Optimal. Leaf size=210 \[ \frac{6 e^2 (a+b x) (b d-a e)^2 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 e (b d-a e)^3}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^4}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 x (a+b x) (4 b d-3 a e)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^4 x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
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Rubi [A] time = 0.352573, antiderivative size = 210, 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 \[ \frac{6 e^2 (a+b x) (b d-a e)^2 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 e (b d-a e)^3}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^4}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 x (a+b x) (4 b d-3 a e)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^4 x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 33.6807, size = 202, normalized size = 0.96 \[ - \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{4}}{4 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2 e \left (d + e x\right )^{3}}{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{3 e^{2} \left (2 a + 2 b x\right ) \left (d + e x\right )^{2}}{2 b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{6 e^{3} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{5}} + \frac{6 e^{2} \left (a + b x\right ) \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.161334, size = 174, normalized size = 0.83 \[ \frac{7 a^4 e^4+2 a^3 b e^3 (e x-10 d)+a^2 b^2 e^2 \left (18 d^2-16 d e x-11 e^2 x^2\right )-4 a b^3 e \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+12 e^2 (a+b x)^2 (b d-a e)^2 \log (a+b x)+b^4 \left (-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right )}{2 b^5 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.026, size = 341, normalized size = 1.6 \[{\frac{ \left ({x}^{4}{b}^{4}{e}^{4}+12\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}-24\,\ln \left ( bx+a \right ){x}^{2}a{b}^{3}d{e}^{3}+12\,\ln \left ( bx+a \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}-4\,{x}^{3}a{b}^{3}{e}^{4}+8\,{x}^{3}{b}^{4}d{e}^{3}+24\,\ln \left ( bx+a \right ) x{a}^{3}b{e}^{4}-48\,\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}d{e}^{3}+24\,\ln \left ( bx+a \right ) xa{b}^{3}{d}^{2}{e}^{2}-11\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+16\,{x}^{2}a{b}^{3}d{e}^{3}+12\,\ln \left ( bx+a \right ){a}^{4}{e}^{4}-24\,\ln \left ( bx+a \right ){a}^{3}bd{e}^{3}+12\,\ln \left ( bx+a \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}+2\,x{a}^{3}b{e}^{4}-16\,x{a}^{2}{b}^{2}d{e}^{3}+24\,xa{b}^{3}{d}^{2}{e}^{2}-8\,x{b}^{4}{d}^{3}e+7\,{a}^{4}{e}^{4}-20\,{a}^{3}bd{e}^{3}+18\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-4\,a{b}^{3}{d}^{3}e-{b}^{4}{d}^{4} \right ) \left ( bx+a \right ) }{2\,{b}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.722693, size = 657, normalized size = 3.13 \[ \frac{e^{4} x^{3}}{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{4 \, d e^{3} x^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{5 \, a e^{4} x^{2}}{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac{6 \, d^{2} e^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{12 \, a d e^{3} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b} + \frac{6 \, a^{2} e^{4} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{2}} + \frac{9 \, a^{2} b^{2} d^{2} e^{2}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{18 \, a^{3} b d e^{3}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{9 \, a^{4} e^{4}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{12 \, a b d^{2} e^{2} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{24 \, a^{2} d e^{3} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{12 \, a^{3} e^{4} x}{{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \frac{4 \, d^{3} e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{8 \, a^{2} d e^{3}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac{5 \, a^{3} e^{4}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} - \frac{d^{4}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, a d^{3} e}{{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \frac{4 \, a^{3} d e^{3}}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{3}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{5 \, a^{4} e^{4}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b^{4}{\left (x + \frac{a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209513, size = 394, normalized size = 1.88 \[ \frac{b^{4} e^{4} x^{4} - b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 20 \, a^{3} b d e^{3} + 7 \, a^{4} e^{4} + 4 \,{\left (2 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} +{\left (16 \, a b^{3} d e^{3} - 11 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (4 \, b^{4} d^{3} e - 12 \, a b^{3} d^{2} e^{2} + 8 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \,{\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} +{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(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. \[ \int \frac{\left (d + e x\right )^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.580995, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")
[Out]