Optimal. Leaf size=122 \[ -\frac{(b c-a d)^5 \log (c+d x)}{d^6}+\frac{b x (b c-a d)^4}{d^5}-\frac{(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac{(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac{(a+b x)^4 (b c-a d)}{4 d^2}+\frac{(a+b x)^5}{5 d} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.117833, antiderivative size = 122, 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 \[ -\frac{(b c-a d)^5 \log (c+d x)}{d^6}+\frac{b x (b c-a d)^4}{d^5}-\frac{(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac{(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac{(a+b x)^4 (b c-a d)}{4 d^2}+\frac{(a+b x)^5}{5 d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^5/(c + d*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\left (a + b x\right )^{5}}{5 d} + \frac{\left (a + b x\right )^{4} \left (a d - b c\right )}{4 d^{2}} + \frac{\left (a + b x\right )^{3} \left (a d - b c\right )^{2}}{3 d^{3}} + \frac{\left (a + b x\right )^{2} \left (a d - b c\right )^{3}}{2 d^{4}} + \frac{\left (a d - b c\right )^{4} \int b\, dx}{d^{5}} + \frac{\left (a d - b c\right )^{5} \log{\left (c + d x \right )}}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**5/(d*x+c),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.115419, size = 167, normalized size = 1.37 \[ \frac{b d x \left (300 a^4 d^4+300 a^3 b d^3 (d x-2 c)+100 a^2 b^2 d^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )+25 a b^3 d \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )+b^4 \left (60 c^4-30 c^3 d x+20 c^2 d^2 x^2-15 c d^3 x^3+12 d^4 x^4\right )\right )-60 (b c-a d)^5 \log (c+d x)}{60 d^6} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^5/(c + d*x),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.006, size = 302, normalized size = 2.5 \[{\frac{{b}^{5}{x}^{5}}{5\,d}}+{\frac{5\,a{b}^{4}{x}^{4}}{4\,d}}-{\frac{{b}^{5}{x}^{4}c}{4\,{d}^{2}}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{3}}{3\,d}}-{\frac{5\,a{b}^{4}{x}^{3}c}{3\,{d}^{2}}}+{\frac{{b}^{5}{x}^{3}{c}^{2}}{3\,{d}^{3}}}+5\,{\frac{{a}^{3}{b}^{2}{x}^{2}}{d}}-5\,{\frac{{a}^{2}{b}^{3}{x}^{2}c}{{d}^{2}}}+{\frac{5\,a{b}^{4}{x}^{2}{c}^{2}}{2\,{d}^{3}}}-{\frac{{b}^{5}{x}^{2}{c}^{3}}{2\,{d}^{4}}}+5\,{\frac{{a}^{4}bx}{d}}-10\,{\frac{{a}^{3}{b}^{2}cx}{{d}^{2}}}+10\,{\frac{{a}^{2}{b}^{3}{c}^{2}x}{{d}^{3}}}-5\,{\frac{a{b}^{4}{c}^{3}x}{{d}^{4}}}+{\frac{{b}^{5}{c}^{4}x}{{d}^{5}}}+{\frac{\ln \left ( dx+c \right ){a}^{5}}{d}}-5\,{\frac{\ln \left ( dx+c \right ){a}^{4}bc}{{d}^{2}}}+10\,{\frac{\ln \left ( dx+c \right ){a}^{3}{b}^{2}{c}^{2}}{{d}^{3}}}-10\,{\frac{\ln \left ( dx+c \right ){a}^{2}{b}^{3}{c}^{3}}{{d}^{4}}}+5\,{\frac{\ln \left ( dx+c \right ) a{b}^{4}{c}^{4}}{{d}^{5}}}-{\frac{\ln \left ( dx+c \right ){b}^{5}{c}^{5}}{{d}^{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^5/(d*x+c),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.35435, size = 348, normalized size = 2.85 \[ \frac{12 \, b^{5} d^{4} x^{5} - 15 \,{\left (b^{5} c d^{3} - 5 \, a b^{4} d^{4}\right )} x^{4} + 20 \,{\left (b^{5} c^{2} d^{2} - 5 \, a b^{4} c d^{3} + 10 \, a^{2} b^{3} d^{4}\right )} x^{3} - 30 \,{\left (b^{5} c^{3} d - 5 \, a b^{4} c^{2} d^{2} + 10 \, a^{2} b^{3} c d^{3} - 10 \, a^{3} b^{2} d^{4}\right )} x^{2} + 60 \,{\left (b^{5} c^{4} - 5 \, a b^{4} c^{3} d + 10 \, a^{2} b^{3} c^{2} d^{2} - 10 \, a^{3} b^{2} c d^{3} + 5 \, a^{4} b d^{4}\right )} x}{60 \, d^{5}} - \frac{{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (d x + c\right )}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^5/(d*x + c),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.203055, size = 350, normalized size = 2.87 \[ \frac{12 \, b^{5} d^{5} x^{5} - 15 \,{\left (b^{5} c d^{4} - 5 \, a b^{4} d^{5}\right )} x^{4} + 20 \,{\left (b^{5} c^{2} d^{3} - 5 \, a b^{4} c d^{4} + 10 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \,{\left (b^{5} c^{3} d^{2} - 5 \, a b^{4} c^{2} d^{3} + 10 \, a^{2} b^{3} c d^{4} - 10 \, a^{3} b^{2} d^{5}\right )} x^{2} + 60 \,{\left (b^{5} c^{4} d - 5 \, a b^{4} c^{3} d^{2} + 10 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x - 60 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (d x + c\right )}{60 \, d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^5/(d*x + c),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.21211, size = 202, normalized size = 1.66 \[ \frac{b^{5} x^{5}}{5 d} + \frac{x^{4} \left (5 a b^{4} d - b^{5} c\right )}{4 d^{2}} + \frac{x^{3} \left (10 a^{2} b^{3} d^{2} - 5 a b^{4} c d + b^{5} c^{2}\right )}{3 d^{3}} + \frac{x^{2} \left (10 a^{3} b^{2} d^{3} - 10 a^{2} b^{3} c d^{2} + 5 a b^{4} c^{2} d - b^{5} c^{3}\right )}{2 d^{4}} + \frac{x \left (5 a^{4} b d^{4} - 10 a^{3} b^{2} c d^{3} + 10 a^{2} b^{3} c^{2} d^{2} - 5 a b^{4} c^{3} d + b^{5} c^{4}\right )}{d^{5}} + \frac{\left (a d - b c\right )^{5} \log{\left (c + d x \right )}}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**5/(d*x+c),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.221767, size = 369, normalized size = 3.02 \[ \frac{12 \, b^{5} d^{4} x^{5} - 15 \, b^{5} c d^{3} x^{4} + 75 \, a b^{4} d^{4} x^{4} + 20 \, b^{5} c^{2} d^{2} x^{3} - 100 \, a b^{4} c d^{3} x^{3} + 200 \, a^{2} b^{3} d^{4} x^{3} - 30 \, b^{5} c^{3} d x^{2} + 150 \, a b^{4} c^{2} d^{2} x^{2} - 300 \, a^{2} b^{3} c d^{3} x^{2} + 300 \, a^{3} b^{2} d^{4} x^{2} + 60 \, b^{5} c^{4} x - 300 \, a b^{4} c^{3} d x + 600 \, a^{2} b^{3} c^{2} d^{2} x - 600 \, a^{3} b^{2} c d^{3} x + 300 \, a^{4} b d^{4} x}{60 \, d^{5}} - \frac{{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^5/(d*x + c),x, algorithm="giac")
[Out]