3.119 \(\int \frac{(a+b x)^7}{x^{13}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{b^4 (a+b x)^8}{3960 a^5 x^8}+\frac{b^3 (a+b x)^8}{495 a^4 x^9}-\frac{b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac{b (a+b x)^8}{33 a^2 x^{11}}-\frac{(a+b x)^8}{12 a x^{12}} \]

[Out]

-(a + b*x)^8/(12*a*x^12) + (b*(a + b*x)^8)/(33*a^2*x^11) - (b^2*(a + b*x)^8)/(110*a^3*x^10) + (b^3*(a + b*x)^8
)/(495*a^4*x^9) - (b^4*(a + b*x)^8)/(3960*a^5*x^8)

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Rubi [A]  time = 0.0255781, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ -\frac{b^4 (a+b x)^8}{3960 a^5 x^8}+\frac{b^3 (a+b x)^8}{495 a^4 x^9}-\frac{b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac{b (a+b x)^8}{33 a^2 x^{11}}-\frac{(a+b x)^8}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^13,x]

[Out]

-(a + b*x)^8/(12*a*x^12) + (b*(a + b*x)^8)/(33*a^2*x^11) - (b^2*(a + b*x)^8)/(110*a^3*x^10) + (b^3*(a + b*x)^8
)/(495*a^4*x^9) - (b^4*(a + b*x)^8)/(3960*a^5*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^{13}} \, dx &=-\frac{(a+b x)^8}{12 a x^{12}}-\frac{b \int \frac{(a+b x)^7}{x^{12}} \, dx}{3 a}\\ &=-\frac{(a+b x)^8}{12 a x^{12}}+\frac{b (a+b x)^8}{33 a^2 x^{11}}+\frac{b^2 \int \frac{(a+b x)^7}{x^{11}} \, dx}{11 a^2}\\ &=-\frac{(a+b x)^8}{12 a x^{12}}+\frac{b (a+b x)^8}{33 a^2 x^{11}}-\frac{b^2 (a+b x)^8}{110 a^3 x^{10}}-\frac{b^3 \int \frac{(a+b x)^7}{x^{10}} \, dx}{55 a^3}\\ &=-\frac{(a+b x)^8}{12 a x^{12}}+\frac{b (a+b x)^8}{33 a^2 x^{11}}-\frac{b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac{b^3 (a+b x)^8}{495 a^4 x^9}+\frac{b^4 \int \frac{(a+b x)^7}{x^9} \, dx}{495 a^4}\\ &=-\frac{(a+b x)^8}{12 a x^{12}}+\frac{b (a+b x)^8}{33 a^2 x^{11}}-\frac{b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac{b^3 (a+b x)^8}{495 a^4 x^9}-\frac{b^4 (a+b x)^8}{3960 a^5 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0059421, size = 93, normalized size = 0.97 \[ -\frac{21 a^5 b^2}{10 x^{10}}-\frac{35 a^4 b^3}{9 x^9}-\frac{35 a^3 b^4}{8 x^8}-\frac{3 a^2 b^5}{x^7}-\frac{7 a^6 b}{11 x^{11}}-\frac{a^7}{12 x^{12}}-\frac{7 a b^6}{6 x^6}-\frac{b^7}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^13,x]

[Out]

-a^7/(12*x^12) - (7*a^6*b)/(11*x^11) - (21*a^5*b^2)/(10*x^10) - (35*a^4*b^3)/(9*x^9) - (35*a^3*b^4)/(8*x^8) -
(3*a^2*b^5)/x^7 - (7*a*b^6)/(6*x^6) - b^7/(5*x^5)

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Maple [A]  time = 0.006, size = 80, normalized size = 0.8 \begin{align*} -{\frac{21\,{a}^{5}{b}^{2}}{10\,{x}^{10}}}-{\frac{{a}^{7}}{12\,{x}^{12}}}-{\frac{{b}^{7}}{5\,{x}^{5}}}-{\frac{7\,{a}^{6}b}{11\,{x}^{11}}}-{\frac{7\,a{b}^{6}}{6\,{x}^{6}}}-{\frac{35\,{a}^{3}{b}^{4}}{8\,{x}^{8}}}-3\,{\frac{{a}^{2}{b}^{5}}{{x}^{7}}}-{\frac{35\,{a}^{4}{b}^{3}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^13,x)

[Out]

-21/10*a^5*b^2/x^10-1/12*a^7/x^12-1/5*b^7/x^5-7/11*a^6*b/x^11-7/6*a*b^6/x^6-35/8*a^3*b^4/x^8-3*a^2*b^5/x^7-35/
9*a^4*b^3/x^9

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Maxima [A]  time = 1.01831, size = 107, normalized size = 1.11 \begin{align*} -\frac{792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^13,x, algorithm="maxima")

[Out]

-1/3960*(792*b^7*x^7 + 4620*a*b^6*x^6 + 11880*a^2*b^5*x^5 + 17325*a^3*b^4*x^4 + 15400*a^4*b^3*x^3 + 8316*a^5*b
^2*x^2 + 2520*a^6*b*x + 330*a^7)/x^12

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Fricas [A]  time = 1.75261, size = 201, normalized size = 2.09 \begin{align*} -\frac{792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^13,x, algorithm="fricas")

[Out]

-1/3960*(792*b^7*x^7 + 4620*a*b^6*x^6 + 11880*a^2*b^5*x^5 + 17325*a^3*b^4*x^4 + 15400*a^4*b^3*x^3 + 8316*a^5*b
^2*x^2 + 2520*a^6*b*x + 330*a^7)/x^12

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Sympy [A]  time = 0.958756, size = 85, normalized size = 0.89 \begin{align*} - \frac{330 a^{7} + 2520 a^{6} b x + 8316 a^{5} b^{2} x^{2} + 15400 a^{4} b^{3} x^{3} + 17325 a^{3} b^{4} x^{4} + 11880 a^{2} b^{5} x^{5} + 4620 a b^{6} x^{6} + 792 b^{7} x^{7}}{3960 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**13,x)

[Out]

-(330*a**7 + 2520*a**6*b*x + 8316*a**5*b**2*x**2 + 15400*a**4*b**3*x**3 + 17325*a**3*b**4*x**4 + 11880*a**2*b*
*5*x**5 + 4620*a*b**6*x**6 + 792*b**7*x**7)/(3960*x**12)

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Giac [A]  time = 1.18725, size = 107, normalized size = 1.11 \begin{align*} -\frac{792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^13,x, algorithm="giac")

[Out]

-1/3960*(792*b^7*x^7 + 4620*a*b^6*x^6 + 11880*a^2*b^5*x^5 + 17325*a^3*b^4*x^4 + 15400*a^4*b^3*x^3 + 8316*a^5*b
^2*x^2 + 2520*a^6*b*x + 330*a^7)/x^12