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

Optimal. Leaf size=66 \[ \frac{b^2 (c d-b e) \log (b+c x)}{c^4}-\frac{b x (c d-b e)}{c^3}+\frac{x^2 (c d-b e)}{2 c^2}+\frac{e x^3}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.123854, antiderivative size = 66, 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 \[ \frac{b^2 (c d-b e) \log (b+c x)}{c^4}-\frac{b x (c d-b e)}{c^3}+\frac{x^2 (c d-b e)}{2 c^2}+\frac{e x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b^{2} \left (b e - c d\right ) \log{\left (b + c x \right )}}{c^{4}} + \frac{e x^{3}}{3 c} - \frac{\left (b e - c d\right ) \int x\, dx}{c^{2}} + \frac{\left (b e - c d\right ) \int b\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0478861, size = 61, normalized size = 0.92 \[ \frac{c x \left (6 b^2 e-3 b c (2 d+e x)+c^2 x (3 d+2 e x)\right )+6 b^2 (c d-b e) \log (b+c x)}{6 c^4} \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(6*b^2*e - 3*b*c*(2*d + e*x) + c^2*x*(3*d + 2*e*x)) + 6*b^2*(c*d - b*e)*Log
[b + c*x])/(6*c^4)

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Maple [A]  time = 0.003, size = 76, normalized size = 1.2 \[{\frac{e{x}^{3}}{3\,c}}-{\frac{b{x}^{2}e}{2\,{c}^{2}}}+{\frac{d{x}^{2}}{2\,c}}+{\frac{{b}^{2}ex}{{c}^{3}}}-{\frac{bdx}{{c}^{2}}}-{\frac{{b}^{3}\ln \left ( cx+b \right ) e}{{c}^{4}}}+{\frac{{b}^{2}\ln \left ( cx+b \right ) d}{{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(e*x+d)/(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 0.701933, size = 93, normalized size = 1.41 \[ \frac{2 \, c^{2} e x^{3} + 3 \,{\left (c^{2} d - b c e\right )} x^{2} - 6 \,{\left (b c d - b^{2} e\right )} x}{6 \, c^{3}} + \frac{{\left (b^{2} c d - b^{3} e\right )} \log \left (c x + b\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*c^2*e*x^3 + 3*(c^2*d - b*c*e)*x^2 - 6*(b*c*d - b^2*e)*x)/c^3 + (b^2*c*d -
b^3*e)*log(c*x + b)/c^4

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Fricas [A]  time = 0.272646, size = 96, normalized size = 1.45 \[ \frac{2 \, c^{3} e x^{3} + 3 \,{\left (c^{3} d - b c^{2} e\right )} x^{2} - 6 \,{\left (b c^{2} d - b^{2} c e\right )} x + 6 \,{\left (b^{2} c d - b^{3} e\right )} \log \left (c x + b\right )}{6 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*c^3*e*x^3 + 3*(c^3*d - b*c^2*e)*x^2 - 6*(b*c^2*d - b^2*c*e)*x + 6*(b^2*c*
d - b^3*e)*log(c*x + b))/c^4

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Sympy [A]  time = 1.454, size = 58, normalized size = 0.88 \[ - \frac{b^{2} \left (b e - c d\right ) \log{\left (b + c x \right )}}{c^{4}} + \frac{e x^{3}}{3 c} - \frac{x^{2} \left (b e - c d\right )}{2 c^{2}} + \frac{x \left (b^{2} e - b c d\right )}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.266984, size = 100, normalized size = 1.52 \[ \frac{2 \, c^{2} x^{3} e + 3 \, c^{2} d x^{2} - 3 \, b c x^{2} e - 6 \, b c d x + 6 \, b^{2} x e}{6 \, c^{3}} + \frac{{\left (b^{2} c d - b^{3} e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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