3.3.71 \(\int \frac {d+e x}{(b x+c x^2)^2} \, dx\) [271]

Optimal. Leaf size=65 \[ -\frac {d}{b^2 x}-\frac {c d-b e}{b^2 (b+c x)}-\frac {(2 c d-b e) \log (x)}{b^3}+\frac {(2 c d-b e) \log (b+c x)}{b^3} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {645} \begin {gather*} -\frac {\log (x) (2 c d-b e)}{b^3}+\frac {(2 c d-b e) \log (b+c x)}{b^3}-\frac {c d-b e}{b^2 (b+c x)}-\frac {d}{b^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(b*x + c*x^2)^2,x]

[Out]

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

Rule 645

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

\begin {align*} \int \frac {d+e x}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {d}{b^2 x^2}+\frac {-2 c d+b e}{b^3 x}-\frac {c (-c d+b e)}{b^2 (b+c x)^2}-\frac {c (-2 c d+b e)}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac {d}{b^2 x}-\frac {c d-b e}{b^2 (b+c x)}-\frac {(2 c d-b e) \log (x)}{b^3}+\frac {(2 c d-b e) \log (b+c x)}{b^3}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 56, normalized size = 0.86 \begin {gather*} \frac {-\frac {b d}{x}+\frac {b (-c d+b e)}{b+c x}+(-2 c d+b e) \log (x)+(2 c d-b e) \log (b+c x)}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.43, size = 63, normalized size = 0.97

method	result	size
default	\(-\frac {\left (b e -2 c d \right ) \ln \left (c x +b \right )}{b^{3}}+\frac {b e -c d}{b^{2} \left (c x +b \right )}-\frac {d}{b^{2} x}+\frac {\left (b e -2 c d \right ) \ln \left (x \right )}{b^{3}}\)	\(63\)
norman	\(\frac {\frac {c \left (-b e +2 c d \right ) x^{2}}{b^{3}}-\frac {d}{b}}{x \left (c x +b \right )}+\frac {\left (b e -2 c d \right ) \ln \left (x \right )}{b^{3}}-\frac {\left (b e -2 c d \right ) \ln \left (c x +b \right )}{b^{3}}\)	\(70\)
risch	\(\frac {\frac {\left (b e -2 c d \right ) x}{b^{2}}-\frac {d}{b}}{x \left (c x +b \right )}+\frac {\ln \left (-x \right ) e}{b^{2}}-\frac {2 \ln \left (-x \right ) c d}{b^{3}}-\frac {\ln \left (c x +b \right ) e}{b^{2}}+\frac {2 \ln \left (c x +b \right ) c d}{b^{3}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 72, normalized size = 1.11 \begin {gather*} -\frac {b d + {\left (2 \, c d - b e\right )} x}{b^{2} c x^{2} + b^{3} x} + \frac {{\left (2 \, c d - b e\right )} \log \left (c x + b\right )}{b^{3}} - \frac {{\left (2 \, c d - b e\right )} \log \left (x\right )}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.01, size = 113, normalized size = 1.74 \begin {gather*} -\frac {2 \, b c d x - b^{2} x e + b^{2} d - {\left (2 \, c^{2} d x^{2} + 2 \, b c d x - {\left (b c x^{2} + b^{2} x\right )} e\right )} \log \left (c x + b\right ) + {\left (2 \, c^{2} d x^{2} + 2 \, b c d x - {\left (b c x^{2} + b^{2} x\right )} e\right )} \log \left (x\right )}{b^{3} c x^{2} + b^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (54) = 108\).
time = 0.25, size = 128, normalized size = 1.97 \begin {gather*} \frac {- b d + x \left (b e - 2 c d\right )}{b^{3} x + b^{2} c x^{2}} + \frac {\left (b e - 2 c d\right ) \log {\left (x + \frac {b^{2} e - 2 b c d - b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} - \frac {\left (b e - 2 c d\right ) \log {\left (x + \frac {b^{2} e - 2 b c d + b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.86, size = 77, normalized size = 1.18 \begin {gather*} -\frac {{\left (2 \, c d - b e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac {2 \, c d x - b x e + b d}{{\left (c x^{2} + b x\right )} b^{2}} + \frac {{\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.22, size = 57, normalized size = 0.88 \begin {gather*} -\frac {\frac {d}{b}-\frac {x\,\left (b\,e-2\,c\,d\right )}{b^2}}{c\,x^2+b\,x}-\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )\,\left (b\,e-2\,c\,d\right )}{b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(b*x + c*x^2)^2,x)

[Out]

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

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