### 3.278 $$\int \frac{(d+e x)^4}{(b x+c x^2)^3} \, dx$$

Optimal. Leaf size=136 $\frac{(c d-b e)^3 (b e+3 c d)}{b^4 c^2 (b+c x)}+\frac{(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}+\frac{d^3 (3 c d-4 b e)}{b^4 x}+\frac{6 d^2 \log (x) (c d-b e)^2}{b^5}-\frac{6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}-\frac{d^4}{2 b^3 x^2}$

[Out]

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

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Rubi [A]  time = 0.151547, antiderivative size = 136, 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} $\frac{(c d-b e)^3 (b e+3 c d)}{b^4 c^2 (b+c x)}+\frac{(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}+\frac{d^3 (3 c d-4 b e)}{b^4 x}+\frac{6 d^2 \log (x) (c d-b e)^2}{b^5}-\frac{6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}-\frac{d^4}{2 b^3 x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4/(b*x + c*x^2)^3,x]

[Out]

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

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 )^3} \, dx &=\int \left (\frac{d^4}{b^3 x^3}+\frac{d^3 (-3 c d+4 b e)}{b^4 x^2}+\frac{6 d^2 (-c d+b e)^2}{b^5 x}-\frac{(-c d+b e)^4}{b^3 c (b+c x)^3}+\frac{(-c d+b e)^3 (3 c d+b e)}{b^4 c (b+c x)^2}-\frac{6 c d^2 (-c d+b e)^2}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac{d^4}{2 b^3 x^2}+\frac{d^3 (3 c d-4 b e)}{b^4 x}+\frac{(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}+\frac{(c d-b e)^3 (3 c d+b e)}{b^4 c^2 (b+c x)}+\frac{6 d^2 (c d-b e)^2 \log (x)}{b^5}-\frac{6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0835923, size = 130, normalized size = 0.96 $-\frac{-\frac{b^2 (c d-b e)^4}{c^2 (b+c x)^2}+\frac{b^2 d^4}{x^2}+\frac{2 b (b e-c d)^3 (b e+3 c d)}{c^2 (b+c x)}+\frac{2 b d^3 (4 b e-3 c d)}{x}-12 d^2 \log (x) (c d-b e)^2+12 d^2 (c d-b e)^2 \log (b+c x)}{2 b^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^4/(b*x + c*x^2)^3,x]

[Out]

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

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Maple [B]  time = 0.062, size = 278, normalized size = 2. \begin{align*} -{\frac{{d}^{4}}{2\,{b}^{3}{x}^{2}}}-4\,{\frac{{d}^{3}e}{{b}^{3}x}}+3\,{\frac{{d}^{4}c}{{b}^{4}x}}+6\,{\frac{{d}^{2}\ln \left ( x \right ){e}^{2}}{{b}^{3}}}-12\,{\frac{{d}^{3}\ln \left ( x \right ) ce}{{b}^{4}}}+6\,{\frac{{d}^{4}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}-{\frac{{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+6\,{\frac{{d}^{2}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-8\,{\frac{{d}^{3}ec}{{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}{d}^{4}}{{b}^{4} \left ( cx+b \right ) }}+{\frac{{e}^{4}b}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}-2\,{\frac{d{e}^{3}}{c \left ( cx+b \right ) ^{2}}}+3\,{\frac{{d}^{2}{e}^{2}}{b \left ( cx+b \right ) ^{2}}}-2\,{\frac{{d}^{3}ec}{{b}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}{d}^{4}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-6\,{\frac{{d}^{2}\ln \left ( cx+b \right ){e}^{2}}{{b}^{3}}}+12\,{\frac{{d}^{3}\ln \left ( cx+b \right ) ce}{{b}^{4}}}-6\,{\frac{{d}^{4}\ln \left ( cx+b \right ){c}^{2}}{{b}^{5}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(c*x^2+b*x)^3,x)

[Out]

-1/2*d^4/b^3/x^2-4*d^3/b^3/x*e+3*d^4/b^4/x*c+6*d^2/b^3*ln(x)*e^2-12*d^3/b^4*ln(x)*c*e+6*d^4/b^5*ln(x)*c^2-1/c^
2/(c*x+b)*e^4+6/b^2/(c*x+b)*d^2*e^2-8/b^3*c/(c*x+b)*d^3*e+3/b^4*c^2/(c*x+b)*d^4+1/2*b/c^2/(c*x+b)^2*e^4-2/c/(c
*x+b)^2*d*e^3+3/b/(c*x+b)^2*d^2*e^2-2/b^2*c/(c*x+b)^2*d^3*e+1/2/b^3*c^2/(c*x+b)^2*d^4-6*d^2/b^3*ln(c*x+b)*e^2+
12*d^3/b^4*ln(c*x+b)*c*e-6*d^4/b^5*ln(c*x+b)*c^2

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Maxima [A]  time = 1.02357, size = 338, normalized size = 2.49 \begin{align*} -\frac{b^{3} c^{2} d^{4} - 2 \,{\left (6 \, c^{5} d^{4} - 12 \, b c^{4} d^{3} e + 6 \, b^{2} c^{3} d^{2} e^{2} - b^{4} c e^{4}\right )} x^{3} -{\left (18 \, b c^{4} d^{4} - 36 \, b^{2} c^{3} d^{3} e + 18 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} - b^{5} e^{4}\right )} x^{2} - 4 \,{\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )} x}{2 \,{\left (b^{4} c^{4} x^{4} + 2 \, b^{5} c^{3} x^{3} + b^{6} c^{2} x^{2}\right )}} - \frac{6 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac{6 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (x\right )}{b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-1/2*(b^3*c^2*d^4 - 2*(6*c^5*d^4 - 12*b*c^4*d^3*e + 6*b^2*c^3*d^2*e^2 - b^4*c*e^4)*x^3 - (18*b*c^4*d^4 - 36*b^
2*c^3*d^3*e + 18*b^3*c^2*d^2*e^2 - 4*b^4*c*d*e^3 - b^5*e^4)*x^2 - 4*(b^2*c^3*d^4 - 2*b^3*c^2*d^3*e)*x)/(b^4*c^
4*x^4 + 2*b^5*c^3*x^3 + b^6*c^2*x^2) - 6*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*log(c*x + b)/b^5 + 6*(c^2*d^4 -
2*b*c*d^3*e + b^2*d^2*e^2)*log(x)/b^5

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Fricas [B]  time = 1.8601, size = 834, normalized size = 6.13 \begin{align*} -\frac{b^{4} c^{2} d^{4} - 2 \,{\left (6 \, b c^{5} d^{4} - 12 \, b^{2} c^{4} d^{3} e + 6 \, b^{3} c^{3} d^{2} e^{2} - b^{5} c e^{4}\right )} x^{3} -{\left (18 \, b^{2} c^{4} d^{4} - 36 \, b^{3} c^{3} d^{3} e + 18 \, b^{4} c^{2} d^{2} e^{2} - 4 \, b^{5} c d e^{3} - b^{6} e^{4}\right )} x^{2} - 4 \,{\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e\right )} x + 12 \,{\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \,{\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} +{\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 12 \,{\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \,{\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} +{\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

-1/2*(b^4*c^2*d^4 - 2*(6*b*c^5*d^4 - 12*b^2*c^4*d^3*e + 6*b^3*c^3*d^2*e^2 - b^5*c*e^4)*x^3 - (18*b^2*c^4*d^4 -
36*b^3*c^3*d^3*e + 18*b^4*c^2*d^2*e^2 - 4*b^5*c*d*e^3 - b^6*e^4)*x^2 - 4*(b^3*c^3*d^4 - 2*b^4*c^2*d^3*e)*x +
12*((c^6*d^4 - 2*b*c^5*d^3*e + b^2*c^4*d^2*e^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c^4*d^3*e + b^3*c^3*d^2*e^2)*x^3 +
(b^2*c^4*d^4 - 2*b^3*c^3*d^3*e + b^4*c^2*d^2*e^2)*x^2)*log(c*x + b) - 12*((c^6*d^4 - 2*b*c^5*d^3*e + b^2*c^4*d
^2*e^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c^4*d^3*e + b^3*c^3*d^2*e^2)*x^3 + (b^2*c^4*d^4 - 2*b^3*c^3*d^3*e + b^4*c^2
*d^2*e^2)*x^2)*log(x))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2)

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Sympy [B]  time = 8.51847, size = 389, normalized size = 2.86 \begin{align*} - \frac{b^{3} c^{2} d^{4} + x^{3} \left (2 b^{4} c e^{4} - 12 b^{2} c^{3} d^{2} e^{2} + 24 b c^{4} d^{3} e - 12 c^{5} d^{4}\right ) + x^{2} \left (b^{5} e^{4} + 4 b^{4} c d e^{3} - 18 b^{3} c^{2} d^{2} e^{2} + 36 b^{2} c^{3} d^{3} e - 18 b c^{4} d^{4}\right ) + x \left (8 b^{3} c^{2} d^{3} e - 4 b^{2} c^{3} d^{4}\right )}{2 b^{6} c^{2} x^{2} + 4 b^{5} c^{3} x^{3} + 2 b^{4} c^{4} x^{4}} + \frac{6 d^{2} \left (b e - c d\right )^{2} \log{\left (x + \frac{6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} - 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} - \frac{6 d^{2} \left (b e - c d\right )^{2} \log{\left (x + \frac{6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} + 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(c*x**2+b*x)**3,x)

[Out]

-(b**3*c**2*d**4 + x**3*(2*b**4*c*e**4 - 12*b**2*c**3*d**2*e**2 + 24*b*c**4*d**3*e - 12*c**5*d**4) + x**2*(b**
5*e**4 + 4*b**4*c*d*e**3 - 18*b**3*c**2*d**2*e**2 + 36*b**2*c**3*d**3*e - 18*b*c**4*d**4) + x*(8*b**3*c**2*d**
3*e - 4*b**2*c**3*d**4))/(2*b**6*c**2*x**2 + 4*b**5*c**3*x**3 + 2*b**4*c**4*x**4) + 6*d**2*(b*e - c*d)**2*log(
x + (6*b**3*d**2*e**2 - 12*b**2*c*d**3*e + 6*b*c**2*d**4 - 6*b*d**2*(b*e - c*d)**2)/(12*b**2*c*d**2*e**2 - 24*
b*c**2*d**3*e + 12*c**3*d**4))/b**5 - 6*d**2*(b*e - c*d)**2*log(x + (6*b**3*d**2*e**2 - 12*b**2*c*d**3*e + 6*b
*c**2*d**4 + 6*b*d**2*(b*e - c*d)**2)/(12*b**2*c*d**2*e**2 - 24*b*c**2*d**3*e + 12*c**3*d**4))/b**5

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Giac [A]  time = 1.2758, size = 343, normalized size = 2.52 \begin{align*} \frac{6 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{6 \,{\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac{12 \, c^{5} d^{4} x^{3} - 24 \, b c^{4} d^{3} x^{3} e + 18 \, b c^{4} d^{4} x^{2} + 12 \, b^{2} c^{3} d^{2} x^{3} e^{2} - 36 \, b^{2} c^{3} d^{3} x^{2} e + 4 \, b^{2} c^{3} d^{4} x + 18 \, b^{3} c^{2} d^{2} x^{2} e^{2} - 8 \, b^{3} c^{2} d^{3} x e - b^{3} c^{2} d^{4} - 2 \, b^{4} c x^{3} e^{4} - 4 \, b^{4} c d x^{2} e^{3} - b^{5} x^{2} e^{4}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4} c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

6*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*log(abs(x))/b^5 - 6*(c^3*d^4 - 2*b*c^2*d^3*e + b^2*c*d^2*e^2)*log(abs(
c*x + b))/(b^5*c) + 1/2*(12*c^5*d^4*x^3 - 24*b*c^4*d^3*x^3*e + 18*b*c^4*d^4*x^2 + 12*b^2*c^3*d^2*x^3*e^2 - 36*
b^2*c^3*d^3*x^2*e + 4*b^2*c^3*d^4*x + 18*b^3*c^2*d^2*x^2*e^2 - 8*b^3*c^2*d^3*x*e - b^3*c^2*d^4 - 2*b^4*c*x^3*e
^4 - 4*b^4*c*d*x^2*e^3 - b^5*x^2*e^4)/((c*x^2 + b*x)^2*b^4*c^2)