3.8.62 \(\int \frac {A+B x}{x^{7/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

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

________________________________________________________________________________________

Rubi [A]  time = 0.21, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {770, 78, 51, 63, 205} \begin {gather*} -\frac {231 b (a+b x) (13 A b-5 a B)}{64 a^7 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (a+b x) (13 A b-5 a B)}{64 a^6 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (a+b x) (13 A b-5 a B)}{320 a^5 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b^{3/2} (a+b x) (13 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

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

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]

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

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 80, normalized size = 0.20 \begin {gather*} \frac {5 a^4 (A b-a B)-(a+b x)^4 (13 A b-5 a B) \, _2F_1\left (-\frac {5}{2},4;-\frac {3}{2};-\frac {b x}{a}\right )}{20 a^5 b x^{5/2} (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(5*a^4*(A*b - a*B) - (13*A*b - 5*a*B)*(a + b*x)^4*Hypergeometric2F1[-5/2, 4, -3/2, -((b*x)/a)])/(20*a^5*b*x^(5
/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 43.64, size = 214, normalized size = 0.53 \begin {gather*} \frac {(a+b x) \left (\frac {231 \left (5 a b^{3/2} B-13 A b^{5/2}\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{15/2}}+\frac {-384 a^6 A-640 a^6 B x+1664 a^5 A b x+7040 a^5 b B x^2-18304 a^4 A b^2 x^2+46035 a^4 b^2 B x^3-119691 a^3 A b^3 x^3+84315 a^3 b^3 B x^4-219219 a^2 A b^4 x^4+63525 a^2 b^4 B x^5-165165 a A b^5 x^5+17325 a b^5 B x^6-45045 A b^6 x^6}{960 a^7 x^{5/2} (a+b x)^4}\right )}{\sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(x^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 673, normalized size = 1.67 \begin {gather*} \left [-\frac {3465 \, {\left ({\left (5 \, B a b^{5} - 13 \, A b^{6}\right )} x^{7} + 4 \, {\left (5 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{6} + 6 \, {\left (5 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{5} + 4 \, {\left (5 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + {\left (5 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{3}\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 (384 \, A a^{6} - 3465 \, {\left (5 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} - 12705 \, {\left (5 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} - 16863 \, {\left (5 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} - 9207 \, {\left (5 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} - 1408 \, {\left (5 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (5 \, B a^{6} - 13 \, A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{7} b^{4} x^{7} + 4 \, a^{8} b^{3} x^{6} + 6 \, a^{9} b^{2} x^{5} + 4 \, a^{10} b x^{4} + a^{11} x^{3}\right )}}, -\frac {3465 \, {\left ({\left (5 \, B a b^{5} - 13 \, A b^{6}\right )} x^{7} + 4 \, {\left (5 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{6} + 6 \, {\left (5 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{5} + 4 \, {\left (5 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + {\left (5 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (384 \, A a^{6} - 3465 \, {\left (5 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} - 12705 \, {\left (5 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} - 16863 \, {\left (5 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} - 9207 \, {\left (5 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} - 1408 \, {\left (5 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (5 \, B a^{6} - 13 \, A a^{5} b\right )} x\right )} \sqrt {x}}{960 \, {\left (a^{7} b^{4} x^{7} + 4 \, a^{8} b^{3} x^{6} + 6 \, a^{9} b^{2} x^{5} + 4 \, a^{10} b x^{4} + a^{11} x^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.23, size = 207, normalized size = 0.51 \begin {gather*} \frac {231 \, {\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (75 \, B a b x^{2} - 225 \, A b^{2} x^{2} - 5 \, B a^{2} x + 25 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{7} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right )} + \frac {1545 \, B a b^{5} x^{\frac {7}{2}} - 3249 \, A b^{6} x^{\frac {7}{2}} + 5153 \, B a^{2} b^{4} x^{\frac {5}{2}} - 10633 \, A a b^{5} x^{\frac {5}{2}} + 5855 \, B a^{3} b^{3} x^{\frac {3}{2}} - 11767 \, A a^{2} b^{4} x^{\frac {3}{2}} + 2295 \, B a^{4} b^{2} \sqrt {x} - 4431 \, A a^{3} b^{3} \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a^{7} \mathrm {sgn}\left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.08, size = 449, normalized size = 1.11 \begin {gather*} -\frac {\left (45045 A \,b^{7} x^{\frac {13}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-17325 B a \,b^{6} x^{\frac {13}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+180180 A a \,b^{6} x^{\frac {11}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-69300 B \,a^{2} b^{5} x^{\frac {11}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+270270 A \,a^{2} b^{5} x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-103950 B \,a^{3} b^{4} x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+45045 \sqrt {a b}\, A \,b^{6} x^{6}-17325 \sqrt {a b}\, B a \,b^{5} x^{6}+180180 A \,a^{3} b^{4} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-69300 B \,a^{4} b^{3} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+165165 \sqrt {a b}\, A a \,b^{5} x^{5}-63525 \sqrt {a b}\, B \,a^{2} b^{4} x^{5}+45045 A \,a^{4} b^{3} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-17325 B \,a^{5} b^{2} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+219219 \sqrt {a b}\, A \,a^{2} b^{4} x^{4}-84315 \sqrt {a b}\, B \,a^{3} b^{3} x^{4}+119691 \sqrt {a b}\, A \,a^{3} b^{3} x^{3}-46035 \sqrt {a b}\, B \,a^{4} b^{2} x^{3}+18304 \sqrt {a b}\, A \,a^{4} b^{2} x^{2}-7040 \sqrt {a b}\, B \,a^{5} b \,x^{2}-1664 \sqrt {a b}\, A \,a^{5} b x +640 \sqrt {a b}\, B \,a^{6} x +384 \sqrt {a b}\, A \,a^{6}\right ) \left (b x +a \right )}{960 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{7} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 2.12, size = 550, normalized size = 1.36 \begin {gather*} \frac {1155 \, {\left ({\left (3 \, B a b^{8} - 13 \, A b^{9}\right )} x^{2} + 39 \, {\left (B a^{2} b^{7} - 3 \, A a b^{8}\right )} x\right )} x^{\frac {9}{2}} + 2310 \, {\left ({\left (3 \, B a^{2} b^{7} - 13 \, A a b^{8}\right )} x^{2} + 117 \, {\left (B a^{3} b^{6} - 3 \, A a^{2} b^{7}\right )} x\right )} x^{\frac {7}{2}} - 4620 \, {\left (2 \, {\left (3 \, B a^{3} b^{6} - 13 \, A a^{2} b^{7}\right )} x^{2} - 143 \, {\left (B a^{4} b^{5} - 3 \, A a^{3} b^{6}\right )} x\right )} x^{\frac {5}{2}} - 462 \, {\left (85 \, {\left (3 \, B a^{4} b^{5} - 13 \, A a^{3} b^{6}\right )} x^{2} - 1807 \, {\left (B a^{5} b^{4} - 3 \, A a^{4} b^{5}\right )} x\right )} x^{\frac {3}{2}} - 33 \, {\left (1771 \, {\left (3 \, B a^{5} b^{4} - 13 \, A a^{4} b^{5}\right )} x^{2} - 17095 \, {\left (B a^{6} b^{3} - 3 \, A a^{5} b^{4}\right )} x\right )} \sqrt {x} - \frac {14080 \, {\left (3 \, {\left (3 \, B a^{6} b^{3} - 13 \, A a^{5} b^{4}\right )} x^{2} - 13 \, {\left (B a^{7} b^{2} - 3 \, A a^{6} b^{3}\right )} x\right )}}{\sqrt {x}} - \frac {1280 \, {\left (11 \, {\left (3 \, B a^{7} b^{2} - 13 \, A a^{6} b^{3}\right )} x^{2} - 13 \, {\left (B a^{8} b - 3 \, A a^{7} b^{2}\right )} x\right )}}{x^{\frac {3}{2}}} - \frac {1280 \, {\left ({\left (3 \, B a^{8} b - 13 \, A a^{7} b^{2}\right )} x^{2} + {\left (B a^{9} - 3 \, A a^{8} b\right )} x\right )}}{x^{\frac {5}{2}}} - \frac {256 \, {\left (5 \, A a^{8} b x^{2} + 3 \, A a^{9} x\right )}}{x^{\frac {7}{2}}}}{1920 \, {\left (a^{9} b^{5} x^{5} + 5 \, a^{10} b^{4} x^{4} + 10 \, a^{11} b^{3} x^{3} + 10 \, a^{12} b^{2} x^{2} + 5 \, a^{13} b x + a^{14}\right )}} + \frac {231 \, {\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{7}} - \frac {77 \, {\left ({\left (3 \, B a b^{3} - 13 \, A b^{4}\right )} x^{\frac {3}{2}} + 6 \, {\left (5 \, B a^{2} b^{2} - 13 \, A a b^{3}\right )} \sqrt {x}\right )}}{128 \, a^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/1920*(1155*((3*B*a*b^8 - 13*A*b^9)*x^2 + 39*(B*a^2*b^7 - 3*A*a*b^8)*x)*x^(9/2) + 2310*((3*B*a^2*b^7 - 13*A*a
*b^8)*x^2 + 117*(B*a^3*b^6 - 3*A*a^2*b^7)*x)*x^(7/2) - 4620*(2*(3*B*a^3*b^6 - 13*A*a^2*b^7)*x^2 - 143*(B*a^4*b
^5 - 3*A*a^3*b^6)*x)*x^(5/2) - 462*(85*(3*B*a^4*b^5 - 13*A*a^3*b^6)*x^2 - 1807*(B*a^5*b^4 - 3*A*a^4*b^5)*x)*x^
(3/2) - 33*(1771*(3*B*a^5*b^4 - 13*A*a^4*b^5)*x^2 - 17095*(B*a^6*b^3 - 3*A*a^5*b^4)*x)*sqrt(x) - 14080*(3*(3*B
*a^6*b^3 - 13*A*a^5*b^4)*x^2 - 13*(B*a^7*b^2 - 3*A*a^6*b^3)*x)/sqrt(x) - 1280*(11*(3*B*a^7*b^2 - 13*A*a^6*b^3)
*x^2 - 13*(B*a^8*b - 3*A*a^7*b^2)*x)/x^(3/2) - 1280*((3*B*a^8*b - 13*A*a^7*b^2)*x^2 + (B*a^9 - 3*A*a^8*b)*x)/x
^(5/2) - 256*(5*A*a^8*b*x^2 + 3*A*a^9*x)/x^(7/2))/(a^9*b^5*x^5 + 5*a^10*b^4*x^4 + 10*a^11*b^3*x^3 + 10*a^12*b^
2*x^2 + 5*a^13*b*x + a^14) + 231/64*(5*B*a*b^2 - 13*A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7) - 77/12
8*((3*B*a*b^3 - 13*A*b^4)*x^(3/2) + 6*(5*B*a^2*b^2 - 13*A*a*b^3)*sqrt(x))/a^9

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^{7/2}\,{\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((A + B*x)/(x^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)

[Out]

int((A + B*x)/(x^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)

________________________________________________________________________________________

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**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________