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

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

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Rubi [A]  time = 0.06, antiderivative size = 66, 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) \log (b+c x)}{c^4}+\frac {x^2 (c d-b e)}{2 c^2}-\frac {b x (c d-b e)}{c^3}+\frac {e x^3}{3 c} \end {gather*}

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

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

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Mathematica [A]  time = 0.02, size = 61, normalized size = 0.92 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (d+e x)}{b x+c x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 71, normalized size = 1.08 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(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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giac [A]  time = 0.16, size = 74, normalized size = 1.12 \begin {gather*} \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 )} \log \left ({\left | c x + b \right |}\right )}{c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(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)*log(abs(c*x + b)
)/c^4

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maple [A]  time = 0.04, size = 76, normalized size = 1.15 \begin {gather*} \frac {e \,x^{3}}{3 c}-\frac {b e \,x^{2}}{2 c^{2}}+\frac {d \,x^{2}}{2 c}-\frac {b^{3} e \ln \left (c x +b \right )}{c^{4}}+\frac {b^{2} d \ln \left (c x +b \right )}{c^{3}}+\frac {b^{2} e x}{c^{3}}-\frac {b d x}{c^{2}} \end {gather*}

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*x*b^2*e-1/c^2*x*b*d-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 = 0.95, size = 69, normalized size = 1.05 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(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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mupad [B]  time = 1.01, size = 72, normalized size = 1.09 \begin {gather*} x^2\,\left (\frac {d}{2\,c}-\frac {b\,e}{2\,c^2}\right )-\frac {\ln \left (b+c\,x\right )\,\left (b^3\,e-b^2\,c\,d\right )}{c^4}+\frac {e\,x^3}{3\,c}-\frac {b\,x\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(d + e*x))/(b*x + c*x^2),x)

[Out]

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

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sympy [A]  time = 0.22, size = 61, normalized size = 0.92 \begin {gather*} - \frac {b^{2} \left (b e - c d\right ) \log {\left (b + c x \right )}}{c^{4}} + x^{2} \left (- \frac {b e}{2 c^{2}} + \frac {d}{2 c}\right ) + x \left (\frac {b^{2} e}{c^{3}} - \frac {b d}{c^{2}}\right ) + \frac {e x^{3}}{3 c} \end {gather*}

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

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