3.43 \(\int \frac{x^4 (d+e x)}{b x+c x^2} \, dx\)

Optimal. Leaf size=87 \[ \frac{b^2 x (c d-b e)}{c^4}-\frac{b^3 (c d-b e) \log (b+c x)}{c^5}+\frac{x^3 (c d-b e)}{3 c^2}-\frac{b x^2 (c d-b e)}{2 c^3}+\frac{e x^4}{4 c} \]

[Out]

(b^2*(c*d - b*e)*x)/c^4 - (b*(c*d - b*e)*x^2)/(2*c^3) + ((c*d - b*e)*x^3)/(3*c^2) + (e*x^4)/(4*c) - (b^3*(c*d
- b*e)*Log[b + c*x])/c^5

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Rubi [A]  time = 0.0893381, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{b^2 x (c d-b e)}{c^4}-\frac{b^3 (c d-b e) \log (b+c x)}{c^5}+\frac{x^3 (c d-b e)}{3 c^2}-\frac{b x^2 (c d-b e)}{2 c^3}+\frac{e x^4}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(d + e*x))/(b*x + c*x^2),x]

[Out]

(b^2*(c*d - b*e)*x)/c^4 - (b*(c*d - b*e)*x^2)/(2*c^3) + ((c*d - b*e)*x^3)/(3*c^2) + (e*x^4)/(4*c) - (b^3*(c*d
- b*e)*Log[b + c*x])/c^5

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{x^4 (d+e x)}{b x+c x^2} \, dx &=\int \left (-\frac{b^2 (-c d+b e)}{c^4}+\frac{b (-c d+b e) x}{c^3}+\frac{(c d-b e) x^2}{c^2}+\frac{e x^3}{c}+\frac{b^3 (-c d+b e)}{c^4 (b+c x)}\right ) \, dx\\ &=\frac{b^2 (c d-b e) x}{c^4}-\frac{b (c d-b e) x^2}{2 c^3}+\frac{(c d-b e) x^3}{3 c^2}+\frac{e x^4}{4 c}-\frac{b^3 (c d-b e) \log (b+c x)}{c^5}\\ \end{align*}

Mathematica [A]  time = 0.0314028, size = 80, normalized size = 0.92 \[ \frac{c x \left (6 b^2 c (2 d+e x)-12 b^3 e-2 b c^2 x (3 d+2 e x)+c^3 x^2 (4 d+3 e x)\right )+12 b^3 (b e-c d) \log (b+c x)}{12 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(-12*b^3*e + 6*b^2*c*(2*d + e*x) - 2*b*c^2*x*(3*d + 2*e*x) + c^3*x^2*(4*d + 3*e*x)) + 12*b^3*(-(c*d) + b*
e)*Log[b + c*x])/(12*c^5)

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Maple [A]  time = 0.004, size = 100, normalized size = 1.2 \begin{align*}{\frac{e{x}^{4}}{4\,c}}-{\frac{b{x}^{3}e}{3\,{c}^{2}}}+{\frac{d{x}^{3}}{3\,c}}+{\frac{{x}^{2}{b}^{2}e}{2\,{c}^{3}}}-{\frac{b{x}^{2}d}{2\,{c}^{2}}}-{\frac{{b}^{3}ex}{{c}^{4}}}+{\frac{{b}^{2}dx}{{c}^{3}}}+{\frac{{b}^{4}\ln \left ( cx+b \right ) e}{{c}^{5}}}-{\frac{{b}^{3}\ln \left ( cx+b \right ) d}{{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)/(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 1.12445, size = 126, normalized size = 1.45 \begin{align*} \frac{3 \, c^{3} e x^{4} + 4 \,{\left (c^{3} d - b c^{2} e\right )} x^{3} - 6 \,{\left (b c^{2} d - b^{2} c e\right )} x^{2} + 12 \,{\left (b^{2} c d - b^{3} e\right )} x}{12 \, c^{4}} - \frac{{\left (b^{3} c d - b^{4} e\right )} \log \left (c x + b\right )}{c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*c^3*e*x^4 + 4*(c^3*d - b*c^2*e)*x^3 - 6*(b*c^2*d - b^2*c*e)*x^2 + 12*(b^2*c*d - b^3*e)*x)/c^4 - (b^3*c
*d - b^4*e)*log(c*x + b)/c^5

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Fricas [A]  time = 1.83993, size = 196, normalized size = 2.25 \begin{align*} \frac{3 \, c^{4} e x^{4} + 4 \,{\left (c^{4} d - b c^{3} e\right )} x^{3} - 6 \,{\left (b c^{3} d - b^{2} c^{2} e\right )} x^{2} + 12 \,{\left (b^{2} c^{2} d - b^{3} c e\right )} x - 12 \,{\left (b^{3} c d - b^{4} e\right )} \log \left (c x + b\right )}{12 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*c^4*e*x^4 + 4*(c^4*d - b*c^3*e)*x^3 - 6*(b*c^3*d - b^2*c^2*e)*x^2 + 12*(b^2*c^2*d - b^3*c*e)*x - 12*(b
^3*c*d - b^4*e)*log(c*x + b))/c^5

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Sympy [A]  time = 0.72876, size = 78, normalized size = 0.9 \begin{align*} \frac{b^{3} \left (b e - c d\right ) \log{\left (b + c x \right )}}{c^{5}} + \frac{e x^{4}}{4 c} - \frac{x^{3} \left (b e - c d\right )}{3 c^{2}} + \frac{x^{2} \left (b^{2} e - b c d\right )}{2 c^{3}} - \frac{x \left (b^{3} e - b^{2} c d\right )}{c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)/(c*x**2+b*x),x)

[Out]

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

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Giac [A]  time = 1.13781, size = 135, normalized size = 1.55 \begin{align*} \frac{3 \, c^{3} x^{4} e + 4 \, c^{3} d x^{3} - 4 \, b c^{2} x^{3} e - 6 \, b c^{2} d x^{2} + 6 \, b^{2} c x^{2} e + 12 \, b^{2} c d x - 12 \, b^{3} x e}{12 \, c^{4}} - \frac{{\left (b^{3} c d - b^{4} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*c^3*x^4*e + 4*c^3*d*x^3 - 4*b*c^2*x^3*e - 6*b*c^2*d*x^2 + 6*b^2*c*x^2*e + 12*b^2*c*d*x - 12*b^3*x*e)/c
^4 - (b^3*c*d - b^4*e)*log(abs(c*x + b))/c^5