3.3.12 \(\int \frac {x^5}{(a+b x)^7} \, dx\) [212]

Optimal. Leaf size=17 \[ \frac {x^6}{6 a (a+b x)^6} \]

[Out]

1/6*x^6/a/(b*x+a)^6

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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^6}{6 a (a+b x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x)^7,x]

[Out]

x^6/(6*a*(a + b*x)^6)

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 {x^5}{(a+b x)^7} \, dx &=\frac {x^6}{6 a (a+b x)^6}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(17)=34\).
time = 0.01, size = 64, normalized size = 3.76 \begin {gather*} -\frac {a^5+6 a^4 b x+15 a^3 b^2 x^2+20 a^2 b^3 x^3+15 a b^4 x^4+6 b^5 x^5}{6 b^6 (a+b x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x)^7,x]

[Out]

-1/6*(a^5 + 6*a^4*b*x + 15*a^3*b^2*x^2 + 20*a^2*b^3*x^3 + 15*a*b^4*x^4 + 6*b^5*x^5)/(b^6*(a + b*x)^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(15)=30\).
time = 0.08, size = 87, normalized size = 5.12

method	result	size
gosper	\(-\frac {6 b^{5} x^{5}+15 a \,b^{4} x^{4}+20 a^{2} b^{3} x^{3}+15 a^{3} b^{2} x^{2}+6 a^{4} b x +a^{5}}{6 \left (b x +a \right )^{6} b^{6}}\)	\(63\)
norman	\(\frac {-\frac {x^{5}}{b}-\frac {5 a \,x^{4}}{2 b^{2}}-\frac {10 a^{2} x^{3}}{3 b^{3}}-\frac {5 a^{3} x^{2}}{2 b^{4}}-\frac {a^{4} x}{b^{5}}-\frac {a^{5}}{6 b^{6}}}{\left (b x +a \right )^{6}}\)	\(66\)
risch	\(\frac {-\frac {x^{5}}{b}-\frac {5 a \,x^{4}}{2 b^{2}}-\frac {10 a^{2} x^{3}}{3 b^{3}}-\frac {5 a^{3} x^{2}}{2 b^{4}}-\frac {a^{4} x}{b^{5}}-\frac {a^{5}}{6 b^{6}}}{\left (b x +a \right )^{6}}\)	\(66\)
default	\(-\frac {1}{b^{6} \left (b x +a \right )}+\frac {5 a^{3}}{2 b^{6} \left (b x +a \right )^{4}}+\frac {5 a}{2 b^{6} \left (b x +a \right )^{2}}+\frac {a^{5}}{6 b^{6} \left (b x +a \right )^{6}}-\frac {10 a^{2}}{3 b^{6} \left (b x +a \right )^{3}}-\frac {a^{4}}{b^{6} \left (b x +a \right )^{5}}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (15) = 30\).
time = 0.28, size = 120, normalized size = 7.06 \begin {gather*} -\frac {6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, {\left (b^{12} x^{6} + 6 \, a b^{11} x^{5} + 15 \, a^{2} b^{10} x^{4} + 20 \, a^{3} b^{9} x^{3} + 15 \, a^{4} b^{8} x^{2} + 6 \, a^{5} b^{7} x + a^{6} b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/(b^12*x^6 + 6*a*b^11*x^5 +
15*a^2*b^10*x^4 + 20*a^3*b^9*x^3 + 15*a^4*b^8*x^2 + 6*a^5*b^7*x + a^6*b^6)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (15) = 30\).
time = 0.63, size = 120, normalized size = 7.06 \begin {gather*} -\frac {6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, {\left (b^{12} x^{6} + 6 \, a b^{11} x^{5} + 15 \, a^{2} b^{10} x^{4} + 20 \, a^{3} b^{9} x^{3} + 15 \, a^{4} b^{8} x^{2} + 6 \, a^{5} b^{7} x + a^{6} b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/(b^12*x^6 + 6*a*b^11*x^5 +
15*a^2*b^10*x^4 + 20*a^3*b^9*x^3 + 15*a^4*b^8*x^2 + 6*a^5*b^7*x + a^6*b^6)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (12) = 24\).
time = 0.23, size = 128, normalized size = 7.53 \begin {gather*} \frac {- a^{5} - 6 a^{4} b x - 15 a^{3} b^{2} x^{2} - 20 a^{2} b^{3} x^{3} - 15 a b^{4} x^{4} - 6 b^{5} x^{5}}{6 a^{6} b^{6} + 36 a^{5} b^{7} x + 90 a^{4} b^{8} x^{2} + 120 a^{3} b^{9} x^{3} + 90 a^{2} b^{10} x^{4} + 36 a b^{11} x^{5} + 6 b^{12} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x+a)**7,x)

[Out]

(-a**5 - 6*a**4*b*x - 15*a**3*b**2*x**2 - 20*a**2*b**3*x**3 - 15*a*b**4*x**4 - 6*b**5*x**5)/(6*a**6*b**6 + 36*
a**5*b**7*x + 90*a**4*b**8*x**2 + 120*a**3*b**9*x**3 + 90*a**2*b**10*x**4 + 36*a*b**11*x**5 + 6*b**12*x**6)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (15) = 30\).
time = 1.03, size = 62, normalized size = 3.65 \begin {gather*} -\frac {6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, {\left (b x + a\right )}^{6} b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^7,x, algorithm="giac")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/((b*x + a)^6*b^6)

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Mupad [B]
time = 0.12, size = 72, normalized size = 4.24 \begin {gather*} \frac {\frac {5\,a}{2\,{\left (a+b\,x\right )}^2}-\frac {1}{a+b\,x}-\frac {10\,a^2}{3\,{\left (a+b\,x\right )}^3}+\frac {5\,a^3}{2\,{\left (a+b\,x\right )}^4}-\frac {a^4}{{\left (a+b\,x\right )}^5}+\frac {a^5}{6\,{\left (a+b\,x\right )}^6}}{b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b*x)^7,x)

[Out]

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

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