\(\int \frac {x^5}{(a+b x)^3} \, dx\) [182]

Optimal result

Integrand size = 11, antiderivative size = 77 \[ \int \frac {x^5}{(a+b x)^3} \, dx=\frac {6 a^2 x}{b^5}-\frac {3 a x^2}{2 b^4}+\frac {x^3}{3 b^3}+\frac {a^5}{2 b^6 (a+b x)^2}-\frac {5 a^4}{b^6 (a+b x)}-\frac {10 a^3 \log (a+b x)}{b^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, 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} \[ \int \frac {x^5}{(a+b x)^3} \, dx=\frac {a^5}{2 b^6 (a+b x)^2}-\frac {5 a^4}{b^6 (a+b x)}-\frac {10 a^3 \log (a+b x)}{b^6}+\frac {6 a^2 x}{b^5}-\frac {3 a x^2}{2 b^4}+\frac {x^3}{3 b^3} \]

[In]

Int[x^5/(a + b*x)^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int \frac {x^5}{(a+b x)^3} \, dx=\frac {36 a^2 b x-9 a b^2 x^2+2 b^3 x^3+\frac {3 a^5}{(a+b x)^2}-\frac {30 a^4}{a+b x}-60 a^3 \log (a+b x)}{6 b^6} \]

[In]

Integrate[x^5/(a + b*x)^3,x]

[Out]

(36*a^2*b*x - 9*a*b^2*x^2 + 2*b^3*x^3 + (3*a^5)/(a + b*x)^2 - (30*a^4)/(a + b*x) - 60*a^3*Log[a + b*x])/(6*b^6
)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {x^{3}}{3 b^{3}}-\frac {3 a \,x^{2}}{2 b^{4}}+\frac {6 a^{2} x}{b^{5}}+\frac {-5 a^{4} x -\frac {9 a^{5}}{2 b}}{b^{5} \left (b x +a \right )^{2}}-\frac {10 a^{3} \ln \left (b x +a \right )}{b^{6}}\)	\(68\)
norman	\(\frac {\frac {x^{5}}{3 b}-\frac {5 a \,x^{4}}{6 b^{2}}+\frac {10 a^{2} x^{3}}{3 b^{3}}-\frac {15 a^{5}}{b^{6}}-\frac {20 a^{4} x}{b^{5}}}{\left (b x +a \right )^{2}}-\frac {10 a^{3} \ln \left (b x +a \right )}{b^{6}}\)	\(70\)
default	\(\frac {\frac {1}{3} b^{2} x^{3}-\frac {3}{2} a b \,x^{2}+6 a^{2} x}{b^{5}}-\frac {10 a^{3} \ln \left (b x +a \right )}{b^{6}}+\frac {a^{5}}{2 b^{6} \left (b x +a \right )^{2}}-\frac {5 a^{4}}{b^{6} \left (b x +a \right )}\)	\(72\)
parallelrisch	\(-\frac {-2 b^{5} x^{5}+5 a \,b^{4} x^{4}+60 \ln \left (b x +a \right ) x^{2} a^{3} b^{2}-20 a^{2} b^{3} x^{3}+120 \ln \left (b x +a \right ) x \,a^{4} b +60 a^{5} \ln \left (b x +a \right )+120 a^{4} b x +90 a^{5}}{6 b^{6} \left (b x +a \right )^{2}}\)	\(95\)

[In]

int(x^5/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/3*x^3/b^3-3/2*a*x^2/b^4+6*a^2*x/b^5+(-5*a^4*x-9/2*a^5/b)/b^5/(b*x+a)^2-10*a^3*ln(b*x+a)/b^6

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.39 \[ \int \frac {x^5}{(a+b x)^3} \, dx=\frac {2 \, b^{5} x^{5} - 5 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 63 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x - 27 \, a^{5} - 60 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]

[In]

integrate(x^5/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/6*(2*b^5*x^5 - 5*a*b^4*x^4 + 20*a^2*b^3*x^3 + 63*a^3*b^2*x^2 + 6*a^4*b*x - 27*a^5 - 60*(a^3*b^2*x^2 + 2*a^4*
b*x + a^5)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6)

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int \frac {x^5}{(a+b x)^3} \, dx=- \frac {10 a^{3} \log {\left (a + b x \right )}}{b^{6}} + \frac {6 a^{2} x}{b^{5}} - \frac {3 a x^{2}}{2 b^{4}} + \frac {- 9 a^{5} - 10 a^{4} b x}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} + \frac {x^{3}}{3 b^{3}} \]

[In]

integrate(x**5/(b*x+a)**3,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {x^5}{(a+b x)^3} \, dx=-\frac {10 \, a^{4} b x + 9 \, a^{5}}{2 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} - \frac {10 \, a^{3} \log \left (b x + a\right )}{b^{6}} + \frac {2 \, b^{2} x^{3} - 9 \, a b x^{2} + 36 \, a^{2} x}{6 \, b^{5}} \]

[In]

integrate(x^5/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*(10*a^4*b*x + 9*a^5)/(b^8*x^2 + 2*a*b^7*x + a^2*b^6) - 10*a^3*log(b*x + a)/b^6 + 1/6*(2*b^2*x^3 - 9*a*b*x
^2 + 36*a^2*x)/b^5

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95 \[ \int \frac {x^5}{(a+b x)^3} \, dx=-\frac {10 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {10 \, a^{4} b x + 9 \, a^{5}}{2 \, {\left (b x + a\right )}^{2} b^{6}} + \frac {2 \, b^{6} x^{3} - 9 \, a b^{5} x^{2} + 36 \, a^{2} b^{4} x}{6 \, b^{9}} \]

[In]

integrate(x^5/(b*x+a)^3,x, algorithm="giac")

[Out]

-10*a^3*log(abs(b*x + a))/b^6 - 1/2*(10*a^4*b*x + 9*a^5)/((b*x + a)^2*b^6) + 1/6*(2*b^6*x^3 - 9*a*b^5*x^2 + 36
*a^2*b^4*x)/b^9

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int \frac {x^5}{(a+b x)^3} \, dx=-\frac {\frac {5\,a\,{\left (a+b\,x\right )}^2}{2}-\frac {{\left (a+b\,x\right )}^3}{3}+\frac {5\,a^4}{a+b\,x}-\frac {a^5}{2\,{\left (a+b\,x\right )}^2}+10\,a^3\,\ln \left (a+b\,x\right )-10\,a^2\,b\,x}{b^6} \]

[In]

int(x^5/(a + b*x)^3,x)

[Out]

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