### 3.260 $$\int \frac{(d+e x)^3}{b x+c x^2} \, dx$$

Optimal. Leaf size=64 $\frac{e^2 x (3 c d-b e)}{c^2}-\frac{(c d-b e)^3 \log (b+c x)}{b c^3}+\frac{d^3 \log (x)}{b}+\frac{e^3 x^2}{2 c}$

[Out]

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

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Rubi [A]  time = 0.0574003, antiderivative size = 64, 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} $\frac{e^2 x (3 c d-b e)}{c^2}-\frac{(c d-b e)^3 \log (b+c x)}{b c^3}+\frac{d^3 \log (x)}{b}+\frac{e^3 x^2}{2 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3/(b*x + c*x^2),x]

[Out]

(e^2*(3*c*d - b*e)*x)/c^2 + (e^3*x^2)/(2*c) + (d^3*Log[x])/b - ((c*d - b*e)^3*Log[b + c*x])/(b*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)^3}{b x+c x^2} \, dx &=\int \left (\frac{e^2 (3 c d-b e)}{c^2}+\frac{d^3}{b x}+\frac{e^3 x}{c}+\frac{(-c d+b e)^3}{b c^2 (b+c x)}\right ) \, dx\\ &=\frac{e^2 (3 c d-b e) x}{c^2}+\frac{e^3 x^2}{2 c}+\frac{d^3 \log (x)}{b}-\frac{(c d-b e)^3 \log (b+c x)}{b c^3}\\ \end{align*}

Mathematica [A]  time = 0.0275231, size = 59, normalized size = 0.92 $\frac{b c e^2 x (-2 b e+6 c d+c e x)-2 (c d-b e)^3 \log (b+c x)+2 c^3 d^3 \log (x)}{2 b c^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.048, size = 103, normalized size = 1.6 \begin{align*}{\frac{{e}^{3}{x}^{2}}{2\,c}}-{\frac{{e}^{3}xb}{{c}^{2}}}+3\,{\frac{d{e}^{2}x}{c}}+{\frac{{d}^{3}\ln \left ( x \right ) }{b}}+{\frac{{b}^{2}\ln \left ( cx+b \right ){e}^{3}}{{c}^{3}}}-3\,{\frac{b\ln \left ( cx+b \right ) d{e}^{2}}{{c}^{2}}}+3\,{\frac{\ln \left ( cx+b \right ){d}^{2}e}{c}}-{\frac{\ln \left ( cx+b \right ){d}^{3}}{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3/(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 1.09648, size = 123, normalized size = 1.92 \begin{align*} \frac{d^{3} \log \left (x\right )}{b} + \frac{c e^{3} x^{2} + 2 \,{\left (3 \, c d e^{2} - b e^{3}\right )} x}{2 \, c^{2}} - \frac{{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (c x + b\right )}{b c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.70464, size = 204, normalized size = 3.19 \begin{align*} \frac{b c^{2} e^{3} x^{2} + 2 \, c^{3} d^{3} \log \left (x\right ) + 2 \,{\left (3 \, b c^{2} d e^{2} - b^{2} c e^{3}\right )} x - 2 \,{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (c x + b\right )}{2 \, b c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 5.58811, size = 112, normalized size = 1.75 \begin{align*} \frac{e^{3} x^{2}}{2 c} - \frac{x \left (b e^{3} - 3 c d e^{2}\right )}{c^{2}} + \frac{d^{3} \log{\left (x \right )}}{b} + \frac{\left (b e - c d\right )^{3} \log{\left (x + \frac{- b c^{2} d^{3} + \frac{b \left (b e - c d\right )^{3}}{c}}{b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - 2 c^{3} d^{3}} \right )}}{b c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(c*x**2+b*x),x)

[Out]

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

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Giac [A]  time = 1.30506, size = 117, normalized size = 1.83 \begin{align*} \frac{d^{3} \log \left ({\left | x \right |}\right )}{b} + \frac{c x^{2} e^{3} + 6 \, c d x e^{2} - 2 \, b x e^{3}}{2 \, c^{2}} - \frac{{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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