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

Optimal. Leaf size=101 \[ \frac{2 \sqrt{b c-a d} (d e-c f) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{c d^{3/2}}-\frac{2 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{c}+\frac{2 f \sqrt{a+b x}}{d} \]

[Out]

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

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Rubi [A]  time = 0.379897, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \sqrt{b c-a d} (d e-c f) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{c d^{3/2}}-\frac{2 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{c}+\frac{2 f \sqrt{a+b x}}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.5042, size = 88, normalized size = 0.87 \[ - \frac{2 \sqrt{a} e \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{c} + \frac{2 f \sqrt{a + b x}}{d} - \frac{2 \sqrt{a d - b c} \left (c f - d e\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{a d - b c}} \right )}}{c d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.399403, size = 101, normalized size = 1. \[ -\frac{2 \sqrt{a d-b c} (c f-d e) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )}{c d^{3/2}}-\frac{2 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{c}+\frac{2 f \sqrt{a+b x}}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.024, size = 103, normalized size = 1. \[ 2\,{\frac{f\sqrt{bx+a}}{d}}-2\,{\frac{e\sqrt{a}}{c}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) }-2\,{\frac{acdf-a{d}^{2}e-b{c}^{2}f+bcde}{dc\sqrt{ \left ( ad-bc \right ) d}}{\it Artanh} \left ({\frac{\sqrt{bx+a}d}{\sqrt{ \left ( ad-bc \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284557, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{a} d e \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{b x + a} c f -{\left (d e - c f\right )} \sqrt{-\frac{b c - a d}{d}} \log \left (\frac{b d x - b c + 2 \, a d - 2 \, \sqrt{b x + a} d \sqrt{-\frac{b c - a d}{d}}}{d x + c}\right )}{c d}, -\frac{2 \, \sqrt{-a} d e \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) - 2 \, \sqrt{b x + a} c f +{\left (d e - c f\right )} \sqrt{-\frac{b c - a d}{d}} \log \left (\frac{b d x - b c + 2 \, a d - 2 \, \sqrt{b x + a} d \sqrt{-\frac{b c - a d}{d}}}{d x + c}\right )}{c d}, \frac{\sqrt{a} d e \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{b x + a} c f + 2 \,{\left (d e - c f\right )} \sqrt{\frac{b c - a d}{d}} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{\frac{b c - a d}{d}}}\right )}{c d}, -\frac{2 \,{\left (\sqrt{-a} d e \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) - \sqrt{b x + a} c f -{\left (d e - c f\right )} \sqrt{\frac{b c - a d}{d}} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{\frac{b c - a d}{d}}}\right )\right )}}{c d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 37.9135, size = 287, normalized size = 2.84 \[ - \frac{2 a e \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right )}{c} + \frac{2 f \sqrt{a + b x}}{d} - \frac{2 \left (a d - b c\right ) \left (c f - d e\right ) \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- \frac{a d - b c}{d}}} \right )}}{d \sqrt{- \frac{a d - b c}{d}}} & \text{for}\: - \frac{a d - b c}{d} > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{- \frac{- a d + b c}{d}}} \right )}}{d \sqrt{- \frac{- a d + b c}{d}}} & \text{for}\: a + b x > - \frac{- a d + b c}{d} \wedge - \frac{a d - b c}{d} < 0 \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{- \frac{- a d + b c}{d}}} \right )}}{d \sqrt{- \frac{- a d + b c}{d}}} & \text{for}\: - \frac{a d - b c}{d} < 0 \wedge a + b x < - \frac{- a d + b c}{d} \end{cases}\right )}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21754, size = 151, normalized size = 1.5 \[ \frac{2 \, a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) e}{\sqrt{-a} c} + \frac{2 \, \sqrt{b x + a} f}{d} - \frac{2 \,{\left (b c^{2} f - a c d f - b c d e + a d^{2} e\right )} \arctan \left (\frac{\sqrt{b x + a} d}{\sqrt{b c d - a d^{2}}}\right )}{\sqrt{b c d - a d^{2}} c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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