3.1145 \(\int \frac{A+B x}{(d+e x)^4 (b x+c x^2)} \, dx\)

Optimal. Leaf size=245 \[ \frac{B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )}{d^3 (d+e x) (c d-b e)^3}-\frac{\log (d+e x) \left (B c^3 d^4-A e \left (4 b^2 c d e^2-b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d^4 (c d-b e)^4}+\frac{c^3 (b B-A c) \log (b+c x)}{b (c d-b e)^4}+\frac{B c d^2-A e (2 c d-b e)}{2 d^2 (d+e x)^2 (c d-b e)^2}+\frac{B d-A e}{3 d (d+e x)^3 (c d-b e)}+\frac{A \log (x)}{b d^4} \]

[Out]

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

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Rubi [A]  time = 0.301861, antiderivative size = 245, 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} \[ \frac{B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )}{d^3 (d+e x) (c d-b e)^3}-\frac{\log (d+e x) \left (B c^3 d^4-A e \left (4 b^2 c d e^2-b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d^4 (c d-b e)^4}+\frac{c^3 (b B-A c) \log (b+c x)}{b (c d-b e)^4}+\frac{B c d^2-A e (2 c d-b e)}{2 d^2 (d+e x)^2 (c d-b e)^2}+\frac{B d-A e}{3 d (d+e x)^3 (c d-b e)}+\frac{A \log (x)}{b d^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^4*(b*x + c*x^2)),x]

[Out]

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

Mathematica [A]  time = 0.38206, size = 241, normalized size = 0.98 \[ \frac{B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )}{d^3 (d+e x) (c d-b e)^3}-\frac{\log (d+e x) \left (A e \left (-4 b^2 c d e^2+b^3 e^3+6 b c^2 d^2 e-4 c^3 d^3\right )+B c^3 d^4\right )}{d^4 (c d-b e)^4}+\frac{c^3 (b B-A c) \log (b+c x)}{b (c d-b e)^4}+\frac{A e (b e-2 c d)+B c d^2}{2 d^2 (d+e x)^2 (c d-b e)^2}+\frac{B d-A e}{3 d (d+e x)^3 (c d-b e)}+\frac{A \log (x)}{b d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^4*(b*x + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.016, size = 415, normalized size = 1.7 \begin{align*}{\frac{A\ln \left ( x \right ) }{{d}^{4}b}}+{\frac{Ab{e}^{2}}{2\,{d}^{2} \left ( be-cd \right ) ^{2} \left ( ex+d \right ) ^{2}}}-{\frac{Ace}{d \left ( be-cd \right ) ^{2} \left ( ex+d \right ) ^{2}}}+{\frac{Bc}{2\, \left ( be-cd \right ) ^{2} \left ( ex+d \right ) ^{2}}}+{\frac{A{b}^{2}{e}^{3}}{{d}^{3} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{Abc{e}^{2}}{{d}^{2} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}+3\,{\frac{A{c}^{2}e}{d \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}-{\frac{B{c}^{2}}{ \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}-{\frac{\ln \left ( ex+d \right ) A{b}^{3}{e}^{4}}{{d}^{4} \left ( be-cd \right ) ^{4}}}+4\,{\frac{\ln \left ( ex+d \right ) A{b}^{2}c{e}^{3}}{{d}^{3} \left ( be-cd \right ) ^{4}}}-6\,{\frac{\ln \left ( ex+d \right ) Ab{c}^{2}{e}^{2}}{{d}^{2} \left ( be-cd \right ) ^{4}}}+4\,{\frac{\ln \left ( ex+d \right ) A{c}^{3}e}{d \left ( be-cd \right ) ^{4}}}-{\frac{\ln \left ( ex+d \right ) B{c}^{3}}{ \left ( be-cd \right ) ^{4}}}+{\frac{Ae}{3\,d \left ( be-cd \right ) \left ( ex+d \right ) ^{3}}}-{\frac{B}{ \left ( 3\,be-3\,cd \right ) \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{4}\ln \left ( cx+b \right ) A}{b \left ( be-cd \right ) ^{4}}}+{\frac{{c}^{3}\ln \left ( cx+b \right ) B}{ \left ( be-cd \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^4/(c*x^2+b*x),x)

[Out]

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

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Maxima [B]  time = 1.23894, size = 747, normalized size = 3.05 \begin{align*} \frac{{\left (B b c^{3} - A c^{4}\right )} \log \left (c x + b\right )}{b c^{4} d^{4} - 4 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} + b^{5} e^{4}} - \frac{{\left (B c^{3} d^{4} - 4 \, A c^{3} d^{3} e + 6 \, A b c^{2} d^{2} e^{2} - 4 \, A b^{2} c d e^{3} + A b^{3} e^{4}\right )} \log \left (e x + d\right )}{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}} + \frac{11 \, B c^{2} d^{5} - 11 \, A b^{2} d^{2} e^{3} -{\left (7 \, B b c + 26 \, A c^{2}\right )} d^{4} e +{\left (2 \, B b^{2} + 31 \, A b c\right )} d^{3} e^{2} + 6 \,{\left (B c^{2} d^{3} e^{2} - 3 \, A c^{2} d^{2} e^{3} + 3 \, A b c d e^{4} - A b^{2} e^{5}\right )} x^{2} + 3 \,{\left (5 \, B c^{2} d^{4} e + 15 \, A b c d^{2} e^{3} - 5 \, A b^{2} d e^{4} -{\left (B b c + 14 \, A c^{2}\right )} d^{3} e^{2}\right )} x}{6 \,{\left (c^{3} d^{9} - 3 \, b c^{2} d^{8} e + 3 \, b^{2} c d^{7} e^{2} - b^{3} d^{6} e^{3} +{\left (c^{3} d^{6} e^{3} - 3 \, b c^{2} d^{5} e^{4} + 3 \, b^{2} c d^{4} e^{5} - b^{3} d^{3} e^{6}\right )} x^{3} + 3 \,{\left (c^{3} d^{7} e^{2} - 3 \, b c^{2} d^{6} e^{3} + 3 \, b^{2} c d^{5} e^{4} - b^{3} d^{4} e^{5}\right )} x^{2} + 3 \,{\left (c^{3} d^{8} e - 3 \, b c^{2} d^{7} e^{2} + 3 \, b^{2} c d^{6} e^{3} - b^{3} d^{5} e^{4}\right )} x\right )}} + \frac{A \log \left (x\right )}{b d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x),x, algorithm="maxima")

[Out]

(B*b*c^3 - A*c^4)*log(c*x + b)/(b*c^4*d^4 - 4*b^2*c^3*d^3*e + 6*b^3*c^2*d^2*e^2 - 4*b^4*c*d*e^3 + b^5*e^4) - (
B*c^3*d^4 - 4*A*c^3*d^3*e + 6*A*b*c^2*d^2*e^2 - 4*A*b^2*c*d*e^3 + A*b^3*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) + 1/6*(11*B*c^2*d^5 - 11*A*b^2*d^2*e^3 - (7*B*b*c +
26*A*c^2)*d^4*e + (2*B*b^2 + 31*A*b*c)*d^3*e^2 + 6*(B*c^2*d^3*e^2 - 3*A*c^2*d^2*e^3 + 3*A*b*c*d*e^4 - A*b^2*e^
5)*x^2 + 3*(5*B*c^2*d^4*e + 15*A*b*c*d^2*e^3 - 5*A*b^2*d*e^4 - (B*b*c + 14*A*c^2)*d^3*e^2)*x)/(c^3*d^9 - 3*b*c
^2*d^8*e + 3*b^2*c*d^7*e^2 - b^3*d^6*e^3 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 + 3*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^3
+ 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 + 3*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 + 3*
b^2*c*d^6*e^3 - b^3*d^5*e^4)*x) + A*log(x)/(b*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)^4/(c*x^2+b*x),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)**4/(c*x**2+b*x),x)

[Out]

Timed out

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Giac [A]  time = 1.2933, size = 641, normalized size = 2.62 \begin{align*} \frac{{\left (B b c^{4} - A c^{5}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{5} d^{4} - 4 \, b^{2} c^{4} d^{3} e + 6 \, b^{3} c^{3} d^{2} e^{2} - 4 \, b^{4} c^{2} d e^{3} + b^{5} c e^{4}} - \frac{{\left (B c^{3} d^{4} e - 4 \, A c^{3} d^{3} e^{2} + 6 \, A b c^{2} d^{2} e^{3} - 4 \, A b^{2} c d e^{4} + A b^{3} e^{5}\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{A \log \left ({\left | x \right |}\right )}{b d^{4}} + \frac{11 \, B c^{3} d^{7} - 18 \, B b c^{2} d^{6} e - 26 \, A c^{3} d^{6} e + 9 \, B b^{2} c d^{5} e^{2} + 57 \, A b c^{2} d^{5} e^{2} - 2 \, B b^{3} d^{4} e^{3} - 42 \, A b^{2} c d^{4} e^{3} + 11 \, A b^{3} d^{3} e^{4} + 6 \,{\left (B c^{3} d^{5} e^{2} - B b c^{2} d^{4} e^{3} - 3 \, A c^{3} d^{4} e^{3} + 6 \, A b c^{2} d^{3} e^{4} - 4 \, A b^{2} c d^{2} e^{5} + A b^{3} d e^{6}\right )} x^{2} + 3 \,{\left (5 \, B c^{3} d^{6} e - 6 \, B b c^{2} d^{5} e^{2} - 14 \, A c^{3} d^{5} e^{2} + B b^{2} c d^{4} e^{3} + 29 \, A b c^{2} d^{4} e^{3} - 20 \, A b^{2} c d^{3} e^{4} + 5 \, A b^{3} d^{2} e^{5}\right )} x}{6 \,{\left (c d - b e\right )}^{4}{\left (x e + d\right )}^{3} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x),x, algorithm="giac")

[Out]

(B*b*c^4 - A*c^5)*log(abs(c*x + b))/(b*c^5*d^4 - 4*b^2*c^4*d^3*e + 6*b^3*c^3*d^2*e^2 - 4*b^4*c^2*d*e^3 + b^5*c
*e^4) - (B*c^3*d^4*e - 4*A*c^3*d^3*e^2 + 6*A*b*c^2*d^2*e^3 - 4*A*b^2*c*d*e^4 + A*b^3*e^5)*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) + A*log(abs(x))/(b*d^4) + 1/6*
(11*B*c^3*d^7 - 18*B*b*c^2*d^6*e - 26*A*c^3*d^6*e + 9*B*b^2*c*d^5*e^2 + 57*A*b*c^2*d^5*e^2 - 2*B*b^3*d^4*e^3 -
42*A*b^2*c*d^4*e^3 + 11*A*b^3*d^3*e^4 + 6*(B*c^3*d^5*e^2 - B*b*c^2*d^4*e^3 - 3*A*c^3*d^4*e^3 + 6*A*b*c^2*d^3*
e^4 - 4*A*b^2*c*d^2*e^5 + A*b^3*d*e^6)*x^2 + 3*(5*B*c^3*d^6*e - 6*B*b*c^2*d^5*e^2 - 14*A*c^3*d^5*e^2 + B*b^2*c
*d^4*e^3 + 29*A*b*c^2*d^4*e^3 - 20*A*b^2*c*d^3*e^4 + 5*A*b^3*d^2*e^5)*x)/((c*d - b*e)^4*(x*e + d)^3*d^4)