Optimal. Leaf size=173 \[ \frac{(a+b x) (d+e x)^2 (b d-a e)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^3}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^3 \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e x (a+b x) (b d-a e)^2}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
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Rubi [A] time = 0.182978, antiderivative size = 173, 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 \[ \frac{(a+b x) (d+e x)^2 (b d-a e)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^3}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^3 \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e x (a+b x) (b d-a e)^2}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 32.1987, size = 163, normalized size = 0.94 \[ \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{3}}{6 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{2} \left (a e - b d\right )}{4 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{e \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{4}} - \frac{\left (a + b x\right ) \left (a e - b d\right )^{3} \log{\left (a + b x \right )}}{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0992776, size = 90, normalized size = 0.52 \[ \frac{(a+b x) \left (b e x \left (6 a^2 e^2-3 a b e (6 d+e x)+b^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 (b d-a e)^3 \log (a+b x)\right )}{6 b^4 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.013, size = 147, normalized size = 0.9 \[ -{\frac{ \left ( bx+a \right ) \left ( -2\,{x}^{3}{b}^{3}{e}^{3}+3\,{x}^{2}a{b}^{2}{e}^{3}-9\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( bx+a \right ){a}^{3}{e}^{3}-18\,\ln \left ( bx+a \right ){a}^{2}bd{e}^{2}+18\,\ln \left ( bx+a \right ) a{b}^{2}{d}^{2}e-6\,\ln \left ( bx+a \right ){b}^{3}{d}^{3}-6\,x{a}^{2}b{e}^{3}+18\,xa{b}^{2}d{e}^{2}-18\,x{b}^{3}{d}^{2}e \right ) }{6\,{b}^{4}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/((b*x+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.6961, size = 346, normalized size = 2. \[ \frac{3 \, a^{2} b^{2} d e^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{5 \, a^{3} b e^{3} \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{3 \, a b d e^{2} x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{5 \, a^{2} e^{3} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \, d e^{2} x^{2}}{2 \, \sqrt{b^{2}}} - \frac{5 \, a e^{3} x^{2}}{6 \, \sqrt{b^{2}} b} + \sqrt{\frac{1}{b^{2}}} d^{3} \log \left (x + \frac{a}{b}\right ) - \frac{3 \, a \sqrt{\frac{1}{b^{2}}} d^{2} e \log \left (x + \frac{a}{b}\right )}{b} + \frac{2 \, a^{3} \sqrt{\frac{1}{b^{2}}} e^{3} \log \left (x + \frac{a}{b}\right )}{3 \, b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e^{3} x^{2}}{3 \, b^{2}} + \frac{3 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} d^{2} e}{b^{2}} - \frac{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} e^{3}}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/sqrt((b*x + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205344, size = 157, normalized size = 0.91 \[ \frac{2 \, b^{3} e^{3} x^{3} + 3 \,{\left (3 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/sqrt((b*x + a)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.86646, size = 82, normalized size = 0.47 \[ \frac{e^{3} x^{3}}{3 b} - \frac{x^{2} \left (a e^{3} - 3 b d e^{2}\right )}{2 b^{2}} + \frac{x \left (a^{2} e^{3} - 3 a b d e^{2} + 3 b^{2} d^{2} e\right )}{b^{3}} - \frac{\left (a e - b d\right )^{3} \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.213725, size = 230, normalized size = 1.33 \[ \frac{2 \, b^{2} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 9 \, b^{2} d x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 18 \, b^{2} d^{2} x e{\rm sign}\left (b x + a\right ) - 3 \, a b x^{2} e^{3}{\rm sign}\left (b x + a\right ) - 18 \, a b d x e^{2}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} x e^{3}{\rm sign}\left (b x + a\right )}{6 \, b^{3}} + \frac{{\left (b^{3} d^{3}{\rm sign}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) - a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/sqrt((b*x + a)^2),x, algorithm="giac")
[Out]