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

Optimal. Leaf size=25 \[ \frac{(c d-b e) \log (b+c x)}{c^2}+\frac{e x}{c} \]

[Out]

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

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Rubi [A]  time = 0.0483811, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(c d-b e) \log (b+c x)}{c^2}+\frac{e x}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0125238, size = 25, normalized size = 1. \[ \frac{(c d-b e) \log (b+c x)}{c^2}+\frac{e x}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 32, normalized size = 1.3 \[{\frac{ex}{c}}-{\frac{\ln \left ( cx+b \right ) be}{{c}^{2}}}+{\frac{\ln \left ( cx+b \right ) d}{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x+d)/(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 0.702677, size = 34, normalized size = 1.36 \[ \frac{e x}{c} + \frac{{\left (c d - b e\right )} \log \left (c x + b\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*x/c + (c*d - b*e)*log(c*x + b)/c^2

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Fricas [A]  time = 0.287526, size = 32, normalized size = 1.28 \[ \frac{c e x +{\left (c d - b e\right )} \log \left (c x + b\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.2625, size = 20, normalized size = 0.8 \[ \frac{e x}{c} - \frac{\left (b e - c d\right ) \log{\left (b + c x \right )}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*x/c - (b*e - c*d)*log(b + c*x)/c**2

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GIAC/XCAS [A]  time = 0.267613, size = 38, normalized size = 1.52 \[ \frac{x e}{c} + \frac{{\left (c d - b e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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