3.11.47 \(\int \frac {(a+b x)^2}{(a c-b c x)^6} \, dx\) [1047]

Optimal. Leaf size=57 \[ \frac {4 a^2}{5 b c^6 (a-b x)^5}-\frac {a}{b c^6 (a-b x)^4}+\frac {1}{3 b c^6 (a-b x)^3} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {45} \begin {gather*} \frac {4 a^2}{5 b c^6 (a-b x)^5}-\frac {a}{b c^6 (a-b x)^4}+\frac {1}{3 b c^6 (a-b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^6,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {(a+b x)^2}{(a c-b c x)^6} \, dx &=\int \left (\frac {4 a^2}{c^6 (a-b x)^6}-\frac {4 a}{c^6 (a-b x)^5}+\frac {1}{c^6 (a-b x)^4}\right ) \, dx\\ &=\frac {4 a^2}{5 b c^6 (a-b x)^5}-\frac {a}{b c^6 (a-b x)^4}+\frac {1}{3 b c^6 (a-b x)^3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 38, normalized size = 0.67 \begin {gather*} -\frac {2 a^2+5 a b x+5 b^2 x^2}{15 b c^6 (-a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(a*c - b*c*x)^6,x]

[Out]

-1/15*(2*a^2 + 5*a*b*x + 5*b^2*x^2)/(b*c^6*(-a + b*x)^5)

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Maple [A]
time = 0.15, size = 49, normalized size = 0.86

method	result	size
risch	\(\frac {\frac {x^{2} b}{3}+\frac {a x}{3}+\frac {2 a^{2}}{15 b}}{c^{6} \left (-b x +a \right )^{5}}\)	\(32\)
gosper	\(\frac {5 x^{2} b^{2}+5 a b x +2 a^{2}}{15 \left (-b x +a \right )^{5} c^{6} b}\)	\(36\)
norman	\(\frac {\frac {2 a^{2}}{15 b c}+\frac {b \,x^{2}}{3 c}+\frac {a x}{3 c}}{c^{5} \left (-b x +a \right )^{5}}\)	\(41\)
default	\(\frac {\frac {1}{3 b \left (-b x +a \right )^{3}}+\frac {4 a^{2}}{5 b \left (-b x +a \right )^{5}}-\frac {a}{b \left (-b x +a \right )^{4}}}{c^{6}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/(-b*c*x+a*c)^6,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 95, normalized size = 1.67 \begin {gather*} -\frac {5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \, {\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(-b*c*x+a*c)^6,x, algorithm="maxima")

[Out]

-1/15*(5*b^2*x^2 + 5*a*b*x + 2*a^2)/(b^6*c^6*x^5 - 5*a*b^5*c^6*x^4 + 10*a^2*b^4*c^6*x^3 - 10*a^3*b^3*c^6*x^2 +
5*a^4*b^2*c^6*x - a^5*b*c^6)

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Fricas [A]
time = 0.42, size = 95, normalized size = 1.67 \begin {gather*} -\frac {5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \, {\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(-b*c*x+a*c)^6,x, algorithm="fricas")

[Out]

-1/15*(5*b^2*x^2 + 5*a*b*x + 2*a^2)/(b^6*c^6*x^5 - 5*a*b^5*c^6*x^4 + 10*a^2*b^4*c^6*x^3 - 10*a^3*b^3*c^6*x^2 +
5*a^4*b^2*c^6*x - a^5*b*c^6)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (46) = 92\).
time = 0.26, size = 100, normalized size = 1.75 \begin {gather*} \frac {- 2 a^{2} - 5 a b x - 5 b^{2} x^{2}}{- 15 a^{5} b c^{6} + 75 a^{4} b^{2} c^{6} x - 150 a^{3} b^{3} c^{6} x^{2} + 150 a^{2} b^{4} c^{6} x^{3} - 75 a b^{5} c^{6} x^{4} + 15 b^{6} c^{6} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/(-b*c*x+a*c)**6,x)

[Out]

(-2*a**2 - 5*a*b*x - 5*b**2*x**2)/(-15*a**5*b*c**6 + 75*a**4*b**2*c**6*x - 150*a**3*b**3*c**6*x**2 + 150*a**2*
b**4*c**6*x**3 - 75*a*b**5*c**6*x**4 + 15*b**6*c**6*x**5)

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Giac [A]
time = 1.33, size = 36, normalized size = 0.63 \begin {gather*} -\frac {5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \, {\left (b x - a\right )}^{5} b c^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(-b*c*x+a*c)^6,x, algorithm="giac")

[Out]

-1/15*(5*b^2*x^2 + 5*a*b*x + 2*a^2)/((b*x - a)^5*b*c^6)

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Mupad [B]
time = 0.19, size = 91, normalized size = 1.60 \begin {gather*} \frac {\frac {a\,x}{3}+\frac {b\,x^2}{3}+\frac {2\,a^2}{15\,b}}{a^5\,c^6-5\,a^4\,b\,c^6\,x+10\,a^3\,b^2\,c^6\,x^2-10\,a^2\,b^3\,c^6\,x^3+5\,a\,b^4\,c^6\,x^4-b^5\,c^6\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/(a*c - b*c*x)^6,x)

[Out]

((a*x)/3 + (b*x^2)/3 + (2*a^2)/(15*b))/(a^5*c^6 - b^5*c^6*x^5 + 5*a*b^4*c^6*x^4 + 10*a^3*b^2*c^6*x^2 - 10*a^2*
b^3*c^6*x^3 - 5*a^4*b*c^6*x)

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