3.11.45 \(\int \frac {(a+b x)^2}{(a c-b c x)^4} \, dx\) [1045]

Optimal. Leaf size=28 \[ \frac {(a+b x)^3}{6 a b c^4 (a-b x)^3} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \begin {gather*} \frac {(a+b x)^3}{6 a b c^4 (a-b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^4,x]

[Out]

(a + b*x)^3/(6*a*b*c^4*(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 {align*} \int \frac {(a+b x)^2}{(a c-b c x)^4} \, dx &=\frac {(a+b x)^3}{6 a b c^4 (a-b x)^3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.11 \begin {gather*} -\frac {a^2+3 b^2 x^2}{3 b c^4 (-a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(a*c - b*c*x)^4,x]

[Out]

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

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Maple [A]
time = 0.14, size = 48, normalized size = 1.71

method	result	size
risch	\(\frac {x^{2} b +\frac {a^{2}}{3 b}}{c^{4} \left (-b x +a \right )^{3}}\)	\(27\)
gosper	\(\frac {3 x^{2} b^{2}+a^{2}}{3 \left (-b x +a \right )^{3} c^{4} b}\)	\(29\)
norman	\(\frac {\frac {a^{2}}{3 b c}+\frac {b \,x^{2}}{c}}{c^{3} \left (-b x +a \right )^{3}}\)	\(33\)
default	\(\frac {-\frac {2 a}{b \left (-b x +a \right )^{2}}+\frac {4 a^{2}}{3 b \left (-b x +a \right )^{3}}+\frac {1}{b \left (-b x +a \right )}}{c^{4}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).
time = 0.30, size = 60, normalized size = 2.14 \begin {gather*} -\frac {3 \, b^{2} x^{2} + a^{2}}{3 \, {\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).
time = 0.41, size = 60, normalized size = 2.14 \begin {gather*} -\frac {3 \, b^{2} x^{2} + a^{2}}{3 \, {\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
time = 0.15, size = 61, normalized size = 2.18 \begin {gather*} \frac {- a^{2} - 3 b^{2} x^{2}}{- 3 a^{3} b c^{4} + 9 a^{2} b^{2} c^{4} x - 9 a b^{3} c^{4} x^{2} + 3 b^{4} c^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.08, size = 29, normalized size = 1.04 \begin {gather*} -\frac {3 \, b^{2} x^{2} + a^{2}}{3 \, {\left (b x - a\right )}^{3} b c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.05, size = 58, normalized size = 2.07 \begin {gather*} \frac {b\,x^2+\frac {a^2}{3\,b}}{a^3\,c^4-3\,a^2\,b\,c^4\,x+3\,a\,b^2\,c^4\,x^2-b^3\,c^4\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/(a*c - b*c*x)^4,x)

[Out]

(b*x^2 + a^2/(3*b))/(a^3*c^4 - b^3*c^4*x^3 + 3*a*b^2*c^4*x^2 - 3*a^2*b*c^4*x)

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