3.1752 \(\int \frac{a+b x}{(c+d x) \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=74 \[ \frac{2 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} \sqrt{d e-c f}}+\frac{2 b \sqrt{e+f x}}{d f} \]

[Out]

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

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Rubi [A]  time = 0.120619, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} \sqrt{d e-c f}}+\frac{2 b \sqrt{e+f x}}{d f} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((c + d*x)*Sqrt[e + f*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.7004, size = 63, normalized size = 0.85 \[ \frac{2 b \sqrt{e + f x}}{d f} + \frac{2 \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{3}{2}} \sqrt{c f - d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(d*x+c)/(f*x+e)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.134985, size = 74, normalized size = 1. \[ \frac{2 b \sqrt{e+f x}}{d f}-\frac{2 (a d-b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} \sqrt{d e-c f}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((c + d*x)*Sqrt[e + f*x]),x]

[Out]

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

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Maple [A]  time = 0.013, size = 96, normalized size = 1.3 \[ 2\,{\frac{b\sqrt{fx+e}}{df}}+2\,{\frac{a}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{bc}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(d*x+c)/(f*x+e)^(1/2),x)

[Out]

2*b*(f*x+e)^(1/2)/d/f+2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)
^(1/2))*a-2/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b*
c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((d*x + c)*sqrt(f*x + e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219801, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c - a d\right )} f \log \left (\frac{\sqrt{d^{2} e - c d f}{\left (d f x + 2 \, d e - c f\right )} - 2 \,{\left (d^{2} e - c d f\right )} \sqrt{f x + e}}{d x + c}\right ) - 2 \, \sqrt{d^{2} e - c d f} \sqrt{f x + e} b}{\sqrt{d^{2} e - c d f} d f}, \frac{2 \,{\left ({\left (b c - a d\right )} f \arctan \left (-\frac{d e - c f}{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ) + \sqrt{-d^{2} e + c d f} \sqrt{f x + e} b\right )}}{\sqrt{-d^{2} e + c d f} d f}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((d*x + c)*sqrt(f*x + e)),x, algorithm="fricas")

[Out]

[-((b*c - a*d)*f*log((sqrt(d^2*e - c*d*f)*(d*f*x + 2*d*e - c*f) - 2*(d^2*e - c*d
*f)*sqrt(f*x + e))/(d*x + c)) - 2*sqrt(d^2*e - c*d*f)*sqrt(f*x + e)*b)/(sqrt(d^2
*e - c*d*f)*d*f), 2*((b*c - a*d)*f*arctan(-(d*e - c*f)/(sqrt(-d^2*e + c*d*f)*sqr
t(f*x + e))) + sqrt(-d^2*e + c*d*f)*sqrt(f*x + e)*b)/(sqrt(-d^2*e + c*d*f)*d*f)]

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Sympy [A]  time = 12.8869, size = 211, normalized size = 2.85 \[ \frac{2 b \sqrt{e + f x}}{d f} - \frac{2 \left (a d - b c\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{\frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{d}{c f - d e} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{- \frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{1}{e + f x} > - \frac{d}{c f - d e} \wedge \frac{d}{c f - d e} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{- \frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{d}{c f - d e} < 0 \wedge \frac{1}{e + f x} < - \frac{d}{c f - d e} \end{cases}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(d*x+c)/(f*x+e)**(1/2),x)

[Out]

2*b*sqrt(e + f*x)/(d*f) - 2*(a*d - b*c)*Piecewise((atan(1/(sqrt(d/(c*f - d*e))*s
qrt(e + f*x)))/(sqrt(d/(c*f - d*e))*(c*f - d*e)), d/(c*f - d*e) > 0), (-acoth(1/
(sqrt(-d/(c*f - d*e))*sqrt(e + f*x)))/(sqrt(-d/(c*f - d*e))*(c*f - d*e)), (d/(c*
f - d*e) < 0) & (1/(e + f*x) > -d/(c*f - d*e))), (-atanh(1/(sqrt(-d/(c*f - d*e))
*sqrt(e + f*x)))/(sqrt(-d/(c*f - d*e))*(c*f - d*e)), (d/(c*f - d*e) < 0) & (1/(e
+ f*x) < -d/(c*f - d*e))))/d

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GIAC/XCAS [A]  time = 0.213691, size = 95, normalized size = 1.28 \[ -\frac{2 \,{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d} + \frac{2 \, \sqrt{f x + e} b}{d f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((d*x + c)*sqrt(f*x + e)),x, algorithm="giac")

[Out]

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