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

Optimal. Leaf size=240 \[ -\frac {231 \sqrt {a} (3 A b-13 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 b^{15/2}}+\frac {231 \sqrt {x} (3 A b-13 a B)}{128 b^7}-\frac {77 x^{3/2} (3 A b-13 a B)}{128 a b^6}+\frac {231 x^{5/2} (3 A b-13 a B)}{640 a b^5 (a+b x)}+\frac {33 x^{7/2} (3 A b-13 a B)}{320 a b^4 (a+b x)^2}+\frac {11 x^{9/2} (3 A b-13 a B)}{240 a b^3 (a+b x)^3}+\frac {x^{11/2} (3 A b-13 a B)}{40 a b^2 (a+b x)^4}+\frac {x^{13/2} (A b-a B)}{5 a b (a+b x)^5} \]

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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 = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {27, 78, 47, 50, 63, 205} \begin {gather*} \frac {x^{11/2} (3 A b-13 a B)}{40 a b^2 (a+b x)^4}+\frac {11 x^{9/2} (3 A b-13 a B)}{240 a b^3 (a+b x)^3}+\frac {33 x^{7/2} (3 A b-13 a B)}{320 a b^4 (a+b x)^2}+\frac {231 x^{5/2} (3 A b-13 a B)}{640 a b^5 (a+b x)}-\frac {77 x^{3/2} (3 A b-13 a B)}{128 a b^6}+\frac {231 \sqrt {x} (3 A b-13 a B)}{128 b^7}-\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}}+\frac {x^{13/2} (A b-a B)}{5 a b (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

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 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 {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\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)) \operatorname {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*}

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Mathematica [C]  time = 0.03, size = 61, normalized size = 0.25 \begin {gather*} \frac {x^{13/2} \left (\frac {13 a^5 (A b-a B)}{(a+b x)^5}+(13 a B-3 A b) \, _2F_1\left (5,\frac {13}{2};\frac {15}{2};-\frac {b x}{a}\right )\right )}{65 a^6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(13/2)*((13*a^5*(A*b - a*B))/(a + b*x)^5 + (-3*A*b + 13*a*B)*Hypergeometric2F1[5, 13/2, 15/2, -((b*x)/a)]))
/(65*a^6*b)

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IntegrateAlgebraic [A]  time = 0.41, size = 194, normalized size = 0.81 \begin {gather*} \frac {231 \left (13 a^{3/2} B-3 \sqrt {a} A b\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 b^{15/2}}+\frac {\sqrt {x} \left (-45045 a^6 B+10395 a^5 A b-210210 a^5 b B x+48510 a^4 A b^2 x-384384 a^4 b^2 B x^2+88704 a^3 A b^3 x^2-338910 a^3 b^3 B x^3+78210 a^2 A b^4 x^3-137995 a^2 b^4 B x^4+31845 a A b^5 x^4-16640 a b^5 B x^5+3840 A b^6 x^5+1280 b^6 B x^6\right )}{1920 b^7 (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 703, normalized size = 2.93 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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giac [A]  time = 0.18, size = 191, normalized size = 0.80 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.08, size = 266, normalized size = 1.11 \begin {gather*} \frac {843 A a \,x^{\frac {9}{2}}}{128 \left (b x +a \right )^{5} b^{2}}-\frac {2373 B \,a^{2} x^{\frac {9}{2}}}{128 \left (b x +a \right )^{5} b^{3}}+\frac {1327 A \,a^{2} x^{\frac {7}{2}}}{64 \left (b x +a \right )^{5} b^{3}}-\frac {12131 B \,a^{3} x^{\frac {7}{2}}}{192 \left (b x +a \right )^{5} b^{4}}+\frac {131 A \,a^{3} x^{\frac {5}{2}}}{5 \left (b x +a \right )^{5} b^{4}}-\frac {1253 B \,a^{4} x^{\frac {5}{2}}}{15 \left (b x +a \right )^{5} b^{5}}+\frac {977 A \,a^{4} x^{\frac {3}{2}}}{64 \left (b x +a \right )^{5} b^{5}}-\frac {9629 B \,a^{5} x^{\frac {3}{2}}}{192 \left (b x +a \right )^{5} b^{6}}+\frac {437 A \,a^{5} \sqrt {x}}{128 \left (b x +a \right )^{5} b^{6}}-\frac {1467 B \,a^{6} \sqrt {x}}{128 \left (b x +a \right )^{5} b^{7}}-\frac {693 A a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, b^{6}}+\frac {3003 B \,a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, b^{7}}+\frac {2 B \,x^{\frac {3}{2}}}{3 b^{6}}+\frac {2 A \sqrt {x}}{b^{6}}-\frac {12 B a \sqrt {x}}{b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.18, size = 231, normalized size = 0.96 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 1.24, size = 246, normalized size = 1.02 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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

[Out]

Timed out

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