\(\int \frac {x^6}{(a+b x)^7} \, dx\) [211]

Optimal result

Integrand size = 11, antiderivative size = 109 \[ \int \frac {x^6}{(a+b x)^7} \, dx=-\frac {a^6}{6 b^7 (a+b x)^6}+\frac {6 a^5}{5 b^7 (a+b x)^5}-\frac {15 a^4}{4 b^7 (a+b x)^4}+\frac {20 a^3}{3 b^7 (a+b x)^3}-\frac {15 a^2}{2 b^7 (a+b x)^2}+\frac {6 a}{b^7 (a+b x)}+\frac {\log (a+b x)}{b^7} \]

[Out]

-1/6*a^6/b^7/(b*x+a)^6+6/5*a^5/b^7/(b*x+a)^5-15/4*a^4/b^7/(b*x+a)^4+20/3*a^3/b^7/(b*x+a)^3-15/2*a^2/b^7/(b*x+a
)^2+6*a/b^7/(b*x+a)+ln(b*x+a)/b^7

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^6}{(a+b x)^7} \, dx=-\frac {a^6}{6 b^7 (a+b x)^6}+\frac {6 a^5}{5 b^7 (a+b x)^5}-\frac {15 a^4}{4 b^7 (a+b x)^4}+\frac {20 a^3}{3 b^7 (a+b x)^3}-\frac {15 a^2}{2 b^7 (a+b x)^2}+\frac {6 a}{b^7 (a+b x)}+\frac {\log (a+b x)}{b^7} \]

[In]

Int[x^6/(a + b*x)^7,x]

[Out]

-1/6*a^6/(b^7*(a + b*x)^6) + (6*a^5)/(5*b^7*(a + b*x)^5) - (15*a^4)/(4*b^7*(a + b*x)^4) + (20*a^3)/(3*b^7*(a +
b*x)^3) - (15*a^2)/(2*b^7*(a + b*x)^2) + (6*a)/(b^7*(a + b*x)) + Log[a + b*x]/b^7

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*} \text {integral}& = \int \left (\frac {a^6}{b^6 (a+b x)^7}-\frac {6 a^5}{b^6 (a+b x)^6}+\frac {15 a^4}{b^6 (a+b x)^5}-\frac {20 a^3}{b^6 (a+b x)^4}+\frac {15 a^2}{b^6 (a+b x)^3}-\frac {6 a}{b^6 (a+b x)^2}+\frac {1}{b^6 (a+b x)}\right ) \, dx \\ & = -\frac {a^6}{6 b^7 (a+b x)^6}+\frac {6 a^5}{5 b^7 (a+b x)^5}-\frac {15 a^4}{4 b^7 (a+b x)^4}+\frac {20 a^3}{3 b^7 (a+b x)^3}-\frac {15 a^2}{2 b^7 (a+b x)^2}+\frac {6 a}{b^7 (a+b x)}+\frac {\log (a+b x)}{b^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int \frac {x^6}{(a+b x)^7} \, dx=\frac {\frac {a \left (147 a^5+822 a^4 b x+1875 a^3 b^2 x^2+2200 a^2 b^3 x^3+1350 a b^4 x^4+360 b^5 x^5\right )}{(a+b x)^6}+60 \log (a+b x)}{60 b^7} \]

[In]

Integrate[x^6/(a + b*x)^7,x]

[Out]

((a*(147*a^5 + 822*a^4*b*x + 1875*a^3*b^2*x^2 + 2200*a^2*b^3*x^3 + 1350*a*b^4*x^4 + 360*b^5*x^5))/(a + b*x)^6
+ 60*Log[a + b*x])/(60*b^7)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.73

method	result	size
norman	\(\frac {\frac {49 a^{6}}{20 b^{7}}+\frac {6 a \,x^{5}}{b^{2}}+\frac {45 a^{2} x^{4}}{2 b^{3}}+\frac {125 a^{4} x^{2}}{4 b^{5}}+\frac {137 a^{5} x}{10 b^{6}}+\frac {110 a^{3} x^{3}}{3 b^{4}}}{\left (b x +a \right )^{6}}+\frac {\ln \left (b x +a \right )}{b^{7}}\)	\(80\)
risch	\(\frac {\frac {49 a^{6}}{20 b^{7}}+\frac {6 a \,x^{5}}{b^{2}}+\frac {45 a^{2} x^{4}}{2 b^{3}}+\frac {125 a^{4} x^{2}}{4 b^{5}}+\frac {137 a^{5} x}{10 b^{6}}+\frac {110 a^{3} x^{3}}{3 b^{4}}}{\left (b x +a \right )^{6}}+\frac {\ln \left (b x +a \right )}{b^{7}}\)	\(80\)
default	\(-\frac {a^{6}}{6 b^{7} \left (b x +a \right )^{6}}+\frac {6 a^{5}}{5 b^{7} \left (b x +a \right )^{5}}-\frac {15 a^{4}}{4 b^{7} \left (b x +a \right )^{4}}+\frac {20 a^{3}}{3 b^{7} \left (b x +a \right )^{3}}-\frac {15 a^{2}}{2 b^{7} \left (b x +a \right )^{2}}+\frac {6 a}{b^{7} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{7}}\)	\(100\)
parallelrisch	\(\frac {60 \ln \left (b x +a \right ) x^{6} b^{6}+360 \ln \left (b x +a \right ) x^{5} a \,b^{5}+900 \ln \left (b x +a \right ) x^{4} a^{2} b^{4}+360 a \,x^{5} b^{5}+1200 \ln \left (b x +a \right ) x^{3} a^{3} b^{3}+1350 a^{2} x^{4} b^{4}+900 \ln \left (b x +a \right ) x^{2} a^{4} b^{2}+2200 a^{3} x^{3} b^{3}+360 \ln \left (b x +a \right ) x \,a^{5} b +1875 a^{4} x^{2} b^{2}+60 \ln \left (b x +a \right ) a^{6}+822 a^{5} x b +147 a^{6}}{60 b^{7} \left (b x +a \right )^{6}}\)	\(172\)

[In]

int(x^6/(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

(49/20*a^6/b^7+6*a/b^2*x^5+45/2*a^2/b^3*x^4+125/4*a^4/b^5*x^2+137/10*a^5/b^6*x+110/3*a^3/b^4*x^3)/(b*x+a)^6+ln
(b*x+a)/b^7

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.77 \[ \int \frac {x^6}{(a+b x)^7} \, dx=\frac {360 \, a b^{5} x^{5} + 1350 \, a^{2} b^{4} x^{4} + 2200 \, a^{3} b^{3} x^{3} + 1875 \, a^{4} b^{2} x^{2} + 822 \, a^{5} b x + 147 \, a^{6} + 60 \, {\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{60 \, {\left (b^{13} x^{6} + 6 \, a b^{12} x^{5} + 15 \, a^{2} b^{11} x^{4} + 20 \, a^{3} b^{10} x^{3} + 15 \, a^{4} b^{9} x^{2} + 6 \, a^{5} b^{8} x + a^{6} b^{7}\right )}} \]

[In]

integrate(x^6/(b*x+a)^7,x, algorithm="fricas")

[Out]

1/60*(360*a*b^5*x^5 + 1350*a^2*b^4*x^4 + 2200*a^3*b^3*x^3 + 1875*a^4*b^2*x^2 + 822*a^5*b*x + 147*a^6 + 60*(b^6
*x^6 + 6*a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3*x^3 + 15*a^4*b^2*x^2 + 6*a^5*b*x + a^6)*log(b*x + a))/(b^13*x
^6 + 6*a*b^12*x^5 + 15*a^2*b^11*x^4 + 20*a^3*b^10*x^3 + 15*a^4*b^9*x^2 + 6*a^5*b^8*x + a^6*b^7)

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.29 \[ \int \frac {x^6}{(a+b x)^7} \, dx=\frac {147 a^{6} + 822 a^{5} b x + 1875 a^{4} b^{2} x^{2} + 2200 a^{3} b^{3} x^{3} + 1350 a^{2} b^{4} x^{4} + 360 a b^{5} x^{5}}{60 a^{6} b^{7} + 360 a^{5} b^{8} x + 900 a^{4} b^{9} x^{2} + 1200 a^{3} b^{10} x^{3} + 900 a^{2} b^{11} x^{4} + 360 a b^{12} x^{5} + 60 b^{13} x^{6}} + \frac {\log {\left (a + b x \right )}}{b^{7}} \]

[In]

integrate(x**6/(b*x+a)**7,x)

[Out]

(147*a**6 + 822*a**5*b*x + 1875*a**4*b**2*x**2 + 2200*a**3*b**3*x**3 + 1350*a**2*b**4*x**4 + 360*a*b**5*x**5)/
(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**1
2*x**5 + 60*b**13*x**6) + log(a + b*x)/b**7

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.25 \[ \int \frac {x^6}{(a+b x)^7} \, dx=\frac {360 \, a b^{5} x^{5} + 1350 \, a^{2} b^{4} x^{4} + 2200 \, a^{3} b^{3} x^{3} + 1875 \, a^{4} b^{2} x^{2} + 822 \, a^{5} b x + 147 \, a^{6}}{60 \, {\left (b^{13} x^{6} + 6 \, a b^{12} x^{5} + 15 \, a^{2} b^{11} x^{4} + 20 \, a^{3} b^{10} x^{3} + 15 \, a^{4} b^{9} x^{2} + 6 \, a^{5} b^{8} x + a^{6} b^{7}\right )}} + \frac {\log \left (b x + a\right )}{b^{7}} \]

[In]

integrate(x^6/(b*x+a)^7,x, algorithm="maxima")

[Out]

1/60*(360*a*b^5*x^5 + 1350*a^2*b^4*x^4 + 2200*a^3*b^3*x^3 + 1875*a^4*b^2*x^2 + 822*a^5*b*x + 147*a^6)/(b^13*x^
6 + 6*a*b^12*x^5 + 15*a^2*b^11*x^4 + 20*a^3*b^10*x^3 + 15*a^4*b^9*x^2 + 6*a^5*b^8*x + a^6*b^7) + log(b*x + a)/
b^7

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72 \[ \int \frac {x^6}{(a+b x)^7} \, dx=\frac {\log \left ({\left | b x + a \right |}\right )}{b^{7}} + \frac {360 \, a b^{4} x^{5} + 1350 \, a^{2} b^{3} x^{4} + 2200 \, a^{3} b^{2} x^{3} + 1875 \, a^{4} b x^{2} + 822 \, a^{5} x + \frac {147 \, a^{6}}{b}}{60 \, {\left (b x + a\right )}^{6} b^{6}} \]

[In]

integrate(x^6/(b*x+a)^7,x, algorithm="giac")

[Out]

log(abs(b*x + a))/b^7 + 1/60*(360*a*b^4*x^5 + 1350*a^2*b^3*x^4 + 2200*a^3*b^2*x^3 + 1875*a^4*b*x^2 + 822*a^5*x
+ 147*a^6/b)/((b*x + a)^6*b^6)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.74 \[ \int \frac {x^6}{(a+b x)^7} \, dx=\frac {\ln \left (a+b\,x\right )+\frac {6\,a}{a+b\,x}-\frac {15\,a^2}{2\,{\left (a+b\,x\right )}^2}+\frac {20\,a^3}{3\,{\left (a+b\,x\right )}^3}-\frac {15\,a^4}{4\,{\left (a+b\,x\right )}^4}+\frac {6\,a^5}{5\,{\left (a+b\,x\right )}^5}-\frac {a^6}{6\,{\left (a+b\,x\right )}^6}}{b^7} \]

[In]

int(x^6/(a + b*x)^7,x)

[Out]

(log(a + b*x) + (6*a)/(a + b*x) - (15*a^2)/(2*(a + b*x)^2) + (20*a^3)/(3*(a + b*x)^3) - (15*a^4)/(4*(a + b*x)^
4) + (6*a^5)/(5*(a + b*x)^5) - a^6/(6*(a + b*x)^6))/b^7