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

Optimal. Leaf size=94 \[ -\frac{(c d-b e)^4}{b^2 c^3 (b+c x)}+\frac{2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}-\frac{2 d^3 \log (x) (c d-2 b e)}{b^3}-\frac{d^4}{b^2 x}+\frac{e^4 x}{c^2} \]

[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)

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Rubi [A]  time = 0.100295, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{(c d-b e)^4}{b^2 c^3 (b+c x)}+\frac{2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}-\frac{2 d^3 \log (x) (c d-2 b e)}{b^3}-\frac{d^4}{b^2 x}+\frac{e^4 x}{c^2} \]

Antiderivative was successfully verified.

[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 698

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*} \int \frac{(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx &=\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]  time = 0.0943638, size = 95, normalized size = 1.01 \[ -\frac{(c d-b e)^4}{b^2 c^3 (b+c x)}+\frac{2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}+\frac{2 d^3 \log (x) (2 b e-c d)}{b^3}-\frac{d^4}{b^2 x}+\frac{e^4 x}{c^2} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.055, size = 188, normalized size = 2. \begin{align*}{\frac{{e}^{4}x}{{c}^{2}}}-{\frac{{d}^{4}}{{b}^{2}x}}+4\,{\frac{{d}^{3}\ln \left ( x \right ) e}{{b}^{2}}}-2\,{\frac{{d}^{4}\ln \left ( x \right ) c}{{b}^{3}}}-2\,{\frac{b\ln \left ( cx+b \right ){e}^{4}}{{c}^{3}}}+4\,{\frac{\ln \left ( cx+b \right ) d{e}^{3}}{{c}^{2}}}-4\,{\frac{\ln \left ( cx+b \right ){d}^{3}e}{{b}^{2}}}+2\,{\frac{c\ln \left ( cx+b \right ){d}^{4}}{{b}^{3}}}-{\frac{{b}^{2}{e}^{4}}{{c}^{3} \left ( cx+b \right ) }}+4\,{\frac{bd{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-6\,{\frac{{d}^{2}{e}^{2}}{c \left ( cx+b \right ) }}+4\,{\frac{{d}^{3}e}{b \left ( cx+b \right ) }}-{\frac{c{d}^{4}}{{b}^{2} \left ( cx+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12418, size = 220, normalized size = 2.34 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.74641, size = 504, normalized size = 5.36 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 7.36321, size = 306, normalized size = 3.26 \begin{align*} - \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} \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} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.36, size = 216, normalized size = 2.3 \begin{align*} \frac{x e^{4}}{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^4/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)