3.2.68 \(\int \frac {x^6}{(a+b x)^2} \, dx\) [168]

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

[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

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Rubi [A]
time = 0.04, 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 = {45} \begin {gather*} -\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} \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 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*} \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*}

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Mathematica [A]
time = 0.01, size = 77, normalized size = 0.95 \begin {gather*} \frac {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-\frac {10 a^6}{a+b x}-60 a^5 \log (a+b x)}{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)

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Mathics [A]
time = 2.26, size = 98, normalized size = 1.21 \begin {gather*} \frac {-6 a^6 \text {Log}\left [a+b x\right ]-a^6-6 a^5 b x \text {Log}\left [a+b x\right ]+5 a^5 b x+3 a^4 b^2 x^2-a^3 b^3 x^3+\frac {a^2 b^4 x^4}{2}-\frac {3 a b^5 x^5}{10}+\frac {b^6 x^6}{5}}{b^7 \left (a+b x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^6/(a + b*x)^2,x]')

[Out]

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

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Maple [A]
time = 0.08, size = 78, normalized size = 0.96

method	result	size
default	\(\frac {\frac {1}{5} b^{4} x^{5}-\frac {1}{2} a \,b^{3} x^{4}+a^{2} b^{2} x^{3}-2 a^{3} b \,x^{2}+5 a^{4} x}{b^{6}}-\frac {a^{6}}{b^{7} \left (b x +a \right )}-\frac {6 a^{5} \ln \left (b x +a \right )}{b^{7}}\)	\(78\)
risch	\(\frac {5 a^{4} x}{b^{6}}-\frac {2 a^{3} x^{2}}{b^{5}}+\frac {a^{2} x^{3}}{b^{4}}-\frac {a \,x^{4}}{2 b^{3}}+\frac {x^{5}}{5 b^{2}}-\frac {a^{6}}{b^{7} \left (b x +a \right )}-\frac {6 a^{5} \ln \left (b x +a \right )}{b^{7}}\)	\(78\)
norman	\(\frac {\frac {x^{6}}{5 b}-\frac {3 a \,x^{5}}{10 b^{2}}-\frac {6 a^{6}}{b^{7}}-\frac {a^{3} x^{3}}{b^{4}}+\frac {3 a^{4} x^{2}}{b^{5}}+\frac {a^{2} x^{4}}{2 b^{3}}}{b x +a}-\frac {6 a^{5} \ln \left (b x +a \right )}{b^{7}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.24, 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

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Fricas [A]
time = 0.32, 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)

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Sympy [A]
time = 0.12, 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)

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Giac [A]
time = 0.00, size = 92, normalized size = 1.14 \begin {gather*} \frac {\frac {1}{5} x^{5} b^{8}-\frac {1}{2} x^{4} b^{7} a+x^{3} b^{6} a^{2}-2 x^{2} b^{5} a^{3}+5 x b^{4} a^{4}}{b^{10}}-\frac {a^{6}}{b^{7} \left (x b+a\right )}-\frac {6 a^{5} \ln \left |x b+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)

[Out]

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

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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))

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