3.1382 \(\int \frac{\sqrt{c+d x}}{(a+b x)^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x}}{b (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.030299, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {47, 63, 208} \[ -\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x}}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^2,x]

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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{c+d x}}{(a+b x)^2} \, dx &=-\frac{\sqrt{c+d x}}{b (a+b x)}+\frac{d \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{2 b}\\ &=-\frac{\sqrt{c+d x}}{b (a+b x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{b}\\ &=-\frac{\sqrt{c+d x}}{b (a+b x)}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}\\ \end{align*}

Mathematica [A]  time = 0.0737274, size = 69, normalized size = 0.99 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{a d-b c}}\right )}{b^{3/2} \sqrt{a d-b c}}-\frac{\sqrt{c+d x}}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 64, normalized size = 0.9 \begin{align*} -{\frac{d}{b \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{d}{b}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.15632, size = 498, normalized size = 7.11 \begin{align*} \left [\frac{\sqrt{b^{2} c - a b d}{\left (b d x + a d\right )} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{b^{2} c - a b d} \sqrt{d x + c}}{b x + a}\right ) - 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{2 \,{\left (a b^{3} c - a^{2} b^{2} d +{\left (b^{4} c - a b^{3} d\right )} x\right )}}, \frac{\sqrt{-b^{2} c + a b d}{\left (b d x + a d\right )} \arctan \left (\frac{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}{b d x + b c}\right ) -{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{a b^{3} c - a^{2} b^{2} d +{\left (b^{4} c - a b^{3} d\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 28.5851, size = 573, normalized size = 8.19 \begin{align*} - \frac{2 a d^{2} \sqrt{c + d x}}{2 a^{2} b d^{2} - 2 a b^{2} c d + 2 a b^{2} d^{2} x - 2 b^{3} c d x} + \frac{a d^{2} \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} \log{\left (- a^{2} d^{2} \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} + \sqrt{c + d x} \right )}}{2 b} - \frac{a d^{2} \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} \log{\left (a^{2} d^{2} \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} + \sqrt{c + d x} \right )}}{2 b} - \frac{c d \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} \log{\left (- a^{2} d^{2} \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} + \sqrt{c + d x} \right )}}{2} + \frac{c d \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} \log{\left (a^{2} d^{2} \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt{- \frac{1}{b \left (a d - b c\right )^{3}}} + \sqrt{c + d x} \right )}}{2} + \frac{2 c d \sqrt{c + d x}}{2 a^{2} d^{2} - 2 a b c d + 2 a b d^{2} x - 2 b^{2} c d x} + \frac{2 d \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d}{b} - c}} \right )}}{b^{2} \sqrt{\frac{a d}{b} - c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*d**2*sqrt(c + d*x)/(2*a**2*b*d**2 - 2*a*b**2*c*d + 2*a*b**2*d**2*x - 2*b**3*c*d*x) + a*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*b) - a*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*b) - 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 + 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 + 2*c*d*sqrt(c + d*x)/(2*a**2*d**2 - 2*a*b*c*d + 2*a*b*d**2*x - 2*b**2*c*d*x
) + 2*d*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b**2*sqrt(a*d/b - c))

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Giac [A]  time = 1.08943, size = 97, normalized size = 1.39 \begin{align*} \frac{d \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b} - \frac{\sqrt{d x + c} d}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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