3.61 \(\int \frac{x^4 (d+e x)}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=55 \[ -\frac{c d-2 b e}{c^3 (b+c x)}+\frac{b (c d-b e)}{2 c^3 (b+c x)^2}+\frac{e \log (b+c x)}{c^3} \]

[Out]

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

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Rubi [A]  time = 0.102945, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{c d-2 b e}{c^3 (b+c x)}+\frac{b (c d-b e)}{2 c^3 (b+c x)^2}+\frac{e \log (b+c x)}{c^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(d + e*x))/(b*x + c*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.2689, size = 48, normalized size = 0.87 \[ - \frac{b \left (b e - c d\right )}{2 c^{3} \left (b + c x\right )^{2}} + \frac{e \log{\left (b + c x \right )}}{c^{3}} + \frac{2 b e - c d}{c^{3} \left (b + c x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(e*x+d)/(c*x**2+b*x)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0296426, size = 54, normalized size = 0.98 \[ \frac{3 b^2 e-b c (d-4 e x)+2 e (b+c x)^2 \log (b+c x)-2 c^2 d x}{2 c^3 (b+c x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(d + e*x))/(b*x + c*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 70, normalized size = 1.3 \[{\frac{e\ln \left ( cx+b \right ) }{{c}^{3}}}-{\frac{{b}^{2}e}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{bd}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}+2\,{\frac{be}{{c}^{3} \left ( cx+b \right ) }}-{\frac{d}{{c}^{2} \left ( cx+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(e*x+d)/(c*x^2+b*x)^3,x)

[Out]

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

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Maxima [A]  time = 0.700559, size = 85, normalized size = 1.55 \[ -\frac{b c d - 3 \, b^{2} e + 2 \,{\left (c^{2} d - 2 \, b c e\right )} x}{2 \,{\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\right )}} + \frac{e \log \left (c x + b\right )}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^4/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

-1/2*(b*c*d - 3*b^2*e + 2*(c^2*d - 2*b*c*e)*x)/(c^5*x^2 + 2*b*c^4*x + b^2*c^3) +
e*log(c*x + b)/c^3

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Fricas [A]  time = 0.27748, size = 107, normalized size = 1.95 \[ -\frac{b c d - 3 \, b^{2} e + 2 \,{\left (c^{2} d - 2 \, b c e\right )} x - 2 \,{\left (c^{2} e x^{2} + 2 \, b c e x + b^{2} e\right )} \log \left (c x + b\right )}{2 \,{\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^4/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

-1/2*(b*c*d - 3*b^2*e + 2*(c^2*d - 2*b*c*e)*x - 2*(c^2*e*x^2 + 2*b*c*e*x + b^2*e
)*log(c*x + b))/(c^5*x^2 + 2*b*c^4*x + b^2*c^3)

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Sympy [A]  time = 2.18768, size = 63, normalized size = 1.15 \[ \frac{3 b^{2} e - b c d + x \left (4 b c e - 2 c^{2} d\right )}{2 b^{2} c^{3} + 4 b c^{4} x + 2 c^{5} x^{2}} + \frac{e \log{\left (b + c x \right )}}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(e*x+d)/(c*x**2+b*x)**3,x)

[Out]

(3*b**2*e - b*c*d + x*(4*b*c*e - 2*c**2*d))/(2*b**2*c**3 + 4*b*c**4*x + 2*c**5*x
**2) + e*log(b + c*x)/c**3

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GIAC/XCAS [A]  time = 0.272253, size = 74, normalized size = 1.35 \[ \frac{e{\rm ln}\left ({\left | c x + b \right |}\right )}{c^{3}} - \frac{2 \,{\left (c d - 2 \, b e\right )} x + \frac{b c d - 3 \, b^{2} e}{c}}{2 \,{\left (c x + b\right )}^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^4/(c*x^2 + b*x)^3,x, algorithm="giac")

[Out]

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