\(\int \frac {x^5}{(a+b x)^2} \, dx\) [169]

Optimal result

Integrand size = 11, antiderivative size = 72 \[ \int \frac {x^5}{(a+b x)^2} \, dx=-\frac {4 a^3 x}{b^5}+\frac {3 a^2 x^2}{2 b^4}-\frac {2 a x^3}{3 b^3}+\frac {x^4}{4 b^2}+\frac {a^5}{b^6 (a+b x)}+\frac {5 a^4 \log (a+b x)}{b^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 72, 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} \[ \int \frac {x^5}{(a+b x)^2} \, dx=\frac {a^5}{b^6 (a+b x)}+\frac {5 a^4 \log (a+b x)}{b^6}-\frac {4 a^3 x}{b^5}+\frac {3 a^2 x^2}{2 b^4}-\frac {2 a x^3}{3 b^3}+\frac {x^4}{4 b^2} \]

[In]

Int[x^5/(a + b*x)^2,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.92 \[ \int \frac {x^5}{(a+b x)^2} \, dx=\frac {-48 a^3 b x+18 a^2 b^2 x^2-8 a b^3 x^3+3 b^4 x^4+\frac {12 a^5}{a+b x}+60 a^4 \log (a+b x)}{12 b^6} \]

[In]

Integrate[x^5/(a + b*x)^2,x]

[Out]

(-48*a^3*b*x + 18*a^2*b^2*x^2 - 8*a*b^3*x^3 + 3*b^4*x^4 + (12*a^5)/(a + b*x) + 60*a^4*Log[a + b*x])/(12*b^6)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {4 a^{3} x}{b^{5}}+\frac {3 a^{2} x^{2}}{2 b^{4}}-\frac {2 a \,x^{3}}{3 b^{3}}+\frac {x^{4}}{4 b^{2}}+\frac {a^{5}}{b^{6} \left (b x +a \right )}+\frac {5 a^{4} \ln \left (b x +a \right )}{b^{6}}\)	\(67\)
default	\(-\frac {-\frac {1}{4} b^{3} x^{4}+\frac {2}{3} a \,b^{2} x^{3}-\frac {3}{2} a^{2} b \,x^{2}+4 a^{3} x}{b^{5}}+\frac {5 a^{4} \ln \left (b x +a \right )}{b^{6}}+\frac {a^{5}}{b^{6} \left (b x +a \right )}\)	\(68\)
norman	\(\frac {\frac {5 a^{5}}{b^{6}}+\frac {x^{5}}{4 b}-\frac {5 a \,x^{4}}{12 b^{2}}+\frac {5 a^{2} x^{3}}{6 b^{3}}-\frac {5 a^{3} x^{2}}{2 b^{4}}}{b x +a}+\frac {5 a^{4} \ln \left (b x +a \right )}{b^{6}}\)	\(72\)
parallelrisch	\(\frac {3 b^{5} x^{5}-5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+60 \ln \left (b x +a \right ) x \,a^{4} b -30 a^{3} b^{2} x^{2}+60 a^{5} \ln \left (b x +a \right )+60 a^{5}}{12 b^{6} \left (b x +a \right )}\)	\(82\)

[In]

int(x^5/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.18 \[ \int \frac {x^5}{(a+b x)^2} \, dx=\frac {3 \, b^{5} x^{5} - 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} - 30 \, a^{3} b^{2} x^{2} - 48 \, a^{4} b x + 12 \, a^{5} + 60 \, {\left (a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \]

[In]

integrate(x^5/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/12*(3*b^5*x^5 - 5*a*b^4*x^4 + 10*a^2*b^3*x^3 - 30*a^3*b^2*x^2 - 48*a^4*b*x + 12*a^5 + 60*(a^4*b*x + a^5)*log
(b*x + a))/(b^7*x + a*b^6)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99 \[ \int \frac {x^5}{(a+b x)^2} \, dx=\frac {a^{5}}{a b^{6} + b^{7} x} + \frac {5 a^{4} \log {\left (a + b x \right )}}{b^{6}} - \frac {4 a^{3} x}{b^{5}} + \frac {3 a^{2} x^{2}}{2 b^{4}} - \frac {2 a x^{3}}{3 b^{3}} + \frac {x^{4}}{4 b^{2}} \]

[In]

integrate(x**5/(b*x+a)**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \frac {x^5}{(a+b x)^2} \, dx=\frac {a^{5}}{b^{7} x + a b^{6}} + \frac {5 \, a^{4} \log \left (b x + a\right )}{b^{6}} + \frac {3 \, b^{3} x^{4} - 8 \, a b^{2} x^{3} + 18 \, a^{2} b x^{2} - 48 \, a^{3} x}{12 \, b^{5}} \]

[In]

integrate(x^5/(b*x+a)^2,x, algorithm="maxima")

[Out]

a^5/(b^7*x + a*b^6) + 5*a^4*log(b*x + a)/b^6 + 1/12*(3*b^3*x^4 - 8*a*b^2*x^3 + 18*a^2*b*x^2 - 48*a^3*x)/b^5

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25 \[ \int \frac {x^5}{(a+b x)^2} \, dx=-\frac {{\left (b x + a\right )}^{4} {\left (\frac {20 \, a}{b x + a} - \frac {60 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac {120 \, a^{3}}{{\left (b x + a\right )}^{3}} - 3\right )}}{12 \, b^{6}} - \frac {5 \, a^{4} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{6}} + \frac {a^{5}}{{\left (b x + a\right )} b^{6}} \]

[In]

integrate(x^5/(b*x+a)^2,x, algorithm="giac")

[Out]

-1/12*(b*x + a)^4*(20*a/(b*x + a) - 60*a^2/(b*x + a)^2 + 120*a^3/(b*x + a)^3 - 3)/b^6 - 5*a^4*log(abs(b*x + a)
/((b*x + a)^2*abs(b)))/b^6 + a^5/((b*x + a)*b^6)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{(a+b x)^2} \, dx=\frac {x^4}{4\,b^2}+\frac {5\,a^4\,\ln \left (a+b\,x\right )}{b^6}-\frac {2\,a\,x^3}{3\,b^3}-\frac {4\,a^3\,x}{b^5}+\frac {3\,a^2\,x^2}{2\,b^4}+\frac {a^5}{b\,\left (x\,b^6+a\,b^5\right )} \]

[In]

int(x^5/(a + b*x)^2,x)

[Out]

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