3.17.34 \(\int \frac {x^3}{(a+\frac {b}{x})^3} \, dx\) [1634]

Optimal. Leaf size=86 \[ -\frac {10 b^3 x}{a^6}+\frac {3 b^2 x^2}{a^5}-\frac {b x^3}{a^4}+\frac {x^4}{4 a^3}-\frac {b^6}{2 a^7 (b+a x)^2}+\frac {6 b^5}{a^7 (b+a x)}+\frac {15 b^4 \log (b+a x)}{a^7} \]

[Out]

-10*b^3*x/a^6+3*b^2*x^2/a^5-b*x^3/a^4+1/4*x^4/a^3-1/2*b^6/a^7/(a*x+b)^2+6*b^5/a^7/(a*x+b)+15*b^4*ln(a*x+b)/a^7

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Rubi [A]
time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 45} \begin {gather*} -\frac {b^6}{2 a^7 (a x+b)^2}+\frac {6 b^5}{a^7 (a x+b)}+\frac {15 b^4 \log (a x+b)}{a^7}-\frac {10 b^3 x}{a^6}+\frac {3 b^2 x^2}{a^5}-\frac {b x^3}{a^4}+\frac {x^4}{4 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b/x)^3,x]

[Out]

(-10*b^3*x)/a^6 + (3*b^2*x^2)/a^5 - (b*x^3)/a^4 + x^4/(4*a^3) - b^6/(2*a^7*(b + a*x)^2) + (6*b^5)/(a^7*(b + a*
x)) + (15*b^4*Log[b + a*x])/a^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])

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rubi steps

\begin {align*} \int \frac {x^3}{\left (a+\frac {b}{x}\right )^3} \, dx &=\int \frac {x^6}{(b+a x)^3} \, dx\\ &=\int \left (-\frac {10 b^3}{a^6}+\frac {6 b^2 x}{a^5}-\frac {3 b x^2}{a^4}+\frac {x^3}{a^3}+\frac {b^6}{a^6 (b+a x)^3}-\frac {6 b^5}{a^6 (b+a x)^2}+\frac {15 b^4}{a^6 (b+a x)}\right ) \, dx\\ &=-\frac {10 b^3 x}{a^6}+\frac {3 b^2 x^2}{a^5}-\frac {b x^3}{a^4}+\frac {x^4}{4 a^3}-\frac {b^6}{2 a^7 (b+a x)^2}+\frac {6 b^5}{a^7 (b+a x)}+\frac {15 b^4 \log (b+a x)}{a^7}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 73, normalized size = 0.85 \begin {gather*} \frac {-40 a b^3 x+12 a^2 b^2 x^2-4 a^3 b x^3+a^4 x^4+\frac {2 b^5 (11 b+12 a x)}{(b+a x)^2}+60 b^4 \log (b+a x)}{4 a^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b/x)^3,x]

[Out]

(-40*a*b^3*x + 12*a^2*b^2*x^2 - 4*a^3*b*x^3 + a^4*x^4 + (2*b^5*(11*b + 12*a*x))/(b + a*x)^2 + 60*b^4*Log[b + a
*x])/(4*a^7)

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Maple [A]
time = 0.03, size = 83, normalized size = 0.97

method	result	size
risch	\(\frac {x^{4}}{4 a^{3}}-\frac {b \,x^{3}}{a^{4}}+\frac {3 b^{2} x^{2}}{a^{5}}-\frac {10 b^{3} x}{a^{6}}+\frac {6 b^{5} x +\frac {11 b^{6}}{2 a}}{a^{6} \left (a x +b \right )^{2}}+\frac {15 b^{4} \ln \left (a x +b \right )}{a^{7}}\)	\(79\)
norman	\(\frac {\frac {x^{6}}{4 a}-\frac {b \,x^{5}}{2 a^{2}}+\frac {5 b^{2} x^{4}}{4 a^{3}}+\frac {45 b^{6}}{2 a^{7}}-\frac {5 b^{3} x^{3}}{a^{4}}+\frac {30 b^{5} x}{a^{6}}}{\left (a x +b \right )^{2}}+\frac {15 b^{4} \ln \left (a x +b \right )}{a^{7}}\)	\(81\)
default	\(\frac {\frac {1}{4} a^{3} x^{4}-a^{2} b \,x^{3}+3 a \,b^{2} x^{2}-10 b^{3} x}{a^{6}}-\frac {b^{6}}{2 a^{7} \left (a x +b \right )^{2}}+\frac {6 b^{5}}{a^{7} \left (a x +b \right )}+\frac {15 b^{4} \ln \left (a x +b \right )}{a^{7}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+1/x*b)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^6*(1/4*a^3*x^4-a^2*b*x^3+3*a*b^2*x^2-10*b^3*x)-1/2*b^6/a^7/(a*x+b)^2+6*b^5/a^7/(a*x+b)+15*b^4*ln(a*x+b)/a^
7

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Maxima [A]
time = 0.29, size = 91, normalized size = 1.06 \begin {gather*} \frac {12 \, a b^{5} x + 11 \, b^{6}}{2 \, {\left (a^{9} x^{2} + 2 \, a^{8} b x + a^{7} b^{2}\right )}} + \frac {15 \, b^{4} \log \left (a x + b\right )}{a^{7}} + \frac {a^{3} x^{4} - 4 \, a^{2} b x^{3} + 12 \, a b^{2} x^{2} - 40 \, b^{3} x}{4 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(12*a*b^5*x + 11*b^6)/(a^9*x^2 + 2*a^8*b*x + a^7*b^2) + 15*b^4*log(a*x + b)/a^7 + 1/4*(a^3*x^4 - 4*a^2*b*x
^3 + 12*a*b^2*x^2 - 40*b^3*x)/a^6

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Fricas [A]
time = 0.36, size = 117, normalized size = 1.36 \begin {gather*} \frac {a^{6} x^{6} - 2 \, a^{5} b x^{5} + 5 \, a^{4} b^{2} x^{4} - 20 \, a^{3} b^{3} x^{3} - 68 \, a^{2} b^{4} x^{2} - 16 \, a b^{5} x + 22 \, b^{6} + 60 \, {\left (a^{2} b^{4} x^{2} + 2 \, a b^{5} x + b^{6}\right )} \log \left (a x + b\right )}{4 \, {\left (a^{9} x^{2} + 2 \, a^{8} b x + a^{7} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a^6*x^6 - 2*a^5*b*x^5 + 5*a^4*b^2*x^4 - 20*a^3*b^3*x^3 - 68*a^2*b^4*x^2 - 16*a*b^5*x + 22*b^6 + 60*(a^2*b
^4*x^2 + 2*a*b^5*x + b^6)*log(a*x + b))/(a^9*x^2 + 2*a^8*b*x + a^7*b^2)

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Sympy [A]
time = 0.15, size = 92, normalized size = 1.07 \begin {gather*} \frac {12 a b^{5} x + 11 b^{6}}{2 a^{9} x^{2} + 4 a^{8} b x + 2 a^{7} b^{2}} + \frac {x^{4}}{4 a^{3}} - \frac {b x^{3}}{a^{4}} + \frac {3 b^{2} x^{2}}{a^{5}} - \frac {10 b^{3} x}{a^{6}} + \frac {15 b^{4} \log {\left (a x + b \right )}}{a^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(12*a*b**5*x + 11*b**6)/(2*a**9*x**2 + 4*a**8*b*x + 2*a**7*b**2) + x**4/(4*a**3) - b*x**3/a**4 + 3*b**2*x**2/a
**5 - 10*b**3*x/a**6 + 15*b**4*log(a*x + b)/a**7

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Giac [A]
time = 0.52, size = 83, normalized size = 0.97 \begin {gather*} \frac {15 \, b^{4} \log \left ({\left | a x + b \right |}\right )}{a^{7}} + \frac {12 \, a b^{5} x + 11 \, b^{6}}{2 \, {\left (a x + b\right )}^{2} a^{7}} + \frac {a^{9} x^{4} - 4 \, a^{8} b x^{3} + 12 \, a^{7} b^{2} x^{2} - 40 \, a^{6} b^{3} x}{4 \, a^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15*b^4*log(abs(a*x + b))/a^7 + 1/2*(12*a*b^5*x + 11*b^6)/((a*x + b)^2*a^7) + 1/4*(a^9*x^4 - 4*a^8*b*x^3 + 12*a
^7*b^2*x^2 - 40*a^6*b^3*x)/a^12

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Mupad [B]
time = 0.06, size = 92, normalized size = 1.07 \begin {gather*} \frac {6\,b^5\,x+\frac {11\,b^6}{2\,a}}{a^8\,x^2+2\,a^7\,b\,x+a^6\,b^2}+\frac {x^4}{4\,a^3}+\frac {15\,b^4\,\ln \left (b+a\,x\right )}{a^7}-\frac {b\,x^3}{a^4}-\frac {10\,b^3\,x}{a^6}+\frac {3\,b^2\,x^2}{a^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b/x)^3,x)

[Out]

(6*b^5*x + (11*b^6)/(2*a))/(a^6*b^2 + a^8*x^2 + 2*a^7*b*x) + x^4/(4*a^3) + (15*b^4*log(b + a*x))/a^7 - (b*x^3)
/a^4 - (10*b^3*x)/a^6 + (3*b^2*x^2)/a^5

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