3.277 \(\int \frac{(d+e x)^5}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=171 \[ \frac{d^3 \log (x) \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right )}{b^5}-\frac{(c d-b e)^3 \left (b^2 e^2+3 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 c^3}+\frac{(c d-b e)^4 (2 b e+3 c d)}{b^4 c^3 (b+c x)}+\frac{(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}+\frac{d^4 (3 c d-5 b e)}{b^4 x}-\frac{d^5}{2 b^3 x^2} \]

[Out]

-d^5/(2*b^3*x^2) + (d^4*(3*c*d - 5*b*e))/(b^4*x) + (c*d - b*e)^5/(2*b^3*c^3*(b + c*x)^2) + ((c*d - b*e)^4*(3*c
*d + 2*b*e))/(b^4*c^3*(b + c*x)) + (d^3*(6*c^2*d^2 - 15*b*c*d*e + 10*b^2*e^2)*Log[x])/b^5 - ((c*d - b*e)^3*(6*
c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[b + c*x])/(b^5*c^3)

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Rubi [A]  time = 0.178273, antiderivative size = 171, 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} \[ \frac{d^3 \log (x) \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right )}{b^5}-\frac{(c d-b e)^3 \left (b^2 e^2+3 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 c^3}+\frac{(c d-b e)^4 (2 b e+3 c d)}{b^4 c^3 (b+c x)}+\frac{(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}+\frac{d^4 (3 c d-5 b e)}{b^4 x}-\frac{d^5}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(b*x + c*x^2)^3,x]

[Out]

-d^5/(2*b^3*x^2) + (d^4*(3*c*d - 5*b*e))/(b^4*x) + (c*d - b*e)^5/(2*b^3*c^3*(b + c*x)^2) + ((c*d - b*e)^4*(3*c
*d + 2*b*e))/(b^4*c^3*(b + c*x)) + (d^3*(6*c^2*d^2 - 15*b*c*d*e + 10*b^2*e^2)*Log[x])/b^5 - ((c*d - b*e)^3*(6*
c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[b + c*x])/(b^5*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)^5}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{d^5}{b^3 x^3}+\frac{d^4 (-3 c d+5 b e)}{b^4 x^2}+\frac{d^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right )}{b^5 x}+\frac{(-c d+b e)^5}{b^3 c^2 (b+c x)^3}-\frac{(-c d+b e)^4 (3 c d+2 b e)}{b^4 c^2 (b+c x)^2}+\frac{(-c d+b e)^3 \left (6 c^2 d^2+3 b c d e+b^2 e^2\right )}{b^5 c^2 (b+c x)}\right ) \, dx\\ &=-\frac{d^5}{2 b^3 x^2}+\frac{d^4 (3 c d-5 b e)}{b^4 x}+\frac{(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}+\frac{(c d-b e)^4 (3 c d+2 b e)}{b^4 c^3 (b+c x)}+\frac{d^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (x)}{b^5}-\frac{(c d-b e)^3 \left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^3}\\ \end{align*}

Mathematica [A]  time = 0.108152, size = 165, normalized size = 0.96 \[ -\frac{-2 d^3 \log (x) \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right )+\frac{2 (c d-b e)^3 \left (b^2 e^2+3 b c d e+6 c^2 d^2\right ) \log (b+c x)}{c^3}+\frac{b^2 (b e-c d)^5}{c^3 (b+c x)^2}+\frac{b^2 d^5}{x^2}-\frac{2 b (c d-b e)^4 (2 b e+3 c d)}{c^3 (b+c x)}+\frac{2 b d^4 (5 b e-3 c d)}{x}}{2 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(b*x + c*x^2)^3,x]

[Out]

-((b^2*d^5)/x^2 + (2*b*d^4*(-3*c*d + 5*b*e))/x + (b^2*(-(c*d) + b*e)^5)/(c^3*(b + c*x)^2) - (2*b*(c*d - b*e)^4
*(3*c*d + 2*b*e))/(c^3*(b + c*x)) - 2*d^3*(6*c^2*d^2 - 15*b*c*d*e + 10*b^2*e^2)*Log[x] + (2*(c*d - b*e)^3*(6*c
^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[b + c*x])/c^3)/(2*b^5)

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Maple [A]  time = 0.06, size = 329, normalized size = 1.9 \begin{align*} -{\frac{{d}^{5}}{2\,{b}^{3}{x}^{2}}}+10\,{\frac{{d}^{3}\ln \left ( x \right ){e}^{2}}{{b}^{3}}}-15\,{\frac{{d}^{4}\ln \left ( x \right ) ce}{{b}^{4}}}+6\,{\frac{{d}^{5}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}-5\,{\frac{{d}^{4}e}{{b}^{3}x}}+3\,{\frac{{d}^{5}c}{{b}^{4}x}}+{\frac{\ln \left ( cx+b \right ){e}^{5}}{{c}^{3}}}-10\,{\frac{\ln \left ( cx+b \right ){d}^{3}{e}^{2}}{{b}^{3}}}+15\,{\frac{c\ln \left ( cx+b \right ){d}^{4}e}{{b}^{4}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ){d}^{5}}{{b}^{5}}}+2\,{\frac{b{e}^{5}}{{c}^{3} \left ( cx+b \right ) }}-5\,{\frac{d{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+10\,{\frac{{d}^{3}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-10\,{\frac{{d}^{4}ec}{{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{d}^{5}{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}-{\frac{{b}^{2}{e}^{5}}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{5\,bd{e}^{4}}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}-5\,{\frac{{d}^{2}{e}^{3}}{c \left ( cx+b \right ) ^{2}}}+5\,{\frac{{d}^{3}{e}^{2}}{b \left ( cx+b \right ) ^{2}}}-{\frac{5\,{d}^{4}ec}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{{d}^{5}{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*x^2+b*x)^3,x)

[Out]

-1/2*d^5/b^3/x^2+10*d^3/b^3*ln(x)*e^2-15*d^4/b^4*ln(x)*c*e+6*d^5/b^5*ln(x)*c^2-5*d^4/b^3/x*e+3*d^5/b^4/x*c+1/c
^3*ln(c*x+b)*e^5-10/b^3*ln(c*x+b)*d^3*e^2+15/b^4*c*ln(c*x+b)*d^4*e-6/b^5*c^2*ln(c*x+b)*d^5+2/c^3*b/(c*x+b)*e^5
-5/c^2/(c*x+b)*d*e^4+10/b^2/(c*x+b)*d^3*e^2-10*c/b^3/(c*x+b)*d^4*e+3*c^2/b^4/(c*x+b)*d^5-1/2/c^3*b^2/(c*x+b)^2
*e^5+5/2/c^2*b/(c*x+b)^2*d*e^4-5/c/(c*x+b)^2*d^2*e^3+5/b/(c*x+b)^2*d^3*e^2-5/2*c/b^2/(c*x+b)^2*d^4*e+1/2*c^2/b
^3/(c*x+b)^2*d^5

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Maxima [A]  time = 1.13554, size = 400, normalized size = 2.34 \begin{align*} -\frac{b^{3} c^{3} d^{5} - 2 \,{\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - 5 \, b^{4} c^{2} d e^{4} + 2 \, b^{5} c e^{5}\right )} x^{3} -{\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} - 5 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e\right )} x}{2 \,{\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}} + \frac{{\left (6 \, c^{2} d^{5} - 15 \, b c d^{4} e + 10 \, b^{2} d^{3} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac{{\left (6 \, c^{5} d^{5} - 15 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - b^{5} e^{5}\right )} \log \left (c x + b\right )}{b^{5} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-1/2*(b^3*c^3*d^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 - 5*b^4*c^2*d*e^4 + 2*b^5*c*e^5)*x^3 -
(18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 - 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^4 + 3*b^6*e^5)*x^2 -
2*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^4*e)*x)/(b^4*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2) + (6*c^2*d^5 - 15*b*c*d^4*e
+ 10*b^2*d^3*e^2)*log(x)/b^5 - (6*c^5*d^5 - 15*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2 - b^5*e^5)*log(c*x + b)/(b^5*
c^3)

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Fricas [B]  time = 2.01836, size = 991, normalized size = 5.8 \begin{align*} -\frac{b^{4} c^{3} d^{5} - 2 \,{\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - 5 \, b^{5} c^{2} d e^{4} + 2 \, b^{6} c e^{5}\right )} x^{3} -{\left (18 \, b^{2} c^{5} d^{5} - 45 \, b^{3} c^{4} d^{4} e + 30 \, b^{4} c^{3} d^{3} e^{2} - 10 \, b^{5} c^{2} d^{2} e^{3} - 5 \, b^{6} c d e^{4} + 3 \, b^{7} e^{5}\right )} x^{2} - 2 \,{\left (2 \, b^{3} c^{4} d^{5} - 5 \, b^{4} c^{3} d^{4} e\right )} x + 2 \,{\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2} - b^{5} c^{2} e^{5}\right )} x^{4} + 2 \,{\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{6} c e^{5}\right )} x^{3} +{\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2} - b^{7} e^{5}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2}\right )} x^{4} + 2 \,{\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2}\right )} x^{3} +{\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{5} x^{4} + 2 \, b^{6} c^{4} x^{3} + b^{7} c^{3} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

-1/2*(b^4*c^3*d^5 - 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - 5*b^5*c^2*d*e^4 + 2*b^6*c*e^5)*x^
3 - (18*b^2*c^5*d^5 - 45*b^3*c^4*d^4*e + 30*b^4*c^3*d^3*e^2 - 10*b^5*c^2*d^2*e^3 - 5*b^6*c*d*e^4 + 3*b^7*e^5)*
x^2 - 2*(2*b^3*c^4*d^5 - 5*b^4*c^3*d^4*e)*x + 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2 - b^5*c^2*e^
5)*x^4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - b^6*c*e^5)*x^3 + (6*b^2*c^5*d^5 - 15*b^3*c^4
*d^4*e + 10*b^4*c^3*d^3*e^2 - b^7*e^5)*x^2)*log(c*x + b) - 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2
)*x^4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2)*x^3 + (6*b^2*c^5*d^5 - 15*b^3*c^4*d^4*e + 10*b
^4*c^3*d^3*e^2)*x^2)*log(x))/(b^5*c^5*x^4 + 2*b^6*c^4*x^3 + b^7*c^3*x^2)

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Sympy [B]  time = 12.1009, size = 524, normalized size = 3.06 \begin{align*} \frac{- b^{3} c^{3} d^{5} + x^{3} \left (4 b^{5} c e^{5} - 10 b^{4} c^{2} d e^{4} + 20 b^{2} c^{4} d^{3} e^{2} - 30 b c^{5} d^{4} e + 12 c^{6} d^{5}\right ) + x^{2} \left (3 b^{6} e^{5} - 5 b^{5} c d e^{4} - 10 b^{4} c^{2} d^{2} e^{3} + 30 b^{3} c^{3} d^{3} e^{2} - 45 b^{2} c^{4} d^{4} e + 18 b c^{5} d^{5}\right ) + x \left (- 10 b^{3} c^{3} d^{4} e + 4 b^{2} c^{4} d^{5}\right )}{2 b^{6} c^{3} x^{2} + 4 b^{5} c^{4} x^{3} + 2 b^{4} c^{5} x^{4}} + \frac{d^{3} \left (10 b^{2} e^{2} - 15 b c d e + 6 c^{2} d^{2}\right ) \log{\left (x + \frac{- 10 b^{3} c^{2} d^{3} e^{2} + 15 b^{2} c^{3} d^{4} e - 6 b c^{4} d^{5} + b c^{2} d^{3} \left (10 b^{2} e^{2} - 15 b c d e + 6 c^{2} d^{2}\right )}{b^{5} e^{5} - 20 b^{2} c^{3} d^{3} e^{2} + 30 b c^{4} d^{4} e - 12 c^{5} d^{5}} \right )}}{b^{5}} + \frac{\left (b e - c d\right )^{3} \left (b^{2} e^{2} + 3 b c d e + 6 c^{2} d^{2}\right ) \log{\left (x + \frac{- 10 b^{3} c^{2} d^{3} e^{2} + 15 b^{2} c^{3} d^{4} e - 6 b c^{4} d^{5} + \frac{b \left (b e - c d\right )^{3} \left (b^{2} e^{2} + 3 b c d e + 6 c^{2} d^{2}\right )}{c}}{b^{5} e^{5} - 20 b^{2} c^{3} d^{3} e^{2} + 30 b c^{4} d^{4} e - 12 c^{5} d^{5}} \right )}}{b^{5} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(c*x**2+b*x)**3,x)

[Out]

(-b**3*c**3*d**5 + x**3*(4*b**5*c*e**5 - 10*b**4*c**2*d*e**4 + 20*b**2*c**4*d**3*e**2 - 30*b*c**5*d**4*e + 12*
c**6*d**5) + x**2*(3*b**6*e**5 - 5*b**5*c*d*e**4 - 10*b**4*c**2*d**2*e**3 + 30*b**3*c**3*d**3*e**2 - 45*b**2*c
**4*d**4*e + 18*b*c**5*d**5) + x*(-10*b**3*c**3*d**4*e + 4*b**2*c**4*d**5))/(2*b**6*c**3*x**2 + 4*b**5*c**4*x*
*3 + 2*b**4*c**5*x**4) + d**3*(10*b**2*e**2 - 15*b*c*d*e + 6*c**2*d**2)*log(x + (-10*b**3*c**2*d**3*e**2 + 15*
b**2*c**3*d**4*e - 6*b*c**4*d**5 + b*c**2*d**3*(10*b**2*e**2 - 15*b*c*d*e + 6*c**2*d**2))/(b**5*e**5 - 20*b**2
*c**3*d**3*e**2 + 30*b*c**4*d**4*e - 12*c**5*d**5))/b**5 + (b*e - c*d)**3*(b**2*e**2 + 3*b*c*d*e + 6*c**2*d**2
)*log(x + (-10*b**3*c**2*d**3*e**2 + 15*b**2*c**3*d**4*e - 6*b*c**4*d**5 + b*(b*e - c*d)**3*(b**2*e**2 + 3*b*c
*d*e + 6*c**2*d**2)/c)/(b**5*e**5 - 20*b**2*c**3*d**3*e**2 + 30*b*c**4*d**4*e - 12*c**5*d**5))/(b**5*c**3)

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Giac [A]  time = 1.32663, size = 371, normalized size = 2.17 \begin{align*} \frac{{\left (6 \, c^{2} d^{5} - 15 \, b c d^{4} e + 10 \, b^{2} d^{3} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{{\left (6 \, c^{5} d^{5} - 15 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - b^{5} e^{5}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{3}} - \frac{b^{3} c^{3} d^{5} - 2 \,{\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - 5 \, b^{4} c^{2} d e^{4} + 2 \, b^{5} c e^{5}\right )} x^{3} -{\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} - 5 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e\right )} x}{2 \,{\left (c x + b\right )}^{2} b^{4} c^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

(6*c^2*d^5 - 15*b*c*d^4*e + 10*b^2*d^3*e^2)*log(abs(x))/b^5 - (6*c^5*d^5 - 15*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2
- b^5*e^5)*log(abs(c*x + b))/(b^5*c^3) - 1/2*(b^3*c^3*d^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^
2 - 5*b^4*c^2*d*e^4 + 2*b^5*c*e^5)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 - 10*b^4*c^2*d^
2*e^3 - 5*b^5*c*d*e^4 + 3*b^6*e^5)*x^2 - 2*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^4*e)*x)/((c*x + b)^2*b^4*c^3*x^2)