Optimal. Leaf size=106 \[ -\frac{(d+e x)^2}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 e (a+b x) (b d-a e) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 e^2 \sqrt{a^2+2 a b x+b^2 x^2}}{b^3} \]
[Out]
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Rubi [A] time = 0.199085, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ -\frac{(d+e x)^2}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 e (a+b x) (b d-a e) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 e^2 \sqrt{a^2+2 a b x+b^2 x^2}}{b^3} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(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 = 25.7531, size = 102, normalized size = 0.96 \[ - \frac{\left (d + e x\right )^{2}}{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{2 e^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{3}} - \frac{2 e \left (a + b x\right ) \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(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 = 0.0671126, size = 74, normalized size = 0.7 \[ \frac{-a^2 e^2+a b e (2 d+e x)-2 e (a+b x) (a e-b d) \log (a+b x)+b^2 \left (e^2 x^2-d^2\right )}{b^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(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 = 0.025, size = 116, normalized size = 1.1 \[ -{\frac{ \left ( 2\,\ln \left ( bx+a \right ) xab{e}^{2}-2\,\ln \left ( bx+a \right ) x{b}^{2}de-{x}^{2}{b}^{2}{e}^{2}+2\,\ln \left ( bx+a \right ){a}^{2}{e}^{2}-2\,\ln \left ( bx+a \right ) abde-xab{e}^{2}+{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) \left ( bx+a \right ) ^{2}}{{b}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(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 = 0.702903, size = 437, normalized size = 4.12 \[ \frac{e^{2} x^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b} - \frac{3 \, a e^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{9 \, a^{3} b^{2} e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{6 \, a^{2} b e^{2} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, a^{2} e^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac{{\left (2 \, b d e + a e^{2}\right )} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (2 \, b d e + a e^{2}\right )} a^{2} b^{2}}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{a d^{2}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{a^{3} e^{2}}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{2}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \,{\left (2 \, b d e + a e^{2}\right )} a b x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{b d^{2} + 2 \, a d e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{{\left (b d^{2} + 2 \, a d e\right )} a}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(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 = 0.277722, size = 124, normalized size = 1.17 \[ \frac{b^{2} e^{2} x^{2} + a b e^{2} x - b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} + 2 \,{\left (a b d e - a^{2} e^{2} +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(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. \[ \int \frac{\left (a + b x\right ) \left (d + e x\right )^{2}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}{\left (e x + d\right )}^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")
[Out]