Optimal. Leaf size=100 \[ \frac{(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} b^{3/2}}+\frac{\sqrt{x} (a B+3 A b)}{4 a^2 b (a+b x)}+\frac{\sqrt{x} (A b-a B)}{2 a b (a+b x)^2} \]
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Rubi [A] time = 0.113157, antiderivative size = 100, 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 \[ \frac{(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} b^{3/2}}+\frac{\sqrt{x} (a B+3 A b)}{4 a^2 b (a+b x)}+\frac{\sqrt{x} (A b-a B)}{2 a b (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(Sqrt[x]*(a + b*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 14.177, size = 85, normalized size = 0.85 \[ \frac{\sqrt{x} \left (A b - B a\right )}{2 a b \left (a + b x\right )^{2}} + \frac{\sqrt{x} \left (3 A b + B a\right )}{4 a^{2} b \left (a + b x\right )} + \frac{\left (3 A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b*x+a)**3/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.127217, size = 86, normalized size = 0.86 \[ \frac{(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} b^{3/2}}+\frac{\sqrt{x} \left (a^2 (-B)+a b (5 A+B x)+3 A b^2 x\right )}{4 a^2 b (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(Sqrt[x]*(a + b*x)^3),x]
[Out]
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Maple [A] time = 0.019, size = 95, normalized size = 1. \[ 2\,{\frac{1}{ \left ( bx+a \right ) ^{2}} \left ( 1/8\,{\frac{ \left ( 3\,Ab+Ba \right ){x}^{3/2}}{{a}^{2}}}+1/8\,{\frac{ \left ( 5\,Ab-Ba \right ) \sqrt{x}}{ab}} \right ) }+{\frac{3\,A}{4\,{a}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{4\,ab}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b*x+a)^3/x^(1/2),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)/((b*x + a)^3*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229633, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (B a^{2} - 5 \, A a b -{\left (B a b + 3 \, A b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} -{\left (B a^{3} + 3 \, A a^{2} b +{\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \,{\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{8 \,{\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )} \sqrt{-a b}}, -\frac{{\left (B a^{2} - 5 \, A a b -{\left (B a b + 3 \, A b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} +{\left (B a^{3} + 3 \, A a^{2} b +{\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \,{\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(b*x+a)**3/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212181, size = 111, normalized size = 1.11 \[ \frac{{\left (B a + 3 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{2} b} + \frac{B a b x^{\frac{3}{2}} + 3 \, A b^{2} x^{\frac{3}{2}} - B a^{2} \sqrt{x} + 5 \, A a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*sqrt(x)),x, algorithm="giac")
[Out]