3.11.36 \(\int \frac {a+b x}{(a c-b c x)^5} \, dx\) [1036]

Optimal. Leaf size=38 \[ \frac {a}{2 b c^5 (a-b x)^4}-\frac {1}{3 b c^5 (a-b x)^3} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \begin {gather*} \frac {a}{2 b c^5 (a-b x)^4}-\frac {1}{3 b c^5 (a-b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(a*c - b*c*x)^5,x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 24, normalized size = 0.63 \begin {gather*} \frac {a+2 b x}{6 b c^5 (a-b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(a*c - b*c*x)^5,x]

[Out]

(a + 2*b*x)/(6*b*c^5*(a - b*x)^4)

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Maple [A]
time = 0.13, size = 33, normalized size = 0.87

method	result	size
gosper	\(\frac {2 b x +a}{6 \left (-b x +a \right )^{4} c^{5} b}\)	\(23\)
risch	\(\frac {\frac {x}{3}+\frac {a}{6 b}}{c^{5} \left (-b x +a \right )^{4}}\)	\(23\)
norman	\(\frac {\frac {a}{6 b c}+\frac {x}{3 c}}{c^{4} \left (-b x +a \right )^{4}}\)	\(29\)
default	\(\frac {-\frac {1}{3 b \left (-b x +a \right )^{3}}+\frac {a}{2 b \left (-b x +a \right )^{4}}}{c^{5}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(-b*c*x+a*c)^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 67, normalized size = 1.76 \begin {gather*} \frac {2 \, b x + a}{6 \, {\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.42, size = 67, normalized size = 1.76 \begin {gather*} \frac {2 \, b x + a}{6 \, {\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^5,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (29) = 58\).
time = 0.17, size = 73, normalized size = 1.92 \begin {gather*} - \frac {- a - 2 b x}{6 a^{4} b c^{5} - 24 a^{3} b^{2} c^{5} x + 36 a^{2} b^{3} c^{5} x^{2} - 24 a b^{4} c^{5} x^{3} + 6 b^{5} c^{5} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)**5,x)

[Out]

-(-a - 2*b*x)/(6*a**4*b*c**5 - 24*a**3*b**2*c**5*x + 36*a**2*b**3*c**5*x**2 - 24*a*b**4*c**5*x**3 + 6*b**5*c**
5*x**4)

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Giac [A]
time = 1.53, size = 40, normalized size = 1.05 \begin {gather*} \frac {a}{2 \, {\left (b c x - a c\right )}^{4} b c} + \frac {1}{3 \, {\left (b c x - a c\right )}^{3} b c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^5,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.17, size = 67, normalized size = 1.76 \begin {gather*} \frac {\frac {x}{3}+\frac {a}{6\,b}}{a^4\,c^5-4\,a^3\,b\,c^5\,x+6\,a^2\,b^2\,c^5\,x^2-4\,a\,b^3\,c^5\,x^3+b^4\,c^5\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(a*c - b*c*x)^5,x)

[Out]

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

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