3.245 \(\int \frac{(a+b x)^6}{x^{10}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{b^2 (a+b x)^7}{252 a^3 x^7}+\frac{b (a+b x)^7}{36 a^2 x^8}-\frac{(a+b x)^7}{9 a x^9} \]

[Out]

-(a + b*x)^7/(9*a*x^9) + (b*(a + b*x)^7)/(36*a^2*x^8) - (b^2*(a + b*x)^7)/(252*a^3*x^7)

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Rubi [A]  time = 0.0101443, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ -\frac{b^2 (a+b x)^7}{252 a^3 x^7}+\frac{b (a+b x)^7}{36 a^2 x^8}-\frac{(a+b x)^7}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6/x^10,x]

[Out]

-(a + b*x)^7/(9*a*x^9) + (b*(a + b*x)^7)/(36*a^2*x^8) - (b^2*(a + b*x)^7)/(252*a^3*x^7)

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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)^6}{x^{10}} \, dx &=-\frac{(a+b x)^7}{9 a x^9}-\frac{(2 b) \int \frac{(a+b x)^6}{x^9} \, dx}{9 a}\\ &=-\frac{(a+b x)^7}{9 a x^9}+\frac{b (a+b x)^7}{36 a^2 x^8}+\frac{b^2 \int \frac{(a+b x)^6}{x^8} \, dx}{36 a^2}\\ &=-\frac{(a+b x)^7}{9 a x^9}+\frac{b (a+b x)^7}{36 a^2 x^8}-\frac{b^2 (a+b x)^7}{252 a^3 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0093772, size = 80, normalized size = 1.43 \[ -\frac{15 a^4 b^2}{7 x^7}-\frac{10 a^3 b^3}{3 x^6}-\frac{3 a^2 b^4}{x^5}-\frac{3 a^5 b}{4 x^8}-\frac{a^6}{9 x^9}-\frac{3 a b^5}{2 x^4}-\frac{b^6}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6/x^10,x]

[Out]

-a^6/(9*x^9) - (3*a^5*b)/(4*x^8) - (15*a^4*b^2)/(7*x^7) - (10*a^3*b^3)/(3*x^6) - (3*a^2*b^4)/x^5 - (3*a*b^5)/(
2*x^4) - b^6/(3*x^3)

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Maple [A]  time = 0.005, size = 69, normalized size = 1.2 \begin{align*} -{\frac{{b}^{6}}{3\,{x}^{3}}}-3\,{\frac{{a}^{2}{b}^{4}}{{x}^{5}}}-{\frac{3\,a{b}^{5}}{2\,{x}^{4}}}-{\frac{10\,{a}^{3}{b}^{3}}{3\,{x}^{6}}}-{\frac{3\,{a}^{5}b}{4\,{x}^{8}}}-{\frac{15\,{a}^{4}{b}^{2}}{7\,{x}^{7}}}-{\frac{{a}^{6}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6/x^10,x)

[Out]

-1/3*b^6/x^3-3*a^2*b^4/x^5-3/2*a*b^5/x^4-10/3*a^3*b^3/x^6-3/4*a^5*b/x^8-15/7*a^4*b^2/x^7-1/9*a^6/x^9

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Maxima [A]  time = 1.08964, size = 92, normalized size = 1.64 \begin{align*} -\frac{84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}}{252 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/x^10,x, algorithm="maxima")

[Out]

-1/252*(84*b^6*x^6 + 378*a*b^5*x^5 + 756*a^2*b^4*x^4 + 840*a^3*b^3*x^3 + 540*a^4*b^2*x^2 + 189*a^5*b*x + 28*a^
6)/x^9

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Fricas [A]  time = 1.56025, size = 159, normalized size = 2.84 \begin{align*} -\frac{84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}}{252 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/x^10,x, algorithm="fricas")

[Out]

-1/252*(84*b^6*x^6 + 378*a*b^5*x^5 + 756*a^2*b^4*x^4 + 840*a^3*b^3*x^3 + 540*a^4*b^2*x^2 + 189*a^5*b*x + 28*a^
6)/x^9

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Sympy [A]  time = 0.863066, size = 73, normalized size = 1.3 \begin{align*} - \frac{28 a^{6} + 189 a^{5} b x + 540 a^{4} b^{2} x^{2} + 840 a^{3} b^{3} x^{3} + 756 a^{2} b^{4} x^{4} + 378 a b^{5} x^{5} + 84 b^{6} x^{6}}{252 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6/x**10,x)

[Out]

-(28*a**6 + 189*a**5*b*x + 540*a**4*b**2*x**2 + 840*a**3*b**3*x**3 + 756*a**2*b**4*x**4 + 378*a*b**5*x**5 + 84
*b**6*x**6)/(252*x**9)

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Giac [A]  time = 1.15094, size = 92, normalized size = 1.64 \begin{align*} -\frac{84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}}{252 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/x^10,x, algorithm="giac")

[Out]

-1/252*(84*b^6*x^6 + 378*a*b^5*x^5 + 756*a^2*b^4*x^4 + 840*a^3*b^3*x^3 + 540*a^4*b^2*x^2 + 189*a^5*b*x + 28*a^
6)/x^9