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

Optimal. Leaf size=140 \[ -\frac{2 b \sqrt{c+d x}}{a^2 (a+b x)}-\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 \sqrt{b c-a d}}+\frac{(4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 \sqrt{c}}-\frac{\sqrt{c+d x}}{a x (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.158645, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {99, 151, 156, 63, 208} \[ -\frac{2 b \sqrt{c+d x}}{a^2 (a+b x)}-\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 \sqrt{b c-a d}}+\frac{(4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 \sqrt{c}}-\frac{\sqrt{c+d x}}{a x (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 99

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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{c+d x}}{x^2 (a+b x)^2} \, dx &=-\frac{\sqrt{c+d x}}{a x (a+b x)}+\frac{\int \frac{\frac{1}{2} (-4 b c+a d)-\frac{3 b d x}{2}}{x (a+b x)^2 \sqrt{c+d x}} \, dx}{a}\\ &=-\frac{2 b \sqrt{c+d x}}{a^2 (a+b x)}-\frac{\sqrt{c+d x}}{a x (a+b x)}+\frac{\int \frac{-\frac{1}{2} (b c-a d) (4 b c-a d)-b d (b c-a d) x}{x (a+b x) \sqrt{c+d x}} \, dx}{a^2 (b c-a d)}\\ &=-\frac{2 b \sqrt{c+d x}}{a^2 (a+b x)}-\frac{\sqrt{c+d x}}{a x (a+b x)}+\frac{(b (4 b c-3 a d)) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{2 a^3}-\frac{(4 b c-a d) \int \frac{1}{x \sqrt{c+d x}} \, dx}{2 a^3}\\ &=-\frac{2 b \sqrt{c+d x}}{a^2 (a+b x)}-\frac{\sqrt{c+d x}}{a x (a+b x)}+\frac{(b (4 b c-3 a d)) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{a^3 d}-\frac{(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 )}{a^3 d}\\ &=-\frac{2 b \sqrt{c+d x}}{a^2 (a+b x)}-\frac{\sqrt{c+d x}}{a x (a+b x)}+\frac{(4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 \sqrt{c}}-\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 \sqrt{b c-a d}}\\ \end{align*}

Mathematica [A]  time = 0.224381, size = 169, normalized size = 1.21 \[ \frac{\sqrt{c} \left (a (a+2 b x) \sqrt{c+d x} (b c-a d)+\sqrt{b} x (a+b x) (4 b c-3 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )\right )-x (a+b x) \left (a^2 d^2-5 a b c d+4 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 \sqrt{c} x (a+b x) (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*x*(a + b*x)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]) + Sqrt[c]*(a*(b*c - a*d)*(a +
2*b*x)*Sqrt[c + d*x] + Sqrt[b]*(4*b*c - 3*a*d)*Sqrt[b*c - a*d]*x*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sq
rt[b*c - a*d]]))/(a^3*Sqrt[c]*(-(b*c) + a*d)*x*(a + b*x))

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Maple [A]  time = 0.017, size = 167, normalized size = 1.2 \begin{align*} -{\frac{1}{{a}^{2}x}\sqrt{dx+c}}-{\frac{d}{{a}^{2}}{\it Artanh} \left ({\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+4\,{\frac{\sqrt{c}b}{{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-{\frac{bd}{{a}^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}-3\,{\frac{bd}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+4\,{\frac{{b}^{2}c}{{a}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^2*(d*x+c)^(1/2)/x-d/a^2/c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))+4/a^3*c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2
))*b-d/a^2*b*(d*x+c)^(1/2)/(b*d*x+a*d)-3*d/a^2*b/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2
))+4/a^3*b^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*c

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.31996, size = 1750, normalized size = 12.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(((4*b^2*c^2 - 3*a*b*c*d)*x^2 + (4*a*b*c^2 - 3*a^2*c*d)*x)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d
+ 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) + ((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*d)*x)*
sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(2*a*b*c*x + a^2*c)*sqrt(d*x + c))/(a^3*b*c*x^2 + a^4
*c*x), -1/2*(2*((4*b^2*c^2 - 3*a*b*c*d)*x^2 + (4*a*b*c^2 - 3*a^2*c*d)*x)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a
*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) + ((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*d)*x)*sqrt(c)*
log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(2*a*b*c*x + a^2*c)*sqrt(d*x + c))/(a^3*b*c*x^2 + a^4*c*x), -
1/2*(2*((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*d)*x)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) + ((4*b^2*c^2 -
3*a*b*c*d)*x^2 + (4*a*b*c^2 - 3*a^2*c*d)*x)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d + 2*(b*c - a*d)*sqrt
(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) + 2*(2*a*b*c*x + a^2*c)*sqrt(d*x + c))/(a^3*b*c*x^2 + a^4*c*x), -(((
4*b^2*c^2 - 3*a*b*c*d)*x^2 + (4*a*b*c^2 - 3*a^2*c*d)*x)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(d*x + c)
*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) + ((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*d)*x)*sqrt(-c)*arctan(sqrt(d*x
+ c)*sqrt(-c)/c) + (2*a*b*c*x + a^2*c)*sqrt(d*x + c))/(a^3*b*c*x^2 + a^4*c*x)]

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Sympy [B]  time = 64.7382, size = 790, normalized size = 5.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**2*c*d*sqrt(c + d*x)/(2*a**4*d**2 - 2*a**3*b*c*d + 2*a**3*b*d**2*x - 2*a**2*b**2*c*d*x) - 2*b*d**2*sqrt(c
+ d*x)/(2*a**3*d**2 - 2*a**2*b*c*d + 2*a**2*b*d**2*x - 2*a*b**2*c*d*x) + b*d**2*sqrt(-1/(b*(a*d - b*c)**3))*lo
g(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d -
b*c)**3)) + sqrt(c + d*x))/(2*a) - b*d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3
)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*a) - b*
*2*c*d*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*
c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*a**2) + b**2*c*d*sqrt(-1/(b*(a*d - b*c)**3
))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a
*d - b*c)**3)) + sqrt(c + d*x))/(2*a**2) - c*d*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(c + d*x))/(2*a**2)
+ c*d*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(c + d*x))/(2*a**2) - 2*d*atan(sqrt(c + d*x)/sqrt(a*d/b - c)
)/(a**2*sqrt(a*d/b - c)) + 2*d*atan(sqrt(c + d*x)/sqrt(-c))/(a**2*sqrt(-c)) - sqrt(c + d*x)/(a**2*x) + 4*b*c*a
tan(sqrt(c + d*x)/sqrt(a*d/b - c))/(a**3*sqrt(a*d/b - c)) - 4*b*c*atan(sqrt(c + d*x)/sqrt(-c))/(a**3*sqrt(-c))

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Giac [A]  time = 1.21374, size = 224, normalized size = 1.6 \begin{align*} \frac{{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3}} - \frac{{\left (4 \, b c - a d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c}} - \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b d - 2 \, \sqrt{d x + c} b c d + \sqrt{d x + c} a d^{2}}{{\left ({\left (d x + c\right )}^{2} b - 2 \,{\left (d x + c\right )} b c + b c^{2} +{\left (d x + c\right )} a d - a c d\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*b^2*c - 3*a*b*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^3) - (4*b*c - a*d)*ar
ctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)) - (2*(d*x + c)^(3/2)*b*d - 2*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*a
*d^2)/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 + (d*x + c)*a*d - a*c*d)*a^2)