3.1161 \(\int \frac {A+B x}{(d+e x)^2 (b x+c x^2)^3} \, dx\)

Optimal. Leaf size=331 \[ -\frac {-2 A b e-3 A c d+b B d}{b^4 d^3 x}-\frac {c^3 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac {A}{2 b^3 d^2 x^2}+\frac {\log (x) \left (b^2 (-e) (2 B d-3 A e)-3 b c d (B d-2 A e)+6 A c^2 d^2\right )}{b^5 d^4}-\frac {c^3 \left (5 A b c e-3 A c^2 d-4 b^2 B e+2 b B c d\right )}{b^4 (b+c x) (c d-b e)^3}-\frac {c^3 \log (b+c x) \left (5 b^2 c e (3 A e+2 B d)-3 b c^2 d (6 A e+B d)+6 A c^3 d^2-10 b^3 B e^2\right )}{b^5 (c d-b e)^4}-\frac {e^4 \log (d+e x) (B d (5 c d-2 b e)-3 A e (2 c d-b e))}{d^4 (c d-b e)^4}+\frac {e^4 (B d-A e)}{d^3 (d+e x) (c d-b e)^3} \]

[Out]

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

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Rubi [A]  time = 0.67, antiderivative size = 331, 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} \[ -\frac {c^3 \log (b+c x) \left (5 b^2 c e (3 A e+2 B d)-3 b c^2 d (6 A e+B d)+6 A c^3 d^2-10 b^3 B e^2\right )}{b^5 (c d-b e)^4}+\frac {\log (x) \left (b^2 (-e) (2 B d-3 A e)-3 b c d (B d-2 A e)+6 A c^2 d^2\right )}{b^5 d^4}-\frac {c^3 \left (5 A b c e-3 A c^2 d-4 b^2 B e+2 b B c d\right )}{b^4 (b+c x) (c d-b e)^3}-\frac {c^3 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac {-2 A b e-3 A c d+b B d}{b^4 d^3 x}-\frac {A}{2 b^3 d^2 x^2}+\frac {e^4 (B d-A e)}{d^3 (d+e x) (c d-b e)^3}-\frac {e^4 \log (d+e x) (B d (5 c d-2 b e)-3 A e (2 c d-b e))}{d^4 (c d-b e)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.55, size = 328, normalized size = 0.99 \[ \frac {2 A b e+3 A c d-b B d}{b^4 d^3 x}+\frac {c^3 (A c-b B)}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac {A}{2 b^3 d^2 x^2}-\frac {\log (x) \left (b^2 e (2 B d-3 A e)+3 b c d (B d-2 A e)-6 A c^2 d^2\right )}{b^5 d^4}+\frac {c^3 \left (b c (5 A e+2 B d)-3 A c^2 d-4 b^2 B e\right )}{b^4 (b+c x) (b e-c d)^3}+\frac {c^3 \log (b+c x) \left (-5 b^2 c e (3 A e+2 B d)+3 b c^2 d (6 A e+B d)-6 A c^3 d^2+10 b^3 B e^2\right )}{b^5 (c d-b e)^4}-\frac {e^4 \log (d+e x) (3 A e (b e-2 c d)+B d (5 c d-2 b e))}{d^4 (c d-b e)^4}+\frac {e^4 (B d-A e)}{d^3 (d+e x) (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [B]  time = 0.37, size = 1292, normalized size = 3.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(6*B*b*c^5*d^6*e^2 - 12*A*c^6*d^6*e^2 - 20*B*b^2*c^4*d^5*e^3 + 36*A*b*c^5*d^5*e^3 + 20*B*b^3*c^3*d^4*e^4
- 30*A*b^2*c^4*d^4*e^4 - 5*B*b^5*c*d^2*e^6 + 2*B*b^6*d*e^7 + 6*A*b^5*c*d*e^7 - 3*A*b^6*e^8)*e^(-2)*log(abs(2*c
*d*e - 2*c*d^2*e/(x*e + d) - b*e^2 + 2*b*d*e^2/(x*e + d) - abs(b)*e^2)/abs(2*c*d*e - 2*c*d^2*e/(x*e + d) - b*e
^2 + 2*b*d*e^2/(x*e + d) + abs(b)*e^2))/((b^4*c^4*d^8 - 4*b^5*c^3*d^7*e + 6*b^6*c^2*d^6*e^2 - 4*b^7*c*d^5*e^3
+ b^8*d^4*e^4)*abs(b)) + 1/2*(5*B*c*d^2*e^4 - 2*B*b*d*e^5 - 6*A*c*d*e^5 + 3*A*b*e^6)*log(abs(c - 2*c*d/(x*e +
d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2))/(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) + (B*d*e^10/(x*e + d) - A*e^11/(x*e + d))/(c^3*d^6*e^6 - 3*b*c^2*d^5*e^7 + 3*b^2*
c*d^4*e^8 - b^3*d^3*e^9) - 1/2*(6*B*b*c^6*d^5*e - 12*A*c^7*d^5*e - 17*B*b^2*c^5*d^4*e^2 + 30*A*b*c^6*d^4*e^2 +
12*B*b^3*c^4*d^3*e^3 - 16*A*b^2*c^5*d^3*e^3 - 8*B*b^4*c^3*d^2*e^4 - 6*A*b^3*c^4*d^2*e^4 + 2*B*b^5*c^2*d*e^5 +
14*A*b^4*c^3*d*e^5 - 5*A*b^5*c^2*e^6 - 2*(9*B*b*c^6*d^6*e^2 - 18*A*c^7*d^6*e^2 - 30*B*b^2*c^5*d^5*e^3 + 54*A*
b*c^6*d^5*e^3 + 31*B*b^3*c^4*d^4*e^4 - 47*A*b^2*c^5*d^4*e^4 - 24*B*b^4*c^3*d^3*e^5 + 4*A*b^3*c^4*d^3*e^5 + 11*
B*b^5*c^2*d^2*e^6 + 29*A*b^4*c^3*d^2*e^6 - 2*B*b^6*c*d*e^7 - 22*A*b^5*c^2*d*e^7 + 5*A*b^6*c*e^8)*e^(-1)/(x*e +
d) + (18*B*b*c^6*d^7*e^3 - 36*A*c^7*d^7*e^3 - 69*B*b^2*c^5*d^6*e^4 + 126*A*b*c^6*d^6*e^4 + 90*B*b^3*c^4*d^5*e
^5 - 144*A*b^2*c^5*d^5*e^5 - 80*B*b^4*c^3*d^4*e^6 + 45*A*b^3*c^4*d^4*e^6 + 50*B*b^5*c^2*d^3*e^7 + 70*A*b^4*c^3
*d^3*e^7 - 16*B*b^6*c*d^2*e^8 - 87*A*b^5*c^2*d^2*e^8 + 2*B*b^7*d*e^9 + 36*A*b^6*c*d*e^9 - 5*A*b^7*e^10)*e^(-2)
/(x*e + d)^2 - 2*(3*B*b*c^6*d^8*e^4 - 6*A*c^7*d^8*e^4 - 13*B*b^2*c^5*d^7*e^5 + 24*A*b*c^6*d^7*e^5 + 20*B*b^3*c
^4*d^6*e^6 - 33*A*b^2*c^5*d^6*e^6 - 20*B*b^4*c^3*d^5*e^7 + 15*A*b^3*c^4*d^5*e^7 + 15*B*b^5*c^2*d^4*e^8 + 15*A*
b^4*c^3*d^4*e^8 - 6*B*b^6*c*d^3*e^9 - 27*A*b^5*c^2*d^3*e^9 + B*b^7*d^2*e^10 + 15*A*b^6*c*d^2*e^10 - 3*A*b^7*d*
e^11)*e^(-3)/(x*e + d)^3)/((c*d - b*e)^4*b^4*(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/
(x*e + d)^2)^2*d^4)

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maple [A]  time = 0.08, size = 598, normalized size = 1.81 \[ -\frac {3 A b \,e^{6} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{4}}-\frac {15 A \,c^{4} e^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{3}}+\frac {18 A \,c^{5} d e \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{4}}-\frac {6 A \,c^{6} d^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{5}}+\frac {6 A c \,e^{5} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{3}}+\frac {2 B b \,e^{5} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{3}}+\frac {10 B \,c^{3} e^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{2}}-\frac {10 B \,c^{4} d e \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{3}}+\frac {3 B \,c^{5} d^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{4}}-\frac {5 B c \,e^{4} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{2}}+\frac {5 A \,c^{4} e}{\left (b e -c d \right )^{3} \left (c x +b \right ) b^{3}}-\frac {3 A \,c^{5} d}{\left (b e -c d \right )^{3} \left (c x +b \right ) b^{4}}+\frac {A \,e^{5}}{\left (b e -c d \right )^{3} \left (e x +d \right ) d^{3}}-\frac {4 B \,c^{3} e}{\left (b e -c d \right )^{3} \left (c x +b \right ) b^{2}}+\frac {2 B \,c^{4} d}{\left (b e -c d \right )^{3} \left (c x +b \right ) b^{3}}-\frac {B \,e^{4}}{\left (b e -c d \right )^{3} \left (e x +d \right ) d^{2}}+\frac {A \,c^{4}}{2 \left (b e -c d \right )^{2} \left (c x +b \right )^{2} b^{3}}-\frac {B \,c^{3}}{2 \left (b e -c d \right )^{2} \left (c x +b \right )^{2} b^{2}}+\frac {3 A \,e^{2} \ln \relax (x )}{b^{3} d^{4}}+\frac {6 A c e \ln \relax (x )}{b^{4} d^{3}}+\frac {6 A \,c^{2} \ln \relax (x )}{b^{5} d^{2}}-\frac {2 B e \ln \relax (x )}{b^{3} d^{3}}-\frac {3 B c \ln \relax (x )}{b^{4} d^{2}}+\frac {2 A e}{b^{3} d^{3} x}+\frac {3 A c}{b^{4} d^{2} x}-\frac {B}{b^{3} d^{2} x}-\frac {A}{2 b^{3} d^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b^3/d^2/x*B-10*c^4/(b*e-c*d)^4/b^3*ln(c*x+b)*B*d*e+18*c^5/(b*e-c*d)^4/b^4*ln(c*x+b)*A*d*e-2/d^3/b^3*ln(x)*B
*e-3/d^2/b^4*ln(x)*B*c+1/2*c^4/(b*e-c*d)^2/b^3/(c*x+b)^2*A-1/2*c^3/(b*e-c*d)^2/b^2/(c*x+b)^2*B+e^5/(b*e-c*d)^3
/d^3/(e*x+d)*A-e^4/(b*e-c*d)^3/d^2/(e*x+d)*B+2/b^3/d^3/x*A*e+3/b^4/d^2/x*A*c+2*c^4/(b*e-c*d)^3/b^3/(c*x+b)*B*d
+6/d^3/b^4*ln(x)*A*e*c-3*e^6/(b*e-c*d)^4/d^4*ln(e*x+d)*A*b+6*e^5/(b*e-c*d)^4/d^3*ln(e*x+d)*A*c+2*e^5/(b*e-c*d)
^4/d^3*ln(e*x+d)*B*b-5*e^4/(b*e-c*d)^4/d^2*ln(e*x+d)*B*c-6*c^6/(b*e-c*d)^4/b^5*ln(c*x+b)*A*d^2+10*c^3/(b*e-c*d
)^4/b^2*ln(c*x+b)*B*e^2+3*c^5/(b*e-c*d)^4/b^4*ln(c*x+b)*B*d^2+5*c^4/(b*e-c*d)^3/b^3/(c*x+b)*A*e-3*c^5/(b*e-c*d
)^3/b^4/(c*x+b)*A*d-15*c^4/(b*e-c*d)^4/b^3*ln(c*x+b)*A*e^2-4*c^3/(b*e-c*d)^3/b^2/(c*x+b)*B*e+3/d^4/b^3*ln(x)*A
*e^2+6/d^2/b^5*ln(x)*A*c^2-1/2*A/b^3/d^2/x^2

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maxima [B]  time = 0.91, size = 1043, normalized size = 3.15 \[ \frac {{\left (3 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{2} - 2 \, {\left (5 \, B b^{2} c^{4} - 9 \, A b c^{5}\right )} d e + 5 \, {\left (2 \, B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{4} d^{4} - 4 \, b^{6} c^{3} d^{3} e + 6 \, b^{7} c^{2} d^{2} e^{2} - 4 \, b^{8} c d e^{3} + b^{9} e^{4}} - \frac {{\left (5 \, B c d^{2} e^{4} + 3 \, A b e^{6} - 2 \, {\left (B b + 3 \, A c\right )} d e^{5}\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 {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 \, A b^{4} c^{2} e^{5} + 3 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4} e - {\left (7 \, B b^{2} c^{4} - 12 \, A b c^{5}\right )} d^{3} e^{2} + 3 \, {\left (B b^{3} c^{3} - A b^{2} c^{4}\right )} d^{2} e^{3} - {\left (2 \, B b^{4} c^{2} + 3 \, A b^{3} c^{3}\right )} d e^{4}\right )} x^{4} + {\left (12 \, A b^{5} c e^{5} + 6 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{5} - {\left (5 \, B b^{2} c^{4} - 6 \, A b c^{5}\right )} d^{4} e - 15 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} e^{2} + 5 \, {\left (2 \, B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e^{3} - {\left (8 \, B b^{5} c + 9 \, A b^{4} c^{2}\right )} d e^{4}\right )} x^{3} - {\left (4 \, B b^{6} d e^{4} - 6 \, A b^{6} e^{5} - 9 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{5} + {\left (19 \, B b^{3} c^{3} - 32 \, A b^{2} c^{4}\right )} d^{4} e - {\left (6 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} d^{3} e^{2} - {\left (2 \, B b^{5} c - 13 \, A b^{4} c^{2}\right )} d^{2} e^{3}\right )} x^{2} + {\left (3 \, A b^{6} d e^{4} + 2 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{5} - 3 \, {\left (2 \, B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{4} e + 3 \, {\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} d^{3} e^{2} - {\left (2 \, B b^{6} + 5 \, A b^{5} c\right )} d^{2} e^{3}\right )} x}{2 \, {\left ({\left (b^{4} c^{5} d^{6} e - 3 \, b^{5} c^{4} d^{5} e^{2} + 3 \, b^{6} c^{3} d^{4} e^{3} - b^{7} c^{2} d^{3} e^{4}\right )} x^{5} + {\left (b^{4} c^{5} d^{7} - b^{5} c^{4} d^{6} e - 3 \, b^{6} c^{3} d^{5} e^{2} + 5 \, b^{7} c^{2} d^{4} e^{3} - 2 \, b^{8} c d^{3} e^{4}\right )} x^{4} + {\left (2 \, b^{5} c^{4} d^{7} - 5 \, b^{6} c^{3} d^{6} e + 3 \, b^{7} c^{2} d^{5} e^{2} + b^{8} c d^{4} e^{3} - b^{9} d^{3} e^{4}\right )} x^{3} + {\left (b^{6} c^{3} d^{7} - 3 \, b^{7} c^{2} d^{6} e + 3 \, b^{8} c d^{5} e^{2} - b^{9} d^{4} e^{3}\right )} x^{2}\right )}} + \frac {{\left (3 \, A b^{2} e^{2} - 3 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} - 2 \, {\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \relax (x)}{b^{5} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*(B*b*c^5 - 2*A*c^6)*d^2 - 2*(5*B*b^2*c^4 - 9*A*b*c^5)*d*e + 5*(2*B*b^3*c^3 - 3*A*b^2*c^4)*e^2)*log(c*x + b)
/(b^5*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d*e^3 + b^9*e^4) - (5*B*c*d^2*e^4 + 3*A*b*e^6 -
2*(B*b + 3*A*c)*d*e^5)*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*(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*A*b^4*c^2*e^5 + 3*(B*b
*c^5 - 2*A*c^6)*d^4*e - (7*B*b^2*c^4 - 12*A*b*c^5)*d^3*e^2 + 3*(B*b^3*c^3 - A*b^2*c^4)*d^2*e^3 - (2*B*b^4*c^2
+ 3*A*b^3*c^3)*d*e^4)*x^4 + (12*A*b^5*c*e^5 + 6*(B*b*c^5 - 2*A*c^6)*d^5 - (5*B*b^2*c^4 - 6*A*b*c^5)*d^4*e - 15
*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3*e^2 + 5*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e^3 - (8*B*b^5*c + 9*A*b^4*c^2)*d*e^4)*
x^3 - (4*B*b^6*d*e^4 - 6*A*b^6*e^5 - 9*(B*b^2*c^4 - 2*A*b*c^5)*d^5 + (19*B*b^3*c^3 - 32*A*b^2*c^4)*d^4*e - (6*
B*b^4*c^2 - A*b^3*c^3)*d^3*e^2 - (2*B*b^5*c - 13*A*b^4*c^2)*d^2*e^3)*x^2 + (3*A*b^6*d*e^4 + 2*(B*b^3*c^3 - 2*A
*b^2*c^4)*d^5 - 3*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^4*e + 3*(2*B*b^5*c - A*b^4*c^2)*d^3*e^2 - (2*B*b^6 + 5*A*b^5*c
)*d^2*e^3)*x)/((b^4*c^5*d^6*e - 3*b^5*c^4*d^5*e^2 + 3*b^6*c^3*d^4*e^3 - b^7*c^2*d^3*e^4)*x^5 + (b^4*c^5*d^7 -
b^5*c^4*d^6*e - 3*b^6*c^3*d^5*e^2 + 5*b^7*c^2*d^4*e^3 - 2*b^8*c*d^3*e^4)*x^4 + (2*b^5*c^4*d^7 - 5*b^6*c^3*d^6*
e + 3*b^7*c^2*d^5*e^2 + b^8*c*d^4*e^3 - b^9*d^3*e^4)*x^3 + (b^6*c^3*d^7 - 3*b^7*c^2*d^6*e + 3*b^8*c*d^5*e^2 -
b^9*d^4*e^3)*x^2) + (3*A*b^2*e^2 - 3*(B*b*c - 2*A*c^2)*d^2 - 2*(B*b^2 - 3*A*b*c)*d*e)*log(x)/(b^5*d^4)

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mupad [B]  time = 3.26, size = 879, normalized size = 2.66 \[ \frac {\ln \relax (x)\,\left (b^2\,\left (3\,A\,e^2-2\,B\,d\,e\right )-b\,\left (3\,B\,c\,d^2-6\,A\,c\,d\,e\right )+6\,A\,c^2\,d^2\right )}{b^5\,d^4}-\frac {\ln \left (b+c\,x\right )\,\left (d^2\,\left (6\,A\,c^6-3\,B\,b\,c^5\right )-d\,\left (18\,A\,b\,c^5\,e-10\,B\,b^2\,c^4\,e\right )+15\,A\,b^2\,c^4\,e^2-10\,B\,b^3\,c^3\,e^2\right )}{b^9\,e^4-4\,b^8\,c\,d\,e^3+6\,b^7\,c^2\,d^2\,e^2-4\,b^6\,c^3\,d^3\,e+b^5\,c^4\,d^4}-\frac {\ln \left (d+e\,x\right )\,\left (c\,\left (5\,B\,d^2\,e^4-6\,A\,d\,e^5\right )+b\,\left (3\,A\,e^6-2\,B\,d\,e^5\right )\right )}{b^4\,d^4\,e^4-4\,b^3\,c\,d^5\,e^3+6\,b^2\,c^2\,d^6\,e^2-4\,b\,c^3\,d^7\,e+c^4\,d^8}-\frac {\frac {A}{2\,b\,d}-\frac {x\,\left (3\,A\,b\,e+4\,A\,c\,d-2\,B\,b\,d\right )}{2\,b^2\,d^2}+\frac {x^3\,\left (8\,B\,b^5\,c\,d\,e^4-12\,A\,b^5\,c\,e^5-10\,B\,b^4\,c^2\,d^2\,e^3+9\,A\,b^4\,c^2\,d\,e^4+15\,B\,b^3\,c^3\,d^3\,e^2+15\,A\,b^3\,c^3\,d^2\,e^3+5\,B\,b^2\,c^4\,d^4\,e-30\,A\,b^2\,c^4\,d^3\,e^2-6\,B\,b\,c^5\,d^5-6\,A\,b\,c^5\,d^4\,e+12\,A\,c^6\,d^5\right )}{2\,b^4\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}-\frac {x^2\,\left (-4\,B\,b^5\,d\,e^4+6\,A\,b^5\,e^5+2\,B\,b^4\,c\,d^2\,e^3+6\,B\,b^3\,c^2\,d^3\,e^2-13\,A\,b^3\,c^2\,d^2\,e^3-19\,B\,b^2\,c^3\,d^4\,e-A\,b^2\,c^3\,d^3\,e^2+9\,B\,b\,c^4\,d^5+32\,A\,b\,c^4\,d^4\,e-18\,A\,c^5\,d^5\right )}{2\,b^3\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {c^2\,e\,x^4\,\left (2\,B\,b^4\,d\,e^3-3\,A\,b^4\,e^4-3\,B\,b^3\,c\,d^2\,e^2+3\,A\,b^3\,c\,d\,e^3+7\,B\,b^2\,c^2\,d^3\,e+3\,A\,b^2\,c^2\,d^2\,e^2-3\,B\,b\,c^3\,d^4-12\,A\,b\,c^3\,d^3\,e+6\,A\,c^4\,d^4\right )}{b^4\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}}{x^3\,\left (e\,b^2+2\,c\,d\,b\right )+x^4\,\left (d\,c^2+2\,b\,e\,c\right )+b^2\,d\,x^2+c^2\,e\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(x)*(b^2*(3*A*e^2 - 2*B*d*e) - b*(3*B*c*d^2 - 6*A*c*d*e) + 6*A*c^2*d^2))/(b^5*d^4) - (log(b + c*x)*(d^2*(6
*A*c^6 - 3*B*b*c^5) - d*(18*A*b*c^5*e - 10*B*b^2*c^4*e) + 15*A*b^2*c^4*e^2 - 10*B*b^3*c^3*e^2))/(b^9*e^4 + b^5
*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d*e^3) - (log(d + e*x)*(c*(5*B*d^2*e^4 - 6*A*d*e^5) +
b*(3*A*e^6 - 2*B*d*e^5)))/(c^4*d^8 + b^4*d^4*e^4 - 4*b^3*c*d^5*e^3 + 6*b^2*c^2*d^6*e^2 - 4*b*c^3*d^7*e) - (A/
(2*b*d) - (x*(3*A*b*e + 4*A*c*d - 2*B*b*d))/(2*b^2*d^2) + (x^3*(12*A*c^6*d^5 - 12*A*b^5*c*e^5 - 6*B*b*c^5*d^5
+ 9*A*b^4*c^2*d*e^4 + 5*B*b^2*c^4*d^4*e - 30*A*b^2*c^4*d^3*e^2 + 15*A*b^3*c^3*d^2*e^3 + 15*B*b^3*c^3*d^3*e^2 -
10*B*b^4*c^2*d^2*e^3 - 6*A*b*c^5*d^4*e + 8*B*b^5*c*d*e^4))/(2*b^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^2*(6*A*b^5*e^5 - 18*A*c^5*d^5 + 9*B*b*c^4*d^5 - 4*B*b^5*d*e^4 - 19*B*b^2*c^3*d^4*e + 2*B*b^
4*c*d^2*e^3 - A*b^2*c^3*d^3*e^2 - 13*A*b^3*c^2*d^2*e^3 + 6*B*b^3*c^2*d^3*e^2 + 32*A*b*c^4*d^4*e))/(2*b^3*d^3*(
b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2)) + (c^2*e*x^4*(6*A*c^4*d^4 - 3*A*b^4*e^4 - 3*B*b*c^3*d^4 +
2*B*b^4*d*e^3 + 7*B*b^2*c^2*d^3*e - 3*B*b^3*c*d^2*e^2 + 3*A*b^2*c^2*d^2*e^2 - 12*A*b*c^3*d^3*e + 3*A*b^3*c*d*e
^3))/(b^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^3*(b^2*e + 2*b*c*d) + x^4*(c^2*d + 2*b*
c*e) + b^2*d*x^2 + c^2*e*x^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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