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

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

[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)

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Rubi [A]  time = 0.0328734, antiderivative size = 93, 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} \[ -\frac{21 a^5 b^2}{11 x^{11}}-\frac{7 a^4 b^3}{2 x^{10}}-\frac{35 a^3 b^4}{9 x^9}-\frac{21 a^2 b^5}{8 x^8}-\frac{7 a^6 b}{12 x^{12}}-\frac{a^7}{13 x^{13}}-\frac{a b^6}{x^7}-\frac{b^7}{6 x^6} \]

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*}

Mathematica [A]  time = 0.0072827, size = 93, normalized size = 1. \[ -\frac{21 a^5 b^2}{11 x^{11}}-\frac{7 a^4 b^3}{2 x^{10}}-\frac{35 a^3 b^4}{9 x^9}-\frac{21 a^2 b^5}{8 x^8}-\frac{7 a^6 b}{12 x^{12}}-\frac{a^7}{13 x^{13}}-\frac{a b^6}{x^7}-\frac{b^7}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(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)

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Maple [A]  time = 0.006, size = 80, normalized size = 0.9 \begin{align*} -{\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*}

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 = 1.01975, size = 107, normalized size = 1.15 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.81349, size = 207, normalized size = 2.23 \begin{align*} -\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{align*}

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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Sympy [A]  time = 1.06813, size = 85, normalized size = 0.91 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.18739, size = 107, normalized size = 1.15 \begin{align*} -\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{align*}

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