Optimal. Leaf size=69 \[ -\frac{4 b (d+e x)^{3/2} (b d-a e)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e)^2}{e^3}+\frac{2 b^2 (d+e x)^{5/2}}{5 e^3} \]
[Out]
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Rubi [A] time = 0.0781402, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{4 b (d+e x)^{3/2} (b d-a e)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e)^2}{e^3}+\frac{2 b^2 (d+e x)^{5/2}}{5 e^3} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 29.8487, size = 63, normalized size = 0.91 \[ \frac{2 b^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{3}} + \frac{4 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )}{3 e^{3}} + \frac{2 \sqrt{d + e x} \left (a e - b d\right )^{2}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.056637, size = 60, normalized size = 0.87 \[ \frac{2 \sqrt{d+e x} \left (15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.012, size = 63, normalized size = 0.9 \[{\frac{6\,{x}^{2}{b}^{2}{e}^{2}+20\,xab{e}^{2}-8\,x{b}^{2}de+30\,{a}^{2}{e}^{2}-40\,abde+16\,{b}^{2}{d}^{2}}{15\,{e}^{3}}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.72691, size = 111, normalized size = 1.61 \[ \frac{2 \,{\left (15 \, \sqrt{e x + d} a^{2} + \frac{10 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )} a b}{e} + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} b^{2}}{e^{2}}\right )}}{15 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205675, size = 86, normalized size = 1.25 \[ \frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 20 \, a b d e + 15 \, a^{2} e^{2} - 2 \,{\left (2 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.11834, size = 236, normalized size = 3.42 \[ \begin{cases} - \frac{\frac{2 a^{2} d}{\sqrt{d + e x}} + 2 a^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 a b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{4 a b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{\sqrt{d}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217243, size = 123, normalized size = 1.78 \[ \frac{2}{15} \,{\left (10 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a b e^{\left (-1\right )} +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} b^{2} e^{\left (-10\right )} + 15 \, \sqrt{x e + d} a^{2}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/sqrt(e*x + d),x, algorithm="giac")
[Out]