3.1111 \(\int \frac{(A+B x) \left (b x+c x^2\right )}{(d+e x)^4} \, dx\)

Optimal. Leaf size=111 \[ \frac{d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^3}+\frac{-A c e-b B e+3 B c d}{e^4 (d+e x)}-\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{2 e^4 (d+e x)^2}+\frac{B c \log (d+e x)}{e^4} \]

[Out]

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

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Rubi [A]  time = 0.228099, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^3}+\frac{-A c e-b B e+3 B c d}{e^4 (d+e x)}-\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{2 e^4 (d+e x)^2}+\frac{B c \log (d+e x)}{e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.775, size = 109, normalized size = 0.98 \[ \frac{B c \log{\left (d + e x \right )}}{e^{4}} + \frac{d \left (A e - B d\right ) \left (b e - c d\right )}{3 e^{4} \left (d + e x\right )^{3}} - \frac{A c e + B b e - 3 B c d}{e^{4} \left (d + e x\right )} - \frac{A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}}{2 e^{4} \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*c*log(d + e*x)/e**4 + d*(A*e - B*d)*(b*e - c*d)/(3*e**4*(d + e*x)**3) - (A*c*e
+ B*b*e - 3*B*c*d)/(e**4*(d + e*x)) - (A*b*e**2 - 2*A*c*d*e - 2*B*b*d*e + 3*B*c
*d**2)/(2*e**4*(d + e*x)**2)

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Mathematica [A]  time = 0.0980194, size = 112, normalized size = 1.01 \[ \frac{-A e \left (b e (d+3 e x)+2 c \left (d^2+3 d e x+3 e^2 x^2\right )\right )+B \left (c d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 b e \left (d^2+3 d e x+3 e^2 x^2\right )\right )+6 B c (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-(A*e*(b*e*(d + 3*e*x) + 2*c*(d^2 + 3*d*e*x + 3*e^2*x^2))) + B*(-2*b*e*(d^2 + 3
*d*e*x + 3*e^2*x^2) + c*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) + 6*B*c*(d + e*x)^3*
Log[d + e*x])/(6*e^4*(d + e*x)^3)

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Maple [A]  time = 0.01, size = 182, normalized size = 1.6 \[{\frac{Adb}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{Ac{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{Bb{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{Bc{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{Ab}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{Acd}{{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{Bbd}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,Bc{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{Bc\ln \left ( ex+d \right ) }{{e}^{4}}}-{\frac{Ac}{{e}^{3} \left ( ex+d \right ) }}-{\frac{Bb}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{Bcd}{{e}^{4} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.699109, size = 185, normalized size = 1.67 \[ \frac{11 \, B c d^{3} - A b d e^{2} - 2 \,{\left (B b + A c\right )} d^{2} e + 6 \,{\left (3 \, B c d e^{2} -{\left (B b + A c\right )} e^{3}\right )} x^{2} + 3 \,{\left (9 \, B c d^{2} e - A b e^{3} - 2 \,{\left (B b + A c\right )} d e^{2}\right )} x}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac{B c \log \left (e x + d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/(e*x + d)^4,x, algorithm="maxima")

[Out]

1/6*(11*B*c*d^3 - A*b*d*e^2 - 2*(B*b + A*c)*d^2*e + 6*(3*B*c*d*e^2 - (B*b + A*c)
*e^3)*x^2 + 3*(9*B*c*d^2*e - A*b*e^3 - 2*(B*b + A*c)*d*e^2)*x)/(e^7*x^3 + 3*d*e^
6*x^2 + 3*d^2*e^5*x + d^3*e^4) + B*c*log(e*x + d)/e^4

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Fricas [A]  time = 0.332581, size = 227, normalized size = 2.05 \[ \frac{11 \, B c d^{3} - A b d e^{2} - 2 \,{\left (B b + A c\right )} d^{2} e + 6 \,{\left (3 \, B c d e^{2} -{\left (B b + A c\right )} e^{3}\right )} x^{2} + 3 \,{\left (9 \, B c d^{2} e - A b e^{3} - 2 \,{\left (B b + A c\right )} d e^{2}\right )} x + 6 \,{\left (B c e^{3} x^{3} + 3 \, B c d e^{2} x^{2} + 3 \, B c d^{2} e x + B c d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/(e*x + d)^4,x, algorithm="fricas")

[Out]

1/6*(11*B*c*d^3 - A*b*d*e^2 - 2*(B*b + A*c)*d^2*e + 6*(3*B*c*d*e^2 - (B*b + A*c)
*e^3)*x^2 + 3*(9*B*c*d^2*e - A*b*e^3 - 2*(B*b + A*c)*d*e^2)*x + 6*(B*c*e^3*x^3 +
3*B*c*d*e^2*x^2 + 3*B*c*d^2*e*x + B*c*d^3)*log(e*x + d))/(e^7*x^3 + 3*d*e^6*x^2
+ 3*d^2*e^5*x + d^3*e^4)

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Sympy [A]  time = 13.371, size = 158, normalized size = 1.42 \[ \frac{B c \log{\left (d + e x \right )}}{e^{4}} - \frac{A b d e^{2} + 2 A c d^{2} e + 2 B b d^{2} e - 11 B c d^{3} + x^{2} \left (6 A c e^{3} + 6 B b e^{3} - 18 B c d e^{2}\right ) + x \left (3 A b e^{3} + 6 A c d e^{2} + 6 B b d e^{2} - 27 B c d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*c*log(d + e*x)/e**4 - (A*b*d*e**2 + 2*A*c*d**2*e + 2*B*b*d**2*e - 11*B*c*d**3
+ x**2*(6*A*c*e**3 + 6*B*b*e**3 - 18*B*c*d*e**2) + x*(3*A*b*e**3 + 6*A*c*d*e**2
+ 6*B*b*d*e**2 - 27*B*c*d**2*e))/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2
+ 6*e**7*x**3)

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GIAC/XCAS [A]  time = 0.279382, size = 161, normalized size = 1.45 \[ B c e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (6 \,{\left (3 \, B c d e - B b e^{2} - A c e^{2}\right )} x^{2} + 3 \,{\left (9 \, B c d^{2} - 2 \, B b d e - 2 \, A c d e - A b e^{2}\right )} x +{\left (11 \, B c d^{3} - 2 \, B b d^{2} e - 2 \, A c d^{2} e - A b d e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/(e*x + d)^4,x, algorithm="giac")

[Out]

B*c*e^(-4)*ln(abs(x*e + d)) + 1/6*(6*(3*B*c*d*e - B*b*e^2 - A*c*e^2)*x^2 + 3*(9*
B*c*d^2 - 2*B*b*d*e - 2*A*c*d*e - A*b*e^2)*x + (11*B*c*d^3 - 2*B*b*d^2*e - 2*A*c
*d^2*e - A*b*d*e^2)*e^(-1))*e^(-3)/(x*e + d)^3