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

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

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Rubi [A]  time = 0.30, antiderivative size = 245, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.34, size = 241, normalized size = 0.98 \begin {gather*} \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 (b^3 e^3-4 b^2 c d e^2+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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{(d+e x)^4 \left (b x+c x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 127.91, size = 1133, normalized size = 4.62 \begin {gather*} \frac {11 \, B b c^{3} d^{7} + 11 \, A b^{4} d^{3} e^{4} - 2 \, {\left (9 \, B b^{2} c^{2} + 13 \, A b c^{3}\right )} d^{6} e + 3 \, {\left (3 \, B b^{3} c + 19 \, A b^{2} c^{2}\right )} d^{5} e^{2} - 2 \, {\left (B b^{4} + 21 \, A b^{3} c\right )} d^{4} e^{3} + 6 \, {\left (B b c^{3} d^{5} e^{2} + 6 \, A b^{2} c^{2} d^{3} e^{4} - 4 \, A b^{3} c d^{2} e^{5} + A b^{4} d e^{6} - {\left (B b^{2} c^{2} + 3 \, A b c^{3}\right )} d^{4} e^{3}\right )} x^{2} + 3 \, {\left (5 \, B b c^{3} d^{6} e - 20 \, A b^{3} c d^{3} e^{4} + 5 \, A b^{4} d^{2} e^{5} - 2 \, {\left (3 \, B b^{2} c^{2} + 7 \, A b c^{3}\right )} d^{5} e^{2} + {\left (B b^{3} c + 29 \, A b^{2} c^{2}\right )} d^{4} e^{3}\right )} x + 6 \, {\left ({\left (B b c^{3} - A c^{4}\right )} d^{4} e^{3} x^{3} + 3 \, {\left (B b c^{3} - A c^{4}\right )} d^{5} e^{2} x^{2} + 3 \, {\left (B b c^{3} - A c^{4}\right )} d^{6} e x + {\left (B b c^{3} - A c^{4}\right )} d^{7}\right )} \log \left (c x + b\right ) - 6 \, {\left (B b c^{3} d^{7} - 4 \, A b c^{3} d^{6} e + 6 \, A b^{2} c^{2} d^{5} e^{2} - 4 \, A b^{3} c d^{4} e^{3} + A b^{4} d^{3} e^{4} + {\left (B b c^{3} d^{4} e^{3} - 4 \, A b c^{3} d^{3} e^{4} + 6 \, A b^{2} c^{2} d^{2} e^{5} - 4 \, A b^{3} c d e^{6} + A b^{4} e^{7}\right )} x^{3} + 3 \, {\left (B b c^{3} d^{5} e^{2} - 4 \, A b c^{3} d^{4} e^{3} + 6 \, A b^{2} c^{2} d^{3} e^{4} - 4 \, A b^{3} c d^{2} e^{5} + A b^{4} d e^{6}\right )} x^{2} + 3 \, {\left (B b c^{3} d^{6} e - 4 \, A b c^{3} d^{5} e^{2} + 6 \, A b^{2} c^{2} d^{4} e^{3} - 4 \, A b^{3} c d^{3} e^{4} + A b^{4} d^{2} e^{5}\right )} x\right )} \log \left (e x + d\right ) + 6 \, {\left (A c^{4} d^{7} - 4 \, A b c^{3} d^{6} e + 6 \, A b^{2} c^{2} d^{5} e^{2} - 4 \, A b^{3} c d^{4} e^{3} + A b^{4} d^{3} e^{4} + {\left (A c^{4} d^{4} e^{3} - 4 \, A b c^{3} d^{3} e^{4} + 6 \, A b^{2} c^{2} d^{2} e^{5} - 4 \, A b^{3} c d e^{6} + A b^{4} e^{7}\right )} x^{3} + 3 \, {\left (A c^{4} d^{5} e^{2} - 4 \, A b c^{3} d^{4} e^{3} + 6 \, A b^{2} c^{2} d^{3} e^{4} - 4 \, A b^{3} c d^{2} e^{5} + A b^{4} d e^{6}\right )} x^{2} + 3 \, {\left (A c^{4} d^{6} e - 4 \, A b c^{3} d^{5} e^{2} + 6 \, A b^{2} c^{2} d^{4} e^{3} - 4 \, A b^{3} c d^{3} e^{4} + A b^{4} d^{2} e^{5}\right )} x\right )} \log \relax (x)}{6 \, {\left (b c^{4} d^{11} - 4 \, b^{2} c^{3} d^{10} e + 6 \, b^{3} c^{2} d^{9} e^{2} - 4 \, b^{4} c d^{8} e^{3} + b^{5} d^{7} e^{4} + {\left (b c^{4} d^{8} e^{3} - 4 \, b^{2} c^{3} d^{7} e^{4} + 6 \, b^{3} c^{2} d^{6} e^{5} - 4 \, b^{4} c d^{5} e^{6} + b^{5} d^{4} e^{7}\right )} x^{3} + 3 \, {\left (b c^{4} d^{9} e^{2} - 4 \, b^{2} c^{3} d^{8} e^{3} + 6 \, b^{3} c^{2} d^{7} e^{4} - 4 \, b^{4} c d^{6} e^{5} + b^{5} d^{5} e^{6}\right )} x^{2} + 3 \, {\left (b c^{4} d^{10} e - 4 \, b^{2} c^{3} d^{9} e^{2} + 6 \, b^{3} c^{2} d^{8} e^{3} - 4 \, b^{4} c d^{7} e^{4} + b^{5} d^{6} e^{5}\right )} x\right )}} \end {gather*}

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]

1/6*(11*B*b*c^3*d^7 + 11*A*b^4*d^3*e^4 - 2*(9*B*b^2*c^2 + 13*A*b*c^3)*d^6*e + 3*(3*B*b^3*c + 19*A*b^2*c^2)*d^5
*e^2 - 2*(B*b^4 + 21*A*b^3*c)*d^4*e^3 + 6*(B*b*c^3*d^5*e^2 + 6*A*b^2*c^2*d^3*e^4 - 4*A*b^3*c*d^2*e^5 + A*b^4*d
*e^6 - (B*b^2*c^2 + 3*A*b*c^3)*d^4*e^3)*x^2 + 3*(5*B*b*c^3*d^6*e - 20*A*b^3*c*d^3*e^4 + 5*A*b^4*d^2*e^5 - 2*(3
*B*b^2*c^2 + 7*A*b*c^3)*d^5*e^2 + (B*b^3*c + 29*A*b^2*c^2)*d^4*e^3)*x + 6*((B*b*c^3 - A*c^4)*d^4*e^3*x^3 + 3*(
B*b*c^3 - A*c^4)*d^5*e^2*x^2 + 3*(B*b*c^3 - A*c^4)*d^6*e*x + (B*b*c^3 - A*c^4)*d^7)*log(c*x + b) - 6*(B*b*c^3*
d^7 - 4*A*b*c^3*d^6*e + 6*A*b^2*c^2*d^5*e^2 - 4*A*b^3*c*d^4*e^3 + A*b^4*d^3*e^4 + (B*b*c^3*d^4*e^3 - 4*A*b*c^3
*d^3*e^4 + 6*A*b^2*c^2*d^2*e^5 - 4*A*b^3*c*d*e^6 + A*b^4*e^7)*x^3 + 3*(B*b*c^3*d^5*e^2 - 4*A*b*c^3*d^4*e^3 + 6
*A*b^2*c^2*d^3*e^4 - 4*A*b^3*c*d^2*e^5 + A*b^4*d*e^6)*x^2 + 3*(B*b*c^3*d^6*e - 4*A*b*c^3*d^5*e^2 + 6*A*b^2*c^2
*d^4*e^3 - 4*A*b^3*c*d^3*e^4 + A*b^4*d^2*e^5)*x)*log(e*x + d) + 6*(A*c^4*d^7 - 4*A*b*c^3*d^6*e + 6*A*b^2*c^2*d
^5*e^2 - 4*A*b^3*c*d^4*e^3 + A*b^4*d^3*e^4 + (A*c^4*d^4*e^3 - 4*A*b*c^3*d^3*e^4 + 6*A*b^2*c^2*d^2*e^5 - 4*A*b^
3*c*d*e^6 + A*b^4*e^7)*x^3 + 3*(A*c^4*d^5*e^2 - 4*A*b*c^3*d^4*e^3 + 6*A*b^2*c^2*d^3*e^4 - 4*A*b^3*c*d^2*e^5 +
A*b^4*d*e^6)*x^2 + 3*(A*c^4*d^6*e - 4*A*b*c^3*d^5*e^2 + 6*A*b^2*c^2*d^4*e^3 - 4*A*b^3*c*d^3*e^4 + A*b^4*d^2*e^
5)*x)*log(x))/(b*c^4*d^11 - 4*b^2*c^3*d^10*e + 6*b^3*c^2*d^9*e^2 - 4*b^4*c*d^8*e^3 + b^5*d^7*e^4 + (b*c^4*d^8*
e^3 - 4*b^2*c^3*d^7*e^4 + 6*b^3*c^2*d^6*e^5 - 4*b^4*c*d^5*e^6 + b^5*d^4*e^7)*x^3 + 3*(b*c^4*d^9*e^2 - 4*b^2*c^
3*d^8*e^3 + 6*b^3*c^2*d^7*e^4 - 4*b^4*c*d^6*e^5 + b^5*d^5*e^6)*x^2 + 3*(b*c^4*d^10*e - 4*b^2*c^3*d^9*e^2 + 6*b
^3*c^2*d^8*e^3 - 4*b^4*c*d^7*e^4 + b^5*d^6*e^5)*x)

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giac [A]  time = 0.17, size = 475, normalized size = 1.94 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.08, size = 415, normalized size = 1.69 \begin {gather*} -\frac {A \,b^{3} e^{4} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{4}}+\frac {4 A \,b^{2} c \,e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{3}}-\frac {6 A b \,c^{2} e^{2} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{2}}-\frac {A \,c^{4} \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b}+\frac {4 A \,c^{3} e \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d}+\frac {B \,c^{3} \ln \left (c x +b \right )}{\left (b e -c d \right )^{4}}-\frac {B \,c^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4}}+\frac {A \,b^{2} e^{3}}{\left (b e -c d \right )^{3} \left (e x +d \right ) d^{3}}-\frac {3 A b c \,e^{2}}{\left (b e -c d \right )^{3} \left (e x +d \right ) d^{2}}+\frac {3 A \,c^{2} e}{\left (b e -c d \right )^{3} \left (e x +d \right ) d}-\frac {B \,c^{2}}{\left (b e -c d \right )^{3} \left (e x +d \right )}+\frac {A b \,e^{2}}{2 \left (b e -c d \right )^{2} \left (e x +d \right )^{2} d^{2}}-\frac {A c e}{\left (b e -c d \right )^{2} \left (e x +d \right )^{2} d}+\frac {B c}{2 \left (b e -c d \right )^{2} \left (e x +d \right )^{2}}+\frac {A e}{3 \left (b e -c d \right ) \left (e x +d \right )^{3} d}-\frac {B}{3 \left (b e -c d \right ) \left (e x +d \right )^{3}}+\frac {A \ln \relax (x )}{b \,d^{4}} \end {gather*}

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]

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

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maxima [B]  time = 0.78, size = 553, normalized size = 2.26 \begin {gather*} \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 \relax (x)}{b d^{4}} \end {gather*}

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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mupad [B]  time = 2.25, size = 471, normalized size = 1.92 \begin {gather*} \frac {\frac {-2\,B\,b^2\,d\,e^2+11\,A\,b^2\,e^3+7\,B\,b\,c\,d^2\,e-31\,A\,b\,c\,d\,e^2-11\,B\,c^2\,d^3+26\,A\,c^2\,d^2\,e}{6\,d\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {x^2\,\left (A\,b^2\,e^5-3\,A\,b\,c\,d\,e^4-B\,c^2\,d^3\,e^2+3\,A\,c^2\,d^2\,e^3\right )}{d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {x\,\left (5\,A\,b^2\,e^4+B\,b\,c\,d^2\,e^2-15\,A\,b\,c\,d\,e^3-5\,B\,c^2\,d^3\,e+14\,A\,c^2\,d^2\,e^2\right )}{2\,d^2\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3}-\frac {\ln \left (b+c\,x\right )\,\left (A\,c^4-B\,b\,c^3\right )}{b^5\,e^4-4\,b^4\,c\,d\,e^3+6\,b^3\,c^2\,d^2\,e^2-4\,b^2\,c^3\,d^3\,e+b\,c^4\,d^4}+\frac {A\,\ln \relax (x)}{b\,d^4}-\frac {\ln \left (d+e\,x\right )\,\left (A\,b^3\,e^4-4\,A\,b^2\,c\,d\,e^3+6\,A\,b\,c^2\,d^2\,e^2+B\,c^3\,d^4-4\,A\,c^3\,d^3\,e\right )}{d^4\,{\left (b\,e-c\,d\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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