3.2.22 \(\int \frac {(a+b x)^7}{x^{16}} \, dx\) [122]

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

[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

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Rubi [A]
time = 0.02, antiderivative size = 95, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^16,x]

[Out]

-1/15*a^7/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 45

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*}

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Mathematica [A]
time = 0.00, size = 95, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^16,x]

[Out]

-1/15*a^7/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)

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Maple [A]
time = 0.08, size = 80, normalized size = 0.84

method	result	size
norman	\(\frac {-\frac {1}{15} a^{7}-\frac {1}{2} a^{6} b x -\frac {21}{13} a^{5} b^{2} x^{2}-\frac {35}{12} a^{4} b^{3} x^{3}-\frac {35}{11} a^{3} b^{4} x^{4}-\frac {21}{10} a^{2} b^{5} x^{5}-\frac {7}{9} a \,b^{6} x^{6}-\frac {1}{8} b^{7} x^{7}}{x^{15}}\)	\(79\)
risch	\(\frac {-\frac {1}{15} a^{7}-\frac {1}{2} a^{6} b x -\frac {21}{13} a^{5} b^{2} x^{2}-\frac {35}{12} a^{4} b^{3} x^{3}-\frac {35}{11} a^{3} b^{4} x^{4}-\frac {21}{10} a^{2} b^{5} x^{5}-\frac {7}{9} a \,b^{6} x^{6}-\frac {1}{8} b^{7} x^{7}}{x^{15}}\)	\(79\)
gosper	\(-\frac {6435 b^{7} x^{7}+40040 a \,b^{6} x^{6}+108108 a^{2} b^{5} x^{5}+163800 a^{3} b^{4} x^{4}+150150 a^{4} b^{3} x^{3}+83160 a^{5} b^{2} x^{2}+25740 a^{6} b x +3432 a^{7}}{51480 x^{15}}\)	\(80\)
default	\(-\frac {a^{7}}{15 x^{15}}-\frac {a^{6} b}{2 x^{14}}-\frac {21 a^{5} b^{2}}{13 x^{13}}-\frac {35 a^{4} b^{3}}{12 x^{12}}-\frac {35 a^{3} b^{4}}{11 x^{11}}-\frac {21 a^{2} b^{5}}{10 x^{10}}-\frac {7 a \,b^{6}}{9 x^{9}}-\frac {b^{7}}{8 x^{8}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]
time = 0.28, size = 79, normalized size = 0.83 \begin {gather*} -\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 {gather*}

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

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Fricas [A]
time = 1.06, size = 79, normalized size = 0.83 \begin {gather*} -\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 {gather*}

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

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Sympy [A]
time = 0.33, size = 85, normalized size = 0.89 \begin {gather*} \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 {gather*}

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)

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Giac [A]
time = 1.74, size = 79, normalized size = 0.83 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.11, size = 79, normalized size = 0.83 \begin {gather*} -\frac {\frac {a^7}{15}+\frac {a^6\,b\,x}{2}+\frac {21\,a^5\,b^2\,x^2}{13}+\frac {35\,a^4\,b^3\,x^3}{12}+\frac {35\,a^3\,b^4\,x^4}{11}+\frac {21\,a^2\,b^5\,x^5}{10}+\frac {7\,a\,b^6\,x^6}{9}+\frac {b^7\,x^7}{8}}{x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/x^16,x)

[Out]

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

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