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

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]

-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)

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Rubi [A]  time = 1.49346, antiderivative size = 331, 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 \[ -\frac{-2 A b e-3 A c d+b B d}{b^4 d^3 x}-\frac{c^3 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac{A}{2 b^3 d^2 x^2}+\frac{\log (x) \left (b^2 (-e) (2 B d-3 A e)-3 b c d (B d-2 A e)+6 A c^2 d^2\right )}{b^5 d^4}-\frac{c^3 \left (5 A b c e-3 A c^2 d-4 b^2 B e+2 b B c d\right )}{b^4 (b+c x) (c d-b e)^3}-\frac{c^3 \log (b+c x) \left (5 b^2 c e (3 A e+2 B d)-3 b c^2 d (6 A e+B d)+6 A c^3 d^2-10 b^3 B e^2\right )}{b^5 (c d-b e)^4}-\frac{e^4 \log (d+e x) (B d (5 c d-2 b e)-3 A e (2 c d-b e))}{d^4 (c d-b e)^4}+\frac{e^4 (B d-A e)}{d^3 (d+e x) (c d-b e)^3} \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.15468, 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]

-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*(-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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Maple [A]  time = 0.036, size = 598, normalized size = 1.8 \[ 2\,{\frac{Ae}{{b}^{3}{d}^{3}x}}+3\,{\frac{\ln \left ( x \right ) A{e}^{2}}{{d}^{4}{b}^{3}}}-{\frac{B}{{b}^{3}{d}^{2}x}}-4\,{\frac{B{c}^{3}e}{{b}^{2} \left ( be-cd \right ) ^{3} \left ( cx+b \right ) }}+5\,{\frac{{c}^{4}Ae}{{b}^{3} \left ( be-cd \right ) ^{3} \left ( cx+b \right ) }}-3\,{\frac{{c}^{5}Ad}{{b}^{4} \left ( be-cd \right ) ^{3} \left ( cx+b \right ) }}+2\,{\frac{B{c}^{4}d}{{b}^{3} \left ( be-cd \right ) ^{3} \left ( cx+b \right ) }}+6\,{\frac{Ac\ln \left ( x \right ) e}{{d}^{3}{b}^{4}}}-15\,{\frac{{c}^{4}\ln \left ( cx+b \right ) A{e}^{2}}{{b}^{3} \left ( be-cd \right ) ^{4}}}-6\,{\frac{{c}^{6}\ln \left ( cx+b \right ) A{d}^{2}}{{b}^{5} \left ( be-cd \right ) ^{4}}}+10\,{\frac{{c}^{3}\ln \left ( cx+b \right ) B{e}^{2}}{{b}^{2} \left ( be-cd \right ) ^{4}}}+3\,{\frac{{c}^{5}\ln \left ( cx+b \right ) B{d}^{2}}{{b}^{4} \left ( be-cd \right ) ^{4}}}-3\,{\frac{{e}^{6}\ln \left ( ex+d \right ) Ab}{{d}^{4} \left ( be-cd \right ) ^{4}}}+6\,{\frac{{e}^{5}\ln \left ( ex+d \right ) Ac}{{d}^{3} \left ( be-cd \right ) ^{4}}}+2\,{\frac{{e}^{5}\ln \left ( ex+d \right ) Bb}{{d}^{3} \left ( be-cd \right ) ^{4}}}-5\,{\frac{{e}^{4}\ln \left ( ex+d \right ) Bc}{{d}^{2} \left ( be-cd \right ) ^{4}}}-{\frac{A}{2\,{b}^{3}{d}^{2}{x}^{2}}}+18\,{\frac{{c}^{5}\ln \left ( cx+b \right ) Ade}{{b}^{4} \left ( be-cd \right ) ^{4}}}-10\,{\frac{{c}^{4}\ln \left ( cx+b \right ) Bde}{{b}^{3} \left ( be-cd \right ) ^{4}}}+3\,{\frac{Ac}{{b}^{4}{d}^{2}x}}+6\,{\frac{A\ln \left ( x \right ){c}^{2}}{{d}^{2}{b}^{5}}}-2\,{\frac{\ln \left ( x \right ) Be}{{b}^{3}{d}^{3}}}-3\,{\frac{Bc\ln \left ( x \right ) }{{b}^{4}{d}^{2}}}+{\frac{{c}^{4}A}{2\,{b}^{3} \left ( be-cd \right ) ^{2} \left ( cx+b \right ) ^{2}}}-{\frac{B{c}^{3}}{2\,{b}^{2} \left ( be-cd \right ) ^{2} \left ( cx+b \right ) ^{2}}}+{\frac{{e}^{5}A}{{d}^{3} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}-{\frac{{e}^{4}B}{{d}^{2} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }} \]

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]

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

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Maxima [A]  time = 0.757447, size = 1408, normalized size = 4.25 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)^3*(e*x + d)^2),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 - 1
2*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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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)^2),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)**2/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.324139, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done