\(\int \frac {A+B x}{x^{3/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [833]

Optimal result

Integrand size = 31, antiderivative size = 311 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

-35/64*(9*A*b-B*a)*(b*x+a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(11/2)/b^(1/2)/((b*x+a)^2)^(1/2)+35/192*(9*A*b-B*
a)/a^4/b/x^(1/2)/((b*x+a)^2)^(1/2)+1/4*(A*b-B*a)/a/b/(b*x+a)^3/x^(1/2)/((b*x+a)^2)^(1/2)+1/24*(9*A*b-B*a)/a^2/
b/(b*x+a)^2/x^(1/2)/((b*x+a)^2)^(1/2)+7/96*(9*A*b-B*a)/a^3/b/(b*x+a)/x^(1/2)/((b*x+a)^2)^(1/2)-35/64*(9*A*b-B*
a)*(b*x+a)/a^5/b/x^(1/2)/((b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {784, 79, 44, 53, 65, 211} \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (a+b x) (9 A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (a+b x) (9 A b-a B)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]

[In]

Int[(A + B*x)/(x^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(35*(9*A*b - a*B))/(192*a^4*b*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(4*a*b*Sqrt[x]*(a + b*x)^3*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (9*A*b - a*B)/(24*a^2*b*Sqrt[x]*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(7*(9*A*b - a*B))/(96*a^3*b*Sqrt[x]*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(9*A*b - a*B)*(a + b*x))/(6
4*a^5*b*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(9*A*b - a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]
)/(64*a^(11/2)*Sqrt[b]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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 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]

Rule 784

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*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{x^{3/2} \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 \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \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 \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 b (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \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 \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )^2} \, dx}{192 a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 (9 A b-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^4 b \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 (9 A b-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^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 (9 A b-a B) \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{64 a^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.48 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {\frac {\sqrt {a} \left (-945 A b^4 x^4+105 a b^3 x^3 (-33 A+B x)+7 a^2 b^2 x^2 (-657 A+55 B x)+a^4 (-384 A+279 B x)+a^3 b x (-2511 A+511 B x)\right )}{\sqrt {x}}+\frac {105 (-9 A b+a B) (a+b x)^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b}}}{192 a^{11/2} (a+b x)^3 \sqrt {(a+b x)^2}} \]

[In]

Integrate[(A + B*x)/(x^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

((Sqrt[a]*(-945*A*b^4*x^4 + 105*a*b^3*x^3*(-33*A + B*x) + 7*a^2*b^2*x^2*(-657*A + 55*B*x) + a^4*(-384*A + 279*
B*x) + a^3*b*x*(-2511*A + 511*B*x)))/Sqrt[x] + (105*(-9*A*b + a*B)*(a + b*x)^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a
]])/Sqrt[b])/(192*a^(11/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.52

method	result	size
risch	\(-\frac {2 A \sqrt {\left (b x +a \right )^{2}}}{a^{5} \sqrt {x}\, \left (b x +a \right )}-\frac {\left (\frac {2 \left (\frac {187}{128} A \,b^{4}-\frac {35}{128} B \,b^{3} a \right ) x^{\frac {7}{2}}+\frac {b^{2} a \left (1929 A b -385 B a \right ) x^{\frac {5}{2}}}{192}+2 \left (\frac {765}{128} A \,a^{2} b^{2}-\frac {511}{384} B \,a^{3} b \right ) x^{\frac {3}{2}}+2 \left (\frac {325}{128} A \,a^{3} b -\frac {93}{128} B \,a^{4}\right ) \sqrt {x}}{\left (b x +a \right )^{4}}+\frac {35 \left (9 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{64 \sqrt {b a}}\right ) \sqrt {\left (b x +a \right )^{2}}}{a^{5} \left (b x +a \right )}\)	\(161\)
default	\(-\frac {\left (945 A \,x^{4} \sqrt {b a}\, b^{4}-105 B \,x^{4} \sqrt {b a}\, a \,b^{3}+3465 A \,x^{3} \sqrt {b a}\, a \,b^{3}+945 A \,x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) b^{5}-385 B \,x^{3} \sqrt {b a}\, a^{2} b^{2}-105 B \,x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{4}+3780 A \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{4}-420 B \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{3}+4599 A \,x^{2} \sqrt {b a}\, a^{2} b^{2}+5670 A \,x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{3}-511 B \,x^{2} \sqrt {b a}\, a^{3} b -630 B \,x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{2}+3780 A \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{2}-420 B \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b +2511 A x \sqrt {b a}\, a^{3} b +945 A \sqrt {x}\, \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b -279 B x \sqrt {b a}\, a^{4}-105 B \sqrt {x}\, \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{5}+384 A \sqrt {b a}\, a^{4}\right ) \left (b x +a \right )}{192 \sqrt {x}\, \sqrt {b a}\, a^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(374\)

[In]

int((B*x+A)/x^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2*A/a^5/x^(1/2)*((b*x+a)^2)^(1/2)/(b*x+a)-1/a^5*(2*((187/128*A*b^4-35/128*B*b^3*a)*x^(7/2)+1/384*b^2*a*(1929*
A*b-385*B*a)*x^(5/2)+(765/128*A*a^2*b^2-511/384*B*a^3*b)*x^(3/2)+(325/128*A*a^3*b-93/128*B*a^4)*x^(1/2))/(b*x+
a)^4+35/64*(9*A*b-B*a)/(b*a)^(1/2)*arctan(b*x^(1/2)/(b*a)^(1/2)))*((b*x+a)^2)^(1/2)/(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [\frac {105 \, {\left ({\left (B a b^{4} - 9 \, A b^{5}\right )} x^{5} + 4 \, {\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 6 \, {\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} + 4 \, {\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (384 \, A a^{5} b - 105 \, {\left (B a^{2} b^{4} - 9 \, A a b^{5}\right )} x^{4} - 385 \, {\left (B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x^{3} - 511 \, {\left (B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x^{2} - 279 \, {\left (B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{384 \, {\left (a^{6} b^{5} x^{5} + 4 \, a^{7} b^{4} x^{4} + 6 \, a^{8} b^{3} x^{3} + 4 \, a^{9} b^{2} x^{2} + a^{10} b x\right )}}, -\frac {105 \, {\left ({\left (B a b^{4} - 9 \, A b^{5}\right )} x^{5} + 4 \, {\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 6 \, {\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} + 4 \, {\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (384 \, A a^{5} b - 105 \, {\left (B a^{2} b^{4} - 9 \, A a b^{5}\right )} x^{4} - 385 \, {\left (B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x^{3} - 511 \, {\left (B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x^{2} - 279 \, {\left (B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{192 \, {\left (a^{6} b^{5} x^{5} + 4 \, a^{7} b^{4} x^{4} + 6 \, a^{8} b^{3} x^{3} + 4 \, a^{9} b^{2} x^{2} + a^{10} b x\right )}}\right ] \]

[In]

integrate((B*x+A)/x^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/384*(105*((B*a*b^4 - 9*A*b^5)*x^5 + 4*(B*a^2*b^3 - 9*A*a*b^4)*x^4 + 6*(B*a^3*b^2 - 9*A*a^2*b^3)*x^3 + 4*(B*
a^4*b - 9*A*a^3*b^2)*x^2 + (B*a^5 - 9*A*a^4*b)*x)*sqrt(-a*b)*log((b*x - a + 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) -
2*(384*A*a^5*b - 105*(B*a^2*b^4 - 9*A*a*b^5)*x^4 - 385*(B*a^3*b^3 - 9*A*a^2*b^4)*x^3 - 511*(B*a^4*b^2 - 9*A*a
^3*b^3)*x^2 - 279*(B*a^5*b - 9*A*a^4*b^2)*x)*sqrt(x))/(a^6*b^5*x^5 + 4*a^7*b^4*x^4 + 6*a^8*b^3*x^3 + 4*a^9*b^2
*x^2 + a^10*b*x), -1/192*(105*((B*a*b^4 - 9*A*b^5)*x^5 + 4*(B*a^2*b^3 - 9*A*a*b^4)*x^4 + 6*(B*a^3*b^2 - 9*A*a^
2*b^3)*x^3 + 4*(B*a^4*b - 9*A*a^3*b^2)*x^2 + (B*a^5 - 9*A*a^4*b)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) +
(384*A*a^5*b - 105*(B*a^2*b^4 - 9*A*a*b^5)*x^4 - 385*(B*a^3*b^3 - 9*A*a^2*b^4)*x^3 - 511*(B*a^4*b^2 - 9*A*a^3*
b^3)*x^2 - 279*(B*a^5*b - 9*A*a^4*b^2)*x)*sqrt(x))/(a^6*b^5*x^5 + 4*a^7*b^4*x^4 + 6*a^8*b^3*x^3 + 4*a^9*b^2*x^
2 + a^10*b*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)/x**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (208) = 416\).

Time = 0.37 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {35 \, {\left ({\left (B a b^{6} + 9 \, A b^{7}\right )} x^{2} - 27 \, {\left (B a^{2} b^{5} - 11 \, A a b^{6}\right )} x\right )} x^{\frac {9}{2}} + 70 \, {\left ({\left (B a^{2} b^{5} + 9 \, A a b^{6}\right )} x^{2} - 81 \, {\left (B a^{3} b^{4} - 11 \, A a^{2} b^{5}\right )} x\right )} x^{\frac {7}{2}} - 140 \, {\left (2 \, {\left (B a^{3} b^{4} + 9 \, A a^{2} b^{5}\right )} x^{2} + 99 \, {\left (B a^{4} b^{3} - 11 \, A a^{3} b^{4}\right )} x\right )} x^{\frac {5}{2}} - 14 \, {\left (85 \, {\left (B a^{4} b^{3} + 9 \, A a^{3} b^{4}\right )} x^{2} + 1251 \, {\left (B a^{5} b^{2} - 11 \, A a^{4} b^{3}\right )} x\right )} x^{\frac {3}{2}} - {\left (1771 \, {\left (B a^{5} b^{2} + 9 \, A a^{4} b^{3}\right )} x^{2} + 11835 \, {\left (B a^{6} b - 11 \, A a^{5} b^{2}\right )} x\right )} \sqrt {x} - \frac {1280 \, {\left ({\left (B a^{6} b + 9 \, A a^{5} b^{2}\right )} x^{2} + 3 \, {\left (B a^{7} - 11 \, A a^{6} b\right )} x\right )}}{\sqrt {x}} - \frac {3840 \, {\left (A a^{6} b x^{2} - A a^{7} x\right )}}{x^{\frac {3}{2}}}}{1920 \, {\left (a^{7} b^{5} x^{5} + 5 \, a^{8} b^{4} x^{4} + 10 \, a^{9} b^{3} x^{3} + 10 \, a^{10} b^{2} x^{2} + 5 \, a^{11} b x + a^{12}\right )}} + \frac {35 \, {\left (B a - 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{5}} + \frac {7 \, {\left ({\left (B a b + 9 \, A b^{2}\right )} x^{\frac {3}{2}} - 30 \, {\left (B a^{2} - 9 \, A a b\right )} \sqrt {x}\right )}}{384 \, a^{7}} \]

[In]

integrate((B*x+A)/x^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

-1/1920*(35*((B*a*b^6 + 9*A*b^7)*x^2 - 27*(B*a^2*b^5 - 11*A*a*b^6)*x)*x^(9/2) + 70*((B*a^2*b^5 + 9*A*a*b^6)*x^
2 - 81*(B*a^3*b^4 - 11*A*a^2*b^5)*x)*x^(7/2) - 140*(2*(B*a^3*b^4 + 9*A*a^2*b^5)*x^2 + 99*(B*a^4*b^3 - 11*A*a^3
*b^4)*x)*x^(5/2) - 14*(85*(B*a^4*b^3 + 9*A*a^3*b^4)*x^2 + 1251*(B*a^5*b^2 - 11*A*a^4*b^3)*x)*x^(3/2) - (1771*(
B*a^5*b^2 + 9*A*a^4*b^3)*x^2 + 11835*(B*a^6*b - 11*A*a^5*b^2)*x)*sqrt(x) - 1280*((B*a^6*b + 9*A*a^5*b^2)*x^2 +
3*(B*a^7 - 11*A*a^6*b)*x)/sqrt(x) - 3840*(A*a^6*b*x^2 - A*a^7*x)/x^(3/2))/(a^7*b^5*x^5 + 5*a^8*b^4*x^4 + 10*a
^9*b^3*x^3 + 10*a^10*b^2*x^2 + 5*a^11*b*x + a^12) + 35/64*(B*a - 9*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)
*a^5) + 7/384*((B*a*b + 9*A*b^2)*x^(3/2) - 30*(B*a^2 - 9*A*a*b)*sqrt(x))/a^7

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.51 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {35 \, {\left (B a - 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, A}{a^{5} \sqrt {x} \mathrm {sgn}\left (b x + a\right )} + \frac {105 \, B a b^{3} x^{\frac {7}{2}} - 561 \, A b^{4} x^{\frac {7}{2}} + 385 \, B a^{2} b^{2} x^{\frac {5}{2}} - 1929 \, A a b^{3} x^{\frac {5}{2}} + 511 \, B a^{3} b x^{\frac {3}{2}} - 2295 \, A a^{2} b^{2} x^{\frac {3}{2}} + 279 \, B a^{4} \sqrt {x} - 975 \, A a^{3} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x + a\right )} \]

[In]

integrate((B*x+A)/x^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

35/64*(B*a - 9*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5*sgn(b*x + a)) - 2*A/(a^5*sqrt(x)*sgn(b*x + a))
+ 1/192*(105*B*a*b^3*x^(7/2) - 561*A*b^4*x^(7/2) + 385*B*a^2*b^2*x^(5/2) - 1929*A*a*b^3*x^(5/2) + 511*B*a^3*b*
x^(3/2) - 2295*A*a^2*b^2*x^(3/2) + 279*B*a^4*sqrt(x) - 975*A*a^3*b*sqrt(x))/((b*x + a)^4*a^5*sgn(b*x + a))

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{x^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]

[In]

int((A + B*x)/(x^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)

[Out]

int((A + B*x)/(x^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)