\(\int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx\) [817]

Optimal result

Integrand size = 31, antiderivative size = 238 \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^{5/2} (A b-a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

-2/7*A*(b*x+a)/a/x^(7/2)/((b*x+a)^2)^(1/2)+2/5*(A*b-B*a)*(b*x+a)/a^2/x^(5/2)/((b*x+a)^2)^(1/2)-2/3*b*(A*b-B*a)
*(b*x+a)/a^3/x^(3/2)/((b*x+a)^2)^(1/2)+2*b^(5/2)*(A*b-B*a)*(b*x+a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(9/2)/((b
*x+a)^2)^(1/2)+2*b^2*(A*b-B*a)*(b*x+a)/a^4/x^(1/2)/((b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {784, 79, 53, 65, 211} \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) (A b-a B)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^{5/2} (a+b x) (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (a+b x) (A b-a B)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (a+b x) (A b-a B)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[In]

Int[(A + B*x)/(x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(-2*A*(a + b*x))/(7*a*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*(A*b - a*B)*(a + b*x))/(5*a^2*x^(5/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) - (2*b*(A*b - a*B)*(a + b*x))/(3*a^3*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*b^2*
(A*b - a*B)*(a + b*x))/(a^4*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*b^(5/2)*(A*b - a*B)*(a + b*x)*ArcTan[(
Sqrt[b]*Sqrt[x])/Sqrt[a]])/(a^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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 (a b+b^2 x\right ) \int \frac {A+B x}{x^{9/2} \left (a b+b^2 x\right )} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{7/2} \left (a b+b^2 x\right )} \, dx}{7 a b \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{5/2} \left (a b+b^2 x\right )} \, dx}{7 a^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 b \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )} \, dx}{7 a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 b^2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{7 a^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (4 b^2 \left (-\frac {7 A b^2}{2}+\frac {7 a b B}{2}\right ) \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 )}{7 a^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A (a+b x)}{7 a x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^{5/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.53 \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 (a+b x) \left (\sqrt {a} \left (-105 A b^3 x^3+35 a b^2 x^2 (A+3 B x)-7 a^2 b x (3 A+5 B x)+3 a^3 (5 A+7 B x)\right )-105 b^{5/2} (A b-a B) x^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{105 a^{9/2} x^{7/2} \sqrt {(a+b x)^2}} \]

[In]

Integrate[(A + B*x)/(x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(-2*(a + b*x)*(Sqrt[a]*(-105*A*b^3*x^3 + 35*a*b^2*x^2*(A + 3*B*x) - 7*a^2*b*x*(3*A + 5*B*x) + 3*a^3*(5*A + 7*B
*x)) - 105*b^(5/2)*(A*b - a*B)*x^(7/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(105*a^(9/2)*x^(7/2)*Sqrt[(a + b*x)
^2])

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.57

method	result	size
risch	\(-\frac {2 \left (-105 A \,b^{3} x^{3}+105 B a \,b^{2} x^{3}+35 A a \,b^{2} x^{2}-35 B \,a^{2} b \,x^{2}-21 A \,a^{2} b x +21 a^{3} B x +15 A \,a^{3}\right ) \sqrt {\left (b x +a \right )^{2}}}{105 a^{4} x^{\frac {7}{2}} \left (b x +a \right )}+\frac {2 \left (A b -B a \right ) b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) \sqrt {\left (b x +a \right )^{2}}}{a^{4} \sqrt {b a}\, \left (b x +a \right )}\)	\(135\)
default	\(\frac {2 \left (b x +a \right ) \left (105 A \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) b^{4}-105 B \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{3}+105 A \,x^{3} \sqrt {b a}\, b^{3}-105 B \,x^{3} \sqrt {b a}\, a \,b^{2}-35 A \,x^{2} \sqrt {b a}\, a \,b^{2}+35 B \,x^{2} \sqrt {b a}\, a^{2} b +21 A x \sqrt {b a}\, a^{2} b -21 B x \sqrt {b a}\, a^{3}-15 A \,a^{3} \sqrt {b a}\right )}{105 \sqrt {\left (b x +a \right )^{2}}\, a^{4} \sqrt {b a}\, x^{\frac {7}{2}}}\)	\(165\)

[In]

int((B*x+A)/x^(9/2)/((b*x+a)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/105*(-105*A*b^3*x^3+105*B*a*b^2*x^3+35*A*a*b^2*x^2-35*B*a^2*b*x^2-21*A*a^2*b*x+21*B*a^3*x+15*A*a^3)/a^4/x^(
7/2)*((b*x+a)^2)^(1/2)/(b*x+a)+2*(A*b-B*a)/a^4*b^3/(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 = 0.33 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [-\frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt {x}}{105 \, a^{4} x^{4}}, \frac {2 \, {\left (105 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt {x}\right )}}{105 \, a^{4} x^{4}}\right ] \]

[In]

integrate((B*x+A)/x^(9/2)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/105*(105*(B*a*b^2 - A*b^3)*x^4*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(15*A*a^3
+ 105*(B*a*b^2 - A*b^3)*x^3 - 35*(B*a^2*b - A*a*b^2)*x^2 + 21*(B*a^3 - A*a^2*b)*x)*sqrt(x))/(a^4*x^4), 2/105*(
105*(B*a*b^2 - A*b^3)*x^4*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) - (15*A*a^3 + 105*(B*a*b^2 - A*b^3)*x^3 -
35*(B*a^2*b - A*a*b^2)*x^2 + 21*(B*a^3 - A*a^2*b)*x)*sqrt(x))/(a^4*x^4)]

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)/x**(9/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (159) = 318\).

Time = 0.34 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.50 \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {35 \, {\left ({\left (5 \, B a b^{5} - 7 \, A b^{6}\right )} x^{2} + 3 \, {\left (7 \, B a^{2} b^{4} - 9 \, A a b^{5}\right )} x\right )} \sqrt {x} - \frac {70 \, {\left ({\left (5 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{2} - 3 \, {\left (7 \, B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x\right )}}{\sqrt {x}} - \frac {70 \, {\left (3 \, {\left (5 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{2} - {\left (7 \, B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x\right )}}{x^{\frac {3}{2}}} - \frac {14 \, {\left (5 \, {\left (5 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{2} + {\left (7 \, B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )}}{x^{\frac {5}{2}}} + \frac {2 \, {\left (7 \, {\left (5 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} x^{2} + 3 \, {\left (7 \, B a^{6} - 9 \, A a^{5} b\right )} x\right )}}{x^{\frac {7}{2}}} + \frac {6 \, {\left (7 \, A a^{5} b x^{2} + 5 \, A a^{6} x\right )}}{x^{\frac {9}{2}}}}{105 \, {\left (a^{6} b x + a^{7}\right )}} - \frac {2 \, {\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {{\left (5 \, B a b^{4} - 7 \, A b^{5}\right )} x^{\frac {3}{2}} + 6 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} \sqrt {x}}{3 \, a^{6}} \]

[In]

integrate((B*x+A)/x^(9/2)/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/105*(35*((5*B*a*b^5 - 7*A*b^6)*x^2 + 3*(7*B*a^2*b^4 - 9*A*a*b^5)*x)*sqrt(x) - 70*((5*B*a^2*b^4 - 7*A*a*b^5)
*x^2 - 3*(7*B*a^3*b^3 - 9*A*a^2*b^4)*x)/sqrt(x) - 70*(3*(5*B*a^3*b^3 - 7*A*a^2*b^4)*x^2 - (7*B*a^4*b^2 - 9*A*a
^3*b^3)*x)/x^(3/2) - 14*(5*(5*B*a^4*b^2 - 7*A*a^3*b^3)*x^2 + (7*B*a^5*b - 9*A*a^4*b^2)*x)/x^(5/2) + 2*(7*(5*B*
a^5*b - 7*A*a^4*b^2)*x^2 + 3*(7*B*a^6 - 9*A*a^5*b)*x)/x^(7/2) + 6*(7*A*a^5*b*x^2 + 5*A*a^6*x)/x^(9/2))/(a^6*b*
x + a^7) - 2*(B*a*b^3 - A*b^4)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/3*((5*B*a*b^4 - 7*A*b^5)*x^(3/2
) + 6*(B*a^2*b^3 - A*a*b^4)*sqrt(x))/a^6

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 \, {\left (B a b^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{4} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {2 \, {\left (105 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) - 105 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) - 35 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) - 21 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 15 \, A a^{3} \mathrm {sgn}\left (b x + a\right )\right )}}{105 \, a^{4} x^{\frac {7}{2}}} \]

[In]

integrate((B*x+A)/x^(9/2)/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

-2*(B*a*b^3*sgn(b*x + a) - A*b^4*sgn(b*x + a))*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) - 2/105*(105*B*a*b^
2*x^3*sgn(b*x + a) - 105*A*b^3*x^3*sgn(b*x + a) - 35*B*a^2*b*x^2*sgn(b*x + a) + 35*A*a*b^2*x^2*sgn(b*x + a) +
21*B*a^3*x*sgn(b*x + a) - 21*A*a^2*b*x*sgn(b*x + a) + 15*A*a^3*sgn(b*x + a))/(a^4*x^(7/2))

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {A+B\,x}{x^{9/2}\,\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]

[In]

int((A + B*x)/(x^(9/2)*((a + b*x)^2)^(1/2)),x)

[Out]

int((A + B*x)/(x^(9/2)*((a + b*x)^2)^(1/2)), x)