### 3.281 $$\int \frac{d+e x}{(b x+c x^2)^3} \, dx$$

Optimal. Leaf size=110 $\frac{3 c d-b e}{b^4 x}+\frac{c (3 c d-2 b e)}{b^4 (b+c x)}+\frac{c (c d-b e)}{2 b^3 (b+c x)^2}+\frac{3 c \log (x) (2 c d-b e)}{b^5}-\frac{3 c (2 c d-b e) \log (b+c x)}{b^5}-\frac{d}{2 b^3 x^2}$

[Out]

-d/(2*b^3*x^2) + (3*c*d - b*e)/(b^4*x) + (c*(c*d - b*e))/(2*b^3*(b + c*x)^2) + (c*(3*c*d - 2*b*e))/(b^4*(b + c
*x)) + (3*c*(2*c*d - b*e)*Log[x])/b^5 - (3*c*(2*c*d - b*e)*Log[b + c*x])/b^5

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Rubi [A]  time = 0.103272, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {631} $\frac{3 c d-b e}{b^4 x}+\frac{c (3 c d-2 b e)}{b^4 (b+c x)}+\frac{c (c d-b e)}{2 b^3 (b+c x)^2}+\frac{3 c \log (x) (2 c d-b e)}{b^5}-\frac{3 c (2 c d-b e) \log (b+c x)}{b^5}-\frac{d}{2 b^3 x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/(b*x + c*x^2)^3,x]

[Out]

-d/(2*b^3*x^2) + (3*c*d - b*e)/(b^4*x) + (c*(c*d - b*e))/(2*b^3*(b + c*x)^2) + (c*(3*c*d - 2*b*e))/(b^4*(b + c
*x)) + (3*c*(2*c*d - b*e)*Log[x])/b^5 - (3*c*(2*c*d - b*e)*Log[b + c*x])/b^5

Rule 631

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

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{d}{b^3 x^3}+\frac{-3 c d+b e}{b^4 x^2}-\frac{3 c (-2 c d+b e)}{b^5 x}+\frac{c^2 (-c d+b e)}{b^3 (b+c x)^3}+\frac{c^2 (-3 c d+2 b e)}{b^4 (b+c x)^2}+\frac{3 c^2 (-2 c d+b e)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac{d}{2 b^3 x^2}+\frac{3 c d-b e}{b^4 x}+\frac{c (c d-b e)}{2 b^3 (b+c x)^2}+\frac{c (3 c d-2 b e)}{b^4 (b+c x)}+\frac{3 c (2 c d-b e) \log (x)}{b^5}-\frac{3 c (2 c d-b e) \log (b+c x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.080662, size = 102, normalized size = 0.93 $\frac{-\frac{b \left (b^2 c x (9 e x-4 d)+b^3 (d+2 e x)+6 b c^2 x^2 (e x-3 d)-12 c^3 d x^3\right )}{x^2 (b+c x)^2}+6 c \log (x) (2 c d-b e)+6 c (b e-2 c d) \log (b+c x)}{2 b^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-((b*(-12*c^3*d*x^3 + 6*b*c^2*x^2*(-3*d + e*x) + b^3*(d + 2*e*x) + b^2*c*x*(-4*d + 9*e*x)))/(x^2*(b + c*x)^2)
) + 6*c*(2*c*d - b*e)*Log[x] + 6*c*(-2*c*d + b*e)*Log[b + c*x])/(2*b^5)

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Maple [A]  time = 0.056, size = 138, normalized size = 1.3 \begin{align*} -{\frac{d}{2\,{b}^{3}{x}^{2}}}-{\frac{e}{{b}^{3}x}}+3\,{\frac{cd}{{b}^{4}x}}-3\,{\frac{c\ln \left ( x \right ) e}{{b}^{4}}}+6\,{\frac{{c}^{2}\ln \left ( x \right ) d}{{b}^{5}}}-2\,{\frac{ce}{{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{d{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}-{\frac{ce}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{d{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}+3\,{\frac{c\ln \left ( cx+b \right ) e}{{b}^{4}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ) d}{{b}^{5}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d/b^3/x^2-1/b^3/x*e+3/b^4/x*c*d-3*c/b^4*ln(x)*e+6*c^2/b^5*ln(x)*d-2*c/b^3/(c*x+b)*e+3*c^2/b^4/(c*x+b)*d-1
/2*c/b^2/(c*x+b)^2*e+1/2*c^2/b^3/(c*x+b)^2*d+3*c/b^4*ln(c*x+b)*e-6*c^2/b^5*ln(c*x+b)*d

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Maxima [A]  time = 1.12258, size = 184, normalized size = 1.67 \begin{align*} -\frac{b^{3} d - 6 \,{\left (2 \, c^{3} d - b c^{2} e\right )} x^{3} - 9 \,{\left (2 \, b c^{2} d - b^{2} c e\right )} x^{2} - 2 \,{\left (2 \, b^{2} c d - b^{3} e\right )} x}{2 \,{\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} - \frac{3 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (c x + b\right )}{b^{5}} + \frac{3 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (x\right )}{b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*d - 6*(2*c^3*d - b*c^2*e)*x^3 - 9*(2*b*c^2*d - b^2*c*e)*x^2 - 2*(2*b^2*c*d - b^3*e)*x)/(b^4*c^2*x^4
+ 2*b^5*c*x^3 + b^6*x^2) - 3*(2*c^2*d - b*c*e)*log(c*x + b)/b^5 + 3*(2*c^2*d - b*c*e)*log(x)/b^5

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Fricas [B]  time = 1.87802, size = 467, normalized size = 4.25 \begin{align*} -\frac{b^{4} d - 6 \,{\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} - 9 \,{\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2} - 2 \,{\left (2 \, b^{3} c d - b^{4} e\right )} x + 6 \,{\left ({\left (2 \, c^{4} d - b c^{3} e\right )} x^{4} + 2 \,{\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} +{\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \,{\left ({\left (2 \, c^{4} d - b c^{3} e\right )} x^{4} + 2 \,{\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} +{\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^4*d - 6*(2*b*c^3*d - b^2*c^2*e)*x^3 - 9*(2*b^2*c^2*d - b^3*c*e)*x^2 - 2*(2*b^3*c*d - b^4*e)*x + 6*((2*
c^4*d - b*c^3*e)*x^4 + 2*(2*b*c^3*d - b^2*c^2*e)*x^3 + (2*b^2*c^2*d - b^3*c*e)*x^2)*log(c*x + b) - 6*((2*c^4*d
- b*c^3*e)*x^4 + 2*(2*b*c^3*d - b^2*c^2*e)*x^3 + (2*b^2*c^2*d - b^3*c*e)*x^2)*log(x))/(b^5*c^2*x^4 + 2*b^6*c*
x^3 + b^7*x^2)

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Sympy [B]  time = 1.74491, size = 219, normalized size = 1.99 \begin{align*} - \frac{b^{3} d + x^{3} \left (6 b c^{2} e - 12 c^{3} d\right ) + x^{2} \left (9 b^{2} c e - 18 b c^{2} d\right ) + x \left (2 b^{3} e - 4 b^{2} c d\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} - \frac{3 c \left (b e - 2 c d\right ) \log{\left (x + \frac{3 b^{2} c e - 6 b c^{2} d - 3 b c \left (b e - 2 c d\right )}{6 b c^{2} e - 12 c^{3} d} \right )}}{b^{5}} + \frac{3 c \left (b e - 2 c d\right ) \log{\left (x + \frac{3 b^{2} c e - 6 b c^{2} d + 3 b c \left (b e - 2 c d\right )}{6 b c^{2} e - 12 c^{3} d} \right )}}{b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b**3*d + x**3*(6*b*c**2*e - 12*c**3*d) + x**2*(9*b**2*c*e - 18*b*c**2*d) + x*(2*b**3*e - 4*b**2*c*d))/(2*b**
6*x**2 + 4*b**5*c*x**3 + 2*b**4*c**2*x**4) - 3*c*(b*e - 2*c*d)*log(x + (3*b**2*c*e - 6*b*c**2*d - 3*b*c*(b*e -
2*c*d))/(6*b*c**2*e - 12*c**3*d))/b**5 + 3*c*(b*e - 2*c*d)*log(x + (3*b**2*c*e - 6*b*c**2*d + 3*b*c*(b*e - 2*
c*d))/(6*b*c**2*e - 12*c**3*d))/b**5

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Giac [A]  time = 1.25839, size = 178, normalized size = 1.62 \begin{align*} \frac{3 \,{\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{3 \,{\left (2 \, c^{3} d - b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac{12 \, c^{3} d x^{3} - 6 \, b c^{2} x^{3} e + 18 \, b c^{2} d x^{2} - 9 \, b^{2} c x^{2} e + 4 \, b^{2} c d x - 2 \, b^{3} x e - b^{3} d}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(2*c^2*d - b*c*e)*log(abs(x))/b^5 - 3*(2*c^3*d - b*c^2*e)*log(abs(c*x + b))/(b^5*c) + 1/2*(12*c^3*d*x^3 - 6*
b*c^2*x^3*e + 18*b*c^2*d*x^2 - 9*b^2*c*x^2*e + 4*b^2*c*d*x - 2*b^3*x*e - b^3*d)/((c*x^2 + b*x)^2*b^4)