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

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

[Out]

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

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Rubi [A]  time = 0.10029, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}-\frac{\sqrt{x} (A b-3 a B)}{a b^2}+\frac{x^{3/2} (A b-a B)}{a b (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.914, size = 73, normalized size = 0.86 \[ \frac{x^{\frac{3}{2}} \left (A b - B a\right )}{a b \left (a + b x\right )} - \frac{\sqrt{x} \left (A b - 3 B a\right )}{a b^{2}} + \frac{\left (A b - 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{a} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.104269, size = 67, normalized size = 0.79 \[ \frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}+\frac{\sqrt{x} (3 a B-A b+2 b B x)}{b^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 87, normalized size = 1. \[ 2\,{\frac{B\sqrt{x}}{{b}^{2}}}-{\frac{A}{b \left ( bx+a \right ) }\sqrt{x}}+{\frac{Ba}{{b}^{2} \left ( bx+a \right ) }\sqrt{x}}+{\frac{A}{b}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{Ba}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x^(1/2)/b^2-1/b*x^(1/2)/(b*x+a)*A+1/b^2*x^(1/2)/(b*x+a)*B*a+1/b/(a*b)^(1/2)*
arctan(x^(1/2)*b/(a*b)^(1/2))*A-3/b^2/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*
B*a

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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)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320379, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, B b x + 3 \, B a - A b\right )} \sqrt{-a b} \sqrt{x} -{\left (3 \, B a^{2} - A a b +{\left (3 \, B a b - A b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{2 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{-a b}}, \frac{{\left (2 \, B b x + 3 \, B a - A b\right )} \sqrt{a b} \sqrt{x} +{\left (3 \, B a^{2} - A a b +{\left (3 \, B a b - A b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{{\left (b^{3} x + a b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.44806, size = 428, normalized size = 5.04 \[ - \frac{2 A a \sqrt{x}}{2 a^{2} b + 2 a b^{2} x} + \frac{A a \sqrt{- \frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{2 b} - \frac{A a \sqrt{- \frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{2 b} + \frac{2 A \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}} \right )}}{b \sqrt{\frac{a}{b}}} & \text{for}\: \frac{a}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x > - \frac{a}{b} \wedge \frac{a}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x < - \frac{a}{b} \wedge \frac{a}{b} < 0 \end{cases}\right )}{b} + \frac{2 B a^{2} \sqrt{x}}{2 a^{2} b^{2} + 2 a b^{3} x} - \frac{B a^{2} \sqrt{- \frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{2 b^{2}} + \frac{B a^{2} \sqrt{- \frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{2 b^{2}} - \frac{4 B a \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}} \right )}}{b \sqrt{\frac{a}{b}}} & \text{for}\: \frac{a}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x > - \frac{a}{b} \wedge \frac{a}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x < - \frac{a}{b} \wedge \frac{a}{b} < 0 \end{cases}\right )}{b^{2}} + \frac{2 B \sqrt{x}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a*sqrt(x)/(2*a**2*b + 2*a*b**2*x) + A*a*sqrt(-1/(a**3*b))*log(-a**2*sqrt(-1
/(a**3*b)) + sqrt(x))/(2*b) - A*a*sqrt(-1/(a**3*b))*log(a**2*sqrt(-1/(a**3*b)) +
sqrt(x))/(2*b) + 2*A*Piecewise((atan(sqrt(x)/sqrt(a/b))/(b*sqrt(a/b)), a/b > 0)
, (-acoth(sqrt(x)/sqrt(-a/b))/(b*sqrt(-a/b)), (a/b < 0) & (x > -a/b)), (-atanh(s
qrt(x)/sqrt(-a/b))/(b*sqrt(-a/b)), (a/b < 0) & (x < -a/b)))/b + 2*B*a**2*sqrt(x)
/(2*a**2*b**2 + 2*a*b**3*x) - B*a**2*sqrt(-1/(a**3*b))*log(-a**2*sqrt(-1/(a**3*b
)) + sqrt(x))/(2*b**2) + B*a**2*sqrt(-1/(a**3*b))*log(a**2*sqrt(-1/(a**3*b)) + s
qrt(x))/(2*b**2) - 4*B*a*Piecewise((atan(sqrt(x)/sqrt(a/b))/(b*sqrt(a/b)), a/b >
0), (-acoth(sqrt(x)/sqrt(-a/b))/(b*sqrt(-a/b)), (a/b < 0) & (x > -a/b)), (-atan
h(sqrt(x)/sqrt(-a/b))/(b*sqrt(-a/b)), (a/b < 0) & (x < -a/b)))/b**2 + 2*B*sqrt(x
)/b**2

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GIAC/XCAS [A]  time = 0.269518, size = 88, normalized size = 1.04 \[ \frac{2 \, B \sqrt{x}}{b^{2}} - \frac{{\left (3 \, B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{B a \sqrt{x} - A b \sqrt{x}}{{\left (b x + a\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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