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

Optimal. Leaf size=71 \[ \frac{\sqrt{a+b x} (2 a B+A b)}{a}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{A (a+b x)^{3/2}}{a x} \]

[Out]

((A*b + 2*a*B)*Sqrt[a + b*x])/a - (A*(a + b*x)^(3/2))/(a*x) - ((A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/S
qrt[a]

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Rubi [A]  time = 0.0330898, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 50, 63, 208} \[ \frac{\sqrt{a+b x} (2 a B+A b)}{a}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{A (a+b x)^{3/2}}{a x} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/x^2,x]

[Out]

((A*b + 2*a*B)*Sqrt[a + b*x])/a - (A*(a + b*x)^(3/2))/(a*x) - ((A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/S
qrt[a]

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0309573, size = 53, normalized size = 0.75 \[ \frac{\sqrt{a+b x} (2 B x-A)}{x}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*(-A + 2*B*x))/x - ((A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a]

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Maple [A]  time = 0.011, size = 50, normalized size = 0.7 \begin{align*} 2\,B\sqrt{bx+a}-{\frac{A}{x}\sqrt{bx+a}}-{(Ab+2\,Ba){\it Artanh} \left ({\sqrt{bx+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*(b*x+a)^(1/2)-A*(b*x+a)^(1/2)/x-(A*b+2*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.60306, size = 296, normalized size = 4.17 \begin{align*} \left [\frac{{\left (2 \, B a + A b\right )} \sqrt{a} x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (2 \, B a x - A a\right )} \sqrt{b x + a}}{2 \, a x}, \frac{{\left (2 \, B a + A b\right )} \sqrt{-a} x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (2 \, B a x - A a\right )} \sqrt{b x + a}}{a x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((2*B*a + A*b)*sqrt(a)*x*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(2*B*a*x - A*a)*sqrt(b*x + a))/
(a*x), ((2*B*a + A*b)*sqrt(-a)*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (2*B*a*x - A*a)*sqrt(b*x + a))/(a*x)]

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Sympy [B]  time = 10.7004, size = 155, normalized size = 2.18 \begin{align*} - \frac{A a b \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{A a b \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{2 A b \operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} - \frac{A \sqrt{a + b x}}{x} + \frac{2 B a \operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} + 2 B \sqrt{a + b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*a*b*sqrt(a**(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 + A*a*b*sqrt(a**(-3))*log(a**2*sqrt(a**(-3)) +
sqrt(a + b*x))/2 + 2*A*b*atan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a) - A*sqrt(a + b*x)/x + 2*B*a*atan(sqrt(a + b*x)
/sqrt(-a))/sqrt(-a) + 2*B*sqrt(a + b*x)

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Giac [A]  time = 1.21623, size = 82, normalized size = 1.15 \begin{align*} \frac{2 \, \sqrt{b x + a} B b - \frac{\sqrt{b x + a} A b}{x} + \frac{{\left (2 \, B a b + A b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*sqrt(b*x + a)*B*b - sqrt(b*x + a)*A*b/x + (2*B*a*b + A*b^2)*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a))/b