Optimal. Leaf size=111 \[ -\frac{4 e^3 (b d-a e)}{b^5 (a+b x)}-\frac{3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac{4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac{(b d-a e)^4}{4 b^5 (a+b x)^4}+\frac{e^4 \log (a+b x)}{b^5} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.200321, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{4 e^3 (b d-a e)}{b^5 (a+b x)}-\frac{3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac{4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac{(b d-a e)^4}{4 b^5 (a+b x)^4}+\frac{e^4 \log (a+b x)}{b^5} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 64.6241, size = 100, normalized size = 0.9 \[ \frac{e^{4} \log{\left (a + b x \right )}}{b^{5}} + \frac{4 e^{3} \left (a e - b d\right )}{b^{5} \left (a + b x\right )} - \frac{3 e^{2} \left (a e - b d\right )^{2}}{b^{5} \left (a + b x\right )^{2}} + \frac{4 e \left (a e - b d\right )^{3}}{3 b^{5} \left (a + b x\right )^{3}} - \frac{\left (a e - b d\right )^{4}}{4 b^{5} \left (a + b x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.116243, size = 120, normalized size = 1.08 \[ \frac{e^4 \log (a+b x)}{b^5}-\frac{(b d-a e) \left (25 a^3 e^3+a^2 b e^2 (13 d+88 e x)+a b^2 e \left (7 d^2+40 d e x+108 e^2 x^2\right )+b^3 \left (3 d^3+16 d^2 e x+36 d e^2 x^2+48 e^3 x^3\right )\right )}{12 b^5 (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.011, size = 260, normalized size = 2.3 \[ -3\,{\frac{{a}^{2}{e}^{4}}{{b}^{5} \left ( bx+a \right ) ^{2}}}+6\,{\frac{ad{e}^{3}}{{b}^{4} \left ( bx+a \right ) ^{2}}}-3\,{\frac{{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{4}{e}^{4}}{4\,{b}^{5} \left ( bx+a \right ) ^{4}}}+{\frac{d{e}^{3}{a}^{3}}{{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{3\,{d}^{2}{e}^{2}{a}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{4}}}+{\frac{a{d}^{3}e}{{b}^{2} \left ( bx+a \right ) ^{4}}}-{\frac{{d}^{4}}{4\,b \left ( bx+a \right ) ^{4}}}+{\frac{{e}^{4}\ln \left ( bx+a \right ) }{{b}^{5}}}+4\,{\frac{a{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}-4\,{\frac{d{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}+{\frac{4\,{e}^{4}{a}^{3}}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}-4\,{\frac{d{e}^{3}{a}^{2}}{{b}^{4} \left ( bx+a \right ) ^{3}}}+4\,{\frac{a{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{4\,{d}^{3}e}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.717339, size = 296, normalized size = 2.67 \[ -\frac{3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4} + 48 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} - 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (2 \, b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 11 \, a^{3} b e^{4}\right )} x}{12 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} + \frac{e^{4} \log \left (b x + a\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.282145, size = 360, normalized size = 3.24 \[ -\frac{3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4} + 48 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} - 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (2 \, b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 11 \, a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (b x + a\right )}{12 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 13.2037, size = 230, normalized size = 2.07 \[ \frac{25 a^{4} e^{4} - 12 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e - 3 b^{4} d^{4} + x^{3} \left (48 a b^{3} e^{4} - 48 b^{4} d e^{3}\right ) + x^{2} \left (108 a^{2} b^{2} e^{4} - 72 a b^{3} d e^{3} - 36 b^{4} d^{2} e^{2}\right ) + x \left (88 a^{3} b e^{4} - 48 a^{2} b^{2} d e^{3} - 24 a b^{3} d^{2} e^{2} - 16 b^{4} d^{3} e\right )}{12 a^{4} b^{5} + 48 a^{3} b^{6} x + 72 a^{2} b^{7} x^{2} + 48 a b^{8} x^{3} + 12 b^{9} x^{4}} + \frac{e^{4} \log{\left (a + b x \right )}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.300763, size = 235, normalized size = 2.12 \[ \frac{e^{4}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5}} - \frac{48 \,{\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} x^{3} + 36 \,{\left (b^{3} d^{2} e^{2} + 2 \, a b^{2} d e^{3} - 3 \, a^{2} b e^{4}\right )} x^{2} + 8 \,{\left (2 \, b^{3} d^{3} e + 3 \, a b^{2} d^{2} e^{2} + 6 \, a^{2} b d e^{3} - 11 \, a^{3} e^{4}\right )} x + \frac{3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4}}{b}}{12 \,{\left (b x + a\right )}^{4} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
[Out]