3.1160 \(\int \frac{A+B x}{(d+e x) \left (b x+c x^2\right )^3} \, dx\)

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

[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)

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Rubi [A]  time = 1.1021, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{-A b e-3 A c d+b B d}{b^4 d^2 x}-\frac{c^2 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)}-\frac{A}{2 b^3 d x^2}+\frac{\log (x) \left (b^2 (-e) (B d-A e)-3 b c d (B d-A e)+6 A c^2 d^2\right )}{b^5 d^3}+\frac{c^2 \left (-2 b c (2 A e+B d)+3 A c^2 d+3 b^2 B e\right )}{b^4 (b+c x) (c d-b e)^2}-\frac{c^2 \log (b+c x) \left (2 b^2 c e (5 A e+4 B d)-3 b c^2 d (5 A e+B d)+6 A c^3 d^2-6 b^3 B e^2\right )}{b^5 (c d-b e)^3}-\frac{e^4 (B d-A e) \log (d+e x)}{d^3 (c d-b e)^3} \]

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)

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Rubi in Sympy [A]  time = 174.056, size = 296, normalized size = 1.06 \[ - \frac{A}{2 b^{3} d x^{2}} - \frac{e^{4} \left (A e - B d\right ) \log{\left (d + e x \right )}}{d^{3} \left (b e - c d\right )^{3}} - \frac{c^{2} \left (A c - B b\right )}{2 b^{3} \left (b + c x\right )^{2} \left (b e - c d\right )} + \frac{c^{2} \left (- 4 A b c e + 3 A c^{2} d + 3 B b^{2} e - 2 B b c d\right )}{b^{4} \left (b + c x\right ) \left (b e - c d\right )^{2}} + \frac{A b e + 3 A c d - B b d}{b^{4} d^{2} x} + \frac{c^{2} \left (10 A b^{2} c e^{2} - 15 A b c^{2} d e + 6 A c^{3} d^{2} - 6 B b^{3} e^{2} + 8 B b^{2} c d e - 3 B b c^{2} d^{2}\right ) \log{\left (b + c x \right )}}{b^{5} \left (b e - c d\right )^{3}} + \frac{\left (A b^{2} e^{2} + 3 A b c d e + 6 A c^{2} d^{2} - B b^{2} d e - 3 B b c d^{2}\right ) \log{\left (x \right )}}{b^{5} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(2*b**3*d*x**2) - e**4*(A*e - B*d)*log(d + e*x)/(d**3*(b*e - c*d)**3) - c**2*
(A*c - B*b)/(2*b**3*(b + c*x)**2*(b*e - c*d)) + c**2*(-4*A*b*c*e + 3*A*c**2*d +
3*B*b**2*e - 2*B*b*c*d)/(b**4*(b + c*x)*(b*e - c*d)**2) + (A*b*e + 3*A*c*d - B*b
*d)/(b**4*d**2*x) + c**2*(10*A*b**2*c*e**2 - 15*A*b*c**2*d*e + 6*A*c**3*d**2 - 6
*B*b**3*e**2 + 8*B*b**2*c*d*e - 3*B*b*c**2*d**2)*log(b + c*x)/(b**5*(b*e - c*d)*
*3) + (A*b**2*e**2 + 3*A*b*c*d*e + 6*A*c**2*d**2 - B*b**2*d*e - 3*B*b*c*d**2)*lo
g(x)/(b**5*d**3)

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Mathematica [A]  time = 3.68862, size = 276, normalized size = 0.99 \[ \frac{A b e+3 A c d-b B d}{b^4 d^2 x}+\frac{c^2 (b B-A c)}{2 b^3 (b+c x)^2 (b e-c d)}-\frac{A}{2 b^3 d x^2}-\frac{\log (x) \left (b^2 e (B d-A e)+3 b c d (B d-A e)-6 A c^2 d^2\right )}{b^5 d^3}+\frac{c^2 \left (-2 b c (2 A e+B d)+3 A c^2 d+3 b^2 B e\right )}{b^4 (b+c x) (c d-b e)^2}+\frac{c^2 \log (b+c x) \left (2 b^2 c e (5 A e+4 B d)-3 b c^2 d (5 A e+B d)+6 A c^3 d^2-6 b^3 B e^2\right )}{b^5 (b e-c d)^3}+\frac{e^4 (A e-B d) \log (d+e x)}{d^3 (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(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)

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Maple [A]  time = 0.028, size = 490, normalized size = 1.8 \[ -{\frac{A}{2\,d{b}^{3}{x}^{2}}}+{\frac{Ae}{{b}^{3}{d}^{2}x}}+3\,{\frac{Ac}{d{b}^{4}x}}-{\frac{B}{d{b}^{3}x}}+{\frac{\ln \left ( x \right ) A{e}^{2}}{{b}^{3}{d}^{3}}}+3\,{\frac{Ac\ln \left ( x \right ) e}{{b}^{4}{d}^{2}}}+6\,{\frac{A\ln \left ( x \right ){c}^{2}}{d{b}^{5}}}-{\frac{\ln \left ( x \right ) Be}{{b}^{3}{d}^{2}}}-3\,{\frac{Bc\ln \left ( x \right ) }{d{b}^{4}}}-4\,{\frac{A{c}^{3}e}{{b}^{3} \left ( be-cd \right ) ^{2} \left ( cx+b \right ) }}+3\,{\frac{{c}^{4}Ad}{{b}^{4} \left ( be-cd \right ) ^{2} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}Be}{{b}^{2} \left ( be-cd \right ) ^{2} \left ( cx+b \right ) }}-2\,{\frac{B{c}^{3}d}{{b}^{3} \left ( be-cd \right ) ^{2} \left ( cx+b \right ) }}+10\,{\frac{{c}^{3}\ln \left ( cx+b \right ) A{e}^{2}}{{b}^{3} \left ( be-cd \right ) ^{3}}}-15\,{\frac{{c}^{4}\ln \left ( cx+b \right ) Ade}{{b}^{4} \left ( be-cd \right ) ^{3}}}+6\,{\frac{{c}^{5}\ln \left ( cx+b \right ) A{d}^{2}}{{b}^{5} \left ( be-cd \right ) ^{3}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ) B{e}^{2}}{{b}^{2} \left ( be-cd \right ) ^{3}}}+8\,{\frac{{c}^{3}\ln \left ( cx+b \right ) Bde}{{b}^{3} \left ( be-cd \right ) ^{3}}}-3\,{\frac{{c}^{4}\ln \left ( cx+b \right ) B{d}^{2}}{{b}^{4} \left ( be-cd \right ) ^{3}}}-{\frac{A{c}^{3}}{2\,{b}^{3} \left ( be-cd \right ) \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}B}{2\,{b}^{2} \left ( be-cd \right ) \left ( cx+b \right ) ^{2}}}-{\frac{{e}^{5}\ln \left ( ex+d \right ) A}{{d}^{3} \left ( be-cd \right ) ^{3}}}+{\frac{{e}^{4}\ln \left ( ex+d \right ) B}{{d}^{2} \left ( be-cd \right ) ^{3}}} \]

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]

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

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Maxima [A]  time = 0.732068, size = 826, normalized size = 2.96 \[ \frac{{\left (3 \,{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} -{\left (8 \, B b^{2} c^{3} - 15 \, A b c^{4}\right )} d e + 2 \,{\left (3 \, B b^{3} c^{2} - 5 \, A b^{2} c^{3}\right )} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} - \frac{{\left (B d e^{4} - A e^{5}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac{A b^{3} c^{2} d^{3} - 2 \, A b^{4} c d^{2} e + A b^{5} d e^{2} - 2 \,{\left (A b^{3} c^{2} e^{3} - 3 \,{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} +{\left (5 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} d^{2} e -{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d e^{2}\right )} x^{3} -{\left (4 \, A b^{4} c e^{3} - 9 \,{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} + 3 \,{\left (5 \, B b^{3} c^{2} - 9 \, A b^{2} c^{3}\right )} d^{2} e -{\left (4 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e^{2}\right )} x^{2} + 2 \,{\left (B b^{5} d e^{2} - A b^{5} e^{3} +{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3} -{\left (2 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} d^{2} e\right )} x}{2 \,{\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \,{\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} +{\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac{{\left (A b^{2} e^{2} - 3 \,{\left (B b c - 2 \, A c^{2}\right )} d^{2} -{\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \left (x\right )}{b^{5} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)^3*(e*x + d)),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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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

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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GIAC/XCAS [A]  time = 0.281556, size = 876, normalized size = 3.14 \[ \frac{{\left (3 \, B b c^{5} d^{2} - 6 \, A c^{6} d^{2} - 8 \, B b^{2} c^{4} d e + 15 \, A b c^{5} d e + 6 \, B b^{3} c^{3} e^{2} - 10 \, A b^{2} c^{4} e^{2}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} - \frac{{\left (B d e^{5} - A e^{6}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} - \frac{{\left (3 \, B b c d^{2} - 6 \, A c^{2} d^{2} + B b^{2} d e - 3 \, A b c d e - A b^{2} e^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{5} d^{3}} - \frac{A b^{3} c^{3} d^{5} - 3 \, A b^{4} c^{2} d^{4} e + 3 \, A b^{5} c d^{3} e^{2} - A b^{6} d^{2} e^{3} + 2 \,{\left (3 \, B b c^{5} d^{5} - 6 \, A c^{6} d^{5} - 8 \, B b^{2} c^{4} d^{4} e + 15 \, A b c^{5} d^{4} e + 6 \, B b^{3} c^{3} d^{3} e^{2} - 10 \, A b^{2} c^{4} d^{3} e^{2} - B b^{4} c^{2} d^{2} e^{3} + A b^{4} c^{2} d e^{4}\right )} x^{3} +{\left (9 \, B b^{2} c^{4} d^{5} - 18 \, A b c^{5} d^{5} - 24 \, B b^{3} c^{3} d^{4} e + 45 \, A b^{2} c^{4} d^{4} e + 19 \, B b^{4} c^{2} d^{3} e^{2} - 30 \, A b^{3} c^{3} d^{3} e^{2} - 4 \, B b^{5} c d^{2} e^{3} - A b^{4} c^{2} d^{2} e^{3} + 4 \, A b^{5} c d e^{4}\right )} x^{2} + 2 \,{\left (B b^{3} c^{3} d^{5} - 2 \, A b^{2} c^{4} d^{5} - 3 \, B b^{4} c^{2} d^{4} e + 5 \, A b^{3} c^{3} d^{4} e + 3 \, B b^{5} c d^{3} e^{2} - 3 \, A b^{4} c^{2} d^{3} e^{2} - B b^{6} d^{2} e^{3} - A b^{5} c d^{2} e^{3} + A b^{6} d e^{4}\right )} x}{2 \,{\left (c d - b e\right )}^{3}{\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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