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

Optimal. Leaf size=28 \[ \frac {(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \]

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 37} \begin {gather*} \frac {(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^3/(3*(b*d - a*e)*(d + e*x)^3)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 53, normalized size = 1.89 \begin {gather*} -\frac {a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(a^2*e^2 + a*b*e*(d + 3*e*x) + b^2*(d^2 + 3*d*e*x + 3*e^2*x^2))/(e^3*(d + e*x)^3)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.39, size = 84, normalized size = 3.00 \begin {gather*} -\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.15, size = 58, normalized size = 2.07 \begin {gather*} -\frac {{\left (3 \, b^{2} x^{2} e^{2} + 3 \, b^{2} d x e + b^{2} d^{2} + 3 \, a b x e^{2} + a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*b^2*x^2*e^2 + 3*b^2*d*x*e + b^2*d^2 + 3*a*b*x*e^2 + a*b*d*e + a^2*e^2)*e^(-3)/(x*e + d)^3

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maple [B]  time = 0.05, size = 71, normalized size = 2.54 \begin {gather*} -\frac {b^{2}}{\left (e x +d \right ) e^{3}}-\frac {\left (a e -b d \right ) b}{\left (e x +d \right )^{2} e^{3}}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2/e^3/(e*x+d)-b*(a*e-b*d)/e^3/(e*x+d)^2-1/3*(a^2*e^2-2*a*b*d*e+b^2*d^2)/e^3/(e*x+d)^3

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maxima [B]  time = 1.36, size = 84, normalized size = 3.00 \begin {gather*} -\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 80, normalized size = 2.86 \begin {gather*} -\frac {\frac {a^2\,e^2+a\,b\,d\,e+b^2\,d^2}{3\,e^3}+\frac {b^2\,x^2}{e}+\frac {b\,x\,\left (a\,e+b\,d\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.61, size = 88, normalized size = 3.14 \begin {gather*} \frac {- a^{2} e^{2} - a b d e - b^{2} d^{2} - 3 b^{2} e^{2} x^{2} + x \left (- 3 a b e^{2} - 3 b^{2} d e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a**2*e**2 - a*b*d*e - b**2*d**2 - 3*b**2*e**2*x**2 + x*(-3*a*b*e**2 - 3*b**2*d*e))/(3*d**3*e**3 + 9*d**2*e**
4*x + 9*d*e**5*x**2 + 3*e**6*x**3)

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