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

Optimal. Leaf size=279 \[ -\frac {-A b e-3 A c d+b B d}{b^4 d^2 x}-\frac {c^2 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)}-\frac {A}{2 b^3 d x^2}+\frac {\log (x) \left (b^2 (-e) (B d-A e)-3 b c d (B d-A e)+6 A c^2 d^2\right )}{b^5 d^3}+\frac {c^2 \left (-2 b c (2 A e+B d)+3 A c^2 d+3 b^2 B e\right )}{b^4 (b+c x) (c d-b e)^2}-\frac {c^2 \log (b+c x) \left (2 b^2 c e (5 A e+4 B d)-3 b c^2 d (5 A e+B d)+6 A c^3 d^2-6 b^3 B e^2\right )}{b^5 (c d-b e)^3}-\frac {e^4 (B d-A e) \log (d+e x)}{d^3 (c d-b e)^3} \]

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Rubi [A]  time = 0.50, antiderivative size = 279, 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 {c^2 \log (b+c x) \left (2 b^2 c e (5 A e+4 B d)-3 b c^2 d (5 A e+B d)+6 A c^3 d^2-6 b^3 B e^2\right )}{b^5 (c d-b e)^3}+\frac {\log (x) \left (b^2 (-e) (B d-A e)-3 b c d (B d-A e)+6 A c^2 d^2\right )}{b^5 d^3}+\frac {c^2 \left (-2 b c (2 A e+B d)+3 A c^2 d+3 b^2 B e\right )}{b^4 (b+c x) (c d-b e)^2}-\frac {c^2 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)}-\frac {-A b e-3 A c d+b B d}{b^4 d^2 x}-\frac {A}{2 b^3 d x^2}-\frac {e^4 (B d-A e) \log (d+e x)}{d^3 (c d-b e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 1.05, size = 276, normalized size = 0.99 \begin {gather*} \frac {A b e+3 A c d-b B d}{b^4 d^2 x}+\frac {c^2 (b B-A c)}{2 b^3 (b+c x)^2 (b e-c d)}-\frac {A}{2 b^3 d x^2}-\frac {\log (x) \left (b^2 e (B d-A e)+3 b c d (B d-A e)-6 A c^2 d^2\right )}{b^5 d^3}+\frac {c^2 \left (-2 b c (2 A e+B d)+3 A c^2 d+3 b^2 B e\right )}{b^4 (b+c x) (c d-b e)^2}+\frac {c^2 \log (b+c x) \left (2 b^2 c e (5 A e+4 B d)-3 b c^2 d (5 A e+B d)+6 A c^3 d^2-6 b^3 B e^2\right )}{b^5 (b e-c d)^3}+\frac {e^4 (A e-B d) \log (d+e x)}{d^3 (c d-b e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*A/(b^3*d*x^2) + (-(b*B*d) + 3*A*c*d + A*b*e)/(b^4*d^2*x) + (c^2*(b*B - A*c))/(2*b^3*(-(c*d) + b*e)*(b + c
*x)^2) + (c^2*(3*A*c^2*d + 3*b^2*B*e - 2*b*c*(B*d + 2*A*e)))/(b^4*(c*d - b*e)^2*(b + c*x)) - ((-6*A*c^2*d^2 +
3*b*c*d*(B*d - A*e) + b^2*e*(B*d - A*e))*Log[x])/(b^5*d^3) + (c^2*(6*A*c^3*d^2 - 6*b^3*B*e^2 - 3*b*c^2*d*(B*d
+ 5*A*e) + 2*b^2*c*e*(4*B*d + 5*A*e))*Log[b + c*x])/(b^5*(-(c*d) + b*e)^3) + (e^4*(-(B*d) + A*e)*Log[d + e*x])
/(d^3*(c*d - b*e)^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 )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [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)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.24, size = 649, normalized size = 2.33 \begin {gather*} \frac {{\left (3 \, B b c^{5} d^{2} - 6 \, A c^{6} d^{2} - 8 \, B b^{2} c^{4} d e + 15 \, A b c^{5} d e + 6 \, B b^{3} c^{3} e^{2} - 10 \, A b^{2} c^{4} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} - \frac {{\left (B d e^{5} - A e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} - \frac {{\left (3 \, B b c d^{2} - 6 \, A c^{2} d^{2} + B b^{2} d e - 3 \, A b c d e - A b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5} d^{3}} - \frac {A b^{3} c^{3} d^{5} - 3 \, A b^{4} c^{2} d^{4} e + 3 \, A b^{5} c d^{3} e^{2} - A b^{6} d^{2} e^{3} + 2 \, {\left (3 \, B b c^{5} d^{5} - 6 \, A c^{6} d^{5} - 8 \, B b^{2} c^{4} d^{4} e + 15 \, A b c^{5} d^{4} e + 6 \, B b^{3} c^{3} d^{3} e^{2} - 10 \, A b^{2} c^{4} d^{3} e^{2} - B b^{4} c^{2} d^{2} e^{3} + A b^{4} c^{2} d e^{4}\right )} x^{3} + {\left (9 \, B b^{2} c^{4} d^{5} - 18 \, A b c^{5} d^{5} - 24 \, B b^{3} c^{3} d^{4} e + 45 \, A b^{2} c^{4} d^{4} e + 19 \, B b^{4} c^{2} d^{3} e^{2} - 30 \, A b^{3} c^{3} d^{3} e^{2} - 4 \, B b^{5} c d^{2} e^{3} - A b^{4} c^{2} d^{2} e^{3} + 4 \, A b^{5} c d e^{4}\right )} x^{2} + 2 \, {\left (B b^{3} c^{3} d^{5} - 2 \, A b^{2} c^{4} d^{5} - 3 \, B b^{4} c^{2} d^{4} e + 5 \, A b^{3} c^{3} d^{4} e + 3 \, B b^{5} c d^{3} e^{2} - 3 \, A b^{4} c^{2} d^{3} e^{2} - B b^{6} d^{2} e^{3} - A b^{5} c d^{2} e^{3} + A b^{6} d e^{4}\right )} x}{2 \, {\left (c d - b e\right )}^{3} {\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \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)^3,x, algorithm="giac")

[Out]

(3*B*b*c^5*d^2 - 6*A*c^6*d^2 - 8*B*b^2*c^4*d*e + 15*A*b*c^5*d*e + 6*B*b^3*c^3*e^2 - 10*A*b^2*c^4*e^2)*log(abs(
c*x + b))/(b^5*c^4*d^3 - 3*b^6*c^3*d^2*e + 3*b^7*c^2*d*e^2 - b^8*c*e^3) - (B*d*e^5 - A*e^6)*log(abs(x*e + d))/
(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 - b^3*d^3*e^4) - (3*B*b*c*d^2 - 6*A*c^2*d^2 + B*b^2*d*e - 3*A*b
*c*d*e - A*b^2*e^2)*log(abs(x))/(b^5*d^3) - 1/2*(A*b^3*c^3*d^5 - 3*A*b^4*c^2*d^4*e + 3*A*b^5*c*d^3*e^2 - A*b^6
*d^2*e^3 + 2*(3*B*b*c^5*d^5 - 6*A*c^6*d^5 - 8*B*b^2*c^4*d^4*e + 15*A*b*c^5*d^4*e + 6*B*b^3*c^3*d^3*e^2 - 10*A*
b^2*c^4*d^3*e^2 - B*b^4*c^2*d^2*e^3 + A*b^4*c^2*d*e^4)*x^3 + (9*B*b^2*c^4*d^5 - 18*A*b*c^5*d^5 - 24*B*b^3*c^3*
d^4*e + 45*A*b^2*c^4*d^4*e + 19*B*b^4*c^2*d^3*e^2 - 30*A*b^3*c^3*d^3*e^2 - 4*B*b^5*c*d^2*e^3 - A*b^4*c^2*d^2*e
^3 + 4*A*b^5*c*d*e^4)*x^2 + 2*(B*b^3*c^3*d^5 - 2*A*b^2*c^4*d^5 - 3*B*b^4*c^2*d^4*e + 5*A*b^3*c^3*d^4*e + 3*B*b
^5*c*d^3*e^2 - 3*A*b^4*c^2*d^3*e^2 - B*b^6*d^2*e^3 - A*b^5*c*d^2*e^3 + A*b^6*d*e^4)*x)/((c*d - b*e)^3*(c*x + b
)^2*b^4*d^3*x^2)

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maple [A]  time = 0.07, size = 490, normalized size = 1.76 \begin {gather*} \frac {10 A \,c^{3} e^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{3}}-\frac {15 A \,c^{4} d e \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{4}}+\frac {6 A \,c^{5} d^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{5}}-\frac {A \,e^{5} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{3}}-\frac {6 B \,c^{2} e^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{2}}+\frac {8 B \,c^{3} d e \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{3}}-\frac {3 B \,c^{4} d^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{4}}+\frac {B \,e^{4} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{2}}-\frac {4 A \,c^{3} e}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{3}}+\frac {3 A \,c^{4} d}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{4}}+\frac {3 B \,c^{2} e}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{2}}-\frac {2 B \,c^{3} d}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{3}}-\frac {A \,c^{3}}{2 \left (b e -c d \right ) \left (c x +b \right )^{2} b^{3}}+\frac {B \,c^{2}}{2 \left (b e -c d \right ) \left (c x +b \right )^{2} b^{2}}+\frac {A \,e^{2} \ln \relax (x )}{b^{3} d^{3}}+\frac {3 A c e \ln \relax (x )}{b^{4} d^{2}}+\frac {6 A \,c^{2} \ln \relax (x )}{b^{5} d}-\frac {B e \ln \relax (x )}{b^{3} d^{2}}-\frac {3 B c \ln \relax (x )}{b^{4} d}+\frac {A e}{b^{3} d^{2} x}+\frac {3 A c}{b^{4} d x}-\frac {B}{b^{3} d x}-\frac {A}{2 b^{3} d \,x^{2}} \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)^3,x)

[Out]

-4*c^3/(b*e-c*d)^2/b^3/(c*x+b)*A*e+3*c^4/(b*e-c*d)^2/b^4/(c*x+b)*A*d+3*c^2/(b*e-c*d)^2/b^2/(c*x+b)*B*e-2*c^3/(
b*e-c*d)^2/b^3/(c*x+b)*B*d+10*c^3/(b*e-c*d)^3/b^3*ln(c*x+b)*A*e^2-15*c^4/(b*e-c*d)^3/b^4*ln(c*x+b)*A*d*e+6*c^5
/(b*e-c*d)^3/b^5*ln(c*x+b)*A*d^2-6*c^2/(b*e-c*d)^3/b^2*ln(c*x+b)*B*e^2+8*c^3/(b*e-c*d)^3/b^3*ln(c*x+b)*B*d*e-3
*c^4/(b*e-c*d)^3/b^4*ln(c*x+b)*B*d^2-1/2*c^3/(b*e-c*d)/b^3/(c*x+b)^2*A+1/2*c^2/(b*e-c*d)/b^2/(c*x+b)^2*B-e^5/(
b*e-c*d)^3/d^3*ln(e*x+d)*A+e^4/(b*e-c*d)^3/d^2*ln(e*x+d)*B-1/2*A/b^3/d/x^2+1/b^3/d^2/x*A*e+3/b^4/d/x*A*c-1/b^3
/d/x*B+1/b^3/d^3*ln(x)*A*e^2+3/b^4/d^2*ln(x)*A*e*c+6/b^5/d*ln(x)*A*c^2-1/b^3/d^2*ln(x)*B*e-3/b^4/d*ln(x)*B*c

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maxima [B]  time = 0.71, size = 612, normalized size = 2.19 \begin {gather*} \frac {{\left (3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} - {\left (8 \, B b^{2} c^{3} - 15 \, A b c^{4}\right )} d e + 2 \, {\left (3 \, B b^{3} c^{2} - 5 \, A b^{2} c^{3}\right )} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} - \frac {{\left (B d e^{4} - A e^{5}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {A b^{3} c^{2} d^{3} - 2 \, A b^{4} c d^{2} e + A b^{5} d e^{2} - 2 \, {\left (A b^{3} c^{2} e^{3} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + {\left (5 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} d^{2} e - {\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d e^{2}\right )} x^{3} - {\left (4 \, A b^{4} c e^{3} - 9 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} + 3 \, {\left (5 \, B b^{3} c^{2} - 9 \, A b^{2} c^{3}\right )} d^{2} e - {\left (4 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e^{2}\right )} x^{2} + 2 \, {\left (B b^{5} d e^{2} - A b^{5} e^{3} + {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3} - {\left (2 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} d^{2} e\right )} x}{2 \, {\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} + {\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac {{\left (A b^{2} e^{2} - 3 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} - {\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \relax (x)}{b^{5} d^{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)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.61, size = 512, normalized size = 1.84 \begin {gather*} \frac {\frac {x\,\left (A\,b\,e+2\,A\,c\,d-B\,b\,d\right )}{b^2\,d^2}-\frac {A}{2\,b\,d}+\frac {x^3\,\left (-B\,b^3\,c^2\,d\,e^2+A\,b^3\,c^2\,e^3+5\,B\,b^2\,c^3\,d^2\,e+A\,b^2\,c^3\,d\,e^2-3\,B\,b\,c^4\,d^3-9\,A\,b\,c^4\,d^2\,e+6\,A\,c^5\,d^3\right )}{b^4\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x^2\,\left (-4\,B\,b^3\,c\,d\,e^2+4\,A\,b^3\,c\,e^3+15\,B\,b^2\,c^2\,d^2\,e+3\,A\,b^2\,c^2\,d\,e^2-9\,B\,b\,c^3\,d^3-27\,A\,b\,c^3\,d^2\,e+18\,A\,c^4\,d^3\right )}{2\,b^3\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}+\frac {\ln \left (d+e\,x\right )\,\left (A\,e^5-B\,d\,e^4\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}+\frac {\ln \left (b+c\,x\right )\,\left (d^2\,\left (6\,A\,c^5-3\,B\,b\,c^4\right )-d\,\left (15\,A\,b\,c^4\,e-8\,B\,b^2\,c^3\,e\right )+10\,A\,b^2\,c^3\,e^2-6\,B\,b^3\,c^2\,e^2\right )}{b^8\,e^3-3\,b^7\,c\,d\,e^2+3\,b^6\,c^2\,d^2\,e-b^5\,c^3\,d^3}+\frac {\ln \relax (x)\,\left (d^2\,\left (6\,A\,c^2-3\,B\,b\,c\right )-d\,\left (B\,b^2\,e-3\,A\,b\,c\,e\right )+A\,b^2\,e^2\right )}{b^5\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(A*b*e + 2*A*c*d - B*b*d))/(b^2*d^2) - A/(2*b*d) + (x^3*(6*A*c^5*d^3 - 3*B*b*c^4*d^3 + A*b^3*c^2*e^3 + A*b
^2*c^3*d*e^2 + 5*B*b^2*c^3*d^2*e - B*b^3*c^2*d*e^2 - 9*A*b*c^4*d^2*e))/(b^4*d^2*(b^2*e^2 + c^2*d^2 - 2*b*c*d*e
)) + (x^2*(18*A*c^4*d^3 + 4*A*b^3*c*e^3 - 9*B*b*c^3*d^3 + 3*A*b^2*c^2*d*e^2 + 15*B*b^2*c^2*d^2*e - 27*A*b*c^3*
d^2*e - 4*B*b^3*c*d*e^2))/(2*b^3*d^2*(b^2*e^2 + c^2*d^2 - 2*b*c*d*e)))/(b^2*x^2 + c^2*x^4 + 2*b*c*x^3) + (log(
d + e*x)*(A*e^5 - B*d*e^4))/(c^3*d^6 - b^3*d^3*e^3 + 3*b^2*c*d^4*e^2 - 3*b*c^2*d^5*e) + (log(b + c*x)*(d^2*(6*
A*c^5 - 3*B*b*c^4) - d*(15*A*b*c^4*e - 8*B*b^2*c^3*e) + 10*A*b^2*c^3*e^2 - 6*B*b^3*c^2*e^2))/(b^8*e^3 - b^5*c^
3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2) + (log(x)*(d^2*(6*A*c^2 - 3*B*b*c) - d*(B*b^2*e - 3*A*b*c*e) + A*b^2*
e^2))/(b^5*d^3)

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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)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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