3.1233 \(\int \frac{A+B x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx\)

Optimal. Leaf size=86 \[ -\frac{2 (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c} \sqrt{c d-b e}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b \sqrt{d}} \]

[Out]

(-2*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]) - (2*(b*B - A*c)*ArcTanh[(Sqrt
[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*Sqrt[c*d - b*e])

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Rubi [A]  time = 0.200456, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{2 (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c} \sqrt{c d-b e}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)),x]

[Out]

(-2*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]) - (2*(b*B - A*c)*ArcTanh[(Sqrt
[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*Sqrt[c*d - b*e])

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Rubi in Sympy [A]  time = 22.9647, size = 76, normalized size = 0.88 \[ - \frac{2 A \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b \sqrt{d}} - \frac{2 \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b \sqrt{c} \sqrt{b e - c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*atanh(sqrt(d + e*x)/sqrt(d))/(b*sqrt(d)) - 2*(A*c - B*b)*atan(sqrt(c)*sqrt(
d + e*x)/sqrt(b*e - c*d))/(b*sqrt(c)*sqrt(b*e - c*d))

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Mathematica [A]  time = 0.193641, size = 86, normalized size = 1. \[ -\frac{2 (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c} \sqrt{c d-b e}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)),x]

[Out]

(-2*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]) - (2*(b*B - A*c)*ArcTanh[(Sqrt
[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*Sqrt[c*d - b*e])

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Maple [A]  time = 0.016, size = 101, normalized size = 1.2 \[ -2\,{\frac{Ac}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+2\,{\frac{B}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{A}{b\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*c+2/((b*e
-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B-2*A*arctanh((e*x+d)
^(1/2)/d^(1/2))/b/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((B*x + A)/((c*x^2 + b*x)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.321084, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (B b - A c\right )} \sqrt{d} \log \left (\frac{\sqrt{c^{2} d - b c e}{\left (c e x + 2 \, c d - b e\right )} + 2 \,{\left (c^{2} d - b c e\right )} \sqrt{e x + d}}{c x + b}\right ) - \sqrt{c^{2} d - b c e} A \log \left (\frac{{\left (e x + 2 \, d\right )} \sqrt{d} - 2 \, \sqrt{e x + d} d}{x}\right )}{\sqrt{c^{2} d - b c e} b \sqrt{d}}, -\frac{2 \,{\left (B b - A c\right )} \sqrt{d} \arctan \left (-\frac{c d - b e}{\sqrt{-c^{2} d + b c e} \sqrt{e x + d}}\right ) - \sqrt{-c^{2} d + b c e} A \log \left (\frac{{\left (e x + 2 \, d\right )} \sqrt{d} - 2 \, \sqrt{e x + d} d}{x}\right )}{\sqrt{-c^{2} d + b c e} b \sqrt{d}}, \frac{2 \, \sqrt{c^{2} d - b c e} A \arctan \left (\frac{d}{\sqrt{e x + d} \sqrt{-d}}\right ) -{\left (B b - A c\right )} \sqrt{-d} \log \left (\frac{\sqrt{c^{2} d - b c e}{\left (c e x + 2 \, c d - b e\right )} + 2 \,{\left (c^{2} d - b c e\right )} \sqrt{e x + d}}{c x + b}\right )}{\sqrt{c^{2} d - b c e} b \sqrt{-d}}, -\frac{2 \,{\left ({\left (B b - A c\right )} \sqrt{-d} \arctan \left (-\frac{c d - b e}{\sqrt{-c^{2} d + b c e} \sqrt{e x + d}}\right ) - \sqrt{-c^{2} d + b c e} A \arctan \left (\frac{d}{\sqrt{e x + d} \sqrt{-d}}\right )\right )}}{\sqrt{-c^{2} d + b c e} b \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[-((B*b - A*c)*sqrt(d)*log((sqrt(c^2*d - b*c*e)*(c*e*x + 2*c*d - b*e) + 2*(c^2*d
- b*c*e)*sqrt(e*x + d))/(c*x + b)) - sqrt(c^2*d - b*c*e)*A*log(((e*x + 2*d)*sqr
t(d) - 2*sqrt(e*x + d)*d)/x))/(sqrt(c^2*d - b*c*e)*b*sqrt(d)), -(2*(B*b - A*c)*s
qrt(d)*arctan(-(c*d - b*e)/(sqrt(-c^2*d + b*c*e)*sqrt(e*x + d))) - sqrt(-c^2*d +
b*c*e)*A*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x + d)*d)/x))/(sqrt(-c^2*d + b*c*e
)*b*sqrt(d)), (2*sqrt(c^2*d - b*c*e)*A*arctan(d/(sqrt(e*x + d)*sqrt(-d))) - (B*b
- A*c)*sqrt(-d)*log((sqrt(c^2*d - b*c*e)*(c*e*x + 2*c*d - b*e) + 2*(c^2*d - b*c
*e)*sqrt(e*x + d))/(c*x + b)))/(sqrt(c^2*d - b*c*e)*b*sqrt(-d)), -2*((B*b - A*c)
*sqrt(-d)*arctan(-(c*d - b*e)/(sqrt(-c^2*d + b*c*e)*sqrt(e*x + d))) - sqrt(-c^2*
d + b*c*e)*A*arctan(d/(sqrt(e*x + d)*sqrt(-d))))/(sqrt(-c^2*d + b*c*e)*b*sqrt(-d
))]

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Sympy [A]  time = 18.0295, size = 314, normalized size = 3.65 \[ \frac{2 A \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{d}} \sqrt{d + e x}} \right )}}{d \sqrt{- \frac{1}{d}}} & \text{for}\: - \frac{1}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{d + e x} \sqrt{\frac{1}{d}}} \right )}}{d \sqrt{\frac{1}{d}}} & \text{for}\: - \frac{1}{d} < 0 \wedge \frac{1}{d} < \frac{1}{d + e x} \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{d + e x} \sqrt{\frac{1}{d}}} \right )}}{d \sqrt{\frac{1}{d}}} & \text{for}\: \frac{1}{d} > \frac{1}{d + e x} \wedge - \frac{1}{d} < 0 \end{cases}\right )}{b} - \frac{2 \left (- A c + B b\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{c}{b e - c d}} \sqrt{d + e x}} \right )}}{\sqrt{\frac{c}{b e - c d}} \left (b e - c d\right )} & \text{for}\: \frac{c}{b e - c d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{c}{b e - c d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{c}{b e - c d}} \left (b e - c d\right )} & \text{for}\: \frac{1}{d + e x} > - \frac{c}{b e - c d} \wedge \frac{c}{b e - c d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{c}{b e - c d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{c}{b e - c d}} \left (b e - c d\right )} & \text{for}\: \frac{c}{b e - c d} < 0 \wedge \frac{1}{d + e x} < - \frac{c}{b e - c d} \end{cases}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282487, size = 107, normalized size = 1.24 \[ \frac{2 \,{\left (B b - A c\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b} + \frac{2 \, A \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

2*(B*b - A*c)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)
*b) + 2*A*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d))