3.2.20 \(\int \frac {(a+b x)^7}{x^{14}} \, dx\)

Optimal. Leaf size=93 \[ -\frac {a^7}{13 x^{13}}-\frac {7 a^6 b}{12 x^{12}}-\frac {21 a^5 b^2}{11 x^{11}}-\frac {7 a^4 b^3}{2 x^{10}}-\frac {35 a^3 b^4}{9 x^9}-\frac {21 a^2 b^5}{8 x^8}-\frac {a b^6}{x^7}-\frac {b^7}{6 x^6} \]

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Rubi [A]  time = 0.03, antiderivative size = 93, 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 = {43} \begin {gather*} -\frac {21 a^5 b^2}{11 x^{11}}-\frac {7 a^4 b^3}{2 x^{10}}-\frac {35 a^3 b^4}{9 x^9}-\frac {21 a^2 b^5}{8 x^8}-\frac {7 a^6 b}{12 x^{12}}-\frac {a^7}{13 x^{13}}-\frac {a b^6}{x^7}-\frac {b^7}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^14,x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 93, normalized size = 1.00 \begin {gather*} -\frac {a^7}{13 x^{13}}-\frac {7 a^6 b}{12 x^{12}}-\frac {21 a^5 b^2}{11 x^{11}}-\frac {7 a^4 b^3}{2 x^{10}}-\frac {35 a^3 b^4}{9 x^9}-\frac {21 a^2 b^5}{8 x^8}-\frac {a b^6}{x^7}-\frac {b^7}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^14,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^7}{x^{14}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^7/x^14,x]

[Out]

IntegrateAlgebraic[(a + b*x)^7/x^14, x]

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fricas [A]  time = 0.86, size = 79, normalized size = 0.85 \begin {gather*} -\frac {1716 \, b^{7} x^{7} + 10296 \, a b^{6} x^{6} + 27027 \, a^{2} b^{5} x^{5} + 40040 \, a^{3} b^{4} x^{4} + 36036 \, a^{4} b^{3} x^{3} + 19656 \, a^{5} b^{2} x^{2} + 6006 \, a^{6} b x + 792 \, a^{7}}{10296 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^14,x, algorithm="fricas")

[Out]

-1/10296*(1716*b^7*x^7 + 10296*a*b^6*x^6 + 27027*a^2*b^5*x^5 + 40040*a^3*b^4*x^4 + 36036*a^4*b^3*x^3 + 19656*a
^5*b^2*x^2 + 6006*a^6*b*x + 792*a^7)/x^13

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giac [A]  time = 1.04, size = 79, normalized size = 0.85 \begin {gather*} -\frac {1716 \, b^{7} x^{7} + 10296 \, a b^{6} x^{6} + 27027 \, a^{2} b^{5} x^{5} + 40040 \, a^{3} b^{4} x^{4} + 36036 \, a^{4} b^{3} x^{3} + 19656 \, a^{5} b^{2} x^{2} + 6006 \, a^{6} b x + 792 \, a^{7}}{10296 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^14,x, algorithm="giac")

[Out]

-1/10296*(1716*b^7*x^7 + 10296*a*b^6*x^6 + 27027*a^2*b^5*x^5 + 40040*a^3*b^4*x^4 + 36036*a^4*b^3*x^3 + 19656*a
^5*b^2*x^2 + 6006*a^6*b*x + 792*a^7)/x^13

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maple [A]  time = 0.00, size = 80, normalized size = 0.86 \begin {gather*} -\frac {b^{7}}{6 x^{6}}-\frac {a \,b^{6}}{x^{7}}-\frac {21 a^{2} b^{5}}{8 x^{8}}-\frac {35 a^{3} b^{4}}{9 x^{9}}-\frac {7 a^{4} b^{3}}{2 x^{10}}-\frac {21 a^{5} b^{2}}{11 x^{11}}-\frac {7 a^{6} b}{12 x^{12}}-\frac {a^{7}}{13 x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^14,x)

[Out]

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

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maxima [A]  time = 1.37, size = 79, normalized size = 0.85 \begin {gather*} -\frac {1716 \, b^{7} x^{7} + 10296 \, a b^{6} x^{6} + 27027 \, a^{2} b^{5} x^{5} + 40040 \, a^{3} b^{4} x^{4} + 36036 \, a^{4} b^{3} x^{3} + 19656 \, a^{5} b^{2} x^{2} + 6006 \, a^{6} b x + 792 \, a^{7}}{10296 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^14,x, algorithm="maxima")

[Out]

-1/10296*(1716*b^7*x^7 + 10296*a*b^6*x^6 + 27027*a^2*b^5*x^5 + 40040*a^3*b^4*x^4 + 36036*a^4*b^3*x^3 + 19656*a
^5*b^2*x^2 + 6006*a^6*b*x + 792*a^7)/x^13

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mupad [B]  time = 0.07, size = 78, normalized size = 0.84 \begin {gather*} -\frac {\frac {a^7}{13}+\frac {7\,a^6\,b\,x}{12}+\frac {21\,a^5\,b^2\,x^2}{11}+\frac {7\,a^4\,b^3\,x^3}{2}+\frac {35\,a^3\,b^4\,x^4}{9}+\frac {21\,a^2\,b^5\,x^5}{8}+a\,b^6\,x^6+\frac {b^7\,x^7}{6}}{x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/x^14,x)

[Out]

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

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sympy [A]  time = 0.78, size = 85, normalized size = 0.91 \begin {gather*} \frac {- 792 a^{7} - 6006 a^{6} b x - 19656 a^{5} b^{2} x^{2} - 36036 a^{4} b^{3} x^{3} - 40040 a^{3} b^{4} x^{4} - 27027 a^{2} b^{5} x^{5} - 10296 a b^{6} x^{6} - 1716 b^{7} x^{7}}{10296 x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**14,x)

[Out]

(-792*a**7 - 6006*a**6*b*x - 19656*a**5*b**2*x**2 - 36036*a**4*b**3*x**3 - 40040*a**3*b**4*x**4 - 27027*a**2*b
**5*x**5 - 10296*a*b**6*x**6 - 1716*b**7*x**7)/(10296*x**13)

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