\(\int \frac {x^{3/2} (A+B x)}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [768]

Optimal result

Integrand size = 29, antiderivative size = 130 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {(A b-a B) x^{5/2}}{3 a b (a+b x)^3}-\frac {(A b+5 a B) x^{3/2}}{12 a b^2 (a+b x)^2}-\frac {(A b+5 a B) \sqrt {x}}{8 a b^3 (a+b x)}+\frac {(A b+5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{3/2} b^{7/2}} \]

[Out]

1/3*(A*b-B*a)*x^(5/2)/a/b/(b*x+a)^3-1/12*(A*b+5*B*a)*x^(3/2)/a/b^2/(b*x+a)^2+1/8*(A*b+5*B*a)*arctan(b^(1/2)*x^
(1/2)/a^(1/2))/a^(3/2)/b^(7/2)-1/8*(A*b+5*B*a)*x^(1/2)/a/b^3/(b*x+a)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {27, 79, 43, 65, 211} \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {(5 a B+A b) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{3/2} b^{7/2}}-\frac {\sqrt {x} (5 a B+A b)}{8 a b^3 (a+b x)}-\frac {x^{3/2} (5 a B+A b)}{12 a b^2 (a+b x)^2}+\frac {x^{5/2} (A b-a B)}{3 a b (a+b x)^3} \]

[In]

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

[Out]

((A*b - a*B)*x^(5/2))/(3*a*b*(a + b*x)^3) - ((A*b + 5*a*B)*x^(3/2))/(12*a*b^2*(a + b*x)^2) - ((A*b + 5*a*B)*Sq
rt[x])/(8*a*b^3*(a + b*x)) + ((A*b + 5*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(3/2)*b^(7/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{3/2} (A+B x)}{(a+b x)^4} \, dx \\ & = \frac {(A b-a B) x^{5/2}}{3 a b (a+b x)^3}+\frac {(A b+5 a B) \int \frac {x^{3/2}}{(a+b x)^3} \, dx}{6 a b} \\ & = \frac {(A b-a B) x^{5/2}}{3 a b (a+b x)^3}-\frac {(A b+5 a B) x^{3/2}}{12 a b^2 (a+b x)^2}+\frac {(A b+5 a B) \int \frac {\sqrt {x}}{(a+b x)^2} \, dx}{8 a b^2} \\ & = \frac {(A b-a B) x^{5/2}}{3 a b (a+b x)^3}-\frac {(A b+5 a B) x^{3/2}}{12 a b^2 (a+b x)^2}-\frac {(A b+5 a B) \sqrt {x}}{8 a b^3 (a+b x)}+\frac {(A b+5 a B) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{16 a b^3} \\ & = \frac {(A b-a B) x^{5/2}}{3 a b (a+b x)^3}-\frac {(A b+5 a B) x^{3/2}}{12 a b^2 (a+b x)^2}-\frac {(A b+5 a B) \sqrt {x}}{8 a b^3 (a+b x)}+\frac {(A b+5 a B) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{8 a b^3} \\ & = \frac {(A b-a B) x^{5/2}}{3 a b (a+b x)^3}-\frac {(A b+5 a B) x^{3/2}}{12 a b^2 (a+b x)^2}-\frac {(A b+5 a B) \sqrt {x}}{8 a b^3 (a+b x)}+\frac {(A b+5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{3/2} b^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.81 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\sqrt {x} \left (15 a^3 B-3 A b^3 x^2+a b^2 x (8 A+33 B x)+a^2 b (3 A+40 B x)\right )}{24 a b^3 (a+b x)^3}+\frac {(A b+5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{3/2} b^{7/2}} \]

[In]

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

[Out]

-1/24*(Sqrt[x]*(15*a^3*B - 3*A*b^3*x^2 + a*b^2*x*(8*A + 33*B*x) + a^2*b*(3*A + 40*B*x)))/(a*b^3*(a + b*x)^3) +
((A*b + 5*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(3/2)*b^(7/2))

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {\frac {\left (A b -11 B a \right ) x^{\frac {5}{2}}}{8 b a}-\frac {\left (A b +5 B a \right ) x^{\frac {3}{2}}}{3 b^{2}}-\frac {\left (A b +5 B a \right ) a \sqrt {x}}{8 b^{3}}}{\left (b x +a \right )^{3}}+\frac {\left (A b +5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{8 b^{3} a \sqrt {b a}}\)	\(96\)
default	\(\frac {\frac {\left (A b -11 B a \right ) x^{\frac {5}{2}}}{8 b a}-\frac {\left (A b +5 B a \right ) x^{\frac {3}{2}}}{3 b^{2}}-\frac {\left (A b +5 B a \right ) a \sqrt {x}}{8 b^{3}}}{\left (b x +a \right )^{3}}+\frac {\left (A b +5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{8 b^{3} a \sqrt {b a}}\)	\(96\)

[In]

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

[Out]

2*(1/16*(A*b-11*B*a)/b/a*x^(5/2)-1/6/b^2*(A*b+5*B*a)*x^(3/2)-1/16*(A*b+5*B*a)/b^3*a*x^(1/2))/(b*x+a)^3+1/8*(A*
b+5*B*a)/b^3/a/(b*a)^(1/2)*arctan(b*x^(1/2)/(b*a)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 411, normalized size of antiderivative = 3.16 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [-\frac {3 \, {\left (5 \, B a^{4} + A a^{3} b + {\left (5 \, B a b^{3} + A b^{4}\right )} x^{3} + 3 \, {\left (5 \, B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 3 \, {\left (5 \, B a^{3} b + A a^{2} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (15 \, B a^{4} b + 3 \, A a^{3} b^{2} + 3 \, {\left (11 \, B a^{2} b^{3} - A a b^{4}\right )} x^{2} + 8 \, {\left (5 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x\right )} \sqrt {x}}{48 \, {\left (a^{2} b^{7} x^{3} + 3 \, a^{3} b^{6} x^{2} + 3 \, a^{4} b^{5} x + a^{5} b^{4}\right )}}, -\frac {3 \, {\left (5 \, B a^{4} + A a^{3} b + {\left (5 \, B a b^{3} + A b^{4}\right )} x^{3} + 3 \, {\left (5 \, B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 3 \, {\left (5 \, B a^{3} b + A a^{2} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (15 \, B a^{4} b + 3 \, A a^{3} b^{2} + 3 \, {\left (11 \, B a^{2} b^{3} - A a b^{4}\right )} x^{2} + 8 \, {\left (5 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x\right )} \sqrt {x}}{24 \, {\left (a^{2} b^{7} x^{3} + 3 \, a^{3} b^{6} x^{2} + 3 \, a^{4} b^{5} x + a^{5} b^{4}\right )}}\right ] \]

[In]

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

[Out]

[-1/48*(3*(5*B*a^4 + A*a^3*b + (5*B*a*b^3 + A*b^4)*x^3 + 3*(5*B*a^2*b^2 + A*a*b^3)*x^2 + 3*(5*B*a^3*b + A*a^2*
b^2)*x)*sqrt(-a*b)*log((b*x - a - 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) + 2*(15*B*a^4*b + 3*A*a^3*b^2 + 3*(11*B*a^2
*b^3 - A*a*b^4)*x^2 + 8*(5*B*a^3*b^2 + A*a^2*b^3)*x)*sqrt(x))/(a^2*b^7*x^3 + 3*a^3*b^6*x^2 + 3*a^4*b^5*x + a^5
*b^4), -1/24*(3*(5*B*a^4 + A*a^3*b + (5*B*a*b^3 + A*b^4)*x^3 + 3*(5*B*a^2*b^2 + A*a*b^3)*x^2 + 3*(5*B*a^3*b +
A*a^2*b^2)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (15*B*a^4*b + 3*A*a^3*b^2 + 3*(11*B*a^2*b^3 - A*a*b^4)
*x^2 + 8*(5*B*a^3*b^2 + A*a^2*b^3)*x)*sqrt(x))/(a^2*b^7*x^3 + 3*a^3*b^6*x^2 + 3*a^4*b^5*x + a^5*b^4)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2261 vs. \(2 (119) = 238\).

Time = 50.41 (sec) , antiderivative size = 2261, normalized size of antiderivative = 17.39 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*(-2*A/(3*x**(3/2)) - 2*B/sqrt(x)), Eq(a, 0) & Eq(b, 0)), ((2*A*x**(5/2)/5 + 2*B*x**(7/2)/7)/a**
4, Eq(b, 0)), ((-2*A/(3*x**(3/2)) - 2*B/sqrt(x))/b**4, Eq(a, 0)), (3*A*a**3*b*log(sqrt(x) - sqrt(-a/b))/(48*a*
*4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) -
3*A*a**3*b*log(sqrt(x) + sqrt(-a/b))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x*
*2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) - 6*A*a**2*b**2*sqrt(x)*sqrt(-a/b)/(48*a**4*b**4*sqrt(-a/b) + 144*a
**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) + 9*A*a**2*b**2*x*log(sqrt(
x) - sqrt(-a/b))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*
b**7*x**3*sqrt(-a/b)) - 9*A*a**2*b**2*x*log(sqrt(x) + sqrt(-a/b))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*s
qrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) - 16*A*a*b**3*x**(3/2)*sqrt(-a/b)/(48*a
**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b))
+ 9*A*a*b**3*x**2*log(sqrt(x) - sqrt(-a/b))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b
**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) - 9*A*a*b**3*x**2*log(sqrt(x) + sqrt(-a/b))/(48*a**4*b**4*sqr
t(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) + 6*A*b**4*x
**(5/2)*sqrt(-a/b)/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*
a*b**7*x**3*sqrt(-a/b)) + 3*A*b**4*x**3*log(sqrt(x) - sqrt(-a/b))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*s
qrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) - 3*A*b**4*x**3*log(sqrt(x) + sqrt(-a/b
))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt
(-a/b)) + 15*B*a**4*log(sqrt(x) - sqrt(-a/b))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2
*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) - 15*B*a**4*log(sqrt(x) + sqrt(-a/b))/(48*a**4*b**4*sqrt(-a
/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) - 30*B*a**3*b*sq
rt(x)*sqrt(-a/b)/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*
b**7*x**3*sqrt(-a/b)) + 45*B*a**3*b*x*log(sqrt(x) - sqrt(-a/b))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqr
t(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) - 45*B*a**3*b*x*log(sqrt(x) + sqrt(-a/b))
/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-
a/b)) - 80*B*a**2*b**2*x**(3/2)*sqrt(-a/b)/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b*
*6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) + 45*B*a**2*b**2*x**2*log(sqrt(x) - sqrt(-a/b))/(48*a**4*b**4*
sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) - 45*B*a*
*2*b**2*x**2*log(sqrt(x) + sqrt(-a/b))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x
**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) - 66*B*a*b**3*x**(5/2)*sqrt(-a/b)/(48*a**4*b**4*sqrt(-a/b) + 144*a
**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)) + 15*B*a*b**3*x**3*log(sqrt
(x) - sqrt(-a/b))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a
*b**7*x**3*sqrt(-a/b)) - 15*B*a*b**3*x**3*log(sqrt(x) + sqrt(-a/b))/(48*a**4*b**4*sqrt(-a/b) + 144*a**3*b**5*x
*sqrt(-a/b) + 144*a**2*b**6*x**2*sqrt(-a/b) + 48*a*b**7*x**3*sqrt(-a/b)), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {3 \, {\left (11 \, B a b^{2} - A b^{3}\right )} x^{\frac {5}{2}} + 8 \, {\left (5 \, B a^{2} b + A a b^{2}\right )} x^{\frac {3}{2}} + 3 \, {\left (5 \, B a^{3} + A a^{2} b\right )} \sqrt {x}}{24 \, {\left (a b^{6} x^{3} + 3 \, a^{2} b^{5} x^{2} + 3 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} + \frac {{\left (5 \, B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b^{3}} \]

[In]

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

[Out]

-1/24*(3*(11*B*a*b^2 - A*b^3)*x^(5/2) + 8*(5*B*a^2*b + A*a*b^2)*x^(3/2) + 3*(5*B*a^3 + A*a^2*b)*sqrt(x))/(a*b^
6*x^3 + 3*a^2*b^5*x^2 + 3*a^3*b^4*x + a^4*b^3) + 1/8*(5*B*a + A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a*b^
3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {{\left (5 \, B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b^{3}} - \frac {33 \, B a b^{2} x^{\frac {5}{2}} - 3 \, A b^{3} x^{\frac {5}{2}} + 40 \, B a^{2} b x^{\frac {3}{2}} + 8 \, A a b^{2} x^{\frac {3}{2}} + 15 \, B a^{3} \sqrt {x} + 3 \, A a^{2} b \sqrt {x}}{24 \, {\left (b x + a\right )}^{3} a b^{3}} \]

[In]

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

[Out]

1/8*(5*B*a + A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a*b^3) - 1/24*(33*B*a*b^2*x^(5/2) - 3*A*b^3*x^(5/2) +
40*B*a^2*b*x^(3/2) + 8*A*a*b^2*x^(3/2) + 15*B*a^3*sqrt(x) + 3*A*a^2*b*sqrt(x))/((b*x + a)^3*a*b^3)

Mupad [B] (verification not implemented)

Time = 9.91 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int \frac {x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (A\,b+5\,B\,a\right )}{8\,a^{3/2}\,b^{7/2}}-\frac {\frac {x^{3/2}\,\left (A\,b+5\,B\,a\right )}{3\,b^2}-\frac {x^{5/2}\,\left (A\,b-11\,B\,a\right )}{8\,a\,b}+\frac {a\,\sqrt {x}\,\left (A\,b+5\,B\,a\right )}{8\,b^3}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]

[In]

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

[Out]

(atan((b^(1/2)*x^(1/2))/a^(1/2))*(A*b + 5*B*a))/(8*a^(3/2)*b^(7/2)) - ((x^(3/2)*(A*b + 5*B*a))/(3*b^2) - (x^(5
/2)*(A*b - 11*B*a))/(8*a*b) + (a*x^(1/2)*(A*b + 5*B*a))/(8*b^3))/(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x)