3.2.10 \(\int \frac {(a+b x)^7}{x^4} \, dx\)

Optimal. Leaf size=86 \[ -\frac {a^7}{3 x^3}-\frac {7 a^6 b}{2 x^2}-\frac {21 a^5 b^2}{x}+35 a^4 b^3 \log (x)+35 a^3 b^4 x+\frac {21}{2} a^2 b^5 x^2+\frac {7}{3} a b^6 x^3+\frac {b^7 x^4}{4} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 86, 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}{2} a^2 b^5 x^2-\frac {21 a^5 b^2}{x}+35 a^3 b^4 x+35 a^4 b^3 \log (x)-\frac {7 a^6 b}{2 x^2}-\frac {a^7}{3 x^3}+\frac {7}{3} a b^6 x^3+\frac {b^7 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^4,x]

[Out]

-a^7/(3*x^3) - (7*a^6*b)/(2*x^2) - (21*a^5*b^2)/x + 35*a^3*b^4*x + (21*a^2*b^5*x^2)/2 + (7*a*b^6*x^3)/3 + (b^7
*x^4)/4 + 35*a^4*b^3*Log[x]

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^4} \, dx &=\int \left (35 a^3 b^4+\frac {a^7}{x^4}+\frac {7 a^6 b}{x^3}+\frac {21 a^5 b^2}{x^2}+\frac {35 a^4 b^3}{x}+21 a^2 b^5 x+7 a b^6 x^2+b^7 x^3\right ) \, dx\\ &=-\frac {a^7}{3 x^3}-\frac {7 a^6 b}{2 x^2}-\frac {21 a^5 b^2}{x}+35 a^3 b^4 x+\frac {21}{2} a^2 b^5 x^2+\frac {7}{3} a b^6 x^3+\frac {b^7 x^4}{4}+35 a^4 b^3 \log (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 86, normalized size = 1.00 \begin {gather*} -\frac {a^7}{3 x^3}-\frac {7 a^6 b}{2 x^2}-\frac {21 a^5 b^2}{x}+35 a^4 b^3 \log (x)+35 a^3 b^4 x+\frac {21}{2} a^2 b^5 x^2+\frac {7}{3} a b^6 x^3+\frac {b^7 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^4,x]

[Out]

-1/3*a^7/x^3 - (7*a^6*b)/(2*x^2) - (21*a^5*b^2)/x + 35*a^3*b^4*x + (21*a^2*b^5*x^2)/2 + (7*a*b^6*x^3)/3 + (b^7
*x^4)/4 + 35*a^4*b^3*Log[x]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^7}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^7/x^4,x]

[Out]

IntegrateAlgebraic[(a + b*x)^7/x^4, x]

________________________________________________________________________________________

fricas [A]  time = 1.33, size = 81, normalized size = 0.94 \begin {gather*} \frac {3 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 126 \, a^{2} b^{5} x^{5} + 420 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} \log \relax (x) - 252 \, a^{5} b^{2} x^{2} - 42 \, a^{6} b x - 4 \, a^{7}}{12 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^4,x, algorithm="fricas")

[Out]

1/12*(3*b^7*x^7 + 28*a*b^6*x^6 + 126*a^2*b^5*x^5 + 420*a^3*b^4*x^4 + 420*a^4*b^3*x^3*log(x) - 252*a^5*b^2*x^2
- 42*a^6*b*x - 4*a^7)/x^3

________________________________________________________________________________________

giac [A]  time = 1.09, size = 78, normalized size = 0.91 \begin {gather*} \frac {1}{4} \, b^{7} x^{4} + \frac {7}{3} \, a b^{6} x^{3} + \frac {21}{2} \, a^{2} b^{5} x^{2} + 35 \, a^{3} b^{4} x + 35 \, a^{4} b^{3} \log \left ({\left | x \right |}\right ) - \frac {126 \, a^{5} b^{2} x^{2} + 21 \, a^{6} b x + 2 \, a^{7}}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^4,x, algorithm="giac")

[Out]

1/4*b^7*x^4 + 7/3*a*b^6*x^3 + 21/2*a^2*b^5*x^2 + 35*a^3*b^4*x + 35*a^4*b^3*log(abs(x)) - 1/6*(126*a^5*b^2*x^2
+ 21*a^6*b*x + 2*a^7)/x^3

________________________________________________________________________________________

maple [A]  time = 0.01, size = 77, normalized size = 0.90 \begin {gather*} \frac {b^{7} x^{4}}{4}+\frac {7 a \,b^{6} x^{3}}{3}+\frac {21 a^{2} b^{5} x^{2}}{2}+35 a^{4} b^{3} \ln \relax (x )+35 a^{3} b^{4} x -\frac {21 a^{5} b^{2}}{x}-\frac {7 a^{6} b}{2 x^{2}}-\frac {a^{7}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^4,x)

[Out]

-1/3*a^7/x^3-7/2*a^6*b/x^2-21*a^5*b^2/x+35*a^3*b^4*x+21/2*a^2*b^5*x^2+7/3*a*b^6*x^3+1/4*b^7*x^4+35*a^4*b^3*ln(
x)

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 77, normalized size = 0.90 \begin {gather*} \frac {1}{4} \, b^{7} x^{4} + \frac {7}{3} \, a b^{6} x^{3} + \frac {21}{2} \, a^{2} b^{5} x^{2} + 35 \, a^{3} b^{4} x + 35 \, a^{4} b^{3} \log \relax (x) - \frac {126 \, a^{5} b^{2} x^{2} + 21 \, a^{6} b x + 2 \, a^{7}}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^4,x, algorithm="maxima")

[Out]

1/4*b^7*x^4 + 7/3*a*b^6*x^3 + 21/2*a^2*b^5*x^2 + 35*a^3*b^4*x + 35*a^4*b^3*log(x) - 1/6*(126*a^5*b^2*x^2 + 21*
a^6*b*x + 2*a^7)/x^3

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 77, normalized size = 0.90 \begin {gather*} \frac {b^7\,x^4}{4}-\frac {\frac {a^7}{3}+\frac {7\,a^6\,b\,x}{2}+21\,a^5\,b^2\,x^2}{x^3}+35\,a^3\,b^4\,x+\frac {7\,a\,b^6\,x^3}{3}+\frac {21\,a^2\,b^5\,x^2}{2}+35\,a^4\,b^3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/x^4,x)

[Out]

(b^7*x^4)/4 - (a^7/3 + 21*a^5*b^2*x^2 + (7*a^6*b*x)/2)/x^3 + 35*a^3*b^4*x + (7*a*b^6*x^3)/3 + (21*a^2*b^5*x^2)
/2 + 35*a^4*b^3*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.31, size = 87, normalized size = 1.01 \begin {gather*} 35 a^{4} b^{3} \log {\relax (x )} + 35 a^{3} b^{4} x + \frac {21 a^{2} b^{5} x^{2}}{2} + \frac {7 a b^{6} x^{3}}{3} + \frac {b^{7} x^{4}}{4} + \frac {- 2 a^{7} - 21 a^{6} b x - 126 a^{5} b^{2} x^{2}}{6 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**4,x)

[Out]

35*a**4*b**3*log(x) + 35*a**3*b**4*x + 21*a**2*b**5*x**2/2 + 7*a*b**6*x**3/3 + b**7*x**4/4 + (-2*a**7 - 21*a**
6*b*x - 126*a**5*b**2*x**2)/(6*x**3)

________________________________________________________________________________________