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

Optimal. Leaf size=103 \[ \frac {\log (x) \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b B x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]

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Rubi [A]  time = 0.04, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \begin {gather*} \frac {\log (x) \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b B x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

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]

Rubi steps

\begin {align*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right ) (A+B x)}{x^2} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b^2 B+\frac {a A b}{x^2}+\frac {b (A b+a B)}{x}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b B x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {(A b+a B) \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 44, normalized size = 0.43 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (x \log (x) (a B+A b)-a A+b B x^2\right )}{x (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 1.13, size = 1028, normalized size = 9.98 \begin {gather*} \frac {-2 A x^2 \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right ) b^3-2 a A x \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right ) b^2+2 a A \sqrt {b^2} x b+2 A \sqrt {b^2} x \sqrt {a^2+2 b x a+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right ) b-2 a A \sqrt {a^2+2 b x a+b^2 x^2} b}{\left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )}-\frac {a \sqrt {b^2} B \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )}{2 b}+\frac {-2 \left (b^2\right )^{3/2} B x^3-3 a b \sqrt {b^2} B x^2+2 a b^2 B \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 b x a+b^2 x^2}}{a}\right ) x^2-A \left (b^2\right )^{3/2} \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) x^2-A \left (b^2\right )^{3/2} \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) x^2+2 b^2 B \sqrt {a^2+2 b x a+b^2 x^2} x^2-a^2 \sqrt {b^2} B x+2 a^2 b B \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 b x a+b^2 x^2}}{a}\right ) x-2 a \sqrt {b^2} B \sqrt {a^2+2 b x a+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 b x a+b^2 x^2}}{a}\right ) x-a A b \sqrt {b^2} \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) x+A b^2 \sqrt {a^2+2 b x a+b^2 x^2} \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) x-a A b \sqrt {b^2} \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) x+A b^2 \sqrt {a^2+2 b x a+b^2 x^2} \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) x+a b B \sqrt {a^2+2 b x a+b^2 x^2} x+2 a^2 A \sqrt {b^2}}{\left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )}-\frac {a \sqrt {b^2} B \log \left (-a b-\sqrt {b^2} x b+\sqrt {a^2+2 b x a+b^2 x^2} b\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 26, normalized size = 0.25 \begin {gather*} \frac {B b x^{2} + {\left (B a + A b\right )} x \log \relax (x) - A a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 47, normalized size = 0.46 \begin {gather*} B b x \mathrm {sgn}\left (b x + a\right ) + {\left (B a \mathrm {sgn}\left (b x + a\right ) + A b \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right ) - \frac {A a \mathrm {sgn}\left (b x + a\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b*x*sgn(b*x + a) + (B*a*sgn(b*x + a) + A*b*sgn(b*x + a))*log(abs(x)) - A*a*sgn(b*x + a)/x

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maple [C]  time = 0.06, size = 42, normalized size = 0.41 \begin {gather*} \frac {\left (A b x \ln \left (b x \right )+B a x \ln \left (b x \right )+B b \,x^{2}+B a x -A a \right ) \mathrm {csgn}\left (b x +a \right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

csgn(b*x+a)*(A*ln(b*x)*x*b+B*ln(b*x)*x*a+B*b*x^2+B*a*x-A*a)/x

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maxima [B]  time = 0.72, size = 175, normalized size = 1.70 \begin {gather*} \left (-1\right )^{2 \, b^{2} x + 2 \, a b} B a \log \left (2 \, b^{2} x + 2 \, a b\right ) + \left (-1\right )^{2 \, b^{2} x + 2 \, a b} A b \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B a \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1)^(2*b^2*x + 2*a*b)*B*a*log(2*b^2*x + 2*a*b) + (-1)^(2*b^2*x + 2*a*b)*A*b*log(2*b^2*x + 2*a*b) - (-1)^(2*a*
b*x + 2*a^2)*B*a*log(2*a*b*x/abs(x) + 2*a^2/abs(x)) - (-1)^(2*a*b*x + 2*a^2)*A*b*log(2*a*b*x/abs(x) + 2*a^2/ab
s(x)) + sqrt(b^2*x^2 + 2*a*b*x + a^2)*B - sqrt(b^2*x^2 + 2*a*b*x + a^2)*A/x

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mupad [B]  time = 1.46, size = 207, normalized size = 2.01 \begin {gather*} B\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}-B\,\ln \left (a\,b+\frac {a^2}{x}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}\right )\,\sqrt {a^2}+A\,\ln \left (a\,b+\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {b^2}+b^2\,x\right )\,\sqrt {b^2}-\frac {A\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}+\frac {B\,a\,b\,\ln \left (a\,b+\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {b^2}+b^2\,x\right )}{\sqrt {b^2}}-\frac {A\,a\,b\,\ln \left (a\,b+\frac {a^2}{x}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}\right )}{\sqrt {a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.17, size = 19, normalized size = 0.18 \begin {gather*} - \frac {A a}{x} + B b x + \left (A b + B a\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*a/x + B*b*x + (A*b + B*a)*log(x)

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