3.8.53 \(\int \frac {x^{11/2} (A+B x)}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=404 \[ \frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^{11/2} (5 A b-13 a B)}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 a \sqrt {x} (a+b x) (5 A b-13 a B)}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 x^{3/2} (a+b x) (5 A b-13 a B)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 x^{5/2} (a+b x) (5 A b-13 a B)}{320 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {33 x^{7/2} (5 A b-13 a B)}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 x^{9/2} (5 A b-13 a B)}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 a^{3/2} (a+b x) (5 A b-13 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.20, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {770, 78, 47, 50, 63, 205} \begin {gather*} \frac {x^{13/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^{11/2} (5 A b-13 a B)}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 x^{9/2} (5 A b-13 a B)}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {33 x^{7/2} (5 A b-13 a B)}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 x^{5/2} (a+b x) (5 A b-13 a B)}{320 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 x^{3/2} (a+b x) (5 A b-13 a B)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 a \sqrt {x} (a+b x) (5 A b-13 a B)}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 a^{3/2} (a+b x) (5 A b-13 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(11/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(33*(5*A*b - 13*a*B)*x^(7/2))/(64*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*x^(13/2))/(4*a*b*(a + b*
x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((5*A*b - 13*a*B)*x^(11/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b
^2*x^2]) + (11*(5*A*b - 13*a*B)*x^(9/2))/(96*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*a*(5*A*b -
13*a*B)*Sqrt[x]*(a + b*x))/(64*b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (77*(5*A*b - 13*a*B)*x^(3/2)*(a + b*x))/(6
4*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*(5*A*b - 13*a*B)*x^(5/2)*(a + b*x))/(320*a*b^5*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) + (231*a^(3/2)*(5*A*b - 13*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*b^(15/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])

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]

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {x^{11/2} (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (b^2 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{11/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (11 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{9/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 a \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (33 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{5/2}}{a b+b^2 x} \, dx}{128 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac {x^{3/2}}{a b+b^2 x} \, dx}{128 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (5 A b-13 a B) x^{3/2} (a+b x)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 a (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {x}}{a b+b^2 x} \, dx}{128 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 a (5 A b-13 a B) \sqrt {x} (a+b x)}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (5 A b-13 a B) x^{3/2} (a+b x)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 a^2 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{128 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 a (5 A b-13 a B) \sqrt {x} (a+b x)}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (5 A b-13 a B) x^{3/2} (a+b x)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 a^2 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 a (5 A b-13 a B) \sqrt {x} (a+b x)}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (5 A b-13 a B) x^{3/2} (a+b x)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 a^{3/2} (5 A b-13 a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 80, normalized size = 0.20 \begin {gather*} \frac {x^{13/2} \left (13 a^4 (A b-a B)-(a+b x)^4 (5 A b-13 a B) \, _2F_1\left (4,\frac {13}{2};\frac {15}{2};-\frac {b x}{a}\right )\right )}{52 a^5 b (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(11/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x^(13/2)*(13*a^4*(A*b - a*B) - (5*A*b - 13*a*B)*(a + b*x)^4*Hypergeometric2F1[4, 13/2, 15/2, -((b*x)/a)]))/(5
2*a^5*b*(a + b*x)^3*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [A]  time = 48.62, size = 211, normalized size = 0.52 \begin {gather*} \frac {(a+b x) \left (\frac {\sqrt {x} \left (45045 a^6 B-17325 a^5 A b+165165 a^5 b B x-63525 a^4 A b^2 x+219219 a^4 b^2 B x^2-84315 a^3 A b^3 x^2+119691 a^3 b^3 B x^3-46035 a^2 A b^4 x^3+18304 a^2 b^4 B x^4-7040 a A b^5 x^4-1664 a b^5 B x^5+640 A b^6 x^5+384 b^6 B x^6\right )}{960 b^7 (a+b x)^4}-\frac {231 \left (13 a^{5/2} B-5 a^{3/2} A b\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 b^{15/2}}\right )}{\sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^(11/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((a + b*x)*((Sqrt[x]*(-17325*a^5*A*b + 45045*a^6*B - 63525*a^4*A*b^2*x + 165165*a^5*b*B*x - 84315*a^3*A*b^3*x^
2 + 219219*a^4*b^2*B*x^2 - 46035*a^2*A*b^4*x^3 + 119691*a^3*b^3*B*x^3 - 7040*a*A*b^5*x^4 + 18304*a^2*b^4*B*x^4
+ 640*A*b^6*x^5 - 1664*a*b^5*B*x^5 + 384*b^6*B*x^6))/(960*b^7*(a + b*x)^4) - (231*(-5*a^(3/2)*A*b + 13*a^(5/2
)*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*b^(15/2))))/Sqrt[(a + b*x)^2]

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fricas [A]  time = 0.45, size = 644, normalized size = 1.59 \begin {gather*} \left [-\frac {3465 \, {\left (13 \, B a^{6} - 5 \, A a^{5} b + {\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 4 \, {\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 6 \, {\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 4 \, {\left (13 \, B a^{5} b - 5 \, 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 (384 \, B b^{6} x^{6} + 45045 \, B a^{6} - 17325 \, A a^{5} b - 128 \, {\left (13 \, B a b^{5} - 5 \, A b^{6}\right )} x^{5} + 1408 \, {\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 9207 \, {\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 16863 \, {\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 12705 \, {\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{1920 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}}, -\frac {3465 \, {\left (13 \, B a^{6} - 5 \, A a^{5} b + {\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 4 \, {\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 6 \, {\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 4 \, {\left (13 \, B a^{5} b - 5 \, 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 (384 \, B b^{6} x^{6} + 45045 \, B a^{6} - 17325 \, A a^{5} b - 128 \, {\left (13 \, B a b^{5} - 5 \, A b^{6}\right )} x^{5} + 1408 \, {\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 9207 \, {\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 16863 \, {\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 12705 \, {\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{960 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} 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)^(5/2),x, algorithm="fricas")

[Out]

[-1/1920*(3465*(13*B*a^6 - 5*A*a^5*b + (13*B*a^2*b^4 - 5*A*a*b^5)*x^4 + 4*(13*B*a^3*b^3 - 5*A*a^2*b^4)*x^3 + 6
*(13*B*a^4*b^2 - 5*A*a^3*b^3)*x^2 + 4*(13*B*a^5*b - 5*A*a^4*b^2)*x)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/
b) - a)/(b*x + a)) - 2*(384*B*b^6*x^6 + 45045*B*a^6 - 17325*A*a^5*b - 128*(13*B*a*b^5 - 5*A*b^6)*x^5 + 1408*(1
3*B*a^2*b^4 - 5*A*a*b^5)*x^4 + 9207*(13*B*a^3*b^3 - 5*A*a^2*b^4)*x^3 + 16863*(13*B*a^4*b^2 - 5*A*a^3*b^3)*x^2
+ 12705*(13*B*a^5*b - 5*A*a^4*b^2)*x)*sqrt(x))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^
7), -1/960*(3465*(13*B*a^6 - 5*A*a^5*b + (13*B*a^2*b^4 - 5*A*a*b^5)*x^4 + 4*(13*B*a^3*b^3 - 5*A*a^2*b^4)*x^3 +
6*(13*B*a^4*b^2 - 5*A*a^3*b^3)*x^2 + 4*(13*B*a^5*b - 5*A*a^4*b^2)*x)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a)
- (384*B*b^6*x^6 + 45045*B*a^6 - 17325*A*a^5*b - 128*(13*B*a*b^5 - 5*A*b^6)*x^5 + 1408*(13*B*a^2*b^4 - 5*A*a*b
^5)*x^4 + 9207*(13*B*a^3*b^3 - 5*A*a^2*b^4)*x^3 + 16863*(13*B*a^4*b^2 - 5*A*a^3*b^3)*x^2 + 12705*(13*B*a^5*b -
5*A*a^4*b^2)*x)*sqrt(x))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7)]

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giac [A]  time = 0.29, size = 218, normalized size = 0.54 \begin {gather*} -\frac {231 \, {\left (13 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {4431 \, B a^{3} b^{3} x^{\frac {7}{2}} - 2295 \, A a^{2} b^{4} x^{\frac {7}{2}} + 11767 \, B a^{4} b^{2} x^{\frac {5}{2}} - 5855 \, A a^{3} b^{3} x^{\frac {5}{2}} + 10633 \, B a^{5} b x^{\frac {3}{2}} - 5153 \, A a^{4} b^{2} x^{\frac {3}{2}} + 3249 \, B a^{6} \sqrt {x} - 1545 \, A a^{5} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (3 \, B b^{20} x^{\frac {5}{2}} - 25 \, B a b^{19} x^{\frac {3}{2}} + 5 \, A b^{20} x^{\frac {3}{2}} + 225 \, B a^{2} b^{18} \sqrt {x} - 75 \, A a b^{19} \sqrt {x}\right )}}{15 \, b^{25} \mathrm {sgn}\left (b x + a\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)^(5/2),x, algorithm="giac")

[Out]

-231/64*(13*B*a^3 - 5*A*a^2*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^7*sgn(b*x + a)) + 1/192*(4431*B*a^3*b^
3*x^(7/2) - 2295*A*a^2*b^4*x^(7/2) + 11767*B*a^4*b^2*x^(5/2) - 5855*A*a^3*b^3*x^(5/2) + 10633*B*a^5*b*x^(3/2)
- 5153*A*a^4*b^2*x^(3/2) + 3249*B*a^6*sqrt(x) - 1545*A*a^5*b*sqrt(x))/((b*x + a)^4*b^7*sgn(b*x + a)) + 2/15*(3
*B*b^20*x^(5/2) - 25*B*a*b^19*x^(3/2) + 5*A*b^20*x^(3/2) + 225*B*a^2*b^18*sqrt(x) - 75*A*a*b^19*sqrt(x))/(b^25
*sgn(b*x + a))

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maple [A]  time = 0.10, size = 443, normalized size = 1.10 \begin {gather*} \frac {\left (384 \sqrt {a b}\, B \,b^{6} x^{\frac {13}{2}}+17325 A \,a^{2} b^{5} x^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-45045 B \,a^{3} b^{4} x^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+640 \sqrt {a b}\, A \,b^{6} x^{\frac {11}{2}}-1664 \sqrt {a b}\, B a \,b^{5} x^{\frac {11}{2}}+69300 A \,a^{3} b^{4} x^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-180180 B \,a^{4} b^{3} x^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-7040 \sqrt {a b}\, A a \,b^{5} x^{\frac {9}{2}}+18304 \sqrt {a b}\, B \,a^{2} b^{4} x^{\frac {9}{2}}+103950 A \,a^{4} b^{3} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-270270 B \,a^{5} b^{2} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-46035 \sqrt {a b}\, A \,a^{2} b^{4} x^{\frac {7}{2}}+119691 \sqrt {a b}\, B \,a^{3} b^{3} x^{\frac {7}{2}}+69300 A \,a^{5} b^{2} x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-180180 B \,a^{6} b x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-84315 \sqrt {a b}\, A \,a^{3} b^{3} x^{\frac {5}{2}}+219219 \sqrt {a b}\, B \,a^{4} b^{2} x^{\frac {5}{2}}+17325 A \,a^{6} b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-45045 B \,a^{7} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-63525 \sqrt {a b}\, A \,a^{4} b^{2} x^{\frac {3}{2}}+165165 \sqrt {a b}\, B \,a^{5} b \,x^{\frac {3}{2}}-17325 \sqrt {a b}\, A \,a^{5} b \sqrt {x}+45045 \sqrt {a b}\, B \,a^{6} \sqrt {x}\right ) \left (b x +a \right )}{960 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} 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)^(5/2),x)

[Out]

1/960*(384*B*(a*b)^(1/2)*x^(13/2)*b^6+640*A*(a*b)^(1/2)*x^(11/2)*b^6-1664*B*(a*b)^(1/2)*x^(11/2)*a*b^5-7040*A*
(a*b)^(1/2)*x^(9/2)*a*b^5+18304*B*(a*b)^(1/2)*x^(9/2)*a^2*b^4-46035*A*(a*b)^(1/2)*x^(7/2)*a^2*b^4+119691*B*(a*
b)^(1/2)*x^(7/2)*a^3*b^3-84315*A*(a*b)^(1/2)*x^(5/2)*a^3*b^3+17325*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^4*a^2*b
^5+219219*B*(a*b)^(1/2)*x^(5/2)*a^4*b^2-45045*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^4*a^3*b^4+69300*A*arctan(1/(
a*b)^(1/2)*b*x^(1/2))*x^3*a^3*b^4-180180*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^3*a^4*b^3-63525*A*(a*b)^(1/2)*x^(
3/2)*a^4*b^2+103950*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^2*a^4*b^3+165165*B*(a*b)^(1/2)*x^(3/2)*a^5*b-270270*B*
arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^2*a^5*b^2+69300*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x*a^5*b^2-180180*B*arctan(
1/(a*b)^(1/2)*b*x^(1/2))*x*a^6*b-17325*A*(a*b)^(1/2)*x^(1/2)*a^5*b+17325*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*a^6
*b+45045*B*(a*b)^(1/2)*x^(1/2)*a^6-45045*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*a^7)*(b*x+a)/(a*b)^(1/2)/b^7/((b*x+
a)^2)^(5/2)

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maxima [A]  time = 1.87, size = 401, normalized size = 0.99 \begin {gather*} \frac {256 \, {\left (3 \, B b^{6} x^{2} + 5 \, B a b^{5} x\right )} x^{\frac {11}{2}} + 5 \, {\left (2747 \, {\left (3 \, B a b^{5} - A b^{6}\right )} x^{2} + 437 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x\right )} x^{\frac {9}{2}} + 10 \, {\left (4667 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} x^{2} + 671 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x\right )} x^{\frac {7}{2}} + 2860 \, {\left (22 \, {\left (3 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{2} + 3 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x\right )} x^{\frac {5}{2}} + 66 \, {\left (585 \, {\left (3 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 77 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )} x^{\frac {3}{2}} + 231 \, {\left (39 \, {\left (3 \, B a^{5} b - A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (13 \, B a^{6} - 3 \, A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} - \frac {231 \, {\left (13 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} b^{7}} - \frac {77 \, {\left (13 \, {\left (3 \, B a b - A b^{2}\right )} x^{\frac {3}{2}} - 6 \, {\left (13 \, B a^{2} - 5 \, A a b\right )} \sqrt {x}\right )}}{128 \, 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)^(5/2),x, algorithm="maxima")

[Out]

1/1920*(256*(3*B*b^6*x^2 + 5*B*a*b^5*x)*x^(11/2) + 5*(2747*(3*B*a*b^5 - A*b^6)*x^2 + 437*(13*B*a^2*b^4 - 3*A*a
*b^5)*x)*x^(9/2) + 10*(4667*(3*B*a^2*b^4 - A*a*b^5)*x^2 + 671*(13*B*a^3*b^3 - 3*A*a^2*b^4)*x)*x^(7/2) + 2860*(
22*(3*B*a^3*b^3 - A*a^2*b^4)*x^2 + 3*(13*B*a^4*b^2 - 3*A*a^3*b^3)*x)*x^(5/2) + 66*(585*(3*B*a^4*b^2 - A*a^3*b^
3)*x^2 + 77*(13*B*a^5*b - 3*A*a^4*b^2)*x)*x^(3/2) + 231*(39*(3*B*a^5*b - A*a^4*b^2)*x^2 + 5*(13*B*a^6 - 3*A*a^
5*b)*x)*sqrt(x))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^7*x + a^5*b^6) - 231/64*
(13*B*a^3 - 5*A*a^2*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^7) - 77/128*(13*(3*B*a*b - A*b^2)*x^(3/2) - 6*
(13*B*a^2 - 5*A*a*b)*sqrt(x))/b^7

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^{11/2}\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \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)^(5/2),x)

[Out]

int((x^(11/2)*(A + B*x))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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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)**(5/2),x)

[Out]

Timed out

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