\(\int \frac {x^{11/2} (A+B x)}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [773]

Optimal result

Integrand size = 29, antiderivative size = 240 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {231 (3 A b-13 a B) \sqrt {x}}{128 b^7}-\frac {77 (3 A b-13 a B) x^{3/2}}{128 a b^6}+\frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}-\frac {231 \sqrt {a} (3 A b-13 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 b^{15/2}} \]

[Out]

-77/128*(3*A*b-13*B*a)*x^(3/2)/a/b^6+1/5*(A*b-B*a)*x^(13/2)/a/b/(b*x+a)^5+1/40*(3*A*b-13*B*a)*x^(11/2)/a/b^2/(
b*x+a)^4+11/240*(3*A*b-13*B*a)*x^(9/2)/a/b^3/(b*x+a)^3+33/320*(3*A*b-13*B*a)*x^(7/2)/a/b^4/(b*x+a)^2+231/640*(
3*A*b-13*B*a)*x^(5/2)/a/b^5/(b*x+a)-231/128*(3*A*b-13*B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))*a^(1/2)/b^(15/2)+23
1/128*(3*A*b-13*B*a)*x^(1/2)/b^7

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, 43, 52, 65, 211} \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {231 \sqrt {a} (3 A b-13 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 b^{15/2}}+\frac {231 \sqrt {x} (3 A b-13 a B)}{128 b^7}-\frac {77 x^{3/2} (3 A b-13 a B)}{128 a b^6}+\frac {231 x^{5/2} (3 A b-13 a B)}{640 a b^5 (a+b x)}+\frac {33 x^{7/2} (3 A b-13 a B)}{320 a b^4 (a+b x)^2}+\frac {11 x^{9/2} (3 A b-13 a B)}{240 a b^3 (a+b x)^3}+\frac {x^{11/2} (3 A b-13 a B)}{40 a b^2 (a+b x)^4}+\frac {x^{13/2} (A b-a B)}{5 a b (a+b x)^5} \]

[In]

Int[(x^(11/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(231*(3*A*b - 13*a*B)*Sqrt[x])/(128*b^7) - (77*(3*A*b - 13*a*B)*x^(3/2))/(128*a*b^6) + ((A*b - a*B)*x^(13/2))/
(5*a*b*(a + b*x)^5) + ((3*A*b - 13*a*B)*x^(11/2))/(40*a*b^2*(a + b*x)^4) + (11*(3*A*b - 13*a*B)*x^(9/2))/(240*
a*b^3*(a + b*x)^3) + (33*(3*A*b - 13*a*B)*x^(7/2))/(320*a*b^4*(a + b*x)^2) + (231*(3*A*b - 13*a*B)*x^(5/2))/(6
40*a*b^5*(a + b*x)) - (231*Sqrt[a]*(3*A*b - 13*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*b^(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 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 {x^{11/2} (A+B x)}{(a+b x)^6} \, dx \\ & = \frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}-\frac {\left (\frac {3 A b}{2}-\frac {13 a B}{2}\right ) \int \frac {x^{11/2}}{(a+b x)^5} \, dx}{5 a b} \\ & = \frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}-\frac {(11 (3 A b-13 a B)) \int \frac {x^{9/2}}{(a+b x)^4} \, dx}{80 a b^2} \\ & = \frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}-\frac {(33 (3 A b-13 a B)) \int \frac {x^{7/2}}{(a+b x)^3} \, dx}{160 a b^3} \\ & = \frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}-\frac {(231 (3 A b-13 a B)) \int \frac {x^{5/2}}{(a+b x)^2} \, dx}{640 a b^4} \\ & = \frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}-\frac {(231 (3 A b-13 a B)) \int \frac {x^{3/2}}{a+b x} \, dx}{256 a b^5} \\ & = -\frac {77 (3 A b-13 a B) x^{3/2}}{128 a b^6}+\frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}+\frac {(231 (3 A b-13 a B)) \int \frac {\sqrt {x}}{a+b x} \, dx}{256 b^6} \\ & = \frac {231 (3 A b-13 a B) \sqrt {x}}{128 b^7}-\frac {77 (3 A b-13 a B) x^{3/2}}{128 a b^6}+\frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}-\frac {(231 a (3 A b-13 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 b^7} \\ & = \frac {231 (3 A b-13 a B) \sqrt {x}}{128 b^7}-\frac {77 (3 A b-13 a B) x^{3/2}}{128 a b^6}+\frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}-\frac {(231 a (3 A b-13 a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 b^7} \\ & = \frac {231 (3 A b-13 a B) \sqrt {x}}{128 b^7}-\frac {77 (3 A b-13 a B) x^{3/2}}{128 a b^6}+\frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}-\frac {231 \sqrt {a} (3 A b-13 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 b^{15/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.70 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {\sqrt {b} \sqrt {x} \left (-45045 a^6 B+66 a^3 b^3 x^2 (1344 A-5135 B x)+5 a b^5 x^4 (6369 A-3328 B x)+55 a^2 b^4 x^3 (1422 A-2509 B x)+462 a^4 b^2 x (105 A-832 B x)+1155 a^5 b (9 A-182 B x)+1280 b^6 x^5 (3 A+B x)\right )}{(a+b x)^5}+3465 \sqrt {a} (-3 A b+13 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{1920 b^{15/2}} \]

[In]

Integrate[(x^(11/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

Maple [A] (verified)

Time = 2.14 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.69

method	result	size
risch	\(\frac {2 \left (B b x +3 A b -18 B a \right ) \sqrt {x}}{3 b^{7}}-\frac {a \left (\frac {2 \left (-\frac {843}{256} A \,b^{5}+\frac {2373}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+2 \left (-\frac {1327}{128} A a \,b^{4}+\frac {12131}{384} B \,a^{2} b^{3}\right ) x^{\frac {7}{2}}+2 \left (-\frac {131}{10} A \,a^{2} b^{3}+\frac {1253}{30} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}-\frac {a^{3} b \left (2931 A b -9629 B a \right ) x^{\frac {3}{2}}}{192}+2 \left (-\frac {437}{256} A \,a^{4} b +\frac {1467}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (3 A b -13 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 \sqrt {b a}}\right )}{b^{7}}\)	\(165\)
derivativedivides	\(\frac {\frac {2 B b \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}-12 a B \sqrt {x}}{b^{7}}-\frac {2 a \left (\frac {\left (-\frac {843}{256} A \,b^{5}+\frac {2373}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\left (-\frac {1327}{128} A a \,b^{4}+\frac {12131}{384} B \,a^{2} b^{3}\right ) x^{\frac {7}{2}}+\left (-\frac {131}{10} A \,a^{2} b^{3}+\frac {1253}{30} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}-\frac {a^{3} b \left (2931 A b -9629 B a \right ) x^{\frac {3}{2}}}{384}+\left (-\frac {437}{256} A \,a^{4} b +\frac {1467}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (3 A b -13 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{256 \sqrt {b a}}\right )}{b^{7}}\)	\(169\)
default	\(\frac {\frac {2 B b \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}-12 a B \sqrt {x}}{b^{7}}-\frac {2 a \left (\frac {\left (-\frac {843}{256} A \,b^{5}+\frac {2373}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\left (-\frac {1327}{128} A a \,b^{4}+\frac {12131}{384} B \,a^{2} b^{3}\right ) x^{\frac {7}{2}}+\left (-\frac {131}{10} A \,a^{2} b^{3}+\frac {1253}{30} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}-\frac {a^{3} b \left (2931 A b -9629 B a \right ) x^{\frac {3}{2}}}{384}+\left (-\frac {437}{256} A \,a^{4} b +\frac {1467}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (3 A b -13 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{256 \sqrt {b a}}\right )}{b^{7}}\)	\(169\)

[In]

int(x^(11/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2/3*(B*b*x+3*A*b-18*B*a)*x^(1/2)/b^7-a/b^7*(2*((-843/256*A*b^5+2373/256*B*a*b^4)*x^(9/2)+(-1327/128*A*a*b^4+12
131/384*B*a^2*b^3)*x^(7/2)+(-131/10*A*a^2*b^3+1253/30*B*a^3*b^2)*x^(5/2)-1/384*a^3*b*(2931*A*b-9629*B*a)*x^(3/
2)+(-437/256*A*a^4*b+1467/256*a^5*B)*x^(1/2))/(b*x+a)^5+231/128*(3*A*b-13*B*a)/(b*a)^(1/2)*arctan(b*x^(1/2)/(b
*a)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 703, normalized size of antiderivative = 2.93 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {3465 \, {\left (13 \, B a^{6} - 3 \, A a^{5} b + {\left (13 \, B a b^{5} - 3 \, A b^{6}\right )} x^{5} + 5 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x^{4} + 10 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (1280 \, B b^{6} x^{6} - 45045 \, B a^{6} + 10395 \, A a^{5} b - 1280 \, {\left (13 \, B a b^{5} - 3 \, A b^{6}\right )} x^{5} - 10615 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x^{4} - 26070 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} - 29568 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} - 16170 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{3840 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}}, \frac {3465 \, {\left (13 \, B a^{6} - 3 \, A a^{5} b + {\left (13 \, B a b^{5} - 3 \, A b^{6}\right )} x^{5} + 5 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x^{4} + 10 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (1280 \, B b^{6} x^{6} - 45045 \, B a^{6} + 10395 \, A a^{5} b - 1280 \, {\left (13 \, B a b^{5} - 3 \, A b^{6}\right )} x^{5} - 10615 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x^{4} - 26070 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} - 29568 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} - 16170 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{1920 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}}\right ] \]

[In]

integrate(x^(11/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(3465*(13*B*a^6 - 3*A*a^5*b + (13*B*a*b^5 - 3*A*b^6)*x^5 + 5*(13*B*a^2*b^4 - 3*A*a*b^5)*x^4 + 10*(13*
B*a^3*b^3 - 3*A*a^2*b^4)*x^3 + 10*(13*B*a^4*b^2 - 3*A*a^3*b^3)*x^2 + 5*(13*B*a^5*b - 3*A*a^4*b^2)*x)*sqrt(-a/b
)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(1280*B*b^6*x^6 - 45045*B*a^6 + 10395*A*a^5*b - 1280*(
13*B*a*b^5 - 3*A*b^6)*x^5 - 10615*(13*B*a^2*b^4 - 3*A*a*b^5)*x^4 - 26070*(13*B*a^3*b^3 - 3*A*a^2*b^4)*x^3 - 29
568*(13*B*a^4*b^2 - 3*A*a^3*b^3)*x^2 - 16170*(13*B*a^5*b - 3*A*a^4*b^2)*x)*sqrt(x))/(b^12*x^5 + 5*a*b^11*x^4 +
10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7), 1/1920*(3465*(13*B*a^6 - 3*A*a^5*b + (13*B*a*b^5 -
3*A*b^6)*x^5 + 5*(13*B*a^2*b^4 - 3*A*a*b^5)*x^4 + 10*(13*B*a^3*b^3 - 3*A*a^2*b^4)*x^3 + 10*(13*B*a^4*b^2 - 3*
A*a^3*b^3)*x^2 + 5*(13*B*a^5*b - 3*A*a^4*b^2)*x)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) + (1280*B*b^6*x^6 - 4
5045*B*a^6 + 10395*A*a^5*b - 1280*(13*B*a*b^5 - 3*A*b^6)*x^5 - 10615*(13*B*a^2*b^4 - 3*A*a*b^5)*x^4 - 26070*(1
3*B*a^3*b^3 - 3*A*a^2*b^4)*x^3 - 29568*(13*B*a^4*b^2 - 3*A*a^3*b^3)*x^2 - 16170*(13*B*a^5*b - 3*A*a^4*b^2)*x)*
sqrt(x))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7)]

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(x**(11/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.96 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {45 \, {\left (791 \, B a^{2} b^{4} - 281 \, A a b^{5}\right )} x^{\frac {9}{2}} + 10 \, {\left (12131 \, B a^{3} b^{3} - 3981 \, A a^{2} b^{4}\right )} x^{\frac {7}{2}} + 128 \, {\left (1253 \, B a^{4} b^{2} - 393 \, A a^{3} b^{3}\right )} x^{\frac {5}{2}} + 10 \, {\left (9629 \, B a^{5} b - 2931 \, A a^{4} b^{2}\right )} x^{\frac {3}{2}} + 15 \, {\left (1467 \, B a^{6} - 437 \, A a^{5} b\right )} \sqrt {x}}{1920 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} + \frac {231 \, {\left (13 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} b^{7}} + \frac {2 \, {\left (B b x^{\frac {3}{2}} - 3 \, {\left (6 \, B a - A b\right )} \sqrt {x}\right )}}{3 \, b^{7}} \]

[In]

integrate(x^(11/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/1920*(45*(791*B*a^2*b^4 - 281*A*a*b^5)*x^(9/2) + 10*(12131*B*a^3*b^3 - 3981*A*a^2*b^4)*x^(7/2) + 128*(1253*
B*a^4*b^2 - 393*A*a^3*b^3)*x^(5/2) + 10*(9629*B*a^5*b - 2931*A*a^4*b^2)*x^(3/2) + 15*(1467*B*a^6 - 437*A*a^5*b
)*sqrt(x))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7) + 231/128*(13*
B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^7) + 2/3*(B*b*x^(3/2) - 3*(6*B*a - A*b)*sqrt(x))/b^7

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.80 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {231 \, {\left (13 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} b^{7}} - \frac {35595 \, B a^{2} b^{4} x^{\frac {9}{2}} - 12645 \, A a b^{5} x^{\frac {9}{2}} + 121310 \, B a^{3} b^{3} x^{\frac {7}{2}} - 39810 \, A a^{2} b^{4} x^{\frac {7}{2}} + 160384 \, B a^{4} b^{2} x^{\frac {5}{2}} - 50304 \, A a^{3} b^{3} x^{\frac {5}{2}} + 96290 \, B a^{5} b x^{\frac {3}{2}} - 29310 \, A a^{4} b^{2} x^{\frac {3}{2}} + 22005 \, B a^{6} \sqrt {x} - 6555 \, A a^{5} b \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} b^{7}} + \frac {2 \, {\left (B b^{12} x^{\frac {3}{2}} - 18 \, B a b^{11} \sqrt {x} + 3 \, A b^{12} \sqrt {x}\right )}}{3 \, b^{18}} \]

[In]

integrate(x^(11/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

231/128*(13*B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^7) - 1/1920*(35595*B*a^2*b^4*x^(9/2) - 1
2645*A*a*b^5*x^(9/2) + 121310*B*a^3*b^3*x^(7/2) - 39810*A*a^2*b^4*x^(7/2) + 160384*B*a^4*b^2*x^(5/2) - 50304*A
*a^3*b^3*x^(5/2) + 96290*B*a^5*b*x^(3/2) - 29310*A*a^4*b^2*x^(3/2) + 22005*B*a^6*sqrt(x) - 6555*A*a^5*b*sqrt(x
))/((b*x + a)^5*b^7) + 2/3*(B*b^12*x^(3/2) - 18*B*a*b^11*sqrt(x) + 3*A*b^12*sqrt(x))/b^18

Mupad [B] (verification not implemented)

Time = 10.16 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\sqrt {x}\,\left (\frac {2\,A}{b^6}-\frac {12\,B\,a}{b^7}\right )+\frac {x^{3/2}\,\left (\frac {977\,A\,a^4\,b^2}{64}-\frac {9629\,B\,a^5\,b}{192}\right )-x^{9/2}\,\left (\frac {2373\,B\,a^2\,b^4}{128}-\frac {843\,A\,a\,b^5}{128}\right )-\sqrt {x}\,\left (\frac {1467\,B\,a^6}{128}-\frac {437\,A\,a^5\,b}{128}\right )+x^{5/2}\,\left (\frac {131\,A\,a^3\,b^3}{5}-\frac {1253\,B\,a^4\,b^2}{15}\right )+x^{7/2}\,\left (\frac {1327\,A\,a^2\,b^4}{64}-\frac {12131\,B\,a^3\,b^3}{192}\right )}{a^5\,b^7+5\,a^4\,b^8\,x+10\,a^3\,b^9\,x^2+10\,a^2\,b^{10}\,x^3+5\,a\,b^{11}\,x^4+b^{12}\,x^5}+\frac {2\,B\,x^{3/2}}{3\,b^6}+\frac {231\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\sqrt {x}\,\left (3\,A\,b-13\,B\,a\right )}{13\,B\,a^2-3\,A\,a\,b}\right )\,\left (3\,A\,b-13\,B\,a\right )}{128\,b^{15/2}} \]

[In]

int((x^(11/2)*(A + B*x))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x^(1/2)*((2*A)/b^6 - (12*B*a)/b^7) + (x^(3/2)*((977*A*a^4*b^2)/64 - (9629*B*a^5*b)/192) - x^(9/2)*((2373*B*a^2
*b^4)/128 - (843*A*a*b^5)/128) - x^(1/2)*((1467*B*a^6)/128 - (437*A*a^5*b)/128) + x^(5/2)*((131*A*a^3*b^3)/5 -
(1253*B*a^4*b^2)/15) + x^(7/2)*((1327*A*a^2*b^4)/64 - (12131*B*a^3*b^3)/192))/(a^5*b^7 + b^12*x^5 + 5*a^4*b^8
*x + 5*a*b^11*x^4 + 10*a^3*b^9*x^2 + 10*a^2*b^10*x^3) + (2*B*x^(3/2))/(3*b^6) + (231*a^(1/2)*atan((a^(1/2)*b^(
1/2)*x^(1/2)*(3*A*b - 13*B*a))/(13*B*a^2 - 3*A*a*b))*(3*A*b - 13*B*a))/(128*b^(15/2))