\(\int \frac {(d+e x)^4}{(b x+c x^2)^2} \, dx\) [268]

Optimal result

Integrand size = 19, antiderivative size = 94 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx=-\frac {d^4}{b^2 x}+\frac {e^4 x}{c^2}-\frac {(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac {2 d^3 (c d-2 b e) \log (x)}{b^3}+\frac {2 (c d-b e)^3 (c d+b e) \log (b+c x)}{b^3 c^3} \]

[Out]

-d^4/b^2/x+e^4*x/c^2-(-b*e+c*d)^4/b^2/c^3/(c*x+b)-2*d^3*(-2*b*e+c*d)*ln(x)/b^3+2*(-b*e+c*d)^3*(b*e+c*d)*ln(c*x
+b)/b^3/c^3

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx=\frac {2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}-\frac {2 d^3 \log (x) (c d-2 b e)}{b^3}-\frac {(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac {d^4}{b^2 x}+\frac {e^4 x}{c^2} \]

[In]

Int[(d + e*x)^4/(b*x + c*x^2)^2,x]

[Out]

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

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^4}{c^2}+\frac {d^4}{b^2 x^2}+\frac {2 d^3 (-c d+2 b e)}{b^3 x}+\frac {(-c d+b e)^4}{b^2 c^2 (b+c x)^2}-\frac {2 (-c d+b e)^3 (c d+b e)}{b^3 c^2 (b+c x)}\right ) \, dx \\ & = -\frac {d^4}{b^2 x}+\frac {e^4 x}{c^2}-\frac {(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac {2 d^3 (c d-2 b e) \log (x)}{b^3}+\frac {2 (c d-b e)^3 (c d+b e) \log (b+c x)}{b^3 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx=-\frac {d^4}{b^2 x}+\frac {e^4 x}{c^2}-\frac {(c d-b e)^4}{b^2 c^3 (b+c x)}+\frac {2 d^3 (-c d+2 b e) \log (x)}{b^3}+\frac {2 (c d-b e)^3 (c d+b e) \log (b+c x)}{b^3 c^3} \]

[In]

Integrate[(d + e*x)^4/(b*x + c*x^2)^2,x]

[Out]

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

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.64

method	result	size
default	\(\frac {e^{4} x}{c^{2}}-\frac {d^{4}}{b^{2} x}+\frac {2 d^{3} \left (2 b e -c d \right ) \ln \left (x \right )}{b^{3}}+\frac {\left (-2 b^{4} e^{4}+4 b^{3} c d \,e^{3}-4 b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) \ln \left (c x +b \right )}{c^{3} b^{3}}-\frac {b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}}{c^{3} b^{2} \left (c x +b \right )}\)	\(154\)
norman	\(\frac {\frac {e^{4} x^{3}}{c}+\frac {\left (2 b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) x^{2}}{c^{2} b^{3}}-\frac {d^{4}}{b}}{x \left (c x +b \right )}+\frac {2 d^{3} \left (2 b e -c d \right ) \ln \left (x \right )}{b^{3}}-\frac {2 \left (b^{4} e^{4}-2 b^{3} c d \,e^{3}+2 b \,c^{3} d^{3} e -c^{4} d^{4}\right ) \ln \left (c x +b \right )}{b^{3} c^{3}}\)	\(162\)
risch	\(\frac {e^{4} x}{c^{2}}+\frac {-\frac {\left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) x}{b^{2} c}-\frac {c^{2} d^{4}}{b}}{c^{2} x \left (c x +b \right )}-\frac {2 b \ln \left (c x +b \right ) e^{4}}{c^{3}}+\frac {4 \ln \left (c x +b \right ) d \,e^{3}}{c^{2}}-\frac {4 \ln \left (c x +b \right ) d^{3} e}{b^{2}}+\frac {2 c \ln \left (c x +b \right ) d^{4}}{b^{3}}+\frac {4 d^{3} \ln \left (-x \right ) e}{b^{2}}-\frac {2 d^{4} \ln \left (-x \right ) c}{b^{3}}\)	\(181\)
parallelrisch	\(\frac {4 \ln \left (x \right ) x^{2} b \,c^{4} d^{3} e -2 \ln \left (x \right ) x^{2} c^{5} d^{4}-2 \ln \left (c x +b \right ) x^{2} b^{4} c \,e^{4}+4 \ln \left (c x +b \right ) x^{2} b^{3} c^{2} d \,e^{3}-4 \ln \left (c x +b \right ) x^{2} b \,c^{4} d^{3} e +2 \ln \left (c x +b \right ) x^{2} c^{5} d^{4}+x^{3} b^{3} c^{2} e^{4}+4 \ln \left (x \right ) x \,b^{2} c^{3} d^{3} e -2 \ln \left (x \right ) x b \,c^{4} d^{4}-2 \ln \left (c x +b \right ) x \,b^{5} e^{4}+4 \ln \left (c x +b \right ) x \,b^{4} c d \,e^{3}-4 \ln \left (c x +b \right ) x \,b^{2} c^{3} d^{3} e +2 \ln \left (c x +b \right ) x b \,c^{4} d^{4}+2 x^{2} b^{4} c \,e^{4}-4 x^{2} b^{3} c^{2} d \,e^{3}+6 x^{2} b^{2} c^{3} d^{2} e^{2}-4 x^{2} b \,c^{4} d^{3} e +2 x^{2} c^{5} d^{4}-b^{2} c^{3} d^{4}}{c^{3} b^{3} x \left (c x +b \right )}\)	\(308\)

[In]

int((e*x+d)^4/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (94) = 188\).

Time = 0.27 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.73 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx=\frac {b^{3} c^{2} e^{4} x^{3} + b^{4} c e^{4} x^{2} - b^{2} c^{3} d^{4} - {\left (2 \, b c^{4} d^{4} - 4 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} + b^{5} e^{4}\right )} x + 2 \, {\left ({\left (c^{5} d^{4} - 2 \, b c^{4} d^{3} e + 2 \, b^{3} c^{2} d e^{3} - b^{4} c e^{4}\right )} x^{2} + {\left (b c^{4} d^{4} - 2 \, b^{2} c^{3} d^{3} e + 2 \, b^{4} c d e^{3} - b^{5} e^{4}\right )} x\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (c^{5} d^{4} - 2 \, b c^{4} d^{3} e\right )} x^{2} + {\left (b c^{4} d^{4} - 2 \, b^{2} c^{3} d^{3} e\right )} x\right )} \log \left (x\right )}{b^{3} c^{4} x^{2} + b^{4} c^{3} x} \]

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (87) = 174\).

Time = 1.41 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.26 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx=\frac {- b c^{3} d^{4} + x \left (- b^{4} e^{4} + 4 b^{3} c d e^{3} - 6 b^{2} c^{2} d^{2} e^{2} + 4 b c^{3} d^{3} e - 2 c^{4} d^{4}\right )}{b^{3} c^{3} x + b^{2} c^{4} x^{2}} + \frac {e^{4} x}{c^{2}} + \frac {2 d^{3} \cdot \left (2 b e - c d\right ) \log {\left (x + \frac {4 b^{2} c^{2} d^{3} e - 2 b c^{3} d^{4} - 2 b c^{2} d^{3} \cdot \left (2 b e - c d\right )}{2 b^{4} e^{4} - 4 b^{3} c d e^{3} + 8 b c^{3} d^{3} e - 4 c^{4} d^{4}} \right )}}{b^{3}} - \frac {2 \left (b e - c d\right )^{3} \left (b e + c d\right ) \log {\left (x + \frac {4 b^{2} c^{2} d^{3} e - 2 b c^{3} d^{4} + \frac {2 b \left (b e - c d\right )^{3} \left (b e + c d\right )}{c}}{2 b^{4} e^{4} - 4 b^{3} c d e^{3} + 8 b c^{3} d^{3} e - 4 c^{4} d^{4}} \right )}}{b^{3} c^{3}} \]

[In]

integrate((e*x+d)**4/(c*x**2+b*x)**2,x)

[Out]

(-b*c**3*d**4 + x*(-b**4*e**4 + 4*b**3*c*d*e**3 - 6*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4))/(b**
3*c**3*x + b**2*c**4*x**2) + e**4*x/c**2 + 2*d**3*(2*b*e - c*d)*log(x + (4*b**2*c**2*d**3*e - 2*b*c**3*d**4 -
2*b*c**2*d**3*(2*b*e - c*d))/(2*b**4*e**4 - 4*b**3*c*d*e**3 + 8*b*c**3*d**3*e - 4*c**4*d**4))/b**3 - 2*(b*e -
c*d)**3*(b*e + c*d)*log(x + (4*b**2*c**2*d**3*e - 2*b*c**3*d**4 + 2*b*(b*e - c*d)**3*(b*e + c*d)/c)/(2*b**4*e*
*4 - 4*b**3*c*d*e**3 + 8*b*c**3*d**3*e - 4*c**4*d**4))/(b**3*c**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.73 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx=\frac {e^{4} x}{c^{2}} - \frac {b c^{3} d^{4} + {\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} x}{b^{2} c^{4} x^{2} + b^{3} c^{3} x} - \frac {2 \, {\left (c d^{4} - 2 \, b d^{3} e\right )} \log \left (x\right )}{b^{3}} + \frac {2 \, {\left (c^{4} d^{4} - 2 \, b c^{3} d^{3} e + 2 \, b^{3} c d e^{3} - b^{4} e^{4}\right )} \log \left (c x + b\right )}{b^{3} c^{3}} \]

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

e^4*x/c^2 - (b*c^3*d^4 + (2*c^4*d^4 - 4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 4*b^3*c*d*e^3 + b^4*e^4)*x)/(b^2*c^4
*x^2 + b^3*c^3*x) - 2*(c*d^4 - 2*b*d^3*e)*log(x)/b^3 + 2*(c^4*d^4 - 2*b*c^3*d^3*e + 2*b^3*c*d*e^3 - b^4*e^4)*l
og(c*x + b)/(b^3*c^3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.73 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx=\frac {e^{4} x}{c^{2}} - \frac {2 \, {\left (c d^{4} - 2 \, b d^{3} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {2 \, {\left (c^{4} d^{4} - 2 \, b c^{3} d^{3} e + 2 \, b^{3} c d e^{3} - b^{4} e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{3}} - \frac {b c^{2} d^{4} + \frac {{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} x}{c}}{{\left (c x + b\right )} b^{2} c^{2} x} \]

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

e^4*x/c^2 - 2*(c*d^4 - 2*b*d^3*e)*log(abs(x))/b^3 + 2*(c^4*d^4 - 2*b*c^3*d^3*e + 2*b^3*c*d*e^3 - b^4*e^4)*log(
abs(c*x + b))/(b^3*c^3) - (b*c^2*d^4 + (2*c^4*d^4 - 4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 4*b^3*c*d*e^3 + b^4*e^
4)*x/c)/((c*x + b)*b^2*c^2*x)

Mupad [B] (verification not implemented)

Time = 9.62 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.77 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx=\frac {e^4\,x}{c^2}-\frac {\frac {c^2\,d^4}{b}+\frac {x\,\left (b^4\,e^4-4\,b^3\,c\,d\,e^3+6\,b^2\,c^2\,d^2\,e^2-4\,b\,c^3\,d^3\,e+2\,c^4\,d^4\right )}{b^2\,c}}{c^3\,x^2+b\,c^2\,x}+\frac {2\,d^3\,\ln \left (x\right )\,\left (2\,b\,e-c\,d\right )}{b^3}-\frac {\ln \left (b+c\,x\right )\,\left (2\,b^4\,e^4-4\,b^3\,c\,d\,e^3+4\,b\,c^3\,d^3\,e-2\,c^4\,d^4\right )}{b^3\,c^3} \]

[In]

int((d + e*x)^4/(b*x + c*x^2)^2,x)

[Out]

(e^4*x)/c^2 - ((c^2*d^4)/b + (x*(b^4*e^4 + 2*c^4*d^4 + 6*b^2*c^2*d^2*e^2 - 4*b*c^3*d^3*e - 4*b^3*c*d*e^3))/(b^
2*c))/(c^3*x^2 + b*c^2*x) + (2*d^3*log(x)*(2*b*e - c*d))/b^3 - (log(b + c*x)*(2*b^4*e^4 - 2*c^4*d^4 + 4*b*c^3*
d^3*e - 4*b^3*c*d*e^3))/(b^3*c^3)