\(\int \frac {(d+e x)^4}{b x+c x^2} \, dx\) [259]

Optimal result

Integrand size = 19, antiderivative size = 99 \[ \int \frac {(d+e x)^4}{b x+c x^2} \, dx=\frac {e^2 \left (6 c^2 d^2-4 b c d e+b^2 e^2\right ) x}{c^3}+\frac {e^3 (4 c d-b e) x^2}{2 c^2}+\frac {e^4 x^3}{3 c}+\frac {d^4 \log (x)}{b}-\frac {(c d-b e)^4 \log (b+c x)}{b c^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 99, 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} \[ \int \frac {(d+e x)^4}{b x+c x^2} \, dx=\frac {e^2 x \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )}{c^3}-\frac {(c d-b e)^4 \log (b+c x)}{b c^4}+\frac {e^3 x^2 (4 c d-b e)}{2 c^2}+\frac {d^4 \log (x)}{b}+\frac {e^4 x^3}{3 c} \]

[In]

Int[(d + e*x)^4/(b*x + c*x^2),x]

[Out]

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

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^2 \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )}{c^3}+\frac {d^4}{b x}+\frac {e^3 (4 c d-b e) x}{c^2}+\frac {e^4 x^2}{c}-\frac {(-c d+b e)^4}{b c^3 (b+c x)}\right ) \, dx \\ & = \frac {e^2 \left (6 c^2 d^2-4 b c d e+b^2 e^2\right ) x}{c^3}+\frac {e^3 (4 c d-b e) x^2}{2 c^2}+\frac {e^4 x^3}{3 c}+\frac {d^4 \log (x)}{b}-\frac {(c d-b e)^4 \log (b+c x)}{b c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^4}{b x+c x^2} \, dx=\frac {b c e^2 x \left (6 b^2 e^2-3 b c e (8 d+e x)+2 c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )+6 c^4 d^4 \log (x)-6 (c d-b e)^4 \log (b+c x)}{6 b c^4} \]

[In]

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

[Out]

(b*c*e^2*x*(6*b^2*e^2 - 3*b*c*e*(8*d + e*x) + 2*c^2*(18*d^2 + 6*d*e*x + e^2*x^2)) + 6*c^4*d^4*Log[x] - 6*(c*d
- b*e)^4*Log[b + c*x])/(6*b*c^4)

Maple [A] (verified)

Time = 1.92 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.35

method	result	size
norman	\(\frac {e^{2} \left (b^{2} e^{2}-4 b c d e +6 c^{2} d^{2}\right ) x}{c^{3}}+\frac {e^{4} x^{3}}{3 c}-\frac {e^{3} \left (b e -4 c d \right ) x^{2}}{2 c^{2}}+\frac {d^{4} \ln \left (x \right )}{b}-\frac {\left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right ) \ln \left (c x +b \right )}{b \,c^{4}}\)	\(134\)
default	\(\frac {e^{2} \left (\frac {1}{3} c^{2} e^{2} x^{3}-\frac {1}{2} b c \,e^{2} x^{2}+2 c^{2} d e \,x^{2}+b^{2} e^{2} x -4 b c d e x +6 c^{2} d^{2} x \right )}{c^{3}}+\frac {d^{4} \ln \left (x \right )}{b}+\frac {\left (-b^{4} e^{4}+4 b^{3} c d \,e^{3}-6 b^{2} c^{2} d^{2} e^{2}+4 b \,c^{3} d^{3} e -c^{4} d^{4}\right ) \ln \left (c x +b \right )}{c^{4} b}\)	\(138\)
risch	\(\frac {e^{4} x^{3}}{3 c}-\frac {e^{4} b \,x^{2}}{2 c^{2}}+\frac {2 d \,e^{3} x^{2}}{c}+\frac {e^{4} b^{2} x}{c^{3}}-\frac {4 e^{3} b d x}{c^{2}}+\frac {6 e^{2} d^{2} x}{c}-\frac {b^{3} \ln \left (c x +b \right ) e^{4}}{c^{4}}+\frac {4 b^{2} \ln \left (c x +b \right ) d \,e^{3}}{c^{3}}-\frac {6 b \ln \left (c x +b \right ) d^{2} e^{2}}{c^{2}}+\frac {4 \ln \left (c x +b \right ) d^{3} e}{c}-\frac {\ln \left (c x +b \right ) d^{4}}{b}+\frac {d^{4} \ln \left (-x \right )}{b}\)	\(164\)
parallelrisch	\(\frac {2 e^{4} x^{3} b \,c^{3}-3 x^{2} b^{2} c^{2} e^{4}+12 x^{2} b \,c^{3} d \,e^{3}+6 d^{4} \ln \left (x \right ) c^{4}-6 \ln \left (c x +b \right ) b^{4} e^{4}+24 \ln \left (c x +b \right ) b^{3} c d \,e^{3}-36 \ln \left (c x +b \right ) b^{2} c^{2} d^{2} e^{2}+24 \ln \left (c x +b \right ) b \,c^{3} d^{3} e -6 \ln \left (c x +b \right ) c^{4} d^{4}+6 x \,b^{3} c \,e^{4}-24 x \,b^{2} c^{2} d \,e^{3}+36 x b \,c^{3} d^{2} e^{2}}{6 b \,c^{4}}\)	\(175\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^4}{b x+c x^2} \, dx=\frac {2 \, b c^{3} e^{4} x^{3} + 6 \, c^{4} d^{4} \log \left (x\right ) + 3 \, {\left (4 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} + 6 \, {\left (6 \, b c^{3} d^{2} e^{2} - 4 \, b^{2} c^{2} d e^{3} + b^{3} c e^{4}\right )} x - 6 \, {\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} \log \left (c x + b\right )}{6 \, b c^{4}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.96 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.67 \[ \int \frac {(d+e x)^4}{b x+c x^2} \, dx=x^{2} \left (- \frac {b e^{4}}{2 c^{2}} + \frac {2 d e^{3}}{c}\right ) + x \left (\frac {b^{2} e^{4}}{c^{3}} - \frac {4 b d e^{3}}{c^{2}} + \frac {6 d^{2} e^{2}}{c}\right ) + \frac {e^{4} x^{3}}{3 c} + \frac {d^{4} \log {\left (x \right )}}{b} - \frac {\left (b e - c d\right )^{4} \log {\left (x + \frac {b c^{3} d^{4} + \frac {b \left (b e - c d\right )^{4}}{c}}{b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}} \right )}}{b c^{4}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.43 \[ \int \frac {(d+e x)^4}{b x+c x^2} \, dx=\frac {d^{4} \log \left (x\right )}{b} + \frac {2 \, c^{2} e^{4} x^{3} + 3 \, {\left (4 \, c^{2} d e^{3} - b c e^{4}\right )} x^{2} + 6 \, {\left (6 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3} + b^{2} e^{4}\right )} x}{6 \, c^{3}} - \frac {{\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} \log \left (c x + b\right )}{b c^{4}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.45 \[ \int \frac {(d+e x)^4}{b x+c x^2} \, dx=\frac {d^{4} \log \left ({\left | x \right |}\right )}{b} + \frac {2 \, c^{2} e^{4} x^{3} + 12 \, c^{2} d e^{3} x^{2} - 3 \, b c e^{4} x^{2} + 36 \, c^{2} d^{2} e^{2} x - 24 \, b c d e^{3} x + 6 \, b^{2} e^{4} x}{6 \, c^{3}} - \frac {{\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{4}} \]

[In]

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

[Out]

d^4*log(abs(x))/b + 1/6*(2*c^2*e^4*x^3 + 12*c^2*d*e^3*x^2 - 3*b*c*e^4*x^2 + 36*c^2*d^2*e^2*x - 24*b*c*d*e^3*x
+ 6*b^2*e^4*x)/c^3 - (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)*log(abs(c*x + b))
/(b*c^4)

Mupad [B] (verification not implemented)

Time = 9.61 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^4}{b x+c x^2} \, dx=x\,\left (\frac {b\,\left (\frac {b\,e^4}{c^2}-\frac {4\,d\,e^3}{c}\right )}{c}+\frac {6\,d^2\,e^2}{c}\right )-x^2\,\left (\frac {b\,e^4}{2\,c^2}-\frac {2\,d\,e^3}{c}\right )+\frac {e^4\,x^3}{3\,c}+\frac {d^4\,\ln \left (x\right )}{b}-\frac {\ln \left (b+c\,x\right )\,{\left (b\,e-c\,d\right )}^4}{b\,c^4} \]

[In]

int((d + e*x)^4/(b*x + c*x^2),x)

[Out]

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