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

Optimal. Leaf size=357 \[ \frac {11 A b-3 a B}{24 a^2 b x^{3/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{3/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 \sqrt {b} (a+b x) (11 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 (a+b x) (11 A b-3 a B)}{64 a^6 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (a+b x) (11 A b-3 a B)}{64 a^5 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 (11 A b-3 a B)}{64 a^4 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 (11 A b-3 a B)}{32 a^3 b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.19, antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {105 (a+b x) (11 A b-3 a B)}{64 a^6 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (a+b x) (11 A b-3 a B)}{64 a^5 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 (11 A b-3 a B)}{32 a^3 b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 (11 A b-3 a B)}{64 a^4 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 A b-3 a B}{24 a^2 b x^{3/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{3/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 \sqrt {b} (a+b x) (11 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(21*(11*A*b - 3*a*B))/(64*a^4*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(4*a*b*x^(3/2)*(a + b*x)^
3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*A*b - 3*a*B)/(24*a^2*b*x^(3/2)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) + (3*(11*A*b - 3*a*B))/(32*a^3*b*x^(3/2)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(11*A*b - 3*a*B)*(a
+ b*x))/(64*a^5*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (105*(11*A*b - 3*a*B)*(a + b*x))/(64*a^6*Sqrt[x]*S
qrt[a^2 + 2*a*b*x + b^2*x^2]) + (105*Sqrt[b]*(11*A*b - 3*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64
*a^(13/2)*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^{5/2} \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^{5/2} \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^{3/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (11 A b-3 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{5/2} \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^{3/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 A b-3 a B}{24 a^2 b x^{3/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 b (11 A b-3 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{5/2} \left (a b+b^2 x\right )^3} \, dx}{16 a^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {A b-a B}{4 a b x^{3/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 A b-3 a B}{24 a^2 b x^{3/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 (11 A b-3 a B)}{32 a^3 b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 (11 A b-3 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{5/2} \left (a b+b^2 x\right )^2} \, dx}{64 a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {21 (11 A b-3 a B)}{64 a^4 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{3/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 A b-3 a B}{24 a^2 b x^{3/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 (11 A b-3 a B)}{32 a^3 b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 (11 A b-3 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^4 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {21 (11 A b-3 a B)}{64 a^4 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{3/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 A b-3 a B}{24 a^2 b x^{3/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 (11 A b-3 a B)}{32 a^3 b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (11 A b-3 a B) (a+b x)}{64 a^5 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (105 (11 A b-3 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^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {21 (11 A b-3 a B)}{64 a^4 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{3/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 A b-3 a B}{24 a^2 b x^{3/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 (11 A b-3 a B)}{32 a^3 b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (11 A b-3 a B) (a+b x)}{64 a^5 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 (11 A b-3 a B) (a+b x)}{64 a^6 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 b (11 A b-3 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^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {21 (11 A b-3 a B)}{64 a^4 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{3/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 A b-3 a B}{24 a^2 b x^{3/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 (11 A b-3 a B)}{32 a^3 b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (11 A b-3 a B) (a+b x)}{64 a^5 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 (11 A b-3 a B) (a+b x)}{64 a^6 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 b (11 A b-3 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^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {21 (11 A b-3 a B)}{64 a^4 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{3/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 A b-3 a B}{24 a^2 b x^{3/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 (11 A b-3 a B)}{32 a^3 b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (11 A b-3 a B) (a+b x)}{64 a^5 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 (11 A b-3 a B) (a+b x)}{64 a^6 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 \sqrt {b} (11 A b-3 a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 80, normalized size = 0.22 \begin {gather*} \frac {3 a^4 (A b-a B)-(a+b x)^4 (11 A b-3 a B) \, _2F_1\left (-\frac {3}{2},4;-\frac {1}{2};-\frac {b x}{a}\right )}{12 a^5 b x^{3/2} (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^4*(A*b - a*B) - (11*A*b - 3*a*B)*(a + b*x)^4*Hypergeometric2F1[-3/2, 4, -1/2, -((b*x)/a)])/(12*a^5*b*x^(3
/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [A]  time = 31.43, size = 190, normalized size = 0.53 \begin {gather*} \frac {(a+b x) \left (\frac {-128 a^5 A-384 a^5 B x+1408 a^4 A b x-2511 a^4 b B x^2+9207 a^3 A b^2 x^2-4599 a^3 b^2 B x^3+16863 a^2 A b^3 x^3-3465 a^2 b^3 B x^4+12705 a A b^4 x^4-945 a b^4 B x^5+3465 A b^5 x^5}{192 a^6 x^{3/2} (a+b x)^4}-\frac {105 \left (3 a \sqrt {b} B-11 A b^{3/2}\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{13/2}}\right )}{\sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*((-128*a^5*A + 1408*a^4*A*b*x - 384*a^5*B*x + 9207*a^3*A*b^2*x^2 - 2511*a^4*b*B*x^2 + 16863*a^2*A*b
^3*x^3 - 4599*a^3*b^2*B*x^3 + 12705*a*A*b^4*x^4 - 3465*a^2*b^3*B*x^4 + 3465*A*b^5*x^5 - 945*a*b^4*B*x^5)/(192*
a^6*x^(3/2)*(a + b*x)^4) - (105*(-11*A*b^(3/2) + 3*a*Sqrt[b]*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(13/2
))))/Sqrt[(a + b*x)^2]

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fricas [A]  time = 0.44, size = 616, normalized size = 1.73 \begin {gather*} \left [-\frac {315 \, {\left ({\left (3 \, B a b^{4} - 11 \, A b^{5}\right )} x^{6} + 4 \, {\left (3 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{5} + 6 \, {\left (3 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{4} + 4 \, {\left (3 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{3} + {\left (3 \, B a^{5} - 11 \, A a^{4} 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 (128 \, A a^{5} + 315 \, {\left (3 \, B a b^{4} - 11 \, A b^{5}\right )} x^{5} + 1155 \, {\left (3 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} + 1533 \, {\left (3 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} + 837 \, {\left (3 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} + 128 \, {\left (3 \, B a^{5} - 11 \, A a^{4} b\right )} x\right )} \sqrt {x}}{384 \, {\left (a^{6} b^{4} x^{6} + 4 \, a^{7} b^{3} x^{5} + 6 \, a^{8} b^{2} x^{4} + 4 \, a^{9} b x^{3} + a^{10} x^{2}\right )}}, \frac {315 \, {\left ({\left (3 \, B a b^{4} - 11 \, A b^{5}\right )} x^{6} + 4 \, {\left (3 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{5} + 6 \, {\left (3 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{4} + 4 \, {\left (3 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{3} + {\left (3 \, B a^{5} - 11 \, A a^{4} b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (128 \, A a^{5} + 315 \, {\left (3 \, B a b^{4} - 11 \, A b^{5}\right )} x^{5} + 1155 \, {\left (3 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} + 1533 \, {\left (3 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} + 837 \, {\left (3 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} + 128 \, {\left (3 \, B a^{5} - 11 \, A a^{4} b\right )} x\right )} \sqrt {x}}{192 \, {\left (a^{6} b^{4} x^{6} + 4 \, a^{7} b^{3} x^{5} + 6 \, a^{8} b^{2} x^{4} + 4 \, a^{9} b x^{3} + a^{10} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(315*((3*B*a*b^4 - 11*A*b^5)*x^6 + 4*(3*B*a^2*b^3 - 11*A*a*b^4)*x^5 + 6*(3*B*a^3*b^2 - 11*A*a^2*b^3)*x
^4 + 4*(3*B*a^4*b - 11*A*a^3*b^2)*x^3 + (3*B*a^5 - 11*A*a^4*b)*x^2)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/
a) - a)/(b*x + a)) + 2*(128*A*a^5 + 315*(3*B*a*b^4 - 11*A*b^5)*x^5 + 1155*(3*B*a^2*b^3 - 11*A*a*b^4)*x^4 + 153
3*(3*B*a^3*b^2 - 11*A*a^2*b^3)*x^3 + 837*(3*B*a^4*b - 11*A*a^3*b^2)*x^2 + 128*(3*B*a^5 - 11*A*a^4*b)*x)*sqrt(x
))/(a^6*b^4*x^6 + 4*a^7*b^3*x^5 + 6*a^8*b^2*x^4 + 4*a^9*b*x^3 + a^10*x^2), 1/192*(315*((3*B*a*b^4 - 11*A*b^5)*
x^6 + 4*(3*B*a^2*b^3 - 11*A*a*b^4)*x^5 + 6*(3*B*a^3*b^2 - 11*A*a^2*b^3)*x^4 + 4*(3*B*a^4*b - 11*A*a^3*b^2)*x^3
+ (3*B*a^5 - 11*A*a^4*b)*x^2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) - (128*A*a^5 + 315*(3*B*a*b^4 - 11*A*
b^5)*x^5 + 1155*(3*B*a^2*b^3 - 11*A*a*b^4)*x^4 + 1533*(3*B*a^3*b^2 - 11*A*a^2*b^3)*x^3 + 837*(3*B*a^4*b - 11*A
*a^3*b^2)*x^2 + 128*(3*B*a^5 - 11*A*a^4*b)*x)*sqrt(x))/(a^6*b^4*x^6 + 4*a^7*b^3*x^5 + 6*a^8*b^2*x^4 + 4*a^9*b*
x^3 + a^10*x^2)]

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giac [A]  time = 0.24, size = 180, normalized size = 0.50 \begin {gather*} -\frac {105 \, {\left (3 \, B a b - 11 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, {\left (3 \, B a x - 15 \, A b x + A a\right )}}{3 \, a^{6} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right )} - \frac {561 \, B a b^{4} x^{\frac {7}{2}} - 1545 \, A b^{5} x^{\frac {7}{2}} + 1929 \, B a^{2} b^{3} x^{\frac {5}{2}} - 5153 \, A a b^{4} x^{\frac {5}{2}} + 2295 \, B a^{3} b^{2} x^{\frac {3}{2}} - 5855 \, A a^{2} b^{3} x^{\frac {3}{2}} + 975 \, B a^{4} b \sqrt {x} - 2295 \, A a^{3} b^{2} \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a^{6} \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^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

-105/64*(3*B*a*b - 11*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^6*sgn(b*x + a)) - 2/3*(3*B*a*x - 15*A*b*
x + A*a)/(a^6*x^(3/2)*sgn(b*x + a)) - 1/192*(561*B*a*b^4*x^(7/2) - 1545*A*b^5*x^(7/2) + 1929*B*a^2*b^3*x^(5/2)
- 5153*A*a*b^4*x^(5/2) + 2295*B*a^3*b^2*x^(3/2) - 5855*A*a^2*b^3*x^(3/2) + 975*B*a^4*b*sqrt(x) - 2295*A*a^3*b
^2*sqrt(x))/((b*x + a)^4*a^6*sgn(b*x + a))

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maple [A]  time = 0.08, size = 413, normalized size = 1.16 \begin {gather*} \frac {\left (3465 A \,b^{6} x^{\frac {11}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-945 B a \,b^{5} x^{\frac {11}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+13860 A a \,b^{5} x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-3780 B \,a^{2} b^{4} x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+20790 A \,a^{2} b^{4} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-5670 B \,a^{3} b^{3} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3465 \sqrt {a b}\, A \,b^{5} x^{5}-945 \sqrt {a b}\, B a \,b^{4} x^{5}+13860 A \,a^{3} b^{3} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-3780 B \,a^{4} b^{2} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+12705 \sqrt {a b}\, A a \,b^{4} x^{4}-3465 \sqrt {a b}\, B \,a^{2} b^{3} x^{4}+3465 A \,a^{4} b^{2} x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-945 B \,a^{5} b \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+16863 \sqrt {a b}\, A \,a^{2} b^{3} x^{3}-4599 \sqrt {a b}\, B \,a^{3} b^{2} x^{3}+9207 \sqrt {a b}\, A \,a^{3} b^{2} x^{2}-2511 \sqrt {a b}\, B \,a^{4} b \,x^{2}+1408 \sqrt {a b}\, A \,a^{4} b x -384 \sqrt {a b}\, B \,a^{5} x -128 \sqrt {a b}\, A \,a^{5}\right ) \left (b x +a \right )}{192 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{6} x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(13860*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^(5/2)*a^3*b^3-3780*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^(5/2)*
a^4*b^2-945*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^(11/2)*a*b^5+13860*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^(9/2)*a
*b^5-3780*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^(9/2)*a^2*b^4+20790*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^(7/2)*a^
2*b^4-5670*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^(7/2)*a^3*b^3+3465*A*(a*b)^(1/2)*x^5*b^5-384*B*(a*b)^(1/2)*x*a^
5-128*A*(a*b)^(1/2)*a^5-945*B*(a*b)^(1/2)*x^5*a*b^4+12705*A*(a*b)^(1/2)*x^4*a*b^4+3465*A*arctan(1/(a*b)^(1/2)*
b*x^(1/2))*x^(11/2)*b^6-3465*B*(a*b)^(1/2)*x^4*a^2*b^3+16863*A*(a*b)^(1/2)*x^3*a^2*b^3-4599*B*(a*b)^(1/2)*x^3*
a^3*b^2+9207*A*(a*b)^(1/2)*x^2*a^3*b^2+3465*A*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^(3/2)*a^4*b^2-2511*B*(a*b)^(1/
2)*x^2*a^4*b-945*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*x^(3/2)*a^5*b+1408*A*(a*b)^(1/2)*x*a^4*b)*(b*x+a)/x^(3/2)/(
a*b)^(1/2)/a^6/((b*x+a)^2)^(5/2)

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maxima [B]  time = 2.03, size = 495, normalized size = 1.39 \begin {gather*} -\frac {315 \, {\left ({\left (B a b^{7} - 11 \, A b^{8}\right )} x^{2} + 11 \, {\left (3 \, B a^{2} b^{6} - 13 \, A a b^{7}\right )} x\right )} x^{\frac {9}{2}} + 630 \, {\left ({\left (B a^{2} b^{6} - 11 \, A a b^{7}\right )} x^{2} + 33 \, {\left (3 \, B a^{3} b^{5} - 13 \, A a^{2} b^{6}\right )} x\right )} x^{\frac {7}{2}} - 420 \, {\left (6 \, {\left (B a^{3} b^{5} - 11 \, A a^{2} b^{6}\right )} x^{2} - 121 \, {\left (3 \, B a^{4} b^{4} - 13 \, A a^{3} b^{5}\right )} x\right )} x^{\frac {5}{2}} - 42 \, {\left (255 \, {\left (B a^{4} b^{4} - 11 \, A a^{3} b^{5}\right )} x^{2} - 1529 \, {\left (3 \, B a^{5} b^{3} - 13 \, A a^{4} b^{4}\right )} x\right )} x^{\frac {3}{2}} - 33 \, {\left (483 \, {\left (B a^{5} b^{3} - 11 \, A a^{4} b^{4}\right )} x^{2} - 1315 \, {\left (3 \, B a^{6} b^{2} - 13 \, A a^{5} b^{3}\right )} x\right )} \sqrt {x} - \frac {1280 \, {\left (9 \, {\left (B a^{6} b^{2} - 11 \, A a^{5} b^{3}\right )} x^{2} - 11 \, {\left (3 \, B a^{7} b - 13 \, A a^{6} b^{2}\right )} x\right )}}{\sqrt {x}} - \frac {1280 \, {\left (3 \, {\left (B a^{7} b - 11 \, A a^{6} b^{2}\right )} x^{2} - {\left (3 \, B a^{8} - 13 \, A a^{7} b\right )} x\right )}}{x^{\frac {3}{2}}} + \frac {1280 \, {\left (3 \, A a^{7} b x^{2} + A a^{8} x\right )}}{x^{\frac {5}{2}}}}{1920 \, {\left (a^{8} b^{5} x^{5} + 5 \, a^{9} b^{4} x^{4} + 10 \, a^{10} b^{3} x^{3} + 10 \, a^{11} b^{2} x^{2} + 5 \, a^{12} b x + a^{13}\right )}} - \frac {105 \, {\left (3 \, B a b - 11 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{6}} + \frac {21 \, {\left ({\left (B a b^{2} - 11 \, A b^{3}\right )} x^{\frac {3}{2}} + 10 \, {\left (3 \, B a^{2} b - 11 \, A a b^{2}\right )} \sqrt {x}\right )}}{128 \, a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1920*(315*((B*a*b^7 - 11*A*b^8)*x^2 + 11*(3*B*a^2*b^6 - 13*A*a*b^7)*x)*x^(9/2) + 630*((B*a^2*b^6 - 11*A*a*b
^7)*x^2 + 33*(3*B*a^3*b^5 - 13*A*a^2*b^6)*x)*x^(7/2) - 420*(6*(B*a^3*b^5 - 11*A*a^2*b^6)*x^2 - 121*(3*B*a^4*b^
4 - 13*A*a^3*b^5)*x)*x^(5/2) - 42*(255*(B*a^4*b^4 - 11*A*a^3*b^5)*x^2 - 1529*(3*B*a^5*b^3 - 13*A*a^4*b^4)*x)*x
^(3/2) - 33*(483*(B*a^5*b^3 - 11*A*a^4*b^4)*x^2 - 1315*(3*B*a^6*b^2 - 13*A*a^5*b^3)*x)*sqrt(x) - 1280*(9*(B*a^
6*b^2 - 11*A*a^5*b^3)*x^2 - 11*(3*B*a^7*b - 13*A*a^6*b^2)*x)/sqrt(x) - 1280*(3*(B*a^7*b - 11*A*a^6*b^2)*x^2 -
(3*B*a^8 - 13*A*a^7*b)*x)/x^(3/2) + 1280*(3*A*a^7*b*x^2 + A*a^8*x)/x^(5/2))/(a^8*b^5*x^5 + 5*a^9*b^4*x^4 + 10*
a^10*b^3*x^3 + 10*a^11*b^2*x^2 + 5*a^12*b*x + a^13) - 105/64*(3*B*a*b - 11*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/
(sqrt(a*b)*a^6) + 21/128*((B*a*b^2 - 11*A*b^3)*x^(3/2) + 10*(3*B*a^2*b - 11*A*a*b^2)*sqrt(x))/a^8

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

[Out]

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

[Out]

Timed out

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