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

Optimal. Leaf size=85 \[ \frac{(b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2} \sqrt{c}}+\frac{b B-3 A c}{b^2 c \sqrt{x}}-\frac{b B-A c}{b c \sqrt{x} (b+c x)} \]

[Out]

(b*B - 3*A*c)/(b^2*c*Sqrt[x]) - (b*B - A*c)/(b*c*Sqrt[x]*(b + c*x)) + ((b*B - 3*
A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(b^(5/2)*Sqrt[c])

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Rubi [A]  time = 0.114939, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2} \sqrt{c}}+\frac{b B-3 A c}{b^2 c \sqrt{x}}-\frac{b B-A c}{b c \sqrt{x} (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

(b*B - 3*A*c)/(b^2*c*Sqrt[x]) - (b*B - A*c)/(b*c*Sqrt[x]*(b + c*x)) + ((b*B - 3*
A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(b^(5/2)*Sqrt[c])

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Rubi in Sympy [A]  time = 14.2394, size = 73, normalized size = 0.86 \[ \frac{A c - B b}{b c \sqrt{x} \left (b + c x\right )} - \frac{3 A c - B b}{b^{2} c \sqrt{x}} - \frac{\left (3 A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{5}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*c - B*b)/(b*c*sqrt(x)*(b + c*x)) - (3*A*c - B*b)/(b**2*c*sqrt(x)) - (3*A*c -
B*b)*atan(sqrt(c)*sqrt(x)/sqrt(b))/(b**(5/2)*sqrt(c))

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Mathematica [A]  time = 0.0830775, size = 67, normalized size = 0.79 \[ \frac{(b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2} \sqrt{c}}+\frac{-2 A b-3 A c x+b B x}{b^2 \sqrt{x} (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A*b + b*B*x - 3*A*c*x)/(b^2*Sqrt[x]*(b + c*x)) + ((b*B - 3*A*c)*ArcTan[(Sqrt
[c]*Sqrt[x])/Sqrt[b]])/(b^(5/2)*Sqrt[c])

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Maple [A]  time = 0.022, size = 87, normalized size = 1. \[ -2\,{\frac{A}{{b}^{2}\sqrt{x}}}-{\frac{Ac}{{b}^{2} \left ( cx+b \right ) }\sqrt{x}}+{\frac{B}{b \left ( cx+b \right ) }\sqrt{x}}-3\,{\frac{Ac}{{b}^{2}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }+{\frac{B}{b}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/b^2/x^(1/2)-1/b^2*x^(1/2)/(c*x+b)*A*c+1/b*x^(1/2)/(c*x+b)*B-3/b^2/(b*c)^(1/
2)*arctan(c*x^(1/2)/(b*c)^(1/2))*A*c+1/b/(b*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(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(x)/(c*x^2 + b*x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310375, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (B b^{2} - 3 \, A b c +{\left (B b c - 3 \, A c^{2}\right )} x\right )} \sqrt{x} \log \left (-\frac{2 \, b c \sqrt{x} - \sqrt{-b c}{\left (c x - b\right )}}{c x + b}\right ) + 2 \,{\left (2 \, A b -{\left (B b - 3 \, A c\right )} x\right )} \sqrt{-b c}}{2 \,{\left (b^{2} c x + b^{3}\right )} \sqrt{-b c} \sqrt{x}}, -\frac{{\left (B b^{2} - 3 \, A b c +{\left (B b c - 3 \, A c^{2}\right )} x\right )} \sqrt{x} \arctan \left (\frac{b}{\sqrt{b c} \sqrt{x}}\right ) +{\left (2 \, A b -{\left (B b - 3 \, A c\right )} x\right )} \sqrt{b c}}{{\left (b^{2} c x + b^{3}\right )} \sqrt{b c} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((B*b^2 - 3*A*b*c + (B*b*c - 3*A*c^2)*x)*sqrt(x)*log(-(2*b*c*sqrt(x) - sqr
t(-b*c)*(c*x - b))/(c*x + b)) + 2*(2*A*b - (B*b - 3*A*c)*x)*sqrt(-b*c))/((b^2*c*
x + b^3)*sqrt(-b*c)*sqrt(x)), -((B*b^2 - 3*A*b*c + (B*b*c - 3*A*c^2)*x)*sqrt(x)*
arctan(b/(sqrt(b*c)*sqrt(x))) + (2*A*b - (B*b - 3*A*c)*x)*sqrt(b*c))/((b^2*c*x +
b^3)*sqrt(b*c)*sqrt(x))]

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Sympy [A]  time = 12.8336, size = 309, normalized size = 3.64 \[ - \frac{2 A c \sqrt{x}}{2 b^{3} + 2 b^{2} c x} + \frac{A c \sqrt{- \frac{1}{b^{3} c}} \log{\left (- b^{2} \sqrt{- \frac{1}{b^{3} c}} + \sqrt{x} \right )}}{2 b} - \frac{A c \sqrt{- \frac{1}{b^{3} c}} \log{\left (b^{2} \sqrt{- \frac{1}{b^{3} c}} + \sqrt{x} \right )}}{2 b} - \frac{2 A c \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt{\frac{b}{c}}} \right )}}{c \sqrt{\frac{b}{c}}} & \text{for}\: \frac{b}{c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{b}{c}}} \right )}}{c \sqrt{- \frac{b}{c}}} & \text{for}\: x > - \frac{b}{c} \wedge \frac{b}{c} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{b}{c}}} \right )}}{c \sqrt{- \frac{b}{c}}} & \text{for}\: x < - \frac{b}{c} \wedge \frac{b}{c} < 0 \end{cases}\right )}{b^{2}} - \frac{2 A}{b^{2} \sqrt{x}} + \frac{2 B \sqrt{x}}{2 b^{2} + 2 b c x} - \frac{B \sqrt{- \frac{1}{b^{3} c}} \log{\left (- b^{2} \sqrt{- \frac{1}{b^{3} c}} + \sqrt{x} \right )}}{2} + \frac{B \sqrt{- \frac{1}{b^{3} c}} \log{\left (b^{2} \sqrt{- \frac{1}{b^{3} c}} + \sqrt{x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*c*sqrt(x)/(2*b**3 + 2*b**2*c*x) + A*c*sqrt(-1/(b**3*c))*log(-b**2*sqrt(-1/(
b**3*c)) + sqrt(x))/(2*b) - A*c*sqrt(-1/(b**3*c))*log(b**2*sqrt(-1/(b**3*c)) + s
qrt(x))/(2*b) - 2*A*c*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(s
qrt(x)/sqrt(-b/c))/(c*sqrt(-b/c)), (b/c < 0) & (x < -b/c)))/b**2 - 2*A/(b**2*sqr
t(x)) + 2*B*sqrt(x)/(2*b**2 + 2*b*c*x) - B*sqrt(-1/(b**3*c))*log(-b**2*sqrt(-1/(
b**3*c)) + sqrt(x))/2 + B*sqrt(-1/(b**3*c))*log(b**2*sqrt(-1/(b**3*c)) + sqrt(x)
)/2

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GIAC/XCAS [A]  time = 0.271388, size = 81, normalized size = 0.95 \[ \frac{{\left (B b - 3 \, A c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{2}} + \frac{B b x - 3 \, A c x - 2 \, A b}{{\left (c x^{\frac{3}{2}} + b \sqrt{x}\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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