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

Optimal. Leaf size=49 \[ \frac{2 B \sqrt{x}}{c}-\frac{2 (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}} \]

[Out]

(2*B*Sqrt[x])/c - (2*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*c^(
3/2))

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Rubi [A]  time = 0.0685541, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 B \sqrt{x}}{c}-\frac{2 (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*Sqrt[x])/c - (2*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*c^(
3/2))

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Rubi in Sympy [A]  time = 9.4049, size = 44, normalized size = 0.9 \[ \frac{2 B \sqrt{x}}{c} + \frac{2 \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{b} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.054656, size = 49, normalized size = 1. \[ \frac{2 B \sqrt{x}}{c}-\frac{2 (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*Sqrt[x])/c - (2*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*c^(
3/2))

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Maple [A]  time = 0.011, size = 53, normalized size = 1.1 \[ 2\,{\frac{B\sqrt{x}}{c}}+2\,{\frac{A}{\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }-2\,{\frac{Bb}{c\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x^(1/2)/c+2/(b*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*A-2/c/(b*c)^(1/2)*arct
an(c*x^(1/2)/(b*c)^(1/2))*B*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(x)/(c*x^2 + b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295513, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{-b c} B \sqrt{x} -{\left (B b - A c\right )} \log \left (\frac{2 \, b c \sqrt{x} + \sqrt{-b c}{\left (c x - b\right )}}{c x + b}\right )}{\sqrt{-b c} c}, \frac{2 \,{\left (\sqrt{b c} B \sqrt{x} +{\left (B b - A c\right )} \arctan \left (\frac{b}{\sqrt{b c} \sqrt{x}}\right )\right )}}{\sqrt{b c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.5932, size = 109, normalized size = 2.22 \[ \frac{2 B \sqrt{x}}{c} - \frac{2 \left (- A c + B b\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt{\frac{b}{c}}} \right )}}{c \sqrt{\frac{b}{c}}} & \text{for}\: \frac{b}{c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{b}{c}}} \right )}}{c \sqrt{- \frac{b}{c}}} & \text{for}\: x > - \frac{b}{c} \wedge \frac{b}{c} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{b}{c}}} \right )}}{c \sqrt{- \frac{b}{c}}} & \text{for}\: x < - \frac{b}{c} \wedge \frac{b}{c} < 0 \end{cases}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270747, size = 53, normalized size = 1.08 \[ \frac{2 \, B \sqrt{x}}{c} - \frac{2 \,{\left (B b - A c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(x)/c - 2*(B*b - A*c)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*c)