3.122 \(\int \frac{(a+b x)^7}{x^{16}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{21 a^5 b^2}{13 x^{13}}-\frac{35 a^4 b^3}{12 x^{12}}-\frac{35 a^3 b^4}{11 x^{11}}-\frac{21 a^2 b^5}{10 x^{10}}-\frac{a^6 b}{2 x^{14}}-\frac{a^7}{15 x^{15}}-\frac{7 a b^6}{9 x^9}-\frac{b^7}{8 x^8} \]

[Out]

-a^7/(15*x^15) - (a^6*b)/(2*x^14) - (21*a^5*b^2)/(13*x^13) - (35*a^4*b^3)/(12*x^12) - (35*a^3*b^4)/(11*x^11) -
(21*a^2*b^5)/(10*x^10) - (7*a*b^6)/(9*x^9) - b^7/(8*x^8)

________________________________________________________________________________________

Rubi [A]  time = 0.0304734, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ -\frac{21 a^5 b^2}{13 x^{13}}-\frac{35 a^4 b^3}{12 x^{12}}-\frac{35 a^3 b^4}{11 x^{11}}-\frac{21 a^2 b^5}{10 x^{10}}-\frac{a^6 b}{2 x^{14}}-\frac{a^7}{15 x^{15}}-\frac{7 a b^6}{9 x^9}-\frac{b^7}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^16,x]

[Out]

-a^7/(15*x^15) - (a^6*b)/(2*x^14) - (21*a^5*b^2)/(13*x^13) - (35*a^4*b^3)/(12*x^12) - (35*a^3*b^4)/(11*x^11) -
(21*a^2*b^5)/(10*x^10) - (7*a*b^6)/(9*x^9) - b^7/(8*x^8)

Rule 43

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)^7}{x^{16}} \, dx &=\int \left (\frac{a^7}{x^{16}}+\frac{7 a^6 b}{x^{15}}+\frac{21 a^5 b^2}{x^{14}}+\frac{35 a^4 b^3}{x^{13}}+\frac{35 a^3 b^4}{x^{12}}+\frac{21 a^2 b^5}{x^{11}}+\frac{7 a b^6}{x^{10}}+\frac{b^7}{x^9}\right ) \, dx\\ &=-\frac{a^7}{15 x^{15}}-\frac{a^6 b}{2 x^{14}}-\frac{21 a^5 b^2}{13 x^{13}}-\frac{35 a^4 b^3}{12 x^{12}}-\frac{35 a^3 b^4}{11 x^{11}}-\frac{21 a^2 b^5}{10 x^{10}}-\frac{7 a b^6}{9 x^9}-\frac{b^7}{8 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0036613, size = 95, normalized size = 1. \[ -\frac{21 a^5 b^2}{13 x^{13}}-\frac{35 a^4 b^3}{12 x^{12}}-\frac{35 a^3 b^4}{11 x^{11}}-\frac{21 a^2 b^5}{10 x^{10}}-\frac{a^6 b}{2 x^{14}}-\frac{a^7}{15 x^{15}}-\frac{7 a b^6}{9 x^9}-\frac{b^7}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^16,x]

[Out]

-a^7/(15*x^15) - (a^6*b)/(2*x^14) - (21*a^5*b^2)/(13*x^13) - (35*a^4*b^3)/(12*x^12) - (35*a^3*b^4)/(11*x^11) -
(21*a^2*b^5)/(10*x^10) - (7*a*b^6)/(9*x^9) - b^7/(8*x^8)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 80, normalized size = 0.8 \begin{align*} -{\frac{{a}^{7}}{15\,{x}^{15}}}-{\frac{{a}^{6}b}{2\,{x}^{14}}}-{\frac{21\,{a}^{5}{b}^{2}}{13\,{x}^{13}}}-{\frac{35\,{a}^{4}{b}^{3}}{12\,{x}^{12}}}-{\frac{35\,{a}^{3}{b}^{4}}{11\,{x}^{11}}}-{\frac{21\,{a}^{2}{b}^{5}}{10\,{x}^{10}}}-{\frac{7\,a{b}^{6}}{9\,{x}^{9}}}-{\frac{{b}^{7}}{8\,{x}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^16,x)

[Out]

-1/15*a^7/x^15-1/2*a^6*b/x^14-21/13*a^5*b^2/x^13-35/12*a^4*b^3/x^12-35/11*a^3*b^4/x^11-21/10*a^2*b^5/x^10-7/9*
a*b^6/x^9-1/8*b^7/x^8

________________________________________________________________________________________

Maxima [A]  time = 1.03825, size = 107, normalized size = 1.13 \begin{align*} -\frac{6435 \, b^{7} x^{7} + 40040 \, a b^{6} x^{6} + 108108 \, a^{2} b^{5} x^{5} + 163800 \, a^{3} b^{4} x^{4} + 150150 \, a^{4} b^{3} x^{3} + 83160 \, a^{5} b^{2} x^{2} + 25740 \, a^{6} b x + 3432 \, a^{7}}{51480 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^16,x, algorithm="maxima")

[Out]

-1/51480*(6435*b^7*x^7 + 40040*a*b^6*x^6 + 108108*a^2*b^5*x^5 + 163800*a^3*b^4*x^4 + 150150*a^4*b^3*x^3 + 8316
0*a^5*b^2*x^2 + 25740*a^6*b*x + 3432*a^7)/x^15

________________________________________________________________________________________

Fricas [A]  time = 1.62739, size = 213, normalized size = 2.24 \begin{align*} -\frac{6435 \, b^{7} x^{7} + 40040 \, a b^{6} x^{6} + 108108 \, a^{2} b^{5} x^{5} + 163800 \, a^{3} b^{4} x^{4} + 150150 \, a^{4} b^{3} x^{3} + 83160 \, a^{5} b^{2} x^{2} + 25740 \, a^{6} b x + 3432 \, a^{7}}{51480 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^16,x, algorithm="fricas")

[Out]

-1/51480*(6435*b^7*x^7 + 40040*a*b^6*x^6 + 108108*a^2*b^5*x^5 + 163800*a^3*b^4*x^4 + 150150*a^4*b^3*x^3 + 8316
0*a^5*b^2*x^2 + 25740*a^6*b*x + 3432*a^7)/x^15

________________________________________________________________________________________

Sympy [A]  time = 1.09255, size = 85, normalized size = 0.89 \begin{align*} - \frac{3432 a^{7} + 25740 a^{6} b x + 83160 a^{5} b^{2} x^{2} + 150150 a^{4} b^{3} x^{3} + 163800 a^{3} b^{4} x^{4} + 108108 a^{2} b^{5} x^{5} + 40040 a b^{6} x^{6} + 6435 b^{7} x^{7}}{51480 x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**16,x)

[Out]

-(3432*a**7 + 25740*a**6*b*x + 83160*a**5*b**2*x**2 + 150150*a**4*b**3*x**3 + 163800*a**3*b**4*x**4 + 108108*a
**2*b**5*x**5 + 40040*a*b**6*x**6 + 6435*b**7*x**7)/(51480*x**15)

________________________________________________________________________________________

Giac [A]  time = 1.20802, size = 107, normalized size = 1.13 \begin{align*} -\frac{6435 \, b^{7} x^{7} + 40040 \, a b^{6} x^{6} + 108108 \, a^{2} b^{5} x^{5} + 163800 \, a^{3} b^{4} x^{4} + 150150 \, a^{4} b^{3} x^{3} + 83160 \, a^{5} b^{2} x^{2} + 25740 \, a^{6} b x + 3432 \, a^{7}}{51480 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^16,x, algorithm="giac")

[Out]

-1/51480*(6435*b^7*x^7 + 40040*a*b^6*x^6 + 108108*a^2*b^5*x^5 + 163800*a^3*b^4*x^4 + 150150*a^4*b^3*x^3 + 8316
0*a^5*b^2*x^2 + 25740*a^6*b*x + 3432*a^7)/x^15