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

Optimal. Leaf size=144 \[ \frac {d (3 c d-2 b e)}{b^4 x}+\frac {(c d-b e) (3 c d-b e)}{b^4 (b+c x)}+\frac {(c d-b e)^2}{2 b^3 (b+c x)^2}-\frac {d^2}{2 b^3 x^2}+\frac {\log (x) \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{b^5}-\frac {\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5} \]

[Out]

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

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Rubi [A]  time = 0.13, 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} \[ \frac {\log (x) \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{b^5}-\frac {\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5}+\frac {d (3 c d-2 b e)}{b^4 x}+\frac {(c d-b e) (3 c d-b e)}{b^4 (b+c x)}+\frac {(c d-b e)^2}{2 b^3 (b+c x)^2}-\frac {d^2}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2/(b*x + c*x^2)^3,x]

[Out]

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

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Mathematica [A]  time = 0.09, size = 144, normalized size = 1.00 \[ \frac {\frac {2 b \left (b^2 e^2-4 b c d e+3 c^2 d^2\right )}{b+c x}+2 \log (x) \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (b+c x)+\frac {b^2 (c d-b e)^2}{(b+c x)^2}-\frac {b^2 d^2}{x^2}-\frac {2 b d (2 b e-3 c d)}{x}}{2 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.01, size = 327, normalized size = 2.27 \[ -\frac {b^{4} d^{2} - 2 \, {\left (6 \, b c^{3} d^{2} - 6 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} - 3 \, {\left (6 \, b^{2} c^{2} d^{2} - 6 \, b^{3} c d e + b^{4} e^{2}\right )} x^{2} - 4 \, {\left (b^{3} c d^{2} - b^{4} d e\right )} x + 2 \, {\left ({\left (6 \, c^{4} d^{2} - 6 \, b c^{3} d e + b^{2} c^{2} e^{2}\right )} x^{4} + 2 \, {\left (6 \, b c^{3} d^{2} - 6 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} + {\left (6 \, b^{2} c^{2} d^{2} - 6 \, b^{3} c d e + b^{4} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (6 \, c^{4} d^{2} - 6 \, b c^{3} d e + b^{2} c^{2} e^{2}\right )} x^{4} + 2 \, {\left (6 \, b c^{3} d^{2} - 6 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} + {\left (6 \, b^{2} c^{2} d^{2} - 6 \, b^{3} c d e + b^{4} e^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 182, normalized size = 1.26 \[ \frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {{\left (6 \, c^{3} d^{2} - 6 \, b c^{2} d e + b^{2} c e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac {12 \, c^{3} d^{2} x^{3} - 12 \, b c^{2} d x^{3} e + 18 \, b c^{2} d^{2} x^{2} + 2 \, b^{2} c x^{3} e^{2} - 18 \, b^{2} c d x^{2} e + 4 \, b^{2} c d^{2} x + 3 \, b^{3} x^{2} e^{2} - 4 \, b^{3} d x e - b^{3} d^{2}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 207, normalized size = 1.44 \[ \frac {e^{2}}{2 \left (c x +b \right )^{2} b}-\frac {c d e}{\left (c x +b \right )^{2} b^{2}}+\frac {c^{2} d^{2}}{2 \left (c x +b \right )^{2} b^{3}}+\frac {e^{2}}{\left (c x +b \right ) b^{2}}-\frac {4 c d e}{\left (c x +b \right ) b^{3}}+\frac {e^{2} \ln \relax (x )}{b^{3}}-\frac {e^{2} \ln \left (c x +b \right )}{b^{3}}+\frac {3 c^{2} d^{2}}{\left (c x +b \right ) b^{4}}-\frac {6 c d e \ln \relax (x )}{b^{4}}+\frac {6 c d e \ln \left (c x +b \right )}{b^{4}}+\frac {6 c^{2} d^{2} \ln \relax (x )}{b^{5}}-\frac {6 c^{2} d^{2} \ln \left (c x +b \right )}{b^{5}}-\frac {2 d e}{b^{3} x}+\frac {3 c \,d^{2}}{b^{4} x}-\frac {d^{2}}{2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.41, size = 180, normalized size = 1.25 \[ -\frac {b^{3} d^{2} - 2 \, {\left (6 \, c^{3} d^{2} - 6 \, b c^{2} d e + b^{2} c e^{2}\right )} x^{3} - 3 \, {\left (6 \, b c^{2} d^{2} - 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{2} - 4 \, {\left (b^{2} c d^{2} - b^{3} d e\right )} x}{2 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} - \frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \relax (x)}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.26, size = 149, normalized size = 1.03 \[ -\frac {\frac {d^2}{2\,b}-\frac {3\,x^2\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )}{2\,b^3}-\frac {c\,x^3\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^4}+\frac {2\,d\,x\,\left (b\,e-c\,d\right )}{b^2}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2/(b*x + c*x^2)^3,x)

[Out]

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

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sympy [B]  time = 1.09, size = 345, normalized size = 2.40 \[ \frac {- b^{3} d^{2} + x^{3} \left (2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}\right ) + x^{2} \left (3 b^{3} e^{2} - 18 b^{2} c d e + 18 b c^{2} d^{2}\right ) + x \left (- 4 b^{3} d e + 4 b^{2} c d^{2}\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} + \frac {\left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log {\left (x + \frac {b^{3} e^{2} - 6 b^{2} c d e + 6 b c^{2} d^{2} - b \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}} \right )}}{b^{5}} - \frac {\left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log {\left (x + \frac {b^{3} e^{2} - 6 b^{2} c d e + 6 b c^{2} d^{2} + b \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}} \right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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