3.274 \(\int \frac {1}{(d+e x)^2 (b x+c x^2)^2} \, dx\)

Optimal. Leaf size=144 \[ \frac {2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac {2 \log (x) (b e+c d)}{b^3 d^3}-\frac {c^3}{b^2 (b+c x) (c d-b e)^2}-\frac {1}{b^2 d^2 x}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}-\frac {e^3}{d^2 (d+e x) (c d-b e)^2} \]

[Out]

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

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Rubi [A]  time = 0.17, antiderivative size = 144, normalized size of antiderivative = 1.00, 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} \[ -\frac {c^3}{b^2 (b+c x) (c d-b e)^2}+\frac {2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac {2 \log (x) (b e+c d)}{b^3 d^3}-\frac {1}{b^2 d^2 x}-\frac {e^3}{d^2 (d+e x) (c d-b e)^2}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(b^2*d^2*x)) - c^3/(b^2*(c*d - b*e)^2*(b + c*x)) - e^3/(d^2*(c*d - b*e)^2*(d + e*x)) - (2*(c*d + b*e)*Log[
x])/(b^3*d^3) + (2*c^3*(c*d - 2*b*e)*Log[b + c*x])/(b^3*(c*d - b*e)^3) + (2*e^3*(2*c*d - b*e)*Log[d + e*x])/(d
^3*(c*d - b*e)^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 {1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {1}{b^2 d^2 x^2}-\frac {2 (c d+b e)}{b^3 d^3 x}+\frac {c^4}{b^2 (-c d+b e)^2 (b+c x)^2}+\frac {2 c^4 (-c d+2 b e)}{b^3 (-c d+b e)^3 (b+c x)}+\frac {e^4}{d^2 (c d-b e)^2 (d+e x)^2}+\frac {2 e^4 (2 c d-b e)}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {1}{b^2 d^2 x}-\frac {c^3}{b^2 (c d-b e)^2 (b+c x)}-\frac {e^3}{d^2 (c d-b e)^2 (d+e x)}-\frac {2 (c d+b e) \log (x)}{b^3 d^3}+\frac {2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 145, normalized size = 1.01 \[ \frac {2 c^3 (2 b e-c d) \log (b+c x)}{b^3 (b e-c d)^3}-\frac {2 \log (x) (b e+c d)}{b^3 d^3}-\frac {c^3}{b^2 (b+c x) (c d-b e)^2}-\frac {1}{b^2 d^2 x}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}-\frac {e^3}{d^2 (d+e x) (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 28.53, size = 653, normalized size = 4.53 \[ -\frac {b^{2} c^{3} d^{5} - 3 \, b^{3} c^{2} d^{4} e + 3 \, b^{4} c d^{3} e^{2} - b^{5} d^{2} e^{3} + 2 \, {\left (b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2} + 2 \, b^{3} c^{2} d^{2} e^{3} - b^{4} c d e^{4}\right )} x^{2} + {\left (2 \, b c^{4} d^{5} - 3 \, b^{2} c^{3} d^{4} e + 3 \, b^{4} c d^{2} e^{3} - 2 \, b^{5} d e^{4}\right )} x - 2 \, {\left ({\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2}\right )} x^{3} + {\left (c^{5} d^{5} - b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2}\right )} x^{2} + {\left (b c^{4} d^{5} - 2 \, b^{2} c^{3} d^{4} e\right )} x\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (2 \, b^{3} c^{2} d e^{4} - b^{4} c e^{5}\right )} x^{3} + {\left (2 \, b^{3} c^{2} d^{2} e^{3} + b^{4} c d e^{4} - b^{5} e^{5}\right )} x^{2} + {\left (2 \, b^{4} c d^{2} e^{3} - b^{5} d e^{4}\right )} x\right )} \log \left (e x + d\right ) + 2 \, {\left ({\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2} + 2 \, b^{3} c^{2} d e^{4} - b^{4} c e^{5}\right )} x^{3} + {\left (c^{5} d^{5} - b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2} + 2 \, b^{3} c^{2} d^{2} e^{3} + b^{4} c d e^{4} - b^{5} e^{5}\right )} x^{2} + {\left (b c^{4} d^{5} - 2 \, b^{2} c^{3} d^{4} e + 2 \, b^{4} c d^{2} e^{3} - b^{5} d e^{4}\right )} x\right )} \log \relax (x)}{{\left (b^{3} c^{4} d^{6} e - 3 \, b^{4} c^{3} d^{5} e^{2} + 3 \, b^{5} c^{2} d^{4} e^{3} - b^{6} c d^{3} e^{4}\right )} x^{3} + {\left (b^{3} c^{4} d^{7} - 2 \, b^{4} c^{3} d^{6} e + 2 \, b^{6} c d^{4} e^{3} - b^{7} d^{3} e^{4}\right )} x^{2} + {\left (b^{4} c^{3} d^{7} - 3 \, b^{5} c^{2} d^{6} e + 3 \, b^{6} c d^{5} e^{2} - b^{7} d^{4} e^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.25, size = 555, normalized size = 3.85 \[ \frac {{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 2 \, b^{3} c d e^{5} - b^{4} e^{6}\right )} e^{\left (-2\right )} \log \left (\frac {{\left | -2 \, c d e + \frac {2 \, c d^{2} e}{x e + d} + b e^{2} - \frac {2 \, b d e^{2}}{x e + d} - {\left | b \right |} e^{2} \right |}}{{\left | -2 \, c d e + \frac {2 \, c d^{2} e}{x e + d} + b e^{2} - \frac {2 \, b d e^{2}}{x e + d} + {\left | b \right |} e^{2} \right |}}\right )}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )} {\left | b \right |}} - \frac {{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left ({\left | -c + \frac {2 \, c d}{x e + d} - \frac {c d^{2}}{{\left (x e + d\right )}^{2}} - \frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} \right |}\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {e^{7}}{{\left (c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}\right )} {\left (x e + d\right )}} - \frac {\frac {2 \, c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}}{c d^{2} - b d e} - \frac {{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 6 \, b^{2} c^{2} d^{2} e^{4} - 4 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} - b d e\right )} {\left (x e + d\right )}}}{{\left (c d - b e\right )}^{2} b^{2} {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}\right )} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.11, size = 185, normalized size = 1.28 \[ \frac {2 b \,e^{4} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{3}}+\frac {4 c^{3} e \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{2}}-\frac {2 c^{4} d \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{3}}-\frac {4 c \,e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{2}}-\frac {c^{3}}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{2}}-\frac {e^{3}}{\left (b e -c d \right )^{2} \left (e x +d \right ) d^{2}}-\frac {2 e \ln \relax (x )}{b^{2} d^{3}}-\frac {2 c \ln \relax (x )}{b^{3} d^{2}}-\frac {1}{b^{2} d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(c*x^2+b*x)^2,x)

[Out]

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

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maxima [B]  time = 1.52, size = 373, normalized size = 2.59 \[ \frac {2 \, {\left (c^{4} d - 2 \, b c^{3} e\right )} \log \left (c x + b\right )}{b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}} + \frac {2 \, {\left (2 \, c d e^{3} - b e^{4}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + b^{3} d e^{2} + 2 \, {\left (c^{3} d^{2} e - b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{2} + {\left (2 \, c^{3} d^{3} - b c^{2} d^{2} e - b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} x}{{\left (b^{2} c^{3} d^{4} e - 2 \, b^{3} c^{2} d^{3} e^{2} + b^{4} c d^{2} e^{3}\right )} x^{3} + {\left (b^{2} c^{3} d^{5} - b^{3} c^{2} d^{4} e - b^{4} c d^{3} e^{2} + b^{5} d^{2} e^{3}\right )} x^{2} + {\left (b^{3} c^{2} d^{5} - 2 \, b^{4} c d^{4} e + b^{5} d^{3} e^{2}\right )} x} - \frac {2 \, {\left (c d + b e\right )} \log \relax (x)}{b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.69, size = 304, normalized size = 2.11 \[ -\frac {\frac {1}{b\,d}+\frac {2\,x^2\,\left (b^2\,c\,e^3-b\,c^2\,d\,e^2+c^3\,d^2\,e\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x\,\left (b\,e+c\,d\right )\,\left (2\,b^2\,e^2-3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+b\,d\,x}-\frac {\ln \left (b+c\,x\right )\,\left (2\,c^4\,d-4\,b\,c^3\,e\right )}{b^6\,e^3-3\,b^5\,c\,d\,e^2+3\,b^4\,c^2\,d^2\,e-b^3\,c^3\,d^3}-\frac {\ln \left (d+e\,x\right )\,\left (2\,b\,e^4-4\,c\,d\,e^3\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}-\frac {2\,\ln \relax (x)\,\left (b\,e+c\,d\right )}{b^3\,d^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^2*(d + e*x)^2),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(c*x**2+b*x)**2,x)

[Out]

Timed out

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