\(\int \frac {A+B x}{x^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [781]

Optimal result

Integrand size = 29, antiderivative size = 240 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {231 \sqrt {b} (13 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}} \]

[Out]

-77/128*(13*A*b-3*B*a)/a^6/b/x^(3/2)+1/5*(A*b-B*a)/a/b/x^(3/2)/(b*x+a)^5+1/40*(13*A*b-3*B*a)/a^2/b/x^(3/2)/(b*
x+a)^4+11/240*(13*A*b-3*B*a)/a^3/b/x^(3/2)/(b*x+a)^3+33/320*(13*A*b-3*B*a)/a^4/b/x^(3/2)/(b*x+a)^2+231/640*(13
*A*b-3*B*a)/a^5/b/x^(3/2)/(b*x+a)+231/128*(13*A*b-3*B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))*b^(1/2)/a^(15/2)+231/
128*(13*A*b-3*B*a)/a^7/x^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {27, 79, 44, 53, 65, 211} \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {231 \sqrt {b} (13 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5} \]

[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 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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 65

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 79

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] || L
tQ[p, n]))))

Rule 211

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

Rubi steps \begin{align*} \text {integral}& = \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)) \text {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*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-\frac {\sqrt {a} \left (-45045 A b^6 x^6+1280 a^6 (A+3 B x)+1155 a b^5 x^5 (-182 A+9 B x)+462 a^2 b^4 x^4 (-832 A+105 B x)+66 a^3 b^3 x^3 (-5135 A+1344 B x)+55 a^4 b^2 x^2 (-2509 A+1422 B x)+5 a^5 b x (-3328 A+6369 B x)\right )}{x^{3/2} (a+b x)^5}+3465 \sqrt {b} (13 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{1920 a^{15/2}} \]

[In]

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

[Out]

(-((Sqrt[a]*(-45045*A*b^6*x^6 + 1280*a^6*(A + 3*B*x) + 1155*a*b^5*x^5*(-182*A + 9*B*x) + 462*a^2*b^4*x^4*(-832
*A + 105*B*x) + 66*a^3*b^3*x^3*(-5135*A + 1344*B*x) + 55*a^4*b^2*x^2*(-2509*A + 1422*B*x) + 5*a^5*b*x*(-3328*A
+ 6369*B*x)))/(x^(3/2)*(a + b*x)^5)) + 3465*Sqrt[b]*(13*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(1920
*a^(15/2))

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.69

method	result	size
risch	\(-\frac {2 \left (-18 A b x +3 a B x +a A \right )}{3 a^{7} x^{\frac {3}{2}}}+\frac {b \left (\frac {2 \left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {b^{3} a \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{192}+2 \left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+2 \left (\frac {12131}{384} A \,a^{3} b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+2 \left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 \sqrt {b a}}\right )}{a^{7}}\)	\(165\)
derivativedivides	\(\frac {2 b \left (\frac {\left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {b^{3} a \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+\left (\frac {12131}{384} A \,a^{3} b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+\left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{256 \sqrt {b a}}\right )}{a^{7}}-\frac {2 A}{3 a^{6} x^{\frac {3}{2}}}-\frac {2 \left (-6 A b +B a \right )}{a^{7} \sqrt {x}}\)	\(168\)
default	\(\frac {2 b \left (\frac {\left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {b^{3} a \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+\left (\frac {12131}{384} A \,a^{3} b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+\left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{256 \sqrt {b a}}\right )}{a^{7}}-\frac {2 A}{3 a^{6} x^{\frac {3}{2}}}-\frac {2 \left (-6 A b +B a \right )}{a^{7} \sqrt {x}}\)	\(168\)

[In]

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

[Out]

-2/3*(-18*A*b*x+3*B*a*x+A*a)/a^7/x^(3/2)+1/a^7*b*(2*((1467/256*A*b^5-437/256*B*a*b^4)*x^(9/2)+1/384*b^3*a*(962
9*A*b-2931*B*a)*x^(7/2)+(1253/30*A*a^2*b^3-131/10*B*a^3*b^2)*x^(5/2)+(12131/384*A*a^3*b^2-1327/128*B*a^4*b)*x^
(3/2)+(2373/256*A*a^4*b-843/256*a^5*B)*x^(1/2))/(b*x+a)^5+231/128*(13*A*b-3*B*a)/(b*a)^(1/2)*arctan(b*x^(1/2)/
(b*a)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 734, normalized size of antiderivative = 3.06 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\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 ] \]

[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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x}{1920 \, {\left (a^{7} b^{5} x^{\frac {13}{2}} + 5 \, a^{8} b^{4} x^{\frac {11}{2}} + 10 \, a^{9} b^{3} x^{\frac {9}{2}} + 10 \, a^{10} b^{2} x^{\frac {7}{2}} + 5 \, a^{11} b x^{\frac {5}{2}} + a^{12} x^{\frac {3}{2}}\right )}} - \frac {231 \, {\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{7}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.75 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {231 \, {\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{7}} - \frac {2 \, {\left (3 \, B a x - 18 \, A b x + A a\right )}}{3 \, a^{7} x^{\frac {3}{2}}} - \frac {6555 \, B a b^{5} x^{\frac {9}{2}} - 22005 \, A b^{6} x^{\frac {9}{2}} + 29310 \, B a^{2} b^{4} x^{\frac {7}{2}} - 96290 \, A a b^{5} x^{\frac {7}{2}} + 50304 \, B a^{3} b^{3} x^{\frac {5}{2}} - 160384 \, A a^{2} b^{4} x^{\frac {5}{2}} + 39810 \, B a^{4} b^{2} x^{\frac {3}{2}} - 121310 \, A a^{3} b^{3} x^{\frac {3}{2}} + 12645 \, B a^{5} b \sqrt {x} - 35595 \, A a^{4} b^{2} \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{7}} \]

[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)

Mupad [B] (verification not implemented)

Time = 10.24 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {2\,x\,\left (13\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {2\,A}{3\,a}+\frac {869\,b^2\,x^3\,\left (13\,A\,b-3\,B\,a\right )}{64\,a^4}+\frac {77\,b^3\,x^4\,\left (13\,A\,b-3\,B\,a\right )}{5\,a^5}+\frac {539\,b^4\,x^5\,\left (13\,A\,b-3\,B\,a\right )}{64\,a^6}+\frac {231\,b^5\,x^6\,\left (13\,A\,b-3\,B\,a\right )}{128\,a^7}+\frac {2123\,b\,x^2\,\left (13\,A\,b-3\,B\,a\right )}{384\,a^3}}{a^5\,x^{3/2}+b^5\,x^{13/2}+5\,a^4\,b\,x^{5/2}+5\,a\,b^4\,x^{11/2}+10\,a^3\,b^2\,x^{7/2}+10\,a^2\,b^3\,x^{9/2}}+\frac {231\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (13\,A\,b-3\,B\,a\right )}{128\,a^{15/2}} \]

[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))