6 b \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a^5*(A*b - a*B))/(a + b*x)^5 + (-11*A*b + a*B)*Hypergeometric2F1[-1/2, 5, 1/2, -((b*x)/a)])/(5*a^6*b*Sqrt[x]
)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.39, size = 168, normalized size = 0.77 \begin {gather*} \frac {63 (a B-11 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{13/2} \sqrt {b}}+\frac {-1280 a^5 A+965 a^5 B x-10615 a^4 A b x+2370 a^4 b B x^2-26070 a^3 A b^2 x^2+2688 a^3 b^2 B x^3-29568 a^2 A b^3 x^3+1470 a^2 b^3 B x^4-16170 a A b^4 x^4+315 a b^4 B x^5-3465 A b^5 x^5}{640 a^6 \sqrt {x} (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1280*a^5*A - 10615*a^4*A*b*x + 965*a^5*B*x - 26070*a^3*A*b^2*x^2 + 2370*a^4*b*B*x^2 - 29568*a^2*A*b^3*x^3 +
2688*a^3*b^2*B*x^3 - 16170*a*A*b^4*x^4 + 1470*a^2*b^3*B*x^4 - 3465*A*b^5*x^5 + 315*a*b^4*B*x^5)/(640*a^6*Sqrt[
x]*(a + b*x)^5) + (63*(-11*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(13/2)*Sqrt[b])

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 673, normalized size = 3.07 \begin {gather*} \left [\frac {315 \, {\left ({\left (B a b^{5} - 11 \, A b^{6}\right )} x^{6} + 5 \, {\left (B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + 10 \, {\left (B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4} + 10 \, {\left (B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{3} + 5 \, {\left (B a^{5} b - 11 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} - 11 \, A a^{5} b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (1280 \, A a^{6} b - 315 \, {\left (B a^{2} b^{5} - 11 \, A a b^{6}\right )} x^{5} - 1470 \, {\left (B a^{3} b^{4} - 11 \, A a^{2} b^{5}\right )} x^{4} - 2688 \, {\left (B a^{4} b^{3} - 11 \, A a^{3} b^{4}\right )} x^{3} - 2370 \, {\left (B a^{5} b^{2} - 11 \, A a^{4} b^{3}\right )} x^{2} - 965 \, {\left (B a^{6} b - 11 \, A a^{5} b^{2}\right )} x\right )} \sqrt {x}}{1280 \, {\left (a^{7} b^{6} x^{6} + 5 \, a^{8} b^{5} x^{5} + 10 \, a^{9} b^{4} x^{4} + 10 \, a^{10} b^{3} x^{3} + 5 \, a^{11} b^{2} x^{2} + a^{12} b x\right )}}, -\frac {315 \, {\left ({\left (B a b^{5} - 11 \, A b^{6}\right )} x^{6} + 5 \, {\left (B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + 10 \, {\left (B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4} + 10 \, {\left (B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{3} + 5 \, {\left (B a^{5} b - 11 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} - 11 \, A a^{5} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (1280 \, A a^{6} b - 315 \, {\left (B a^{2} b^{5} - 11 \, A a b^{6}\right )} x^{5} - 1470 \, {\left (B a^{3} b^{4} - 11 \, A a^{2} b^{5}\right )} x^{4} - 2688 \, {\left (B a^{4} b^{3} - 11 \, A a^{3} b^{4}\right )} x^{3} - 2370 \, {\left (B a^{5} b^{2} - 11 \, A a^{4} b^{3}\right )} x^{2} - 965 \, {\left (B a^{6} b - 11 \, A a^{5} b^{2}\right )} x\right )} \sqrt {x}}{640 \, {\left (a^{7} b^{6} x^{6} + 5 \, a^{8} b^{5} x^{5} + 10 \, a^{9} b^{4} x^{4} + 10 \, a^{10} b^{3} x^{3} + 5 \, a^{11} b^{2} x^{2} + a^{12} b x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/1280*(315*((B*a*b^5 - 11*A*b^6)*x^6 + 5*(B*a^2*b^4 - 11*A*a*b^5)*x^5 + 10*(B*a^3*b^3 - 11*A*a^2*b^4)*x^4 +
10*(B*a^4*b^2 - 11*A*a^3*b^3)*x^3 + 5*(B*a^5*b - 11*A*a^4*b^2)*x^2 + (B*a^6 - 11*A*a^5*b)*x)*sqrt(-a*b)*log((b
*x - a + 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) - 2*(1280*A*a^6*b - 315*(B*a^2*b^5 - 11*A*a*b^6)*x^5 - 1470*(B*a^3*b
^4 - 11*A*a^2*b^5)*x^4 - 2688*(B*a^4*b^3 - 11*A*a^3*b^4)*x^3 - 2370*(B*a^5*b^2 - 11*A*a^4*b^3)*x^2 - 965*(B*a^
6*b - 11*A*a^5*b^2)*x)*sqrt(x))/(a^7*b^6*x^6 + 5*a^8*b^5*x^5 + 10*a^9*b^4*x^4 + 10*a^10*b^3*x^3 + 5*a^11*b^2*x
^2 + a^12*b*x), -1/640*(315*((B*a*b^5 - 11*A*b^6)*x^6 + 5*(B*a^2*b^4 - 11*A*a*b^5)*x^5 + 10*(B*a^3*b^3 - 11*A*
a^2*b^4)*x^4 + 10*(B*a^4*b^2 - 11*A*a^3*b^3)*x^3 + 5*(B*a^5*b - 11*A*a^4*b^2)*x^2 + (B*a^6 - 11*A*a^5*b)*x)*sq
rt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (1280*A*a^6*b - 315*(B*a^2*b^5 - 11*A*a*b^6)*x^5 - 1470*(B*a^3*b^4 - 1
1*A*a^2*b^5)*x^4 - 2688*(B*a^4*b^3 - 11*A*a^3*b^4)*x^3 - 2370*(B*a^5*b^2 - 11*A*a^4*b^3)*x^2 - 965*(B*a^6*b -
11*A*a^5*b^2)*x)*sqrt(x))/(a^7*b^6*x^6 + 5*a^8*b^5*x^5 + 10*a^9*b^4*x^4 + 10*a^10*b^3*x^3 + 5*a^11*b^2*x^2 + a
^12*b*x)]

________________________________________________________________________________________

giac [A]  time = 0.21, size = 158, normalized size = 0.72 \begin {gather*} \frac {63 \, {\left (B a - 11 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{6}} - \frac {2 \, A}{a^{6} \sqrt {x}} + \frac {315 \, B a b^{4} x^{\frac {9}{2}} - 2185 \, A b^{5} x^{\frac {9}{2}} + 1470 \, B a^{2} b^{3} x^{\frac {7}{2}} - 9770 \, A a b^{4} x^{\frac {7}{2}} + 2688 \, B a^{3} b^{2} x^{\frac {5}{2}} - 16768 \, A a^{2} b^{3} x^{\frac {5}{2}} + 2370 \, B a^{4} b x^{\frac {3}{2}} - 13270 \, A a^{3} b^{2} x^{\frac {3}{2}} + 965 \, B a^{5} \sqrt {x} - 4215 \, A a^{4} b \sqrt {x}}{640 \, {\left (b x + a\right )}^{5} a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

63/128*(B*a - 11*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^6) - 2*A/(a^6*sqrt(x)) + 1/640*(315*B*a*b^4*x^(
9/2) - 2185*A*b^5*x^(9/2) + 1470*B*a^2*b^3*x^(7/2) - 9770*A*a*b^4*x^(7/2) + 2688*B*a^3*b^2*x^(5/2) - 16768*A*a
^2*b^3*x^(5/2) + 2370*B*a^4*b*x^(3/2) - 13270*A*a^3*b^2*x^(3/2) + 965*B*a^5*sqrt(x) - 4215*A*a^4*b*sqrt(x))/((
b*x + a)^5*a^6)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 239, normalized size = 1.09 \begin {gather*} -\frac {437 A \,b^{5} x^{\frac {9}{2}}}{128 \left (b x +a \right )^{5} a^{6}}+\frac {63 B \,b^{4} x^{\frac {9}{2}}}{128 \left (b x +a \right )^{5} a^{5}}-\frac {977 A \,b^{4} x^{\frac {7}{2}}}{64 \left (b x +a \right )^{5} a^{5}}+\frac {147 B \,b^{3} x^{\frac {7}{2}}}{64 \left (b x +a \right )^{5} a^{4}}-\frac {131 A \,b^{3} x^{\frac {5}{2}}}{5 \left (b x +a \right )^{5} a^{4}}+\frac {21 B \,b^{2} x^{\frac {5}{2}}}{5 \left (b x +a \right )^{5} a^{3}}-\frac {1327 A \,b^{2} x^{\frac {3}{2}}}{64 \left (b x +a \right )^{5} a^{3}}+\frac {237 B b \,x^{\frac {3}{2}}}{64 \left (b x +a \right )^{5} a^{2}}-\frac {843 A b \sqrt {x}}{128 \left (b x +a \right )^{5} a^{2}}+\frac {193 B \sqrt {x}}{128 \left (b x +a \right )^{5} a}-\frac {693 A b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, a^{6}}+\frac {63 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, a^{5}}-\frac {2 A}{a^{6} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-437/128/a^6/(b*x+a)^5*x^(9/2)*A*b^5+63/128/a^5/(b*x+a)^5*x^(9/2)*B*b^4-977/64/a^5/(b*x+a)^5*A*x^(7/2)*b^4+147
/64/a^4/(b*x+a)^5*B*x^(7/2)*b^3-131/5/a^4/(b*x+a)^5*x^(5/2)*A*b^3+21/5/a^3/(b*x+a)^5*x^(5/2)*b^2*B-1327/64/a^3
/(b*x+a)^5*x^(3/2)*A*b^2+237/64/a^2/(b*x+a)^5*x^(3/2)*b*B-843/128/a^2/(b*x+a)^5*x^(1/2)*A*b+193/128/a/(b*x+a)^
5*x^(1/2)*B-693/128/a^6/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))*A*b+63/128/a^5/(a*b)^(1/2)*arctan(1/(a*b)^
(1/2)*b*x^(1/2))*B-2*A/a^6/x^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.25, size = 200, normalized size = 0.91 \begin {gather*} -\frac {1280 \, A a^{5} - 315 \, {\left (B a b^{4} - 11 \, A b^{5}\right )} x^{5} - 1470 \, {\left (B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} - 2688 \, {\left (B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} - 2370 \, {\left (B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} - 965 \, {\left (B a^{5} - 11 \, A a^{4} b\right )} x}{640 \, {\left (a^{6} b^{5} x^{\frac {11}{2}} + 5 \, a^{7} b^{4} x^{\frac {9}{2}} + 10 \, a^{8} b^{3} x^{\frac {7}{2}} + 10 \, a^{9} b^{2} x^{\frac {5}{2}} + 5 \, a^{10} b x^{\frac {3}{2}} + a^{11} \sqrt {x}\right )}} + \frac {63 \, {\left (B a - 11 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/640*(1280*A*a^5 - 315*(B*a*b^4 - 11*A*b^5)*x^5 - 1470*(B*a^2*b^3 - 11*A*a*b^4)*x^4 - 2688*(B*a^3*b^2 - 11*A
*a^2*b^3)*x^3 - 2370*(B*a^4*b - 11*A*a^3*b^2)*x^2 - 965*(B*a^5 - 11*A*a^4*b)*x)/(a^6*b^5*x^(11/2) + 5*a^7*b^4*
x^(9/2) + 10*a^8*b^3*x^(7/2) + 10*a^9*b^2*x^(5/2) + 5*a^10*b*x^(3/2) + a^11*sqrt(x)) + 63/128*(B*a - 11*A*b)*a
rctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^6)

________________________________________________________________________________________

mupad [B]  time = 1.36, size = 209, normalized size = 0.95 \begin {gather*} -\frac {\frac {2\,A}{a}+\frac {193\,x\,\left (11\,A\,b-B\,a\right )}{128\,a^2}+\frac {21\,b^2\,x^3\,\left (11\,A\,b-B\,a\right )}{5\,a^4}+\frac {147\,b^3\,x^4\,\left (11\,A\,b-B\,a\right )}{64\,a^5}+\frac {63\,b^4\,x^5\,\left (11\,A\,b-B\,a\right )}{128\,a^6}+\frac {237\,b\,x^2\,\left (11\,A\,b-B\,a\right )}{64\,a^3}}{a^5\,\sqrt {x}+b^5\,x^{11/2}+5\,a^4\,b\,x^{3/2}+5\,a\,b^4\,x^{9/2}+10\,a^3\,b^2\,x^{5/2}+10\,a^2\,b^3\,x^{7/2}}-\frac {63\,\mathrm {atan}\left (\frac {63\,\sqrt {b}\,\sqrt {x}\,\left (11\,A\,b-B\,a\right )}{\sqrt {a}\,\left (693\,A\,b-63\,B\,a\right )}\right )\,\left (11\,A\,b-B\,a\right )}{128\,a^{13/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*A)/a + (193*x*(11*A*b - B*a))/(128*a^2) + (21*b^2*x^3*(11*A*b - B*a))/(5*a^4) + (147*b^3*x^4*(11*A*b - B
*a))/(64*a^5) + (63*b^4*x^5*(11*A*b - B*a))/(128*a^6) + (237*b*x^2*(11*A*b - B*a))/(64*a^3))/(a^5*x^(1/2) + b^
5*x^(11/2) + 5*a^4*b*x^(3/2) + 5*a*b^4*x^(9/2) + 10*a^3*b^2*x^(5/2) + 10*a^2*b^3*x^(7/2)) - (63*atan((63*b^(1/
2)*x^(1/2)*(11*A*b - B*a))/(a^(1/2)*(693*A*b - 63*B*a)))*(11*A*b - B*a))/(128*a^(13/2)*b^(1/2))

________________________________________________________________________________________

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)/x**(3/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________