### 3.259 $$\int \frac{(d+e x)^4}{b x+c x^2} \, dx$$

Optimal. Leaf size=99 $\frac{e^2 x \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )}{c^3}+\frac{e^3 x^2 (4 c d-b e)}{2 c^2}-\frac{(c d-b e)^4 \log (b+c x)}{b c^4}+\frac{d^4 \log (x)}{b}+\frac{e^4 x^3}{3 c}$

[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)

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Rubi [A]  time = 0.0908252, antiderivative size = 99, 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} $\frac{e^2 x \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )}{c^3}+\frac{e^3 x^2 (4 c d-b e)}{2 c^2}-\frac{(c d-b e)^4 \log (b+c x)}{b c^4}+\frac{d^4 \log (x)}{b}+\frac{e^4 x^3}{3 c}$

Antiderivative was successfully veriﬁed.

[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 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}{b x+c x^2} \, dx &=\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]  time = 0.0454209, size = 90, normalized size = 0.91 $\frac{b c e^2 x \left (6 b^2 e^2-3 b c e (8 d+e x)+2 c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )-6 (c d-b e)^4 \log (b+c x)+6 c^4 d^4 \log (x)}{6 b c^4}$

Antiderivative was successfully veriﬁed.

[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)

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Maple [A]  time = 0.053, size = 162, normalized size = 1.6 \begin{align*}{\frac{{e}^{4}{x}^{3}}{3\,c}}-{\frac{{e}^{4}{x}^{2}b}{2\,{c}^{2}}}+2\,{\frac{d{e}^{3}{x}^{2}}{c}}+{\frac{{e}^{4}{b}^{2}x}{{c}^{3}}}-4\,{\frac{{e}^{3}bdx}{{c}^{2}}}+6\,{\frac{{d}^{2}{e}^{2}x}{c}}+{\frac{{d}^{4}\ln \left ( x \right ) }{b}}-{\frac{{b}^{3}\ln \left ( cx+b \right ){e}^{4}}{{c}^{4}}}+4\,{\frac{{b}^{2}\ln \left ( cx+b \right ) d{e}^{3}}{{c}^{3}}}-6\,{\frac{b\ln \left ( cx+b \right ){d}^{2}{e}^{2}}{{c}^{2}}}+4\,{\frac{\ln \left ( cx+b \right ){d}^{3}e}{c}}-{\frac{\ln \left ( cx+b \right ){d}^{4}}{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14139, size = 192, normalized size = 1.94 \begin{align*} \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.70411, size = 312, normalized size = 3.15 \begin{align*} \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 5.8469, size = 165, normalized size = 1.67 \begin{align*} \frac{e^{4} x^{3}}{3 c} - \frac{x^{2} \left (b e^{4} - 4 c d e^{3}\right )}{2 c^{2}} + \frac{x \left (b^{2} e^{4} - 4 b c d e^{3} + 6 c^{2} d^{2} e^{2}\right )}{c^{3}} + \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.24519, size = 184, normalized size = 1.86 \begin{align*} \frac{d^{4} \log \left ({\left | x \right |}\right )}{b} + \frac{2 \, c^{2} x^{3} e^{4} + 12 \, c^{2} d x^{2} e^{3} + 36 \, c^{2} d^{2} x e^{2} - 3 \, b c x^{2} e^{4} - 24 \, b c d x e^{3} + 6 \, b^{2} x e^{4}}{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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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*x^3*e^4 + 12*c^2*d*x^2*e^3 + 36*c^2*d^2*x*e^2 - 3*b*c*x^2*e^4 - 24*b*c*d*x*e^3
+ 6*b^2*x*e^4)/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)