3.1232 \(\int \frac{(A+B x) \sqrt{d+e x}}{b x+c x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.353731, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 (b B-A c) \sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{3/2}}-\frac{2 A \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 B \sqrt{d+e x}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.2739, size = 88, normalized size = 0.87 \[ - \frac{2 A \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b} + \frac{2 B \sqrt{d + e x}}{c} + \frac{2 \left (A c - B b\right ) \sqrt{b e - c d} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*sqrt(d)*atanh(sqrt(d + e*x)/sqrt(d))/b + 2*B*sqrt(d + e*x)/c + 2*(A*c - B*b
)*sqrt(b*e - c*d)*atan(sqrt(c)*sqrt(d + e*x)/sqrt(b*e - c*d))/(b*c**(3/2))

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Mathematica [A]  time = 0.348006, size = 101, normalized size = 1. \[ \frac{2 (A c-b B) \sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{3/2}}-\frac{2 A \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 B \sqrt{d+e x}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 196, normalized size = 1.9 \[ 2\,{\frac{B\sqrt{ex+d}}{c}}+2\,{\frac{Ae}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{Acd}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{bBe}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+2\,{\frac{Bd}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{A\sqrt{d}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(e*x+d)^(1/2)/c+2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(
1/2))*A*e-2*c/b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*
A*d-2/c*b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*e+2/
((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d-2*A*arctanh(
(e*x+d)^(1/2)/d^(1/2))*d^(1/2)/b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.359076, size = 1, normalized size = 0.01 \[ \left [\frac{A c \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) + 2 \, \sqrt{e x + d} B b -{\left (B b - A c\right )} \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right )}{b c}, \frac{A c \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) + 2 \, \sqrt{e x + d} B b - 2 \,{\left (B b - A c\right )} \sqrt{-\frac{c d - b e}{c}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d - b e}{c}}}\right )}{b c}, -\frac{2 \, A c \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-d}}\right ) - 2 \, \sqrt{e x + d} B b +{\left (B b - A c\right )} \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right )}{b c}, -\frac{2 \,{\left (A c \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-d}}\right ) - \sqrt{e x + d} B b +{\left (B b - A c\right )} \sqrt{-\frac{c d - b e}{c}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d - b e}{c}}}\right )\right )}}{b c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 16.0683, size = 277, normalized size = 2.74 \[ \frac{2 \left (- \frac{A d e \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} & \text{for}\: - d > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{\sqrt{d}} & \text{for}\: - d < 0 \wedge d < d + e x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{\sqrt{d}} & \text{for}\: d > d + e x \wedge - d < 0 \end{cases}\right )}{b} + \frac{B e \sqrt{d + e x}}{c} - \frac{e \left (- A c + B b\right ) \left (b e - c d\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{c \sqrt{\frac{b e - c d}{c}}} & \text{for}\: \frac{b e - c d}{c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- b e + c d}{c}}} \right )}}{c \sqrt{\frac{- b e + c d}{c}}} & \text{for}\: d + e x > \frac{- b e + c d}{c} \wedge \frac{b e - c d}{c} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- b e + c d}{c}}} \right )}}{c \sqrt{\frac{- b e + c d}{c}}} & \text{for}\: \frac{b e - c d}{c} < 0 \wedge d + e x < \frac{- b e + c d}{c} \end{cases}\right )}{b c}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-A*d*e*Piecewise((-atan(sqrt(d + e*x)/sqrt(-d))/sqrt(-d), -d > 0), (acoth(sqr
t(d + e*x)/sqrt(d))/sqrt(d), (-d < 0) & (d < d + e*x)), (atanh(sqrt(d + e*x)/sqr
t(d))/sqrt(d), (-d < 0) & (d > d + e*x)))/b + B*e*sqrt(d + e*x)/c - e*(-A*c + B*
b)*(b*e - c*d)*Piecewise((atan(sqrt(d + e*x)/sqrt((b*e - c*d)/c))/(c*sqrt((b*e -
c*d)/c)), (b*e - c*d)/c > 0), (-acoth(sqrt(d + e*x)/sqrt((-b*e + c*d)/c))/(c*sq
rt((-b*e + c*d)/c)), ((b*e - c*d)/c < 0) & (d + e*x > (-b*e + c*d)/c)), (-atanh(
sqrt(d + e*x)/sqrt((-b*e + c*d)/c))/(c*sqrt((-b*e + c*d)/c)), ((b*e - c*d)/c < 0
) & (d + e*x < (-b*e + c*d)/c)))/(b*c))/e

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GIAC/XCAS [A]  time = 0.281954, size = 157, normalized size = 1.55 \[ \frac{2 \, A d \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} + \frac{2 \, \sqrt{x e + d} B}{c} + \frac{2 \,{\left (B b c d - A c^{2} d - B b^{2} e + A b c e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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