Optimal. Leaf size=132 \[ -\frac{2 b \log (x) (b c-a d) (2 b c-a d)}{a^5}+\frac{2 b (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}-\frac{(b c-a d) (3 b c-a d)}{a^4 x}-\frac{b (b c-a d)^2}{a^4 (a+b x)}+\frac{c (b c-a d)}{a^3 x^2}-\frac{c^2}{3 a^2 x^3} \]
[Out]
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Rubi [A] time = 0.260833, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{2 b \log (x) (b c-a d) (2 b c-a d)}{a^5}+\frac{2 b (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}-\frac{(b c-a d) (3 b c-a d)}{a^4 x}-\frac{b (b c-a d)^2}{a^4 (a+b x)}+\frac{c (b c-a d)}{a^3 x^2}-\frac{c^2}{3 a^2 x^3} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^2/(x^4*(a + b*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 29.4256, size = 119, normalized size = 0.9 \[ - \frac{c^{2}}{3 a^{2} x^{3}} - \frac{c \left (a d - b c\right )}{a^{3} x^{2}} - \frac{b \left (a d - b c\right )^{2}}{a^{4} \left (a + b x\right )} - \frac{\left (a d - 3 b c\right ) \left (a d - b c\right )}{a^{4} x} - \frac{2 b \left (a d - 2 b c\right ) \left (a d - b c\right ) \log{\left (x \right )}}{a^{5}} + \frac{2 b \left (a d - 2 b c\right ) \left (a d - b c\right ) \log{\left (a + b x \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**2/x**4/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.254605, size = 142, normalized size = 1.08 \[ -\frac{\frac{a^3 c^2}{x^3}+\frac{3 a \left (a^2 d^2-4 a b c d+3 b^2 c^2\right )}{x}+6 b \log (x) \left (a^2 d^2-3 a b c d+2 b^2 c^2\right )-6 b \left (a^2 d^2-3 a b c d+2 b^2 c^2\right ) \log (a+b x)+\frac{3 a^2 c (a d-b c)}{x^2}+\frac{3 a b (b c-a d)^2}{a+b x}}{3 a^5} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^2/(x^4*(a + b*x)^2),x]
[Out]
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Maple [A] time = 0.017, size = 205, normalized size = 1.6 \[ -{\frac{{c}^{2}}{3\,{a}^{2}{x}^{3}}}-{\frac{{d}^{2}}{{a}^{2}x}}+4\,{\frac{cdb}{{a}^{3}x}}-3\,{\frac{{b}^{2}{c}^{2}}{{a}^{4}x}}-2\,{\frac{b\ln \left ( x \right ){d}^{2}}{{a}^{3}}}+6\,{\frac{{b}^{2}\ln \left ( x \right ) cd}{{a}^{4}}}-4\,{\frac{{b}^{3}\ln \left ( x \right ){c}^{2}}{{a}^{5}}}-{\frac{cd}{{a}^{2}{x}^{2}}}+{\frac{{c}^{2}b}{{a}^{3}{x}^{2}}}-{\frac{{d}^{2}b}{{a}^{2} \left ( bx+a \right ) }}+2\,{\frac{cd{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-{\frac{{c}^{2}{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}+2\,{\frac{b\ln \left ( bx+a \right ){d}^{2}}{{a}^{3}}}-6\,{\frac{{b}^{2}\ln \left ( bx+a \right ) cd}{{a}^{4}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ){c}^{2}}{{a}^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^2/x^4/(b*x+a)^2,x)
[Out]
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Maxima [A] time = 1.35473, size = 239, normalized size = 1.81 \[ -\frac{a^{3} c^{2} + 6 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 3 \,{\left (2 \, a b^{2} c^{2} - 3 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} -{\left (2 \, a^{2} b c^{2} - 3 \, a^{3} c d\right )} x}{3 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac{2 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (b x + a\right )}{a^{5}} - \frac{2 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (x\right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)^2*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221401, size = 342, normalized size = 2.59 \[ -\frac{a^{4} c^{2} + 6 \,{\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3} + 3 \,{\left (2 \, a^{2} b^{2} c^{2} - 3 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} -{\left (2 \, a^{3} b c^{2} - 3 \, a^{4} c d\right )} x - 6 \,{\left ({\left (2 \, b^{4} c^{2} - 3 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} +{\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3}\right )} \log \left (b x + a\right ) + 6 \,{\left ({\left (2 \, b^{4} c^{2} - 3 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} +{\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)^2*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.55869, size = 326, normalized size = 2.47 \[ - \frac{a^{3} c^{2} + x^{3} \left (6 a^{2} b d^{2} - 18 a b^{2} c d + 12 b^{3} c^{2}\right ) + x^{2} \left (3 a^{3} d^{2} - 9 a^{2} b c d + 6 a b^{2} c^{2}\right ) + x \left (3 a^{3} c d - 2 a^{2} b c^{2}\right )}{3 a^{5} x^{3} + 3 a^{4} b x^{4}} - \frac{2 b \left (a d - 2 b c\right ) \left (a d - b c\right ) \log{\left (x + \frac{2 a^{3} b d^{2} - 6 a^{2} b^{2} c d + 4 a b^{3} c^{2} - 2 a b \left (a d - 2 b c\right ) \left (a d - b c\right )}{4 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 8 b^{4} c^{2}} \right )}}{a^{5}} + \frac{2 b \left (a d - 2 b c\right ) \left (a d - b c\right ) \log{\left (x + \frac{2 a^{3} b d^{2} - 6 a^{2} b^{2} c d + 4 a b^{3} c^{2} + 2 a b \left (a d - 2 b c\right ) \left (a d - b c\right )}{4 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 8 b^{4} c^{2}} \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**2/x**4/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.293305, size = 319, normalized size = 2.42 \[ -\frac{2 \,{\left (2 \, b^{4} c^{2} - 3 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{5} b} - \frac{\frac{b^{7} c^{2}}{b x + a} - \frac{2 \, a b^{6} c d}{b x + a} + \frac{a^{2} b^{5} d^{2}}{b x + a}}{a^{4} b^{4}} + \frac{13 \, b^{3} c^{2} - 15 \, a b^{2} c d + 3 \, a^{2} b d^{2} - \frac{3 \,{\left (10 \, a b^{4} c^{2} - 11 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )}}{{\left (b x + a\right )} b} + \frac{3 \,{\left (6 \, a^{2} b^{5} c^{2} - 6 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{3 \, a^{5}{\left (\frac{a}{b x + a} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)^2*x^4),x, algorithm="giac")
[Out]