3.118 \(\int \frac{(a+b x)^7}{x^{12}} \, dx\)

Optimal. Leaf size=76 \[ \frac{b^3 (a+b x)^8}{1320 a^4 x^8}-\frac{b^2 (a+b x)^8}{165 a^3 x^9}+\frac{3 b (a+b x)^8}{110 a^2 x^{10}}-\frac{(a+b x)^8}{11 a x^{11}} \]

[Out]

-(a + b*x)^8/(11*a*x^11) + (3*b*(a + b*x)^8)/(110*a^2*x^10) - (b^2*(a + b*x)^8)/(165*a^3*x^9) + (b^3*(a + b*x)
^8)/(1320*a^4*x^8)

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Rubi [A]  time = 0.0163322, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ \frac{b^3 (a+b x)^8}{1320 a^4 x^8}-\frac{b^2 (a+b x)^8}{165 a^3 x^9}+\frac{3 b (a+b x)^8}{110 a^2 x^{10}}-\frac{(a+b x)^8}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^12,x]

[Out]

-(a + b*x)^8/(11*a*x^11) + (3*b*(a + b*x)^8)/(110*a^2*x^10) - (b^2*(a + b*x)^8)/(165*a^3*x^9) + (b^3*(a + b*x)
^8)/(1320*a^4*x^8)

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)^7}{x^{12}} \, dx &=-\frac{(a+b x)^8}{11 a x^{11}}-\frac{(3 b) \int \frac{(a+b x)^7}{x^{11}} \, dx}{11 a}\\ &=-\frac{(a+b x)^8}{11 a x^{11}}+\frac{3 b (a+b x)^8}{110 a^2 x^{10}}+\frac{\left (3 b^2\right ) \int \frac{(a+b x)^7}{x^{10}} \, dx}{55 a^2}\\ &=-\frac{(a+b x)^8}{11 a x^{11}}+\frac{3 b (a+b x)^8}{110 a^2 x^{10}}-\frac{b^2 (a+b x)^8}{165 a^3 x^9}-\frac{b^3 \int \frac{(a+b x)^7}{x^9} \, dx}{165 a^3}\\ &=-\frac{(a+b x)^8}{11 a x^{11}}+\frac{3 b (a+b x)^8}{110 a^2 x^{10}}-\frac{b^2 (a+b x)^8}{165 a^3 x^9}+\frac{b^3 (a+b x)^8}{1320 a^4 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0081419, size = 93, normalized size = 1.22 \[ -\frac{7 a^5 b^2}{3 x^9}-\frac{35 a^4 b^3}{8 x^8}-\frac{5 a^3 b^4}{x^7}-\frac{7 a^2 b^5}{2 x^6}-\frac{7 a^6 b}{10 x^{10}}-\frac{a^7}{11 x^{11}}-\frac{7 a b^6}{5 x^5}-\frac{b^7}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^12,x]

[Out]

-a^7/(11*x^11) - (7*a^6*b)/(10*x^10) - (7*a^5*b^2)/(3*x^9) - (35*a^4*b^3)/(8*x^8) - (5*a^3*b^4)/x^7 - (7*a^2*b
^5)/(2*x^6) - (7*a*b^6)/(5*x^5) - b^7/(4*x^4)

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Maple [A]  time = 0.006, size = 80, normalized size = 1.1 \begin{align*} -{\frac{7\,{a}^{6}b}{10\,{x}^{10}}}-{\frac{7\,a{b}^{6}}{5\,{x}^{5}}}-{\frac{{a}^{7}}{11\,{x}^{11}}}-{\frac{{b}^{7}}{4\,{x}^{4}}}-{\frac{7\,{a}^{2}{b}^{5}}{2\,{x}^{6}}}-{\frac{35\,{a}^{4}{b}^{3}}{8\,{x}^{8}}}-5\,{\frac{{a}^{3}{b}^{4}}{{x}^{7}}}-{\frac{7\,{a}^{5}{b}^{2}}{3\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^12,x)

[Out]

-7/10*a^6*b/x^10-7/5*a*b^6/x^5-1/11*a^7/x^11-1/4*b^7/x^4-7/2*a^2*b^5/x^6-35/8*a^4*b^3/x^8-5*a^3*b^4/x^7-7/3*a^
5*b^2/x^9

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Maxima [A]  time = 1.10873, size = 107, normalized size = 1.41 \begin{align*} -\frac{330 \, b^{7} x^{7} + 1848 \, a b^{6} x^{6} + 4620 \, a^{2} b^{5} x^{5} + 6600 \, a^{3} b^{4} x^{4} + 5775 \, a^{4} b^{3} x^{3} + 3080 \, a^{5} b^{2} x^{2} + 924 \, a^{6} b x + 120 \, a^{7}}{1320 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^12,x, algorithm="maxima")

[Out]

-1/1320*(330*b^7*x^7 + 1848*a*b^6*x^6 + 4620*a^2*b^5*x^5 + 6600*a^3*b^4*x^4 + 5775*a^4*b^3*x^3 + 3080*a^5*b^2*
x^2 + 924*a^6*b*x + 120*a^7)/x^11

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Fricas [A]  time = 1.78941, size = 196, normalized size = 2.58 \begin{align*} -\frac{330 \, b^{7} x^{7} + 1848 \, a b^{6} x^{6} + 4620 \, a^{2} b^{5} x^{5} + 6600 \, a^{3} b^{4} x^{4} + 5775 \, a^{4} b^{3} x^{3} + 3080 \, a^{5} b^{2} x^{2} + 924 \, a^{6} b x + 120 \, a^{7}}{1320 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^12,x, algorithm="fricas")

[Out]

-1/1320*(330*b^7*x^7 + 1848*a*b^6*x^6 + 4620*a^2*b^5*x^5 + 6600*a^3*b^4*x^4 + 5775*a^4*b^3*x^3 + 3080*a^5*b^2*
x^2 + 924*a^6*b*x + 120*a^7)/x^11

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Sympy [A]  time = 0.952636, size = 85, normalized size = 1.12 \begin{align*} - \frac{120 a^{7} + 924 a^{6} b x + 3080 a^{5} b^{2} x^{2} + 5775 a^{4} b^{3} x^{3} + 6600 a^{3} b^{4} x^{4} + 4620 a^{2} b^{5} x^{5} + 1848 a b^{6} x^{6} + 330 b^{7} x^{7}}{1320 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**12,x)

[Out]

-(120*a**7 + 924*a**6*b*x + 3080*a**5*b**2*x**2 + 5775*a**4*b**3*x**3 + 6600*a**3*b**4*x**4 + 4620*a**2*b**5*x
**5 + 1848*a*b**6*x**6 + 330*b**7*x**7)/(1320*x**11)

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Giac [A]  time = 1.21208, size = 107, normalized size = 1.41 \begin{align*} -\frac{330 \, b^{7} x^{7} + 1848 \, a b^{6} x^{6} + 4620 \, a^{2} b^{5} x^{5} + 6600 \, a^{3} b^{4} x^{4} + 5775 \, a^{4} b^{3} x^{3} + 3080 \, a^{5} b^{2} x^{2} + 924 \, a^{6} b x + 120 \, a^{7}}{1320 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^12,x, algorithm="giac")

[Out]

-1/1320*(330*b^7*x^7 + 1848*a*b^6*x^6 + 4620*a^2*b^5*x^5 + 6600*a^3*b^4*x^4 + 5775*a^4*b^3*x^3 + 3080*a^5*b^2*
x^2 + 924*a^6*b*x + 120*a^7)/x^11