Optimal. Leaf size=65 \[ \frac{2 (a+b x)^{3/2} (A b-2 a B)}{3 b^3}-\frac{2 a \sqrt{a+b x} (A b-a B)}{b^3}+\frac{2 B (a+b x)^{5/2}}{5 b^3} \]
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Rubi [A] time = 0.0796262, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{2 (a+b x)^{3/2} (A b-2 a B)}{3 b^3}-\frac{2 a \sqrt{a+b x} (A b-a B)}{b^3}+\frac{2 B (a+b x)^{5/2}}{5 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x))/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 11.4027, size = 61, normalized size = 0.94 \[ \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 b^{3}} - \frac{2 a \sqrt{a + b x} \left (A b - B a\right )}{b^{3}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A b - 2 B a\right )}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x+A)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0393419, size = 48, normalized size = 0.74 \[ \frac{2 \sqrt{a+b x} \left (8 a^2 B-2 a b (5 A+2 B x)+b^2 x (5 A+3 B x)\right )}{15 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x))/Sqrt[a + b*x],x]
[Out]
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Maple [A] time = 0.006, size = 47, normalized size = 0.7 \[ -{\frac{-6\,{b}^{2}B{x}^{2}-10\,Ax{b}^{2}+8\,Bxab+20\,Aab-16\,B{a}^{2}}{15\,{b}^{3}}\sqrt{bx+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x+A)/(b*x+a)^(1/2),x)
[Out]
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Maxima [A] time = 1.36451, size = 73, normalized size = 1.12 \[ \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} B - 5 \,{\left (2 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{3}{2}} + 15 \,{\left (B a^{2} - A a b\right )} \sqrt{b x + a}\right )}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/sqrt(b*x + a),x, algorithm="maxima")
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Fricas [A] time = 0.20583, size = 65, normalized size = 1. \[ \frac{2 \,{\left (3 \, B b^{2} x^{2} + 8 \, B a^{2} - 10 \, A a b -{\left (4 \, B a b - 5 \, A b^{2}\right )} x\right )} \sqrt{b x + a}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/sqrt(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.0325, size = 182, normalized size = 2.8 \[ \begin{cases} - \frac{\frac{2 A a \left (- \frac{a}{\sqrt{a + b x}} - \sqrt{a + b x}\right )}{b} + \frac{2 A \left (\frac{a^{2}}{\sqrt{a + b x}} + 2 a \sqrt{a + b x} - \frac{\left (a + b x\right )^{\frac{3}{2}}}{3}\right )}{b} + \frac{2 B a \left (\frac{a^{2}}{\sqrt{a + b x}} + 2 a \sqrt{a + b x} - \frac{\left (a + b x\right )^{\frac{3}{2}}}{3}\right )}{b^{2}} + \frac{2 B \left (- \frac{a^{3}}{\sqrt{a + b x}} - 3 a^{2} \sqrt{a + b x} + a \left (a + b x\right )^{\frac{3}{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}}}{5}\right )}{b^{2}}}{b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{2}}{2} + \frac{B x^{3}}{3}}{\sqrt{a}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x+A)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.227501, size = 103, normalized size = 1.58 \[ \frac{2 \,{\left (\frac{5 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )} A}{b} + \frac{{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{8} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{8} + 15 \, \sqrt{b x + a} a^{2} b^{8}\right )} B}{b^{10}}\right )}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/sqrt(b*x + a),x, algorithm="giac")
[Out]