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

Optimal. Leaf size=69 \[ -\frac {b^2 (c d-b e)}{c^4 (b+c x)}-\frac {b (2 c d-3 b e) \log (b+c x)}{c^4}+\frac {x (c d-2 b e)}{c^3}+\frac {e x^2}{2 c^2} \]

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Rubi [A]  time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} -\frac {b^2 (c d-b e)}{c^4 (b+c x)}+\frac {x (c d-2 b e)}{c^3}-\frac {b (2 c d-3 b e) \log (b+c x)}{c^4}+\frac {e x^2}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.05, size = 66, normalized size = 0.96 \begin {gather*} \frac {\frac {2 b^2 (b e-c d)}{b+c x}+2 c x (c d-2 b e)+2 b (3 b e-2 c d) \log (b+c x)+c^2 e x^2}{2 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 (d+e x)}{\left (b x+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 111, normalized size = 1.61 \begin {gather*} \frac {c^{3} e x^{3} - 2 \, b^{2} c d + 2 \, b^{3} e + {\left (2 \, c^{3} d - 3 \, b c^{2} e\right )} x^{2} + 2 \, {\left (b c^{2} d - 2 \, b^{2} c e\right )} x - 2 \, {\left (2 \, b^{2} c d - 3 \, b^{3} e + {\left (2 \, b c^{2} d - 3 \, b^{2} c e\right )} x\right )} \log \left (c x + b\right )}{2 \, {\left (c^{5} x + b c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 81, normalized size = 1.17 \begin {gather*} -\frac {{\left (2 \, b c d - 3 \, b^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{4}} + \frac {c^{2} x^{2} e + 2 \, c^{2} d x - 4 \, b c x e}{2 \, c^{4}} - \frac {b^{2} c d - b^{3} e}{{\left (c x + b\right )} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 84, normalized size = 1.22 \begin {gather*} \frac {e \,x^{2}}{2 c^{2}}+\frac {b^{3} e}{\left (c x +b \right ) c^{4}}-\frac {b^{2} d}{\left (c x +b \right ) c^{3}}+\frac {3 b^{2} e \ln \left (c x +b \right )}{c^{4}}-\frac {2 b d \ln \left (c x +b \right )}{c^{3}}-\frac {2 b e x}{c^{3}}+\frac {d x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.91, size = 75, normalized size = 1.09 \begin {gather*} -\frac {b^{2} c d - b^{3} e}{c^{5} x + b c^{4}} + \frac {c e x^{2} + 2 \, {\left (c d - 2 \, b e\right )} x}{2 \, c^{3}} - \frac {{\left (2 \, b c d - 3 \, b^{2} e\right )} \log \left (c x + b\right )}{c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 77, normalized size = 1.12 \begin {gather*} x\,\left (\frac {d}{c^2}-\frac {2\,b\,e}{c^3}\right )+\frac {e\,x^2}{2\,c^2}+\frac {b^3\,e-b^2\,c\,d}{c\,\left (x\,c^4+b\,c^3\right )}+\frac {\ln \left (b+c\,x\right )\,\left (3\,b^2\,e-2\,b\,c\,d\right )}{c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.37, size = 68, normalized size = 0.99 \begin {gather*} \frac {b \left (3 b e - 2 c d\right ) \log {\left (b + c x \right )}}{c^{4}} + x \left (- \frac {2 b e}{c^{3}} + \frac {d}{c^{2}}\right ) + \frac {b^{3} e - b^{2} c d}{b c^{4} + c^{5} x} + \frac {e x^{2}}{2 c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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