3.1.40 \(\int \frac {x^2}{(a+b x)^4} \, dx\) [40]

Optimal. Leaf size=34 \[ \frac {-\frac {a^2}{3 b^3}-\frac {a x}{b^2}-\frac {x^2}{b}}{(a+b x)^3} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 0.50, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {37} \begin {gather*} \frac {x^3}{3 a (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^4,x]

[Out]

x^3/(3*a*(a + b*x)^3)

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 {gather*} \begin {aligned} \text {Integral} &=\frac {x^3}{3 a (a+b x)^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.91 \begin {gather*} -\frac {a^2+3 a b x+3 b^2 x^2}{3 b^3 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^4,x]

[Out]

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

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Maple [A]
time = 0.02, size = 41, normalized size = 1.21

method	result	size
gosper	\(-\frac {3 b^{2} x^{2}+3 b a x +a^{2}}{3 b^{3} \left (b x +a \right )^{3}}\)	\(30\)
parallelrisch	\(\frac {-3 b^{2} x^{2}-3 b a x -a^{2}}{3 b^{3} \left (b x +a \right )^{3}}\)	\(32\)
norman	\(\frac {-\frac {x^{2}}{b}-\frac {a x}{b^{2}}-\frac {a^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\)	\(33\)
risch	\(\frac {-\frac {x^{2}}{b}-\frac {a x}{b^{2}}-\frac {a^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\)	\(33\)
default	\(\frac {a}{b^{3} \left (b x +a \right )^{2}}-\frac {a^{2}}{3 b^{3} \left (b x +a \right )^{3}}-\frac {1}{b^{3} \left (b x +a \right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.33, size = 54, normalized size = 1.59 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.54, size = 54, normalized size = 1.59 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (27) = 54\).
time = 0.11, size = 56, normalized size = 1.65 \begin {gather*} \frac {- a^{2} - 3 a b x - 3 b^{2} x^{2}}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.55, size = 29, normalized size = 0.85 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b x + a\right )}^{3} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 56, normalized size = 1.65 \begin {gather*} -\frac {a^2+3\,a\,b\,x+3\,b^2\,x^2}{3\,a^3\,b^3+9\,a^2\,b^4\,x+9\,a\,b^5\,x^2+3\,b^6\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x)^4,x)

[Out]

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

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