Optimal. Leaf size=117 \[ -\frac{2 e (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^2}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
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Rubi [A] time = 0.186788, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{2 e (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^2}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 13.9563, size = 107, normalized size = 0.91 \[ - \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{2}}{4 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{e \left (d + e x\right )}{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{e^{2} \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0666397, size = 67, normalized size = 0.57 \[ \frac{2 e^2 (a+b x)^2 \log (a+b x)-(b d-a e) (3 a e+b (d+4 e x))}{2 b^3 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.019, size = 104, normalized size = 0.9 \[{\frac{ \left ( 2\,\ln \left ( bx+a \right ){x}^{2}{b}^{2}{e}^{2}+4\,\ln \left ( bx+a \right ) xab{e}^{2}+2\,\ln \left ( bx+a \right ){a}^{2}{e}^{2}+4\,xab{e}^{2}-4\,x{b}^{2}de+3\,{a}^{2}{e}^{2}-2\,abde-{b}^{2}{d}^{2} \right ) \left ( bx+a \right ) }{2\,{b}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.72082, size = 176, normalized size = 1.5 \[ \frac{e^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \, a^{2} b^{2} e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, a b e^{2} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{2 \, d e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{d^{2}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{a d e}{{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211327, size = 134, normalized size = 1.15 \[ -\frac{b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2} + 4 \,{\left (b^{2} d e - a b e^{2}\right )} x - 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.607092, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")
[Out]