Optimal. Leaf size=130 \[ -\frac{5 b^4 (c+d x)^3 (b c-a d)}{3 d^6}+\frac{5 b^3 (c+d x)^2 (b c-a d)^2}{d^6}-\frac{10 b^2 x (b c-a d)^3}{d^5}+\frac{(b c-a d)^5}{d^6 (c+d x)}+\frac{5 b (b c-a d)^4 \log (c+d x)}{d^6}+\frac{b^5 (c+d x)^4}{4 d^6} \]
[Out]
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Rubi [A] time = 0.277637, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{5 b^4 (c+d x)^3 (b c-a d)}{3 d^6}+\frac{5 b^3 (c+d x)^2 (b c-a d)^2}{d^6}-\frac{10 b^2 x (b c-a d)^3}{d^5}+\frac{(b c-a d)^5}{d^6 (c+d x)}+\frac{5 b (b c-a d)^4 \log (c+d x)}{d^6}+\frac{b^5 (c+d x)^4}{4 d^6} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^5/(c + d*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 44.6842, size = 119, normalized size = 0.92 \[ \frac{b^{5} \left (c + d x\right )^{4}}{4 d^{6}} + \frac{5 b^{4} \left (c + d x\right )^{3} \left (a d - b c\right )}{3 d^{6}} + \frac{5 b^{3} \left (c + d x\right )^{2} \left (a d - b c\right )^{2}}{d^{6}} + \frac{10 b^{2} x \left (a d - b c\right )^{3}}{d^{5}} + \frac{5 b \left (a d - b c\right )^{4} \log{\left (c + d x \right )}}{d^{6}} - \frac{\left (a d - b c\right )^{5}}{d^{6} \left (c + d x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**5/(d*x+c)**2,x)
[Out]
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Mathematica [A] time = 0.124519, size = 228, normalized size = 1.75 \[ \frac{-12 a^5 d^5+60 a^4 b c d^4+120 a^3 b^2 d^3 \left (-c^2+c d x+d^2 x^2\right )+60 a^2 b^3 d^2 \left (2 c^3-4 c^2 d x-3 c d^2 x^2+d^3 x^3\right )+20 a b^4 d \left (-3 c^4+9 c^3 d x+6 c^2 d^2 x^2-2 c d^3 x^3+d^4 x^4\right )+60 b (c+d x) (b c-a d)^4 \log (c+d x)+b^5 \left (12 c^5-48 c^4 d x-30 c^3 d^2 x^2+10 c^2 d^3 x^3-5 c d^4 x^4+3 d^5 x^5\right )}{12 d^6 (c+d x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^5/(c + d*x)^2,x]
[Out]
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Maple [B] time = 0.013, size = 326, normalized size = 2.5 \[{\frac{{b}^{5}{x}^{4}}{4\,{d}^{2}}}+{\frac{5\,a{b}^{4}{x}^{3}}{3\,{d}^{2}}}-{\frac{2\,{b}^{5}{x}^{3}c}{3\,{d}^{3}}}+5\,{\frac{{a}^{2}{b}^{3}{x}^{2}}{{d}^{2}}}-5\,{\frac{a{b}^{4}{x}^{2}c}{{d}^{3}}}+{\frac{3\,{b}^{5}{x}^{2}{c}^{2}}{2\,{d}^{4}}}+10\,{\frac{{a}^{3}{b}^{2}x}{{d}^{2}}}-20\,{\frac{{a}^{2}{b}^{3}cx}{{d}^{3}}}+15\,{\frac{a{b}^{4}{c}^{2}x}{{d}^{4}}}-4\,{\frac{{b}^{5}{c}^{3}x}{{d}^{5}}}+5\,{\frac{b\ln \left ( dx+c \right ){a}^{4}}{{d}^{2}}}-20\,{\frac{{b}^{2}\ln \left ( dx+c \right ){a}^{3}c}{{d}^{3}}}+30\,{\frac{{b}^{3}\ln \left ( dx+c \right ){a}^{2}{c}^{2}}{{d}^{4}}}-20\,{\frac{{b}^{4}\ln \left ( dx+c \right ) a{c}^{3}}{{d}^{5}}}+5\,{\frac{{b}^{5}\ln \left ( dx+c \right ){c}^{4}}{{d}^{6}}}-{\frac{{a}^{5}}{d \left ( dx+c \right ) }}+5\,{\frac{{a}^{4}bc}{{d}^{2} \left ( dx+c \right ) }}-10\,{\frac{{a}^{3}{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }}+10\,{\frac{{a}^{2}{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) }}-5\,{\frac{a{b}^{4}{c}^{4}}{{d}^{5} \left ( dx+c \right ) }}+{\frac{{b}^{5}{c}^{5}}{{d}^{6} \left ( dx+c \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^5/(d*x+c)^2,x)
[Out]
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Maxima [A] time = 1.3351, size = 356, normalized size = 2.74 \[ \frac{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}}{d^{7} x + c d^{6}} + \frac{3 \, b^{5} d^{3} x^{4} - 4 \,{\left (2 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{3} + 6 \,{\left (3 \, b^{5} c^{2} d - 10 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{2} - 12 \,{\left (4 \, b^{5} c^{3} - 15 \, a b^{4} c^{2} d + 20 \, a^{2} b^{3} c d^{2} - 10 \, a^{3} b^{2} d^{3}\right )} x}{12 \, d^{5}} + \frac{5 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \log \left (d x + c\right )}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^5/(d*x + c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.199912, size = 504, normalized size = 3.88 \[ \frac{3 \, b^{5} d^{5} x^{5} + 12 \, b^{5} c^{5} - 60 \, a b^{4} c^{4} d + 120 \, a^{2} b^{3} c^{3} d^{2} - 120 \, a^{3} b^{2} c^{2} d^{3} + 60 \, a^{4} b c d^{4} - 12 \, a^{5} d^{5} - 5 \,{\left (b^{5} c d^{4} - 4 \, a b^{4} d^{5}\right )} x^{4} + 10 \,{\left (b^{5} c^{2} d^{3} - 4 \, a b^{4} c d^{4} + 6 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \,{\left (b^{5} c^{3} d^{2} - 4 \, a b^{4} c^{2} d^{3} + 6 \, a^{2} b^{3} c d^{4} - 4 \, a^{3} b^{2} d^{5}\right )} x^{2} - 12 \,{\left (4 \, b^{5} c^{4} d - 15 \, a b^{4} c^{3} d^{2} + 20 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4}\right )} x + 60 \,{\left (b^{5} c^{5} - 4 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} - 4 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} +{\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x\right )} \log \left (d x + c\right )}{12 \,{\left (d^{7} x + c d^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^5/(d*x + c)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.13237, size = 224, normalized size = 1.72 \[ \frac{b^{5} x^{4}}{4 d^{2}} + \frac{5 b \left (a d - b c\right )^{4} \log{\left (c + d x \right )}}{d^{6}} - \frac{a^{5} d^{5} - 5 a^{4} b c d^{4} + 10 a^{3} b^{2} c^{2} d^{3} - 10 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d - b^{5} c^{5}}{c d^{6} + d^{7} x} + \frac{x^{3} \left (5 a b^{4} d - 2 b^{5} c\right )}{3 d^{3}} + \frac{x^{2} \left (10 a^{2} b^{3} d^{2} - 10 a b^{4} c d + 3 b^{5} c^{2}\right )}{2 d^{4}} + \frac{x \left (10 a^{3} b^{2} d^{3} - 20 a^{2} b^{3} c d^{2} + 15 a b^{4} c^{2} d - 4 b^{5} c^{3}\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**5/(d*x+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.219628, size = 458, normalized size = 3.52 \[ \frac{{\left (3 \, b^{5} - \frac{20 \,{\left (b^{5} c d - a b^{4} d^{2}\right )}}{{\left (d x + c\right )} d} + \frac{60 \,{\left (b^{5} c^{2} d^{2} - 2 \, a b^{4} c d^{3} + a^{2} b^{3} d^{4}\right )}}{{\left (d x + c\right )}^{2} d^{2}} - \frac{120 \,{\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )}}{{\left (d x + c\right )}^{3} d^{3}}\right )}{\left (d x + c\right )}^{4}}{12 \, d^{6}} - \frac{5 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}{\rm ln}\left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{d^{6}} + \frac{\frac{b^{5} c^{5} d^{4}}{d x + c} - \frac{5 \, a b^{4} c^{4} d^{5}}{d x + c} + \frac{10 \, a^{2} b^{3} c^{3} d^{6}}{d x + c} - \frac{10 \, a^{3} b^{2} c^{2} d^{7}}{d x + c} + \frac{5 \, a^{4} b c d^{8}}{d x + c} - \frac{a^{5} d^{9}}{d x + c}}{d^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^5/(d*x + c)^2,x, algorithm="giac")
[Out]