3.769 \(\int \frac{\sqrt{x} (A+B x)}{(a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=127 \[ \frac{\sqrt{x} (a B+A b)}{8 a^2 b^2 (a+b x)}+\frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2}}-\frac{\sqrt{x} (a B+A b)}{4 a b^2 (a+b x)^2}+\frac{x^{3/2} (A b-a B)}{3 a b (a+b x)^3} \]

[Out]

((A*b - a*B)*x^(3/2))/(3*a*b*(a + b*x)^3) - ((A*b + a*B)*Sqrt[x])/(4*a*b^2*(a + b*x)^2) + ((A*b + a*B)*Sqrt[x]
)/(8*a^2*b^2*(a + b*x)) + ((A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(5/2)*b^(5/2))

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Rubi [A]  time = 0.0570649, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {27, 78, 47, 51, 63, 205} \[ \frac{\sqrt{x} (a B+A b)}{8 a^2 b^2 (a+b x)}+\frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2}}-\frac{\sqrt{x} (a B+A b)}{4 a b^2 (a+b x)^2}+\frac{x^{3/2} (A b-a B)}{3 a b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*x^(3/2))/(3*a*b*(a + b*x)^3) - ((A*b + a*B)*Sqrt[x])/(4*a*b^2*(a + b*x)^2) + ((A*b + a*B)*Sqrt[x]
)/(8*a^2*b^2*(a + b*x)) + ((A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(5/2)*b^(5/2))

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 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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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 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]

Rubi steps

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

Mathematica [C]  time = 0.0259471, size = 60, normalized size = 0.47 \[ \frac{x^{3/2} \left (\frac{3 a^3 (A b-a B)}{(a+b x)^3}+3 (a B+A b) \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};-\frac{b x}{a}\right )\right )}{9 a^4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(3/2)*((3*a^3*(A*b - a*B))/(a + b*x)^3 + 3*(A*b + a*B)*Hypergeometric2F1[3/2, 3, 5/2, -((b*x)/a)]))/(9*a^4*
b)

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Maple [A]  time = 0.018, size = 111, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{ \left ( bx+a \right ) ^{3}} \left ( 1/16\,{\frac{ \left ( Ab+aB \right ){x}^{5/2}}{{a}^{2}}}+1/6\,{\frac{ \left ( Ab-aB \right ){x}^{3/2}}{ab}}-1/16\,{\frac{ \left ( Ab+aB \right ) \sqrt{x}}{{b}^{2}}} \right ) }+{\frac{A}{8\,{a}^{2}b}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{8\,a{b}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89267, size = 846, normalized size = 6.66 \begin{align*} \left [-\frac{3 \,{\left (B a^{4} + A a^{3} b +{\left (B a b^{3} + A b^{4}\right )} x^{3} + 3 \,{\left (B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 3 \,{\left (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 (3 \, B a^{4} b + 3 \, A a^{3} b^{2} - 3 \,{\left (B a^{2} b^{3} + A a b^{4}\right )} x^{2} + 8 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x\right )} \sqrt{x}}{48 \,{\left (a^{3} b^{6} x^{3} + 3 \, a^{4} b^{5} x^{2} + 3 \, a^{5} b^{4} x + a^{6} b^{3}\right )}}, -\frac{3 \,{\left (B a^{4} + A a^{3} b +{\left (B a b^{3} + A b^{4}\right )} x^{3} + 3 \,{\left (B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 3 \,{\left (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 (3 \, B a^{4} b + 3 \, A a^{3} b^{2} - 3 \,{\left (B a^{2} b^{3} + A a b^{4}\right )} x^{2} + 8 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x\right )} \sqrt{x}}{24 \,{\left (a^{3} b^{6} x^{3} + 3 \, a^{4} b^{5} x^{2} + 3 \, a^{5} b^{4} x + a^{6} b^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.18613, size = 143, normalized size = 1.13 \begin{align*} \frac{{\left (B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b^{2}} + \frac{3 \, B a b^{2} x^{\frac{5}{2}} + 3 \, A b^{3} x^{\frac{5}{2}} - 8 \, B a^{2} b x^{\frac{3}{2}} + 8 \, A a b^{2} x^{\frac{3}{2}} - 3 \, B a^{3} \sqrt{x} - 3 \, A a^{2} b \sqrt{x}}{24 \,{\left (b x + a\right )}^{3} a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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