3.1153 \(\int \frac{A+B x}{(d+e x)^3 (b x+c x^2)^2} \, dx\)

Optimal. Leaf size=283 \[ -\frac{e^2 \log (d+e x) \left (B d \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )-A e \left (3 b^2 e^2-10 b c d e+10 c^2 d^2\right )\right )}{d^4 (c d-b e)^4}+\frac{c^3 (b B-A c)}{b^2 (b+c x) (c d-b e)^3}+\frac{c^3 \log (b+c x) \left (-b c (5 A e+B d)+2 A c^2 d+4 b^2 B e\right )}{b^3 (c d-b e)^4}+\frac{\log (x) (-3 A b e-2 A c d+b B d)}{b^3 d^4}-\frac{A}{b^2 d^3 x}-\frac{e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (d+e x) (c d-b e)^3}+\frac{e^2 (B d-A e)}{2 d^2 (d+e x)^2 (c d-b e)^2} \]

[Out]

-(A/(b^2*d^3*x)) + (c^3*(b*B - A*c))/(b^2*(c*d - b*e)^3*(b + c*x)) + (e^2*(B*d - A*e))/(2*d^2*(c*d - b*e)^2*(d
+ e*x)^2) - (e^2*(2*A*e*(2*c*d - b*e) - B*d*(3*c*d - b*e)))/(d^3*(c*d - b*e)^3*(d + e*x)) + ((b*B*d - 2*A*c*d
- 3*A*b*e)*Log[x])/(b^3*d^4) + (c^3*(2*A*c^2*d + 4*b^2*B*e - b*c*(B*d + 5*A*e))*Log[b + c*x])/(b^3*(c*d - b*e
)^4) - (e^2*(B*d*(6*c^2*d^2 - 4*b*c*d*e + b^2*e^2) - A*e*(10*c^2*d^2 - 10*b*c*d*e + 3*b^2*e^2))*Log[d + e*x])/
(d^4*(c*d - b*e)^4)

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Rubi [A]  time = 0.461993, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{e^2 \log (d+e x) \left (B d \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )-A e \left (3 b^2 e^2-10 b c d e+10 c^2 d^2\right )\right )}{d^4 (c d-b e)^4}+\frac{c^3 (b B-A c)}{b^2 (b+c x) (c d-b e)^3}+\frac{c^3 \log (b+c x) \left (-b c (5 A e+B d)+2 A c^2 d+4 b^2 B e\right )}{b^3 (c d-b e)^4}+\frac{\log (x) (-3 A b e-2 A c d+b B d)}{b^3 d^4}-\frac{A}{b^2 d^3 x}-\frac{e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (d+e x) (c d-b e)^3}+\frac{e^2 (B d-A e)}{2 d^2 (d+e x)^2 (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A/(b^2*d^3*x)) + (c^3*(b*B - A*c))/(b^2*(c*d - b*e)^3*(b + c*x)) + (e^2*(B*d - A*e))/(2*d^2*(c*d - b*e)^2*(d
+ e*x)^2) - (e^2*(2*A*e*(2*c*d - b*e) - B*d*(3*c*d - b*e)))/(d^3*(c*d - b*e)^3*(d + e*x)) + ((b*B*d - 2*A*c*d
- 3*A*b*e)*Log[x])/(b^3*d^4) + (c^3*(2*A*c^2*d + 4*b^2*B*e - b*c*(B*d + 5*A*e))*Log[b + c*x])/(b^3*(c*d - b*e
)^4) - (e^2*(B*d*(6*c^2*d^2 - 4*b*c*d*e + b^2*e^2) - A*e*(10*c^2*d^2 - 10*b*c*d*e + 3*b^2*e^2))*Log[d + e*x])/
(d^4*(c*d - b*e)^4)

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)^3 \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{A}{b^2 d^3 x^2}+\frac{b B d-2 A c d-3 A b e}{b^3 d^4 x}+\frac{c^4 (b B-A c)}{b^2 (-c d+b e)^3 (b+c x)^2}+\frac{c^4 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right )}{b^3 (c d-b e)^4 (b+c x)}-\frac{e^3 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)^3}+\frac{e^3 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3 (d+e x)^2}+\frac{e^3 \left (-B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )+A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right )}{d^4 (c d-b e)^4 (d+e x)}\right ) \, dx\\ &=-\frac{A}{b^2 d^3 x}+\frac{c^3 (b B-A c)}{b^2 (c d-b e)^3 (b+c x)}+\frac{e^2 (B d-A e)}{2 d^2 (c d-b e)^2 (d+e x)^2}-\frac{e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3 (d+e x)}+\frac{(b B d-2 A c d-3 A b e) \log (x)}{b^3 d^4}+\frac{c^3 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^4}-\frac{e^2 \left (B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{d^4 (c d-b e)^4}\\ \end{align*}

Mathematica [A]  time = 0.590503, size = 279, normalized size = 0.99 \[ \frac{e^2 \log (d+e x) \left (A e \left (3 b^2 e^2-10 b c d e+10 c^2 d^2\right )-B d \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )\right )}{d^4 (c d-b e)^4}+\frac{c^3 (A c-b B)}{b^2 (b+c x) (b e-c d)^3}+\frac{c^3 \log (b+c x) \left (-b c (5 A e+B d)+2 A c^2 d+4 b^2 B e\right )}{b^3 (c d-b e)^4}+\frac{\log (x) (-3 A b e-2 A c d+b B d)}{b^3 d^4}-\frac{A}{b^2 d^3 x}+\frac{e^2 (2 A e (b e-2 c d)+B d (3 c d-b e))}{d^3 (d+e x) (c d-b e)^3}+\frac{e^2 (B d-A e)}{2 d^2 (d+e x)^2 (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A/(b^2*d^3*x)) + (c^3*(-(b*B) + A*c))/(b^2*(-(c*d) + b*e)^3*(b + c*x)) + (e^2*(B*d - A*e))/(2*d^2*(c*d - b*e
)^2*(d + e*x)^2) + (e^2*(B*d*(3*c*d - b*e) + 2*A*e*(-2*c*d + b*e)))/(d^3*(c*d - b*e)^3*(d + e*x)) + ((b*B*d -
2*A*c*d - 3*A*b*e)*Log[x])/(b^3*d^4) + (c^3*(2*A*c^2*d + 4*b^2*B*e - b*c*(B*d + 5*A*e))*Log[b + c*x])/(b^3*(c*
d - b*e)^4) + (e^2*(-(B*d*(6*c^2*d^2 - 4*b*c*d*e + b^2*e^2)) + A*e*(10*c^2*d^2 - 10*b*c*d*e + 3*b^2*e^2))*Log[
d + e*x])/(d^4*(c*d - b*e)^4)

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Maple [A]  time = 0.02, size = 528, normalized size = 1.9 \begin{align*} -{\frac{A}{{d}^{3}{b}^{2}x}}-3\,{\frac{\ln \left ( x \right ) Ae}{{d}^{4}{b}^{2}}}-2\,{\frac{Ac\ln \left ( x \right ) }{{d}^{3}{b}^{3}}}+{\frac{\ln \left ( x \right ) B}{{d}^{3}{b}^{2}}}-2\,{\frac{{e}^{4}Ab}{{d}^{3} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}+4\,{\frac{A{e}^{3}c}{{d}^{2} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}+{\frac{B{e}^{3}b}{{d}^{2} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{B{e}^{2}c}{d \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}+3\,{\frac{{e}^{5}\ln \left ( ex+d \right ) A{b}^{2}}{{d}^{4} \left ( be-cd \right ) ^{4}}}-10\,{\frac{{e}^{4}\ln \left ( ex+d \right ) Abc}{{d}^{3} \left ( be-cd \right ) ^{4}}}+10\,{\frac{{e}^{3}\ln \left ( ex+d \right ) A{c}^{2}}{{d}^{2} \left ( be-cd \right ) ^{4}}}-{\frac{{e}^{4}\ln \left ( ex+d \right ){b}^{2}B}{{d}^{3} \left ( be-cd \right ) ^{4}}}+4\,{\frac{{e}^{3}\ln \left ( ex+d \right ) Bbc}{{d}^{2} \left ( be-cd \right ) ^{4}}}-6\,{\frac{{e}^{2}\ln \left ( ex+d \right ) B{c}^{2}}{d \left ( be-cd \right ) ^{4}}}-{\frac{A{e}^{3}}{2\,{d}^{2} \left ( be-cd \right ) ^{2} \left ( ex+d \right ) ^{2}}}+{\frac{B{e}^{2}}{2\,d \left ( be-cd \right ) ^{2} \left ( ex+d \right ) ^{2}}}-5\,{\frac{{c}^{4}\ln \left ( cx+b \right ) Ae}{{b}^{2} \left ( be-cd \right ) ^{4}}}+2\,{\frac{{c}^{5}\ln \left ( cx+b \right ) Ad}{{b}^{3} \left ( be-cd \right ) ^{4}}}+4\,{\frac{{c}^{3}\ln \left ( cx+b \right ) Be}{b \left ( be-cd \right ) ^{4}}}-{\frac{{c}^{4}\ln \left ( cx+b \right ) Bd}{{b}^{2} \left ( be-cd \right ) ^{4}}}+{\frac{{c}^{4}A}{{b}^{2} \left ( be-cd \right ) ^{3} \left ( cx+b \right ) }}-{\frac{{c}^{3}B}{b \left ( be-cd \right ) ^{3} \left ( cx+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A/b^2/d^3/x-3/b^2/d^4*ln(x)*A*e-2/b^3/d^3*ln(x)*A*c+1/b^2/d^3*ln(x)*B-2*e^4/d^3/(b*e-c*d)^3/(e*x+d)*A*b+4*e^3
/d^2/(b*e-c*d)^3/(e*x+d)*A*c+e^3/d^2/(b*e-c*d)^3/(e*x+d)*B*b-3*e^2/d/(b*e-c*d)^3/(e*x+d)*B*c+3*e^5/d^4/(b*e-c*
d)^4*ln(e*x+d)*A*b^2-10*e^4/d^3/(b*e-c*d)^4*ln(e*x+d)*A*b*c+10*e^3/d^2/(b*e-c*d)^4*ln(e*x+d)*A*c^2-e^4/d^3/(b*
e-c*d)^4*ln(e*x+d)*b^2*B+4*e^3/d^2/(b*e-c*d)^4*ln(e*x+d)*B*b*c-6*e^2/d/(b*e-c*d)^4*ln(e*x+d)*B*c^2-1/2*e^3/d^2
/(b*e-c*d)^2/(e*x+d)^2*A+1/2*e^2/d/(b*e-c*d)^2/(e*x+d)^2*B-5*c^4/b^2/(b*e-c*d)^4*ln(c*x+b)*A*e+2*c^5/b^3/(b*e-
c*d)^4*ln(c*x+b)*A*d+4*c^3/b/(b*e-c*d)^4*ln(c*x+b)*B*e-c^4/b^2/(b*e-c*d)^4*ln(c*x+b)*B*d+c^4/b^2/(b*e-c*d)^3/(
c*x+b)*A-c^3/b/(b*e-c*d)^3/(c*x+b)*B

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Maxima [B]  time = 1.37269, size = 1098, normalized size = 3.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((B*b*c^4 - 2*A*c^5)*d - (4*B*b^2*c^3 - 5*A*b*c^4)*e)*log(c*x + b)/(b^3*c^4*d^4 - 4*b^4*c^3*d^3*e + 6*b^5*c^2
*d^2*e^2 - 4*b^6*c*d*e^3 + b^7*e^4) - (6*B*c^2*d^3*e^2 - 3*A*b^2*e^5 - 2*(2*B*b*c + 5*A*c^2)*d^2*e^3 + (B*b^2
+ 10*A*b*c)*d*e^4)*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*(2*A*b*c^3*d^5 - 6*A*b^2*c^2*d^4*e + 6*A*b^3*c*d^3*e^2 - 2*A*b^4*d^2*e^3 - 2*(3*A*b^3*c*e^5 + (B*b*c^3 -
2*A*c^4)*d^3*e^2 + 3*(B*b^2*c^2 + A*b*c^3)*d^2*e^3 - (B*b^3*c + 7*A*b^2*c^2)*d*e^4)*x^3 - (6*A*b^4*e^5 + 4*(B
*b*c^3 - 2*A*c^4)*d^4*e + (7*B*b^2*c^2 + 10*A*b*c^3)*d^3*e^2 + 3*(B*b^3*c - 5*A*b^2*c^2)*d^2*e^3 - (2*B*b^4 +
5*A*b^3*c)*d*e^4)*x^2 - (2*A*b*c^3*d^4*e + 9*A*b^4*d*e^4 + 2*(B*b*c^3 - 2*A*c^4)*d^5 + (7*B*b^3*c + 6*A*b^2*c^
2)*d^3*e^2 - (3*B*b^4 + 19*A*b^3*c)*d^2*e^3)*x)/((b^2*c^4*d^6*e^2 - 3*b^3*c^3*d^5*e^3 + 3*b^4*c^2*d^4*e^4 - b^
5*c*d^3*e^5)*x^4 + (2*b^2*c^4*d^7*e - 5*b^3*c^3*d^6*e^2 + 3*b^4*c^2*d^5*e^3 + b^5*c*d^4*e^4 - b^6*d^3*e^5)*x^3
+ (b^2*c^4*d^8 - b^3*c^3*d^7*e - 3*b^4*c^2*d^6*e^2 + 5*b^5*c*d^5*e^3 - 2*b^6*d^4*e^4)*x^2 + (b^3*c^3*d^8 - 3*
b^4*c^2*d^7*e + 3*b^5*c*d^6*e^2 - b^6*d^5*e^3)*x) - (3*A*b*e - (B*b - 2*A*c)*d)*log(x)/(b^3*d^4)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.29803, size = 1006, normalized size = 3.55 \begin{align*} -\frac{{\left (B b c^{5} d - 2 \, A c^{6} d - 4 \, B b^{2} c^{4} e + 5 \, A b c^{5} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{5} d^{4} - 4 \, b^{4} c^{4} d^{3} e + 6 \, b^{5} c^{3} d^{2} e^{2} - 4 \, b^{6} c^{2} d e^{3} + b^{7} c e^{4}} - \frac{{\left (6 \, B c^{2} d^{3} e^{3} - 4 \, B b c d^{2} e^{4} - 10 \, A c^{2} d^{2} e^{4} + B b^{2} d e^{5} + 10 \, A b c d e^{5} - 3 \, A b^{2} e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e - 4 \, b c^{3} d^{7} e^{2} + 6 \, b^{2} c^{2} d^{6} e^{3} - 4 \, b^{3} c d^{5} e^{4} + b^{4} d^{4} e^{5}} + \frac{{\left (B b d - 2 \, A c d - 3 \, A b e\right )} \log \left ({\left | x \right |}\right )}{b^{3} d^{4}} - \frac{2 \, A b c^{4} d^{7} - 8 \, A b^{2} c^{3} d^{6} e + 12 \, A b^{3} c^{2} d^{5} e^{2} - 8 \, A b^{4} c d^{4} e^{3} + 2 \, A b^{5} d^{3} e^{4} - 2 \,{\left (B b c^{4} d^{5} e^{2} - 2 \, A c^{5} d^{5} e^{2} + 2 \, B b^{2} c^{3} d^{4} e^{3} + 5 \, A b c^{4} d^{4} e^{3} - 4 \, B b^{3} c^{2} d^{3} e^{4} - 10 \, A b^{2} c^{3} d^{3} e^{4} + B b^{4} c d^{2} e^{5} + 10 \, A b^{3} c^{2} d^{2} e^{5} - 3 \, A b^{4} c d e^{6}\right )} x^{3} -{\left (4 \, B b c^{4} d^{6} e - 8 \, A c^{5} d^{6} e + 3 \, B b^{2} c^{3} d^{5} e^{2} + 18 \, A b c^{4} d^{5} e^{2} - 4 \, B b^{3} c^{2} d^{4} e^{3} - 25 \, A b^{2} c^{3} d^{4} e^{3} - 5 \, B b^{4} c d^{3} e^{4} + 10 \, A b^{3} c^{2} d^{3} e^{4} + 2 \, B b^{5} d^{2} e^{5} + 11 \, A b^{4} c d^{2} e^{5} - 6 \, A b^{5} d e^{6}\right )} x^{2} -{\left (2 \, B b c^{4} d^{7} - 4 \, A c^{5} d^{7} - 2 \, B b^{2} c^{3} d^{6} e + 6 \, A b c^{4} d^{6} e + 7 \, B b^{3} c^{2} d^{5} e^{2} + 4 \, A b^{2} c^{3} d^{5} e^{2} - 10 \, B b^{4} c d^{4} e^{3} - 25 \, A b^{3} c^{2} d^{4} e^{3} + 3 \, B b^{5} d^{3} e^{4} + 28 \, A b^{4} c d^{3} e^{4} - 9 \, A b^{5} d^{2} e^{5}\right )} x}{2 \,{\left (c d - b e\right )}^{4}{\left (c x + b\right )}{\left (x e + d\right )}^{2} b^{2} d^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B*b*c^5*d - 2*A*c^6*d - 4*B*b^2*c^4*e + 5*A*b*c^5*e)*log(abs(c*x + b))/(b^3*c^5*d^4 - 4*b^4*c^4*d^3*e + 6*b^
5*c^3*d^2*e^2 - 4*b^6*c^2*d*e^3 + b^7*c*e^4) - (6*B*c^2*d^3*e^3 - 4*B*b*c*d^2*e^4 - 10*A*c^2*d^2*e^4 + B*b^2*d
*e^5 + 10*A*b*c*d*e^5 - 3*A*b^2*e^6)*log(abs(x*e + d))/(c^4*d^8*e - 4*b*c^3*d^7*e^2 + 6*b^2*c^2*d^6*e^3 - 4*b^
3*c*d^5*e^4 + b^4*d^4*e^5) + (B*b*d - 2*A*c*d - 3*A*b*e)*log(abs(x))/(b^3*d^4) - 1/2*(2*A*b*c^4*d^7 - 8*A*b^2*
c^3*d^6*e + 12*A*b^3*c^2*d^5*e^2 - 8*A*b^4*c*d^4*e^3 + 2*A*b^5*d^3*e^4 - 2*(B*b*c^4*d^5*e^2 - 2*A*c^5*d^5*e^2
+ 2*B*b^2*c^3*d^4*e^3 + 5*A*b*c^4*d^4*e^3 - 4*B*b^3*c^2*d^3*e^4 - 10*A*b^2*c^3*d^3*e^4 + B*b^4*c*d^2*e^5 + 10*
A*b^3*c^2*d^2*e^5 - 3*A*b^4*c*d*e^6)*x^3 - (4*B*b*c^4*d^6*e - 8*A*c^5*d^6*e + 3*B*b^2*c^3*d^5*e^2 + 18*A*b*c^4
*d^5*e^2 - 4*B*b^3*c^2*d^4*e^3 - 25*A*b^2*c^3*d^4*e^3 - 5*B*b^4*c*d^3*e^4 + 10*A*b^3*c^2*d^3*e^4 + 2*B*b^5*d^2
*e^5 + 11*A*b^4*c*d^2*e^5 - 6*A*b^5*d*e^6)*x^2 - (2*B*b*c^4*d^7 - 4*A*c^5*d^7 - 2*B*b^2*c^3*d^6*e + 6*A*b*c^4*
d^6*e + 7*B*b^3*c^2*d^5*e^2 + 4*A*b^2*c^3*d^5*e^2 - 10*B*b^4*c*d^4*e^3 - 25*A*b^3*c^2*d^4*e^3 + 3*B*b^5*d^3*e^
4 + 28*A*b^4*c*d^3*e^4 - 9*A*b^5*d^2*e^5)*x)/((c*d - b*e)^4*(c*x + b)*(x*e + d)^2*b^2*d^4*x)