3.2.84 \(\int \frac {x^3}{(a+b x)^3} \, dx\)

Optimal. Leaf size=50 \[ \frac {a^3}{2 b^4 (a+b x)^2}-\frac {3 a^2}{b^4 (a+b x)}-\frac {3 a \log (a+b x)}{b^4}+\frac {x}{b^3} \]

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Rubi [A]  time = 0.03, antiderivative size = 50, 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 {a^3}{2 b^4 (a+b x)^2}-\frac {3 a^2}{b^4 (a+b x)}-\frac {3 a \log (a+b x)}{b^4}+\frac {x}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x)^3,x]

[Out]

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

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^3}{(a+b x)^3} \, dx &=\int \left (\frac {1}{b^3}-\frac {a^3}{b^3 (a+b x)^3}+\frac {3 a^2}{b^3 (a+b x)^2}-\frac {3 a}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {x}{b^3}+\frac {a^3}{2 b^4 (a+b x)^2}-\frac {3 a^2}{b^4 (a+b x)}-\frac {3 a \log (a+b x)}{b^4}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 40, normalized size = 0.80 \begin {gather*} -\frac {\frac {a^2 (5 a+6 b x)}{(a+b x)^2}+6 a \log (a+b x)-2 b x}{2 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x)^3,x]

[Out]

-1/2*(-2*b*x + (a^2*(5*a + 6*b*x))/(a + b*x)^2 + 6*a*Log[a + b*x])/b^4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{(a+b x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3/(a + b*x)^3,x]

[Out]

IntegrateAlgebraic[x^3/(a + b*x)^3, x]

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fricas [A]  time = 0.84, size = 83, normalized size = 1.66 \begin {gather*} \frac {2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} - 4 \, a^{2} b x - 5 \, a^{3} - 6 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.96, size = 44, normalized size = 0.88 \begin {gather*} \frac {x}{b^{3}} - \frac {3 \, a \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {6 \, a^{2} b x + 5 \, a^{3}}{2 \, {\left (b x + a\right )}^{2} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 49, normalized size = 0.98 \begin {gather*} \frac {a^{3}}{2 \left (b x +a \right )^{2} b^{4}}-\frac {3 a^{2}}{\left (b x +a \right ) b^{4}}-\frac {3 a \ln \left (b x +a \right )}{b^{4}}+\frac {x}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x+a)^3,x)

[Out]

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

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maxima [A]  time = 1.31, size = 57, normalized size = 1.14 \begin {gather*} -\frac {6 \, a^{2} b x + 5 \, a^{3}}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {x}{b^{3}} - \frac {3 \, a \log \left (b x + a\right )}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.15, size = 43, normalized size = 0.86 \begin {gather*} -\frac {3\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {3\,a^2}{a+b\,x}-\frac {a^3}{2\,{\left (a+b\,x\right )}^2}}{b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x)^3,x)

[Out]

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

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sympy [A]  time = 0.31, size = 58, normalized size = 1.16 \begin {gather*} - \frac {3 a \log {\left (a + b x \right )}}{b^{4}} + \frac {- 5 a^{3} - 6 a^{2} b x}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac {x}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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