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

Optimal. Leaf size=69 \[ -\frac{2 \sqrt{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}+\frac{2 \sqrt{x} (A b-a B)}{b^2}+\frac{2 B x^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0846432, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{2 \sqrt{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}+\frac{2 \sqrt{x} (A b-a B)}{b^2}+\frac{2 B x^{3/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.9137, size = 63, normalized size = 0.91 \[ \frac{2 B x^{\frac{3}{2}}}{3 b} - \frac{2 \sqrt{a} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} + \frac{2 \sqrt{x} \left (A b - B a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0667923, size = 63, normalized size = 0.91 \[ \frac{2 \sqrt{a} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}+\frac{2 \sqrt{x} (-3 a B+3 A b+b B x)}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 78, normalized size = 1.1 \[{\frac{2\,B}{3\,b}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{b}}-2\,{\frac{Ba\sqrt{x}}{{b}^{2}}}-2\,{\frac{Aa}{b\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+2\,{\frac{B{a}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*x^(1/2)/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22367, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B a - A b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt{x}}{3 \, b^{2}}, \frac{2 \,{\left (3 \,{\left (B a - A b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt{x}\right )}}{3 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.21998, size = 131, normalized size = 1.9 \[ \frac{2 B x^{\frac{3}{2}}}{3 b} + \frac{2 a \left (- A b + B a\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}} \right )}}{b \sqrt{\frac{a}{b}}} & \text{for}\: \frac{a}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x > - \frac{a}{b} \wedge \frac{a}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x < - \frac{a}{b} \wedge \frac{a}{b} < 0 \end{cases}\right )}{b^{2}} + \frac{2 \sqrt{x} \left (A b - B a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x**(3/2)/(3*b) + 2*a*(-A*b + 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)), (-atanh(sqrt(x)/sqrt(-a/b))/(b*sqrt(-a/b)), (a/b < 0) & (x < -a/b)))/b**2
+ 2*sqrt(x)*(A*b - B*a)/b**2

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GIAC/XCAS [A]  time = 0.271619, size = 86, normalized size = 1.25 \[ \frac{2 \,{\left (B a^{2} - A a b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{2 \,{\left (B b^{2} x^{\frac{3}{2}} - 3 \, B a b \sqrt{x} + 3 \, A b^{2} \sqrt{x}\right )}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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