Optimal. Leaf size=108 \[ -\frac{\sqrt{a} (3 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}+\frac{\sqrt{x} (3 A b-5 a B)}{b^3}-\frac{x^{3/2} (3 A b-5 a B)}{3 a b^2}+\frac{x^{5/2} (A b-a B)}{a b (a+b x)} \]
[Out]
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Rubi [A] time = 0.129484, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{\sqrt{a} (3 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}+\frac{\sqrt{x} (3 A b-5 a B)}{b^3}-\frac{x^{3/2} (3 A b-5 a B)}{3 a b^2}+\frac{x^{5/2} (A b-a B)}{a b (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(x^(3/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Rubi in Sympy [A] time = 31.0159, size = 97, normalized size = 0.9 \[ - \frac{\sqrt{a} \left (3 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{7}{2}}} + \frac{\sqrt{x} \left (3 A b - 5 B a\right )}{b^{3}} + \frac{x^{\frac{5}{2}} \left (A b - B a\right )}{a b \left (a + b x\right )} - \frac{x^{\frac{3}{2}} \left (3 A b - 5 B a\right )}{3 a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.179351, size = 88, normalized size = 0.81 \[ \frac{\sqrt{x} \left (-15 a^2 B+a b (9 A-10 B x)+2 b^2 x (3 A+B x)\right )}{3 b^3 (a+b x)}+\frac{\sqrt{a} (5 a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(3/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Maple [A] time = 0.022, size = 113, normalized size = 1.1 \[{\frac{2\,B}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{{b}^{2}}}-4\,{\frac{aB\sqrt{x}}{{b}^{3}}}+{\frac{aA}{{b}^{2} \left ( bx+a \right ) }\sqrt{x}}-{\frac{{a}^{2}B}{{b}^{3} \left ( bx+a \right ) }\sqrt{x}}-3\,{\frac{aA}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }+5\,{\frac{{a}^{2}B}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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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)*x^(3/2)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.308695, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, B a^{2} - 3 \, A a b +{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (2 \, B b^{2} x^{2} - 15 \, B a^{2} + 9 \, A a b - 2 \,{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{x}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, \frac{3 \,{\left (5 \, B a^{2} - 3 \, A a b +{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (2 \, B b^{2} x^{2} - 15 \, B a^{2} + 9 \, A a b - 2 \,{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{x}}{3 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}} \left (A + B x\right )}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.272672, size = 128, normalized size = 1.19 \[ \frac{{\left (5 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{B a^{2} \sqrt{x} - A a b \sqrt{x}}{{\left (b x + a\right )} b^{3}} + \frac{2 \,{\left (B b^{4} x^{\frac{3}{2}} - 6 \, B a b^{3} \sqrt{x} + 3 \, A b^{4} \sqrt{x}\right )}}{3 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")
[Out]