3.11.4 \(\int \frac {(A+B x) (d+e x)}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=86 \[ \frac {\log (x) (A b e-2 A c d+b B d)}{b^3}-\frac {\log (b+c x) (A b e-2 A c d+b B d)}{b^3}+\frac {(b B-A c) (c d-b e)}{b^2 c (b+c x)}-\frac {A d}{b^2 x} \]

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Rubi [A]  time = 0.08, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {771} \begin {gather*} \frac {(b B-A c) (c d-b e)}{b^2 c (b+c x)}+\frac {\log (x) (A b e-2 A c d+b B d)}{b^3}-\frac {\log (b+c x) (A b e-2 A c d+b B d)}{b^3}-\frac {A d}{b^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 80, normalized size = 0.93 \begin {gather*} -\frac {\frac {b (b B-A c) (b e-c d)}{c (b+c x)}-\log (x) (A b e-2 A c d+b B d)+\log (b+c x) (A b e-2 A c d+b B d)+\frac {A b d}{x}}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)}{\left (b x+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x))/(b*x + c*x^2)^2,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(d + e*x))/(b*x + c*x^2)^2, x]

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fricas [B]  time = 0.43, size = 184, normalized size = 2.14 \begin {gather*} -\frac {A b^{2} c d - {\left ({\left (B b^{2} c - 2 \, A b c^{2}\right )} d - {\left (B b^{3} - A b^{2} c\right )} e\right )} x + {\left ({\left (A b c^{2} e + {\left (B b c^{2} - 2 \, A c^{3}\right )} d\right )} x^{2} + {\left (A b^{2} c e + {\left (B b^{2} c - 2 \, A b c^{2}\right )} d\right )} x\right )} \log \left (c x + b\right ) - {\left ({\left (A b c^{2} e + {\left (B b c^{2} - 2 \, A c^{3}\right )} d\right )} x^{2} + {\left (A b^{2} c e + {\left (B b^{2} c - 2 \, A b c^{2}\right )} d\right )} x\right )} \log \relax (x)}{b^{3} c^{2} x^{2} + b^{4} c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 112, normalized size = 1.30 \begin {gather*} \frac {{\left (B b d - 2 \, A c d + A b e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac {{\left (B b c d - 2 \, A c^{2} d + A b c e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} + \frac {B b c d x - 2 \, A c^{2} d x - B b^{2} x e + A b c x e - A b c d}{{\left (c x^{2} + b x\right )} b^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 133, normalized size = 1.55 \begin {gather*} \frac {A e}{\left (c x +b \right ) b}-\frac {A c d}{\left (c x +b \right ) b^{2}}+\frac {A e \ln \relax (x )}{b^{2}}-\frac {A e \ln \left (c x +b \right )}{b^{2}}-\frac {2 A c d \ln \relax (x )}{b^{3}}+\frac {2 A c d \ln \left (c x +b \right )}{b^{3}}+\frac {B d}{\left (c x +b \right ) b}+\frac {B d \ln \relax (x )}{b^{2}}-\frac {B d \ln \left (c x +b \right )}{b^{2}}-\frac {B e}{\left (c x +b \right ) c}-\frac {A d}{b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.47, size = 106, normalized size = 1.23 \begin {gather*} -\frac {A b c d - {\left ({\left (B b c - 2 \, A c^{2}\right )} d - {\left (B b^{2} - A b c\right )} e\right )} x}{b^{2} c^{2} x^{2} + b^{3} c x} - \frac {{\left (A b e + {\left (B b - 2 \, A c\right )} d\right )} \log \left (c x + b\right )}{b^{3}} + \frac {{\left (A b e + {\left (B b - 2 \, A c\right )} d\right )} \log \relax (x)}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(A*b*c*d - ((B*b*c - 2*A*c^2)*d - (B*b^2 - A*b*c)*e)*x)/(b^2*c^2*x^2 + b^3*c*x) - (A*b*e + (B*b - 2*A*c)*d)*l
og(c*x + b)/b^3 + (A*b*e + (B*b - 2*A*c)*d)*log(x)/b^3

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mupad [B]  time = 1.44, size = 117, normalized size = 1.36 \begin {gather*} -\frac {\frac {A\,d}{b}+\frac {x\,\left (2\,A\,c^2\,d+B\,b^2\,e-A\,b\,c\,e-B\,b\,c\,d\right )}{b^2\,c}}{c\,x^2+b\,x}-\frac {2\,\mathrm {atanh}\left (\frac {\left (b\,\left (A\,e+B\,d\right )-2\,A\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}\right )\,\left (b\,\left (A\,e+B\,d\right )-2\,A\,c\,d\right )}{b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 1.22, size = 233, normalized size = 2.71 \begin {gather*} \frac {- A b c d + x \left (A b c e - 2 A c^{2} d - B b^{2} e + B b c d\right )}{b^{3} c x + b^{2} c^{2} x^{2}} + \frac {\left (A b e - 2 A c d + B b d\right ) \log {\left (x + \frac {A b^{2} e - 2 A b c d + B b^{2} d - b \left (A b e - 2 A c d + B b d\right )}{2 A b c e - 4 A c^{2} d + 2 B b c d} \right )}}{b^{3}} - \frac {\left (A b e - 2 A c d + B b d\right ) \log {\left (x + \frac {A b^{2} e - 2 A b c d + B b^{2} d + b \left (A b e - 2 A c d + B b d\right )}{2 A b c e - 4 A c^{2} d + 2 B b c d} \right )}}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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