3.2.68 \(\int \frac {x^6}{(a+b x)^2} \, dx\)

Optimal. Leaf size=81 \[ -\frac {a^6}{b^7 (a+b x)}-\frac {6 a^5 \log (a+b x)}{b^7}+\frac {5 a^4 x}{b^6}-\frac {2 a^3 x^2}{b^5}+\frac {a^2 x^3}{b^4}-\frac {a x^4}{2 b^3}+\frac {x^5}{5 b^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 81, 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 {2 a^3 x^2}{b^5}+\frac {a^2 x^3}{b^4}-\frac {a^6}{b^7 (a+b x)}+\frac {5 a^4 x}{b^6}-\frac {6 a^5 \log (a+b x)}{b^7}-\frac {a x^4}{2 b^3}+\frac {x^5}{5 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x)^2,x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 77, normalized size = 0.95 \begin {gather*} \frac {-\frac {10 a^6}{a+b x}-60 a^5 \log (a+b x)+50 a^4 b x-20 a^3 b^2 x^2+10 a^2 b^3 x^3-5 a b^4 x^4+2 b^5 x^5}{10 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x)^2,x]

[Out]

(50*a^4*b*x - 20*a^3*b^2*x^2 + 10*a^2*b^3*x^3 - 5*a*b^4*x^4 + 2*b^5*x^5 - (10*a^6)/(a + b*x) - 60*a^5*Log[a +
b*x])/(10*b^7)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6}{(a+b x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^6/(a + b*x)^2,x]

[Out]

IntegrateAlgebraic[x^6/(a + b*x)^2, x]

________________________________________________________________________________________

fricas [A]  time = 1.31, size = 96, normalized size = 1.19 \begin {gather*} \frac {2 \, b^{6} x^{6} - 3 \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{4} - 10 \, a^{3} b^{3} x^{3} + 30 \, a^{4} b^{2} x^{2} + 50 \, a^{5} b x - 10 \, a^{6} - 60 \, {\left (a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{10 \, {\left (b^{8} x + a b^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/10*(2*b^6*x^6 - 3*a*b^5*x^5 + 5*a^2*b^4*x^4 - 10*a^3*b^3*x^3 + 30*a^4*b^2*x^2 + 50*a^5*b*x - 10*a^6 - 60*(a^
5*b*x + a^6)*log(b*x + a))/(b^8*x + a*b^7)

________________________________________________________________________________________

giac [A]  time = 1.14, size = 103, normalized size = 1.27 \begin {gather*} -\frac {{\left (b x + a\right )}^{5} {\left (\frac {15 \, a}{b x + a} - \frac {50 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac {100 \, a^{3}}{{\left (b x + a\right )}^{3}} - \frac {150 \, a^{4}}{{\left (b x + a\right )}^{4}} - 2\right )}}{10 \, b^{7}} + \frac {6 \, a^{5} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{7}} - \frac {a^{6}}{{\left (b x + a\right )} b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2,x, algorithm="giac")

[Out]

-1/10*(b*x + a)^5*(15*a/(b*x + a) - 50*a^2/(b*x + a)^2 + 100*a^3/(b*x + a)^3 - 150*a^4/(b*x + a)^4 - 2)/b^7 +
6*a^5*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^7 - a^6/((b*x + a)*b^7)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 78, normalized size = 0.96 \begin {gather*} \frac {x^{5}}{5 b^{2}}-\frac {a \,x^{4}}{2 b^{3}}+\frac {a^{2} x^{3}}{b^{4}}-\frac {2 a^{3} x^{2}}{b^{5}}-\frac {a^{6}}{\left (b x +a \right ) b^{7}}-\frac {6 a^{5} \ln \left (b x +a \right )}{b^{7}}+\frac {5 a^{4} x}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x+a)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 82, normalized size = 1.01 \begin {gather*} -\frac {a^{6}}{b^{8} x + a b^{7}} - \frac {6 \, a^{5} \log \left (b x + a\right )}{b^{7}} + \frac {2 \, b^{4} x^{5} - 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} - 20 \, a^{3} b x^{2} + 50 \, a^{4} x}{10 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2,x, algorithm="maxima")

[Out]

-a^6/(b^8*x + a*b^7) - 6*a^5*log(b*x + a)/b^7 + 1/10*(2*b^4*x^5 - 5*a*b^3*x^4 + 10*a^2*b^2*x^3 - 20*a^3*b*x^2
+ 50*a^4*x)/b^6

________________________________________________________________________________________

mupad [B]  time = 0.14, size = 83, normalized size = 1.02 \begin {gather*} \frac {x^5}{5\,b^2}-\frac {6\,a^5\,\ln \left (a+b\,x\right )}{b^7}-\frac {a\,x^4}{2\,b^3}+\frac {5\,a^4\,x}{b^6}+\frac {a^2\,x^3}{b^4}-\frac {2\,a^3\,x^2}{b^5}-\frac {a^6}{b\,\left (x\,b^7+a\,b^6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a + b*x)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.28, size = 78, normalized size = 0.96 \begin {gather*} - \frac {a^{6}}{a b^{7} + b^{8} x} - \frac {6 a^{5} \log {\left (a + b x \right )}}{b^{7}} + \frac {5 a^{4} x}{b^{6}} - \frac {2 a^{3} x^{2}}{b^{5}} + \frac {a^{2} x^{3}}{b^{4}} - \frac {a x^{4}}{2 b^{3}} + \frac {x^{5}}{5 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x+a)**2,x)

[Out]

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

________________________________________________________________________________________