Optimal. Leaf size=122 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^3 (a+b x)}-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e} \]
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Rubi [A] time = 0.195408, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^3 (a+b x)}-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x),x]
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Rubi in Sympy [A] time = 26.4229, size = 102, normalized size = 0.84 \[ \frac{\left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 e} + \frac{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{2}} + \frac{\left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{3} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.054353, size = 61, normalized size = 0.5 \[ \frac{\sqrt{(a+b x)^2} \left (b e x (4 a e-2 b d+b e x)+2 (b d-a e)^2 \log (d+e x)\right )}{2 e^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x),x]
[Out]
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Maple [C] time = 0.017, size = 102, normalized size = 0.8 \[{\frac{{\it csgn} \left ( bx+a \right ) \left ({x}^{2}{b}^{2}{e}^{2}+2\,\ln \left ( bex+bd \right ){a}^{2}{e}^{2}-4\,\ln \left ( bex+bd \right ) abde+2\,\ln \left ( bex+bd \right ){b}^{2}{d}^{2}+4\,xab{e}^{2}-2\,x{b}^{2}de+3\,{a}^{2}{e}^{2}-2\,abde \right ) }{2\,{e}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d),x)
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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(sqrt((b*x + a)^2)*(b*x + a)/(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283, size = 84, normalized size = 0.69 \[ \frac{b^{2} e^{2} x^{2} - 2 \,{\left (b^{2} d e - 2 \, a b e^{2}\right )} x + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.54567, size = 44, normalized size = 0.36 \[ \frac{b^{2} x^{2}}{2 e} + \frac{x \left (2 a b e - b^{2} d\right )}{e^{2}} + \frac{\left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 0.286278, size = 131, normalized size = 1.07 \[{\left (b^{2} d^{2}{\rm sign}\left (b x + a\right ) - 2 \, a b d e{\rm sign}\left (b x + a\right ) + a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{2} x^{2} e{\rm sign}\left (b x + a\right ) - 2 \, b^{2} d x{\rm sign}\left (b x + a\right ) + 4 \, a b x e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)/(e*x + d),x, algorithm="giac")
[Out]