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

Optimal. Leaf size=209 \[ \frac{A b-a B}{2 a b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (a+b x) (5 A b-a B)}{4 a^3 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 A b-a B}{4 a^2 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (a+b x) (5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2} \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.108958, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{A b-a B}{2 a b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (a+b x) (5 A b-a B)}{4 a^3 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 A b-a B}{4 a^2 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (a+b x) (5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2} \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 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{A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{A+B x}{x^{3/2} \left (a b+b^2 x\right )^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{A b-a B}{2 a b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left ((5 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}{4 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{5 A b-a B}{4 a^2 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{2 a b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (3 (5 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}{8 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{5 A b-a B}{4 a^2 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{2 a b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (5 A b-a B) (a+b x)}{4 a^3 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (3 (5 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}{8 a^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{5 A b-a B}{4 a^2 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{2 a b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (5 A b-a B) (a+b x)}{4 a^3 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (3 (5 A b-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 )}{4 a^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{5 A b-a B}{4 a^2 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{2 a b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (5 A b-a B) (a+b x)}{4 a^3 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 (5 A b-a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2} \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.033138, size = 77, normalized size = 0.37 \[ \frac{a^2 (A b-a B)+(a+b x)^2 (a B-5 A b) \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};-\frac{b x}{a}\right )}{2 a^3 b \sqrt{x} (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.019, size = 214, normalized size = 1. \begin{align*} -{\frac{bx+a}{4\,{a}^{3}} \left ( 15\,A\sqrt{ab}{x}^{2}{b}^{2}+15\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{5/2}{b}^{3}-3\,B\sqrt{ab}{x}^{2}ab-3\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{5/2}a{b}^{2}+30\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3/2}a{b}^{2}-6\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3/2}{a}^{2}b+25\,A\sqrt{ab}xab+15\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) \sqrt{x}{a}^{2}b-5\,B\sqrt{ab}x{a}^{2}-3\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) \sqrt{x}{a}^{3}+8\,A{a}^{2}\sqrt{ab} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{x}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(15*A*(a*b)^(1/2)*x^2*b^2+15*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(5/2)*b^3-3*B*(a*b)^(1/2)*x^2*a*b-3*B*arct
an(x^(1/2)*b/(a*b)^(1/2))*x^(5/2)*a*b^2+30*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(3/2)*a*b^2-6*B*arctan(x^(1/2)*b/
(a*b)^(1/2))*x^(3/2)*a^2*b+25*A*(a*b)^(1/2)*x*a*b+15*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(1/2)*a^2*b-5*B*(a*b)^(
1/2)*x*a^2-3*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(1/2)*a^3+8*A*a^2*(a*b)^(1/2))*(b*x+a)/(a*b)^(1/2)/x^(1/2)/a^3/
((b*x+a)^2)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.64242, size = 718, normalized size = 3.44 \begin{align*} \left [\frac{3 \,{\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \,{\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{2} +{\left (B a^{3} - 5 \, A a^{2} 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 (8 \, A a^{3} b - 3 \,{\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} - 5 \,{\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{8 \,{\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\right )}}, -\frac{3 \,{\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \,{\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{2} +{\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (8 \, A a^{3} b - 3 \,{\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} - 5 \,{\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{4 \,{\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.19913, size = 149, normalized size = 0.71 \begin{align*} \frac{3 \,{\left (B a - 5 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{3} \mathrm{sgn}\left (b x + a\right )} - \frac{2 \, A}{a^{3} \sqrt{x} \mathrm{sgn}\left (b x + a\right )} + \frac{3 \, B a b x^{\frac{3}{2}} - 7 \, A b^{2} x^{\frac{3}{2}} + 5 \, B a^{2} \sqrt{x} - 9 \, A a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{3} \mathrm{sgn}\left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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