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

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

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Rubi [A]  time = 0.09, antiderivative size = 87, 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 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} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 80, normalized size = 0.92 \begin {gather*} \frac {12 b^3 (b e-c d) \log (b+c x)+c x \left (-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)\right )}{12 c^5} \end {gather*}

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 94, normalized size = 1.08 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.16, size = 100, normalized size = 1.15 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.05, size = 100, normalized size = 1.15 \begin {gather*} \frac {e \,x^{4}}{4 c}-\frac {b e \,x^{3}}{3 c^{2}}+\frac {d \,x^{3}}{3 c}+\frac {b^{2} e \,x^{2}}{2 c^{3}}-\frac {b d \,x^{2}}{2 c^{2}}+\frac {b^{4} e \ln \left (c x +b \right )}{c^{5}}-\frac {b^{3} d \ln \left (c x +b \right )}{c^{4}}-\frac {b^{3} e x}{c^{4}}+\frac {b^{2} d x}{c^{3}} \end {gather*}

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*x*b^3*e+1/c^3*x*b^2*d+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 = 0.89, size = 93, normalized size = 1.07 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.05, size = 94, normalized size = 1.08 \begin {gather*} x^3\,\left (\frac {d}{3\,c}-\frac {b\,e}{3\,c^2}\right )+\frac {\ln \left (b+c\,x\right )\,\left (b^4\,e-b^3\,c\,d\right )}{c^5}+\frac {e\,x^4}{4\,c}-\frac {b\,x^2\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{2\,c}+\frac {b^2\,x\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(d + e*x))/(b*x + c*x^2),x)

[Out]

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

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sympy [A]  time = 0.24, size = 85, normalized size = 0.98 \begin {gather*} \frac {b^{3} \left (b e - c d\right ) \log {\left (b + c x \right )}}{c^{5}} + x^{3} \left (- \frac {b e}{3 c^{2}} + \frac {d}{3 c}\right ) + x^{2} \left (\frac {b^{2} e}{2 c^{3}} - \frac {b d}{2 c^{2}}\right ) + x \left (- \frac {b^{3} e}{c^{4}} + \frac {b^{2} d}{c^{3}}\right ) + \frac {e x^{4}}{4 c} \end {gather*}

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

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