3.284 \(\int \frac{1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=230 \[ \frac{c^4 (3 c d-5 b e)}{b^4 (b+c x) (c d-b e)^3}+\frac{2 b e+3 c d}{b^4 d^3 x}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac{1}{2 b^3 d^2 x^2}+\frac{3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac{3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3} \]

[Out]

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

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Rubi [A]  time = 0.765562, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{c^4 (3 c d-5 b e)}{b^4 (b+c x) (c d-b e)^3}+\frac{2 b e+3 c d}{b^4 d^3 x}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac{1}{2 b^3 d^2 x^2}+\frac{3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac{3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^2*(b*x + c*x^2)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 153.32, size = 223, normalized size = 0.97 \[ \frac{e^{5}}{d^{3} \left (d + e x\right ) \left (b e - c d\right )^{3}} - \frac{3 e^{5} \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{d^{4} \left (b e - c d\right )^{4}} + \frac{c^{4}}{2 b^{3} \left (b + c x\right )^{2} \left (b e - c d\right )^{2}} - \frac{1}{2 b^{3} d^{2} x^{2}} + \frac{c^{4} \left (5 b e - 3 c d\right )}{b^{4} \left (b + c x\right ) \left (b e - c d\right )^{3}} + \frac{2 b e + 3 c d}{b^{4} d^{3} x} - \frac{3 c^{4} \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right ) \log{\left (b + c x \right )}}{b^{5} \left (b e - c d\right )^{4}} + \frac{3 \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x \right )}}{b^{5} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**2/(c*x**2+b*x)**3,x)

[Out]

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

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Mathematica [A]  time = 0.502875, size = 230, normalized size = 1. \[ \frac{c^4 (5 b e-3 c d)}{b^4 (b+c x) (b e-c d)^3}+\frac{2 b e+3 c d}{b^4 d^3 x}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac{1}{2 b^3 d^2 x^2}+\frac{3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac{3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^2*(b*x + c*x^2)^3),x]

[Out]

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

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Maple [A]  time = 0.029, size = 306, normalized size = 1.3 \[ -{\frac{1}{2\,{d}^{2}{b}^{3}{x}^{2}}}+2\,{\frac{e}{{d}^{3}{b}^{3}x}}+3\,{\frac{c}{{d}^{2}{b}^{4}x}}+3\,{\frac{\ln \left ( x \right ){e}^{2}}{{d}^{4}{b}^{3}}}+6\,{\frac{\ln \left ( x \right ) ce}{{d}^{3}{b}^{4}}}+6\,{\frac{\ln \left ( x \right ){c}^{2}}{{d}^{2}{b}^{5}}}+{\frac{{c}^{4}}{2\, \left ( be-cd \right ) ^{2}{b}^{3} \left ( cx+b \right ) ^{2}}}+5\,{\frac{{c}^{4}e}{ \left ( be-cd \right ) ^{3}{b}^{3} \left ( cx+b \right ) }}-3\,{\frac{{c}^{5}d}{ \left ( be-cd \right ) ^{3}{b}^{4} \left ( cx+b \right ) }}-15\,{\frac{{c}^{4}\ln \left ( cx+b \right ){e}^{2}}{ \left ( be-cd \right ) ^{4}{b}^{3}}}+18\,{\frac{{c}^{5}\ln \left ( cx+b \right ) de}{ \left ( be-cd \right ) ^{4}{b}^{4}}}-6\,{\frac{{c}^{6}\ln \left ( cx+b \right ){d}^{2}}{ \left ( be-cd \right ) ^{4}{b}^{5}}}+{\frac{{e}^{5}}{{d}^{3} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{{e}^{6}\ln \left ( ex+d \right ) b}{{d}^{4} \left ( be-cd \right ) ^{4}}}+6\,{\frac{{e}^{5}\ln \left ( ex+d \right ) c}{{d}^{3} \left ( be-cd \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(c*x^2+b*x)^3,x)

[Out]

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

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Maxima [A]  time = 0.733433, size = 1015, normalized size = 4.41 \[ -\frac{3 \,{\left (2 \, c^{6} d^{2} - 6 \, b c^{5} d e + 5 \, b^{2} c^{4} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{4} d^{4} - 4 \, b^{6} c^{3} d^{3} e + 6 \, b^{7} c^{2} d^{2} e^{2} - 4 \, b^{8} c d e^{3} + b^{9} e^{4}} + \frac{3 \,{\left (2 \, c d e^{5} - b e^{6}\right )} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}} - \frac{b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - 6 \,{\left (2 \, c^{6} d^{4} e - 4 \, b c^{5} d^{3} e^{2} + b^{2} c^{4} d^{2} e^{3} + b^{3} c^{3} d e^{4} - b^{4} c^{2} e^{5}\right )} x^{4} - 3 \,{\left (4 \, c^{6} d^{5} - 2 \, b c^{5} d^{4} e - 10 \, b^{2} c^{4} d^{3} e^{2} + 5 \, b^{3} c^{3} d^{2} e^{3} + 3 \, b^{4} c^{2} d e^{4} - 4 \, b^{5} c e^{5}\right )} x^{3} -{\left (18 \, b c^{5} d^{5} - 32 \, b^{2} c^{4} d^{4} e + b^{3} c^{3} d^{3} e^{2} + 13 \, b^{4} c^{2} d^{2} e^{3} - 6 \, b^{6} e^{5}\right )} x^{2} -{\left (4 \, b^{2} c^{4} d^{5} - 9 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + 5 \, b^{5} c d^{2} e^{3} - 3 \, b^{6} d e^{4}\right )} x}{2 \,{\left ({\left (b^{4} c^{5} d^{6} e - 3 \, b^{5} c^{4} d^{5} e^{2} + 3 \, b^{6} c^{3} d^{4} e^{3} - b^{7} c^{2} d^{3} e^{4}\right )} x^{5} +{\left (b^{4} c^{5} d^{7} - b^{5} c^{4} d^{6} e - 3 \, b^{6} c^{3} d^{5} e^{2} + 5 \, b^{7} c^{2} d^{4} e^{3} - 2 \, b^{8} c d^{3} e^{4}\right )} x^{4} +{\left (2 \, b^{5} c^{4} d^{7} - 5 \, b^{6} c^{3} d^{6} e + 3 \, b^{7} c^{2} d^{5} e^{2} + b^{8} c d^{4} e^{3} - b^{9} d^{3} e^{4}\right )} x^{3} +{\left (b^{6} c^{3} d^{7} - 3 \, b^{7} c^{2} d^{6} e + 3 \, b^{8} c d^{5} e^{2} - b^{9} d^{4} e^{3}\right )} x^{2}\right )}} + \frac{3 \,{\left (2 \, c^{2} d^{2} + 2 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)^2),x, algorithm="maxima")

[Out]

-3*(2*c^6*d^2 - 6*b*c^5*d*e + 5*b^2*c^4*e^2)*log(c*x + b)/(b^5*c^4*d^4 - 4*b^6*c
^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d*e^3 + b^9*e^4) + 3*(2*c*d*e^5 - b*e^6)*
log(e*x + d)/(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^
4*d^4*e^4) - 1/2*(b^3*c^3*d^5 - 3*b^4*c^2*d^4*e + 3*b^5*c*d^3*e^2 - b^6*d^2*e^3
- 6*(2*c^6*d^4*e - 4*b*c^5*d^3*e^2 + b^2*c^4*d^2*e^3 + b^3*c^3*d*e^4 - b^4*c^2*e
^5)*x^4 - 3*(4*c^6*d^5 - 2*b*c^5*d^4*e - 10*b^2*c^4*d^3*e^2 + 5*b^3*c^3*d^2*e^3
+ 3*b^4*c^2*d*e^4 - 4*b^5*c*e^5)*x^3 - (18*b*c^5*d^5 - 32*b^2*c^4*d^4*e + b^3*c^
3*d^3*e^2 + 13*b^4*c^2*d^2*e^3 - 6*b^6*e^5)*x^2 - (4*b^2*c^4*d^5 - 9*b^3*c^3*d^4
*e + 3*b^4*c^2*d^3*e^2 + 5*b^5*c*d^2*e^3 - 3*b^6*d*e^4)*x)/((b^4*c^5*d^6*e - 3*b
^5*c^4*d^5*e^2 + 3*b^6*c^3*d^4*e^3 - b^7*c^2*d^3*e^4)*x^5 + (b^4*c^5*d^7 - b^5*c
^4*d^6*e - 3*b^6*c^3*d^5*e^2 + 5*b^7*c^2*d^4*e^3 - 2*b^8*c*d^3*e^4)*x^4 + (2*b^5
*c^4*d^7 - 5*b^6*c^3*d^6*e + 3*b^7*c^2*d^5*e^2 + b^8*c*d^4*e^3 - b^9*d^3*e^4)*x^
3 + (b^6*c^3*d^7 - 3*b^7*c^2*d^6*e + 3*b^8*c*d^5*e^2 - b^9*d^4*e^3)*x^2) + 3*(2*
c^2*d^2 + 2*b*c*d*e + b^2*e^2)*log(x)/(b^5*d^4)

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Fricas [A]  time = 65.5106, size = 1762, normalized size = 7.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)^2),x, algorithm="fricas")

[Out]

-1/2*(b^4*c^4*d^7 - 4*b^5*c^3*d^6*e + 6*b^6*c^2*d^5*e^2 - 4*b^7*c*d^4*e^3 + b^8*
d^3*e^4 - 6*(2*b*c^7*d^6*e - 6*b^2*c^6*d^5*e^2 + 5*b^3*c^5*d^4*e^3 - 2*b^5*c^3*d
^2*e^5 + b^6*c^2*d*e^6)*x^4 - 3*(4*b*c^7*d^7 - 6*b^2*c^6*d^6*e - 8*b^3*c^5*d^5*e
^2 + 15*b^4*c^4*d^4*e^3 - 2*b^5*c^3*d^3*e^4 - 7*b^6*c^2*d^2*e^5 + 4*b^7*c*d*e^6)
*x^3 - (18*b^2*c^6*d^7 - 50*b^3*c^5*d^6*e + 33*b^4*c^4*d^5*e^2 + 12*b^5*c^3*d^4*
e^3 - 13*b^6*c^2*d^3*e^4 - 6*b^7*c*d^2*e^5 + 6*b^8*d*e^6)*x^2 - (4*b^3*c^5*d^7 -
13*b^4*c^4*d^6*e + 12*b^5*c^3*d^5*e^2 + 2*b^6*c^2*d^4*e^3 - 8*b^7*c*d^3*e^4 + 3
*b^8*d^2*e^5)*x + 6*((2*c^8*d^6*e - 6*b*c^7*d^5*e^2 + 5*b^2*c^6*d^4*e^3)*x^5 + (
2*c^8*d^7 - 2*b*c^7*d^6*e - 7*b^2*c^6*d^5*e^2 + 10*b^3*c^5*d^4*e^3)*x^4 + (4*b*c
^7*d^7 - 10*b^2*c^6*d^6*e + 4*b^3*c^5*d^5*e^2 + 5*b^4*c^4*d^4*e^3)*x^3 + (2*b^2*
c^6*d^7 - 6*b^3*c^5*d^6*e + 5*b^4*c^4*d^5*e^2)*x^2)*log(c*x + b) - 6*((2*b^5*c^3
*d*e^6 - b^6*c^2*e^7)*x^5 + (2*b^5*c^3*d^2*e^5 + 3*b^6*c^2*d*e^6 - 2*b^7*c*e^7)*
x^4 + (4*b^6*c^2*d^2*e^5 - b^8*e^7)*x^3 + (2*b^7*c*d^2*e^5 - b^8*d*e^6)*x^2)*log
(e*x + d) - 6*((2*c^8*d^6*e - 6*b*c^7*d^5*e^2 + 5*b^2*c^6*d^4*e^3 - 2*b^5*c^3*d*
e^6 + b^6*c^2*e^7)*x^5 + (2*c^8*d^7 - 2*b*c^7*d^6*e - 7*b^2*c^6*d^5*e^2 + 10*b^3
*c^5*d^4*e^3 - 2*b^5*c^3*d^2*e^5 - 3*b^6*c^2*d*e^6 + 2*b^7*c*e^7)*x^4 + (4*b*c^7
*d^7 - 10*b^2*c^6*d^6*e + 4*b^3*c^5*d^5*e^2 + 5*b^4*c^4*d^4*e^3 - 4*b^6*c^2*d^2*
e^5 + b^8*e^7)*x^3 + (2*b^2*c^6*d^7 - 6*b^3*c^5*d^6*e + 5*b^4*c^4*d^5*e^2 - 2*b^
7*c*d^2*e^5 + b^8*d*e^6)*x^2)*log(x))/((b^5*c^6*d^8*e - 4*b^6*c^5*d^7*e^2 + 6*b^
7*c^4*d^6*e^3 - 4*b^8*c^3*d^5*e^4 + b^9*c^2*d^4*e^5)*x^5 + (b^5*c^6*d^9 - 2*b^6*
c^5*d^8*e - 2*b^7*c^4*d^7*e^2 + 8*b^8*c^3*d^6*e^3 - 7*b^9*c^2*d^5*e^4 + 2*b^10*c
*d^4*e^5)*x^4 + (2*b^6*c^5*d^9 - 7*b^7*c^4*d^8*e + 8*b^8*c^3*d^7*e^2 - 2*b^9*c^2
*d^6*e^3 - 2*b^10*c*d^5*e^4 + b^11*d^4*e^5)*x^3 + (b^7*c^4*d^9 - 4*b^8*c^3*d^8*e
+ 6*b^9*c^2*d^7*e^2 - 4*b^10*c*d^6*e^3 + b^11*d^5*e^4)*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**2/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231517, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)^2),x, algorithm="giac")

[Out]

Done