3.277 \(\int \frac{(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=171 \[ \frac{(c d-b e)^4 (2 b e+3 c d)}{b^4 c^3 (b+c x)}+\frac{d^4 (3 c d-5 b e)}{b^4 x}+\frac{(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}-\frac{d^5}{2 b^3 x^2}+\frac{d^3 \log (x) \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right )}{b^5}-\frac{(c d-b e)^3 \left (b^2 e^2+3 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 c^3} \]

[Out]

-d^5/(2*b^3*x^2) + (d^4*(3*c*d - 5*b*e))/(b^4*x) + (c*d - b*e)^5/(2*b^3*c^3*(b +
c*x)^2) + ((c*d - b*e)^4*(3*c*d + 2*b*e))/(b^4*c^3*(b + c*x)) + (d^3*(6*c^2*d^2
- 15*b*c*d*e + 10*b^2*e^2)*Log[x])/b^5 - ((c*d - b*e)^3*(6*c^2*d^2 + 3*b*c*d*e
+ b^2*e^2)*Log[b + c*x])/(b^5*c^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.412692, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{(c d-b e)^4 (2 b e+3 c d)}{b^4 c^3 (b+c x)}+\frac{d^4 (3 c d-5 b e)}{b^4 x}+\frac{(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}-\frac{d^5}{2 b^3 x^2}+\frac{d^3 \log (x) \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right )}{b^5}-\frac{(c d-b e)^3 \left (b^2 e^2+3 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 c^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^5/(b*x + c*x^2)^3,x]

[Out]

-d^5/(2*b^3*x^2) + (d^4*(3*c*d - 5*b*e))/(b^4*x) + (c*d - b*e)^5/(2*b^3*c^3*(b +
c*x)^2) + ((c*d - b*e)^4*(3*c*d + 2*b*e))/(b^4*c^3*(b + c*x)) + (d^3*(6*c^2*d^2
- 15*b*c*d*e + 10*b^2*e^2)*Log[x])/b^5 - ((c*d - b*e)^3*(6*c^2*d^2 + 3*b*c*d*e
+ b^2*e^2)*Log[b + c*x])/(b^5*c^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 57.2568, size = 165, normalized size = 0.96 \[ - \frac{d^{5}}{2 b^{3} x^{2}} - \frac{\left (b e - c d\right )^{5}}{2 b^{3} c^{3} \left (b + c x\right )^{2}} - \frac{d^{4} \left (5 b e - 3 c d\right )}{b^{4} x} + \frac{\left (b e - c d\right )^{4} \left (2 b e + 3 c d\right )}{b^{4} c^{3} \left (b + c x\right )} + \frac{d^{3} \left (10 b^{2} e^{2} - 15 b c d e + 6 c^{2} d^{2}\right ) \log{\left (x \right )}}{b^{5}} + \frac{\left (b e - c d\right )^{3} \left (b^{2} e^{2} + 3 b c d e + 6 c^{2} d^{2}\right ) \log{\left (b + c x \right )}}{b^{5} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**5/(c*x**2+b*x)**3,x)

[Out]

-d**5/(2*b**3*x**2) - (b*e - c*d)**5/(2*b**3*c**3*(b + c*x)**2) - d**4*(5*b*e -
3*c*d)/(b**4*x) + (b*e - c*d)**4*(2*b*e + 3*c*d)/(b**4*c**3*(b + c*x)) + d**3*(1
0*b**2*e**2 - 15*b*c*d*e + 6*c**2*d**2)*log(x)/b**5 + (b*e - c*d)**3*(b**2*e**2
+ 3*b*c*d*e + 6*c**2*d**2)*log(b + c*x)/(b**5*c**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.180946, size = 165, normalized size = 0.96 \[ -\frac{\frac{b^2 (b e-c d)^5}{c^3 (b+c x)^2}-2 d^3 \log (x) \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right )+\frac{2 (c d-b e)^3 \left (b^2 e^2+3 b c d e+6 c^2 d^2\right ) \log (b+c x)}{c^3}+\frac{b^2 d^5}{x^2}-\frac{2 b (c d-b e)^4 (2 b e+3 c d)}{c^3 (b+c x)}+\frac{2 b d^4 (5 b e-3 c d)}{x}}{2 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^5/(b*x + c*x^2)^3,x]

[Out]

-((b^2*d^5)/x^2 + (2*b*d^4*(-3*c*d + 5*b*e))/x + (b^2*(-(c*d) + b*e)^5)/(c^3*(b
+ c*x)^2) - (2*b*(c*d - b*e)^4*(3*c*d + 2*b*e))/(c^3*(b + c*x)) - 2*d^3*(6*c^2*d
^2 - 15*b*c*d*e + 10*b^2*e^2)*Log[x] + (2*(c*d - b*e)^3*(6*c^2*d^2 + 3*b*c*d*e +
b^2*e^2)*Log[b + c*x])/c^3)/(2*b^5)

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 329, normalized size = 1.9 \[ -{\frac{{d}^{5}}{2\,{b}^{3}{x}^{2}}}+10\,{\frac{{d}^{3}\ln \left ( x \right ){e}^{2}}{{b}^{3}}}-15\,{\frac{{d}^{4}\ln \left ( x \right ) ce}{{b}^{4}}}+6\,{\frac{{d}^{5}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}-5\,{\frac{{d}^{4}e}{{b}^{3}x}}+3\,{\frac{{d}^{5}c}{{b}^{4}x}}+{\frac{\ln \left ( cx+b \right ){e}^{5}}{{c}^{3}}}-10\,{\frac{\ln \left ( cx+b \right ){d}^{3}{e}^{2}}{{b}^{3}}}+15\,{\frac{c\ln \left ( cx+b \right ){d}^{4}e}{{b}^{4}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ){d}^{5}}{{b}^{5}}}+2\,{\frac{b{e}^{5}}{{c}^{3} \left ( cx+b \right ) }}-5\,{\frac{d{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+10\,{\frac{{d}^{3}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-10\,{\frac{{d}^{4}ec}{{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{d}^{5}{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}-{\frac{{b}^{2}{e}^{5}}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{5\,bd{e}^{4}}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}-5\,{\frac{{d}^{2}{e}^{3}}{c \left ( cx+b \right ) ^{2}}}+5\,{\frac{{d}^{3}{e}^{2}}{b \left ( cx+b \right ) ^{2}}}-{\frac{5\,{d}^{4}ec}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{{d}^{5}{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^5/(c*x^2+b*x)^3,x)

[Out]

-1/2*d^5/b^3/x^2+10*d^3/b^3*ln(x)*e^2-15*d^4/b^4*ln(x)*c*e+6*d^5/b^5*ln(x)*c^2-5
*d^4/b^3/x*e+3*d^5/b^4/x*c+1/c^3*ln(c*x+b)*e^5-10/b^3*ln(c*x+b)*d^3*e^2+15/b^4*c
*ln(c*x+b)*d^4*e-6/b^5*c^2*ln(c*x+b)*d^5+2/c^3*b/(c*x+b)*e^5-5/c^2/(c*x+b)*d*e^4
+10/b^2/(c*x+b)*d^3*e^2-10*c/b^3/(c*x+b)*d^4*e+3*c^2/b^4/(c*x+b)*d^5-1/2/c^3*b^2
/(c*x+b)^2*e^5+5/2/c^2*b/(c*x+b)^2*d*e^4-5/c/(c*x+b)^2*d^2*e^3+5/b/(c*x+b)^2*d^3
*e^2-5/2*c/b^2/(c*x+b)^2*d^4*e+1/2*c^2/b^3/(c*x+b)^2*d^5

_______________________________________________________________________________________

Maxima [A]  time = 0.712359, size = 400, normalized size = 2.34 \[ -\frac{b^{3} c^{3} d^{5} - 2 \,{\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - 5 \, b^{4} c^{2} d e^{4} + 2 \, b^{5} c e^{5}\right )} x^{3} -{\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} - 5 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e\right )} x}{2 \,{\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}} + \frac{{\left (6 \, c^{2} d^{5} - 15 \, b c d^{4} e + 10 \, b^{2} d^{3} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac{{\left (6 \, c^{5} d^{5} - 15 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - b^{5} e^{5}\right )} \log \left (c x + b\right )}{b^{5} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

-1/2*(b^3*c^3*d^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 - 5*b^4*c
^2*d*e^4 + 2*b^5*c*e^5)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*
e^2 - 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^4 + 3*b^6*e^5)*x^2 - 2*(2*b^2*c^4*d^5 - 5
*b^3*c^3*d^4*e)*x)/(b^4*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2) + (6*c^2*d^5 - 15
*b*c*d^4*e + 10*b^2*d^3*e^2)*log(x)/b^5 - (6*c^5*d^5 - 15*b*c^4*d^4*e + 10*b^2*c
^3*d^3*e^2 - b^5*e^5)*log(c*x + b)/(b^5*c^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.23345, size = 666, normalized size = 3.89 \[ -\frac{b^{4} c^{3} d^{5} - 2 \,{\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - 5 \, b^{5} c^{2} d e^{4} + 2 \, b^{6} c e^{5}\right )} x^{3} -{\left (18 \, b^{2} c^{5} d^{5} - 45 \, b^{3} c^{4} d^{4} e + 30 \, b^{4} c^{3} d^{3} e^{2} - 10 \, b^{5} c^{2} d^{2} e^{3} - 5 \, b^{6} c d e^{4} + 3 \, b^{7} e^{5}\right )} x^{2} - 2 \,{\left (2 \, b^{3} c^{4} d^{5} - 5 \, b^{4} c^{3} d^{4} e\right )} x + 2 \,{\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2} - b^{5} c^{2} e^{5}\right )} x^{4} + 2 \,{\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{6} c e^{5}\right )} x^{3} +{\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2} - b^{7} e^{5}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2}\right )} x^{4} + 2 \,{\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2}\right )} x^{3} +{\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{5} x^{4} + 2 \, b^{6} c^{4} x^{3} + b^{7} c^{3} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

-1/2*(b^4*c^3*d^5 - 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - 5*b
^5*c^2*d*e^4 + 2*b^6*c*e^5)*x^3 - (18*b^2*c^5*d^5 - 45*b^3*c^4*d^4*e + 30*b^4*c^
3*d^3*e^2 - 10*b^5*c^2*d^2*e^3 - 5*b^6*c*d*e^4 + 3*b^7*e^5)*x^2 - 2*(2*b^3*c^4*d
^5 - 5*b^4*c^3*d^4*e)*x + 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2 -
b^5*c^2*e^5)*x^4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - b^6*
c*e^5)*x^3 + (6*b^2*c^5*d^5 - 15*b^3*c^4*d^4*e + 10*b^4*c^3*d^3*e^2 - b^7*e^5)*x
^2)*log(c*x + b) - 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2)*x^4 + 2*
(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2)*x^3 + (6*b^2*c^5*d^5 - 15*
b^3*c^4*d^4*e + 10*b^4*c^3*d^3*e^2)*x^2)*log(x))/(b^5*c^5*x^4 + 2*b^6*c^4*x^3 +
b^7*c^3*x^2)

_______________________________________________________________________________________

Sympy [A]  time = 24.1477, size = 524, normalized size = 3.06 \[ \frac{- b^{3} c^{3} d^{5} + x^{3} \left (4 b^{5} c e^{5} - 10 b^{4} c^{2} d e^{4} + 20 b^{2} c^{4} d^{3} e^{2} - 30 b c^{5} d^{4} e + 12 c^{6} d^{5}\right ) + x^{2} \left (3 b^{6} e^{5} - 5 b^{5} c d e^{4} - 10 b^{4} c^{2} d^{2} e^{3} + 30 b^{3} c^{3} d^{3} e^{2} - 45 b^{2} c^{4} d^{4} e + 18 b c^{5} d^{5}\right ) + x \left (- 10 b^{3} c^{3} d^{4} e + 4 b^{2} c^{4} d^{5}\right )}{2 b^{6} c^{3} x^{2} + 4 b^{5} c^{4} x^{3} + 2 b^{4} c^{5} x^{4}} + \frac{d^{3} \left (10 b^{2} e^{2} - 15 b c d e + 6 c^{2} d^{2}\right ) \log{\left (x + \frac{- 10 b^{3} c^{2} d^{3} e^{2} + 15 b^{2} c^{3} d^{4} e - 6 b c^{4} d^{5} + b c^{2} d^{3} \left (10 b^{2} e^{2} - 15 b c d e + 6 c^{2} d^{2}\right )}{b^{5} e^{5} - 20 b^{2} c^{3} d^{3} e^{2} + 30 b c^{4} d^{4} e - 12 c^{5} d^{5}} \right )}}{b^{5}} + \frac{\left (b e - c d\right )^{3} \left (b^{2} e^{2} + 3 b c d e + 6 c^{2} d^{2}\right ) \log{\left (x + \frac{- 10 b^{3} c^{2} d^{3} e^{2} + 15 b^{2} c^{3} d^{4} e - 6 b c^{4} d^{5} + \frac{b \left (b e - c d\right )^{3} \left (b^{2} e^{2} + 3 b c d e + 6 c^{2} d^{2}\right )}{c}}{b^{5} e^{5} - 20 b^{2} c^{3} d^{3} e^{2} + 30 b c^{4} d^{4} e - 12 c^{5} d^{5}} \right )}}{b^{5} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**5/(c*x**2+b*x)**3,x)

[Out]

(-b**3*c**3*d**5 + x**3*(4*b**5*c*e**5 - 10*b**4*c**2*d*e**4 + 20*b**2*c**4*d**3
*e**2 - 30*b*c**5*d**4*e + 12*c**6*d**5) + x**2*(3*b**6*e**5 - 5*b**5*c*d*e**4 -
10*b**4*c**2*d**2*e**3 + 30*b**3*c**3*d**3*e**2 - 45*b**2*c**4*d**4*e + 18*b*c*
*5*d**5) + x*(-10*b**3*c**3*d**4*e + 4*b**2*c**4*d**5))/(2*b**6*c**3*x**2 + 4*b*
*5*c**4*x**3 + 2*b**4*c**5*x**4) + d**3*(10*b**2*e**2 - 15*b*c*d*e + 6*c**2*d**2
)*log(x + (-10*b**3*c**2*d**3*e**2 + 15*b**2*c**3*d**4*e - 6*b*c**4*d**5 + b*c**
2*d**3*(10*b**2*e**2 - 15*b*c*d*e + 6*c**2*d**2))/(b**5*e**5 - 20*b**2*c**3*d**3
*e**2 + 30*b*c**4*d**4*e - 12*c**5*d**5))/b**5 + (b*e - c*d)**3*(b**2*e**2 + 3*b
*c*d*e + 6*c**2*d**2)*log(x + (-10*b**3*c**2*d**3*e**2 + 15*b**2*c**3*d**4*e - 6
*b*c**4*d**5 + b*(b*e - c*d)**3*(b**2*e**2 + 3*b*c*d*e + 6*c**2*d**2)/c)/(b**5*e
**5 - 20*b**2*c**3*d**3*e**2 + 30*b*c**4*d**4*e - 12*c**5*d**5))/(b**5*c**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208753, size = 371, normalized size = 2.17 \[ \frac{{\left (6 \, c^{2} d^{5} - 15 \, b c d^{4} e + 10 \, b^{2} d^{3} e^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} - \frac{{\left (6 \, c^{5} d^{5} - 15 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - b^{5} e^{5}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{5} c^{3}} - \frac{b^{3} c^{3} d^{5} - 2 \,{\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - 5 \, b^{4} c^{2} d e^{4} + 2 \, b^{5} c e^{5}\right )} x^{3} -{\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} - 5 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e\right )} x}{2 \,{\left (c x + b\right )}^{2} b^{4} c^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*x^2 + b*x)^3,x, algorithm="giac")

[Out]

(6*c^2*d^5 - 15*b*c*d^4*e + 10*b^2*d^3*e^2)*ln(abs(x))/b^5 - (6*c^5*d^5 - 15*b*c
^4*d^4*e + 10*b^2*c^3*d^3*e^2 - b^5*e^5)*ln(abs(c*x + b))/(b^5*c^3) - 1/2*(b^3*c
^3*d^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 - 5*b^4*c^2*d*e^4 +
2*b^5*c*e^5)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 - 10*b^
4*c^2*d^2*e^3 - 5*b^5*c*d*e^4 + 3*b^6*e^5)*x^2 - 2*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^
4*e)*x)/((c*x + b)^2*b^4*c^3*x^2)