3.454 \(\int \frac{(a+b x)^2}{x^2 \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{a^2 \sqrt{c+d x}}{c x}-\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{2 b^2 \sqrt{c+d x}}{d} \]

[Out]

(2*b^2*Sqrt[c + d*x])/d - (a^2*Sqrt[c + d*x])/(c*x) - (a*(4*b*c - a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/c^(3/2)

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Rubi [A]  time = 0.0395309, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {89, 80, 63, 208} \[ -\frac{a^2 \sqrt{c+d x}}{c x}-\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{2 b^2 \sqrt{c+d x}}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(x^2*Sqrt[c + d*x]),x]

[Out]

(2*b^2*Sqrt[c + d*x])/d - (a^2*Sqrt[c + d*x])/(c*x) - (a*(4*b*c - a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/c^(3/2)

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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{(a+b x)^2}{x^2 \sqrt{c+d x}} \, dx &=-\frac{a^2 \sqrt{c+d x}}{c x}+\frac{\int \frac{\frac{1}{2} a (4 b c-a d)+b^2 c x}{x \sqrt{c+d x}} \, dx}{c}\\ &=\frac{2 b^2 \sqrt{c+d x}}{d}-\frac{a^2 \sqrt{c+d x}}{c x}+\frac{(a (4 b c-a d)) \int \frac{1}{x \sqrt{c+d x}} \, dx}{2 c}\\ &=\frac{2 b^2 \sqrt{c+d x}}{d}-\frac{a^2 \sqrt{c+d x}}{c x}+\frac{(a (4 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{c d}\\ &=\frac{2 b^2 \sqrt{c+d x}}{d}-\frac{a^2 \sqrt{c+d x}}{c x}-\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0496778, size = 65, normalized size = 0.92 \[ \frac{\sqrt{c+d x} \left (2 b^2 c x-a^2 d\right )}{c d x}+\frac{a (a d-4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(x^2*Sqrt[c + d*x]),x]

[Out]

((-(a^2*d) + 2*b^2*c*x)*Sqrt[c + d*x])/(c*d*x) + (a*(-4*b*c + a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/c^(3/2)

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Maple [A]  time = 0.011, size = 63, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{d} \left ({b}^{2}\sqrt{dx+c}+ad \left ( -1/2\,{\frac{a\sqrt{dx+c}}{cx}}+1/2\,{\frac{ad-4\,bc}{{c}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/x^2/(d*x+c)^(1/2),x)

[Out]

2/d*(b^2*(d*x+c)^(1/2)+a*d*(-1/2*a/c*(d*x+c)^(1/2)/x+1/2*(a*d-4*b*c)/c^(3/2)*arctanh((d*x+c)^(1/2)/c^(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)^2/x^2/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.42041, size = 351, normalized size = 4.94 \begin{align*} \left [-\frac{{\left (4 \, a b c d - a^{2} d^{2}\right )} \sqrt{c} x \log \left (\frac{d x + 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \,{\left (2 \, b^{2} c^{2} x - a^{2} c d\right )} \sqrt{d x + c}}{2 \, c^{2} d x}, \frac{{\left (4 \, a b c d - a^{2} d^{2}\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c}}{c}\right ) +{\left (2 \, b^{2} c^{2} x - a^{2} c d\right )} \sqrt{d x + c}}{c^{2} d x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*((4*a*b*c*d - a^2*d^2)*sqrt(c)*x*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(2*b^2*c^2*x - a^2*c*d
)*sqrt(d*x + c))/(c^2*d*x), ((4*a*b*c*d - a^2*d^2)*sqrt(-c)*x*arctan(sqrt(d*x + c)*sqrt(-c)/c) + (2*b^2*c^2*x
- a^2*c*d)*sqrt(d*x + c))/(c^2*d*x)]

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Sympy [A]  time = 55.5798, size = 109, normalized size = 1.54 \begin{align*} - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x} + 1}}{c \sqrt{x}} + \frac{a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} \sqrt{x}} \right )}}{c^{\frac{3}{2}}} + \frac{4 a b \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{c}} \sqrt{c + d x}} \right )}}{c \sqrt{- \frac{1}{c}}} + b^{2} \left (\begin{cases} \frac{x}{\sqrt{c}} & \text{for}\: d = 0 \\\frac{2 \sqrt{c + d x}}{d} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/x**2/(d*x+c)**(1/2),x)

[Out]

-a**2*sqrt(d)*sqrt(c/(d*x) + 1)/(c*sqrt(x)) + a**2*d*asinh(sqrt(c)/(sqrt(d)*sqrt(x)))/c**(3/2) + 4*a*b*atan(1/
(sqrt(-1/c)*sqrt(c + d*x)))/(c*sqrt(-1/c)) + b**2*Piecewise((x/sqrt(c), Eq(d, 0)), (2*sqrt(c + d*x)/d, True))

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Giac [A]  time = 1.17414, size = 100, normalized size = 1.41 \begin{align*} \frac{2 \, \sqrt{d x + c} b^{2} - \frac{\sqrt{d x + c} a^{2} d}{c x} + \frac{{\left (4 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*sqrt(d*x + c)*b^2 - sqrt(d*x + c)*a^2*d/(c*x) + (4*a*b*c*d - a^2*d^2)*arctan(sqrt(d*x + c)/sqrt(-c))/(sqrt(
-c)*c))/d