\(\int \frac {x^5}{(a+b x)^4} \, dx\) [196]

Optimal result

Integrand size = 11, antiderivative size = 81 \[ \int \frac {x^5}{(a+b x)^4} \, dx=-\frac {4 a x}{b^5}+\frac {x^2}{2 b^4}+\frac {a^5}{3 b^6 (a+b x)^3}-\frac {5 a^4}{2 b^6 (a+b x)^2}+\frac {10 a^3}{b^6 (a+b x)}+\frac {10 a^2 \log (a+b x)}{b^6} \]

[Out]

-4*a*x/b^5+1/2*x^2/b^4+1/3*a^5/b^6/(b*x+a)^3-5/2*a^4/b^6/(b*x+a)^2+10*a^3/b^6/(b*x+a)+10*a^2*ln(b*x+a)/b^6

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 81, 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} \[ \int \frac {x^5}{(a+b x)^4} \, dx=\frac {a^5}{3 b^6 (a+b x)^3}-\frac {5 a^4}{2 b^6 (a+b x)^2}+\frac {10 a^3}{b^6 (a+b x)}+\frac {10 a^2 \log (a+b x)}{b^6}-\frac {4 a x}{b^5}+\frac {x^2}{2 b^4} \]

[In]

Int[x^5/(a + b*x)^4,x]

[Out]

(-4*a*x)/b^5 + x^2/(2*b^4) + a^5/(3*b^6*(a + b*x)^3) - (5*a^4)/(2*b^6*(a + b*x)^2) + (10*a^3)/(b^6*(a + b*x))
+ (10*a^2*Log[a + b*x])/b^6

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 {4 a}{b^5}+\frac {x}{b^4}-\frac {a^5}{b^5 (a+b x)^4}+\frac {5 a^4}{b^5 (a+b x)^3}-\frac {10 a^3}{b^5 (a+b x)^2}+\frac {10 a^2}{b^5 (a+b x)}\right ) \, dx \\ & = -\frac {4 a x}{b^5}+\frac {x^2}{2 b^4}+\frac {a^5}{3 b^6 (a+b x)^3}-\frac {5 a^4}{2 b^6 (a+b x)^2}+\frac {10 a^3}{b^6 (a+b x)}+\frac {10 a^2 \log (a+b x)}{b^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84 \[ \int \frac {x^5}{(a+b x)^4} \, dx=\frac {-24 a b x+3 b^2 x^2+\frac {2 a^5}{(a+b x)^3}-\frac {15 a^4}{(a+b x)^2}+\frac {60 a^3}{a+b x}+60 a^2 \log (a+b x)}{6 b^6} \]

[In]

Integrate[x^5/(a + b*x)^4,x]

[Out]

(-24*a*b*x + 3*b^2*x^2 + (2*a^5)/(a + b*x)^3 - (15*a^4)/(a + b*x)^2 + (60*a^3)/(a + b*x) + 60*a^2*Log[a + b*x]
)/(6*b^6)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81

method	result	size
risch	\(\frac {x^{2}}{2 b^{4}}-\frac {4 a x}{b^{5}}+\frac {10 a^{3} b \,x^{2}+\frac {35 a^{4} x}{2}+\frac {47 a^{5}}{6 b}}{b^{5} \left (b x +a \right )^{3}}+\frac {10 a^{2} \ln \left (b x +a \right )}{b^{6}}\)	\(66\)
norman	\(\frac {\frac {x^{5}}{2 b}-\frac {5 a \,x^{4}}{2 b^{2}}+\frac {55 a^{5}}{3 b^{6}}+\frac {30 a^{3} x^{2}}{b^{4}}+\frac {45 a^{4} x}{b^{5}}}{\left (b x +a \right )^{3}}+\frac {10 a^{2} \ln \left (b x +a \right )}{b^{6}}\)	\(70\)
default	\(-\frac {-\frac {1}{2} b \,x^{2}+4 a x}{b^{5}}+\frac {10 a^{2} \ln \left (b x +a \right )}{b^{6}}+\frac {a^{5}}{3 b^{6} \left (b x +a \right )^{3}}-\frac {5 a^{4}}{2 b^{6} \left (b x +a \right )^{2}}+\frac {10 a^{3}}{b^{6} \left (b x +a \right )}\)	\(77\)
parallelrisch	\(\frac {3 b^{5} x^{5}+60 \ln \left (b x +a \right ) x^{3} a^{2} b^{3}-15 a \,b^{4} x^{4}+180 \ln \left (b x +a \right ) x^{2} a^{3} b^{2}+180 \ln \left (b x +a \right ) x \,a^{4} b +180 a^{3} b^{2} x^{2}+60 a^{5} \ln \left (b x +a \right )+270 a^{4} b x +110 a^{5}}{6 b^{6} \left (b x +a \right )^{3}}\)	\(112\)

[In]

int(x^5/(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

1/2*x^2/b^4-4*a*x/b^5+(10*a^3*b*x^2+35/2*a^4*x+47/6*a^5/b)/b^5/(b*x+a)^3+10*a^2*ln(b*x+a)/b^6

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.59 \[ \int \frac {x^5}{(a+b x)^4} \, dx=\frac {3 \, b^{5} x^{5} - 15 \, a b^{4} x^{4} - 63 \, a^{2} b^{3} x^{3} - 9 \, a^{3} b^{2} x^{2} + 81 \, a^{4} b x + 47 \, a^{5} + 60 \, {\left (a^{2} b^{3} x^{3} + 3 \, a^{3} b^{2} x^{2} + 3 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \]

[In]

integrate(x^5/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/6*(3*b^5*x^5 - 15*a*b^4*x^4 - 63*a^2*b^3*x^3 - 9*a^3*b^2*x^2 + 81*a^4*b*x + 47*a^5 + 60*(a^2*b^3*x^3 + 3*a^3
*b^2*x^2 + 3*a^4*b*x + a^5)*log(b*x + a))/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6)

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.16 \[ \int \frac {x^5}{(a+b x)^4} \, dx=\frac {10 a^{2} \log {\left (a + b x \right )}}{b^{6}} - \frac {4 a x}{b^{5}} + \frac {47 a^{5} + 105 a^{4} b x + 60 a^{3} b^{2} x^{2}}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} + \frac {x^{2}}{2 b^{4}} \]

[In]

integrate(x**5/(b*x+a)**4,x)

[Out]

10*a**2*log(a + b*x)/b**6 - 4*a*x/b**5 + (47*a**5 + 105*a**4*b*x + 60*a**3*b**2*x**2)/(6*a**3*b**6 + 18*a**2*b
**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + x**2/(2*b**4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.12 \[ \int \frac {x^5}{(a+b x)^4} \, dx=\frac {60 \, a^{3} b^{2} x^{2} + 105 \, a^{4} b x + 47 \, a^{5}}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac {10 \, a^{2} \log \left (b x + a\right )}{b^{6}} + \frac {b x^{2} - 8 \, a x}{2 \, b^{5}} \]

[In]

integrate(x^5/(b*x+a)^4,x, algorithm="maxima")

[Out]

1/6*(60*a^3*b^2*x^2 + 105*a^4*b*x + 47*a^5)/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6) + 10*a^2*log(b*x +
a)/b^6 + 1/2*(b*x^2 - 8*a*x)/b^5

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \frac {x^5}{(a+b x)^4} \, dx=\frac {10 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac {b^{4} x^{2} - 8 \, a b^{3} x}{2 \, b^{8}} + \frac {60 \, a^{3} b^{2} x^{2} + 105 \, a^{4} b x + 47 \, a^{5}}{6 \, {\left (b x + a\right )}^{3} b^{6}} \]

[In]

integrate(x^5/(b*x+a)^4,x, algorithm="giac")

[Out]

10*a^2*log(abs(b*x + a))/b^6 + 1/2*(b^4*x^2 - 8*a*b^3*x)/b^8 + 1/6*(60*a^3*b^2*x^2 + 105*a^4*b*x + 47*a^5)/((b
*x + a)^3*b^6)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{(a+b x)^4} \, dx=\frac {\frac {{\left (a+b\,x\right )}^2}{2}+\frac {10\,a^3}{a+b\,x}-\frac {5\,a^4}{2\,{\left (a+b\,x\right )}^2}+\frac {a^5}{3\,{\left (a+b\,x\right )}^3}+10\,a^2\,\ln \left (a+b\,x\right )-5\,a\,b\,x}{b^6} \]

[In]

int(x^5/(a + b*x)^4,x)

[Out]

((a + b*x)^2/2 + (10*a^3)/(a + b*x) - (5*a^4)/(2*(a + b*x)^2) + a^5/(3*(a + b*x)^3) + 10*a^2*log(a + b*x) - 5*
a*b*x)/b^6