### 3.267 $$\int \frac{(d+e x)^5}{(b x+c x^2)^2} \, dx$$

Optimal. Leaf size=118 $-\frac{(c d-b e)^5}{b^2 c^4 (b+c x)}+\frac{(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}-\frac{d^4 \log (x) (2 c d-5 b e)}{b^3}-\frac{d^5}{b^2 x}+\frac{e^4 x (5 c d-2 b e)}{c^3}+\frac{e^5 x^2}{2 c^2}$

[Out]

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

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Rubi [A]  time = 0.13772, antiderivative size = 118, 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} $-\frac{(c d-b e)^5}{b^2 c^4 (b+c x)}+\frac{(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}-\frac{d^4 \log (x) (2 c d-5 b e)}{b^3}-\frac{d^5}{b^2 x}+\frac{e^4 x (5 c d-2 b e)}{c^3}+\frac{e^5 x^2}{2 c^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5/(b*x + c*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0568775, size = 116, normalized size = 0.98 $\frac{(b e-c d)^5}{b^2 c^4 (b+c x)}+\frac{(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}+\frac{d^4 \log (x) (5 b e-2 c d)}{b^3}-\frac{d^5}{b^2 x}+\frac{e^4 x (5 c d-2 b e)}{c^3}+\frac{e^5 x^2}{2 c^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5/(b*x + c*x^2)^2,x]

[Out]

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

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Maple [B]  time = 0.061, size = 251, normalized size = 2.1 \begin{align*}{\frac{{e}^{5}{x}^{2}}{2\,{c}^{2}}}-2\,{\frac{{e}^{5}xb}{{c}^{3}}}+5\,{\frac{d{e}^{4}x}{{c}^{2}}}-{\frac{{d}^{5}}{{b}^{2}x}}+5\,{\frac{{d}^{4}\ln \left ( x \right ) e}{{b}^{2}}}-2\,{\frac{{d}^{5}\ln \left ( x \right ) c}{{b}^{3}}}+3\,{\frac{{b}^{2}\ln \left ( cx+b \right ){e}^{5}}{{c}^{4}}}-10\,{\frac{b\ln \left ( cx+b \right ) d{e}^{4}}{{c}^{3}}}+10\,{\frac{\ln \left ( cx+b \right ){d}^{2}{e}^{3}}{{c}^{2}}}-5\,{\frac{\ln \left ( cx+b \right ){d}^{4}e}{{b}^{2}}}+2\,{\frac{c\ln \left ( cx+b \right ){d}^{5}}{{b}^{3}}}+{\frac{{b}^{3}{e}^{5}}{{c}^{4} \left ( cx+b \right ) }}-5\,{\frac{{b}^{2}d{e}^{4}}{{c}^{3} \left ( cx+b \right ) }}+10\,{\frac{b{d}^{2}{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-10\,{\frac{{d}^{3}{e}^{2}}{c \left ( cx+b \right ) }}+5\,{\frac{{d}^{4}e}{b \left ( cx+b \right ) }}-{\frac{c{d}^{5}}{{b}^{2} \left ( cx+b \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*x^2+b*x)^2,x)

[Out]

1/2*e^5*x^2/c^2-2*e^5/c^3*x*b+5*e^4/c^2*d*x-d^5/b^2/x+5*d^4/b^2*ln(x)*e-2*d^5/b^3*ln(x)*c+3/c^4*b^2*ln(c*x+b)*
e^5-10/c^3*b*ln(c*x+b)*d*e^4+10/c^2*ln(c*x+b)*d^2*e^3-5/b^2*ln(c*x+b)*d^4*e+2*c/b^3*ln(c*x+b)*d^5+b^3/c^4/(c*x
+b)*e^5-5*b^2/c^3/(c*x+b)*d*e^4+10*b/c^2/(c*x+b)*d^2*e^3-10/c/(c*x+b)*d^3*e^2+5/b/(c*x+b)*d^4*e-1/b^2*c/(c*x+b
)*d^5

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Maxima [A]  time = 0.997527, size = 292, normalized size = 2.47 \begin{align*} -\frac{b c^{4} d^{5} +{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - 10 \, b^{3} c^{2} d^{2} e^{3} + 5 \, b^{4} c d e^{4} - b^{5} e^{5}\right )} x}{b^{2} c^{5} x^{2} + b^{3} c^{4} x} - \frac{{\left (2 \, c d^{5} - 5 \, b d^{4} e\right )} \log \left (x\right )}{b^{3}} + \frac{c e^{5} x^{2} + 2 \,{\left (5 \, c d e^{4} - 2 \, b e^{5}\right )} x}{2 \, c^{3}} + \frac{{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{3} c^{2} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 3 \, b^{5} e^{5}\right )} \log \left (c x + b\right )}{b^{3} c^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

-(b*c^4*d^5 + (2*c^5*d^5 - 5*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2 - 10*b^3*c^2*d^2*e^3 + 5*b^4*c*d*e^4 - b^5*e^5)*
x)/(b^2*c^5*x^2 + b^3*c^4*x) - (2*c*d^5 - 5*b*d^4*e)*log(x)/b^3 + 1/2*(c*e^5*x^2 + 2*(5*c*d*e^4 - 2*b*e^5)*x)/
c^3 + (2*c^5*d^5 - 5*b*c^4*d^4*e + 10*b^3*c^2*d^2*e^3 - 10*b^4*c*d*e^4 + 3*b^5*e^5)*log(c*x + b)/(b^3*c^4)

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Fricas [B]  time = 1.75372, size = 707, normalized size = 5.99 \begin{align*} \frac{b^{3} c^{3} e^{5} x^{4} - 2 \, b^{2} c^{4} d^{5} +{\left (10 \, b^{3} c^{3} d e^{4} - 3 \, b^{4} c^{2} e^{5}\right )} x^{3} + 2 \,{\left (5 \, b^{4} c^{2} d e^{4} - 2 \, b^{5} c e^{5}\right )} x^{2} - 2 \,{\left (2 \, b c^{5} d^{5} - 5 \, b^{2} c^{4} d^{4} e + 10 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} + 5 \, b^{5} c d e^{4} - b^{6} e^{5}\right )} x + 2 \,{\left ({\left (2 \, c^{6} d^{5} - 5 \, b c^{5} d^{4} e + 10 \, b^{3} c^{3} d^{2} e^{3} - 10 \, b^{4} c^{2} d e^{4} + 3 \, b^{5} c e^{5}\right )} x^{2} +{\left (2 \, b c^{5} d^{5} - 5 \, b^{2} c^{4} d^{4} e + 10 \, b^{4} c^{2} d^{2} e^{3} - 10 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (2 \, c^{6} d^{5} - 5 \, b c^{5} d^{4} e\right )} x^{2} +{\left (2 \, b c^{5} d^{5} - 5 \, b^{2} c^{4} d^{4} e\right )} x\right )} \log \left (x\right )}{2 \,{\left (b^{3} c^{5} x^{2} + b^{4} c^{4} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

1/2*(b^3*c^3*e^5*x^4 - 2*b^2*c^4*d^5 + (10*b^3*c^3*d*e^4 - 3*b^4*c^2*e^5)*x^3 + 2*(5*b^4*c^2*d*e^4 - 2*b^5*c*e
^5)*x^2 - 2*(2*b*c^5*d^5 - 5*b^2*c^4*d^4*e + 10*b^3*c^3*d^3*e^2 - 10*b^4*c^2*d^2*e^3 + 5*b^5*c*d*e^4 - b^6*e^5
)*x + 2*((2*c^6*d^5 - 5*b*c^5*d^4*e + 10*b^3*c^3*d^2*e^3 - 10*b^4*c^2*d*e^4 + 3*b^5*c*e^5)*x^2 + (2*b*c^5*d^5
- 5*b^2*c^4*d^4*e + 10*b^4*c^2*d^2*e^3 - 10*b^5*c*d*e^4 + 3*b^6*e^5)*x)*log(c*x + b) - 2*((2*c^6*d^5 - 5*b*c^5
*d^4*e)*x^2 + (2*b*c^5*d^5 - 5*b^2*c^4*d^4*e)*x)*log(x))/(b^3*c^5*x^2 + b^4*c^4*x)

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Sympy [B]  time = 8.45732, size = 379, normalized size = 3.21 \begin{align*} \frac{- b c^{4} d^{5} + x \left (b^{5} e^{5} - 5 b^{4} c d e^{4} + 10 b^{3} c^{2} d^{2} e^{3} - 10 b^{2} c^{3} d^{3} e^{2} + 5 b c^{4} d^{4} e - 2 c^{5} d^{5}\right )}{b^{3} c^{4} x + b^{2} c^{5} x^{2}} + \frac{e^{5} x^{2}}{2 c^{2}} - \frac{x \left (2 b e^{5} - 5 c d e^{4}\right )}{c^{3}} + \frac{d^{4} \left (5 b e - 2 c d\right ) \log{\left (x + \frac{- 5 b^{2} c^{3} d^{4} e + 2 b c^{4} d^{5} + b c^{3} d^{4} \left (5 b e - 2 c d\right )}{3 b^{5} e^{5} - 10 b^{4} c d e^{4} + 10 b^{3} c^{2} d^{2} e^{3} - 10 b c^{4} d^{4} e + 4 c^{5} d^{5}} \right )}}{b^{3}} + \frac{\left (b e - c d\right )^{4} \left (3 b e + 2 c d\right ) \log{\left (x + \frac{- 5 b^{2} c^{3} d^{4} e + 2 b c^{4} d^{5} + \frac{b \left (b e - c d\right )^{4} \left (3 b e + 2 c d\right )}{c}}{3 b^{5} e^{5} - 10 b^{4} c d e^{4} + 10 b^{3} c^{2} d^{2} e^{3} - 10 b c^{4} d^{4} e + 4 c^{5} d^{5}} \right )}}{b^{3} c^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(c*x**2+b*x)**2,x)

[Out]

(-b*c**4*d**5 + x*(b**5*e**5 - 5*b**4*c*d*e**4 + 10*b**3*c**2*d**2*e**3 - 10*b**2*c**3*d**3*e**2 + 5*b*c**4*d*
*4*e - 2*c**5*d**5))/(b**3*c**4*x + b**2*c**5*x**2) + e**5*x**2/(2*c**2) - x*(2*b*e**5 - 5*c*d*e**4)/c**3 + d*
*4*(5*b*e - 2*c*d)*log(x + (-5*b**2*c**3*d**4*e + 2*b*c**4*d**5 + b*c**3*d**4*(5*b*e - 2*c*d))/(3*b**5*e**5 -
10*b**4*c*d*e**4 + 10*b**3*c**2*d**2*e**3 - 10*b*c**4*d**4*e + 4*c**5*d**5))/b**3 + (b*e - c*d)**4*(3*b*e + 2*
c*d)*log(x + (-5*b**2*c**3*d**4*e + 2*b*c**4*d**5 + b*(b*e - c*d)**4*(3*b*e + 2*c*d)/c)/(3*b**5*e**5 - 10*b**4
*c*d*e**4 + 10*b**3*c**2*d**2*e**3 - 10*b*c**4*d**4*e + 4*c**5*d**5))/(b**3*c**4)

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Giac [A]  time = 1.30331, size = 282, normalized size = 2.39 \begin{align*} -\frac{{\left (2 \, c d^{5} - 5 \, b d^{4} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{c^{2} x^{2} e^{5} + 10 \, c^{2} d x e^{4} - 4 \, b c x e^{5}}{2 \, c^{4}} + \frac{{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{3} c^{2} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 3 \, b^{5} e^{5}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{4}} - \frac{b c^{4} d^{5} +{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - 10 \, b^{3} c^{2} d^{2} e^{3} + 5 \, b^{4} c d e^{4} - b^{5} e^{5}\right )} x}{{\left (c x + b\right )} b^{2} c^{4} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

-(2*c*d^5 - 5*b*d^4*e)*log(abs(x))/b^3 + 1/2*(c^2*x^2*e^5 + 10*c^2*d*x*e^4 - 4*b*c*x*e^5)/c^4 + (2*c^5*d^5 - 5
*b*c^4*d^4*e + 10*b^3*c^2*d^2*e^3 - 10*b^4*c*d*e^4 + 3*b^5*e^5)*log(abs(c*x + b))/(b^3*c^4) - (b*c^4*d^5 + (2*
c^5*d^5 - 5*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2 - 10*b^3*c^2*d^2*e^3 + 5*b^4*c*d*e^4 - b^5*e^5)*x)/((c*x + b)*b^2
*c^4*x)