3.2.97 \(\int \frac {x^4}{(a+b x)^4} \, dx\)

Optimal. Leaf size=65 \[ -\frac {a^4}{3 b^5 (a+b x)^3}+\frac {2 a^3}{b^5 (a+b x)^2}-\frac {6 a^2}{b^5 (a+b x)}-\frac {4 a \log (a+b x)}{b^5}+\frac {x}{b^4} \]

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Rubi [A]  time = 0.04, antiderivative size = 65, 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^4}{3 b^5 (a+b x)^3}+\frac {2 a^3}{b^5 (a+b x)^2}-\frac {6 a^2}{b^5 (a+b x)}-\frac {4 a \log (a+b x)}{b^5}+\frac {x}{b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x)^4,x]

[Out]

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

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^4}{(a+b x)^4} \, dx &=\int \left (\frac {1}{b^4}+\frac {a^4}{b^4 (a+b x)^4}-\frac {4 a^3}{b^4 (a+b x)^3}+\frac {6 a^2}{b^4 (a+b x)^2}-\frac {4 a}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {x}{b^4}-\frac {a^4}{3 b^5 (a+b x)^3}+\frac {2 a^3}{b^5 (a+b x)^2}-\frac {6 a^2}{b^5 (a+b x)}-\frac {4 a \log (a+b x)}{b^5}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 51, normalized size = 0.78 \begin {gather*} -\frac {\frac {a^2 \left (13 a^2+30 a b x+18 b^2 x^2\right )}{(a+b x)^3}+12 a \log (a+b x)-3 b x}{3 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x)^4,x]

[Out]

-1/3*(-3*b*x + (a^2*(13*a^2 + 30*a*b*x + 18*b^2*x^2))/(a + b*x)^3 + 12*a*Log[a + b*x])/b^5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{(a+b x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4/(a + b*x)^4,x]

[Out]

IntegrateAlgebraic[x^4/(a + b*x)^4, x]

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fricas [A]  time = 0.88, size = 116, normalized size = 1.78 \begin {gather*} \frac {3 \, b^{4} x^{4} + 9 \, a b^{3} x^{3} - 9 \, a^{2} b^{2} x^{2} - 27 \, a^{3} b x - 13 \, a^{4} - 12 \, {\left (a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 3 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{3 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/3*(3*b^4*x^4 + 9*a*b^3*x^3 - 9*a^2*b^2*x^2 - 27*a^3*b*x - 13*a^4 - 12*(a*b^3*x^3 + 3*a^2*b^2*x^2 + 3*a^3*b*x
+ a^4)*log(b*x + a))/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5)

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giac [A]  time = 0.86, size = 55, normalized size = 0.85 \begin {gather*} \frac {x}{b^{4}} - \frac {4 \, a \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {18 \, a^{2} b^{2} x^{2} + 30 \, a^{3} b x + 13 \, a^{4}}{3 \, {\left (b x + a\right )}^{3} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^4,x, algorithm="giac")

[Out]

x/b^4 - 4*a*log(abs(b*x + a))/b^5 - 1/3*(18*a^2*b^2*x^2 + 30*a^3*b*x + 13*a^4)/((b*x + a)^3*b^5)

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maple [A]  time = 0.01, size = 64, normalized size = 0.98 \begin {gather*} -\frac {a^{4}}{3 \left (b x +a \right )^{3} b^{5}}+\frac {2 a^{3}}{\left (b x +a \right )^{2} b^{5}}-\frac {6 a^{2}}{\left (b x +a \right ) b^{5}}-\frac {4 a \ln \left (b x +a \right )}{b^{5}}+\frac {x}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x+a)^4,x)

[Out]

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

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maxima [A]  time = 1.41, size = 79, normalized size = 1.22 \begin {gather*} -\frac {18 \, a^{2} b^{2} x^{2} + 30 \, a^{3} b x + 13 \, a^{4}}{3 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac {x}{b^{4}} - \frac {4 \, a \log \left (b x + a\right )}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/3*(18*a^2*b^2*x^2 + 30*a^3*b*x + 13*a^4)/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5) + x/b^4 - 4*a*log(
b*x + a)/b^5

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mupad [B]  time = 0.17, size = 55, normalized size = 0.85 \begin {gather*} -\frac {4\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {6\,a^2}{a+b\,x}-\frac {2\,a^3}{{\left (a+b\,x\right )}^2}+\frac {a^4}{3\,{\left (a+b\,x\right )}^3}}{b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b*x)^4,x)

[Out]

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

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sympy [A]  time = 0.40, size = 82, normalized size = 1.26 \begin {gather*} - \frac {4 a \log {\left (a + b x \right )}}{b^{5}} + \frac {- 13 a^{4} - 30 a^{3} b x - 18 a^{2} b^{2} x^{2}}{3 a^{3} b^{5} + 9 a^{2} b^{6} x + 9 a b^{7} x^{2} + 3 b^{8} x^{3}} + \frac {x}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a)**4,x)

[Out]

-4*a*log(a + b*x)/b**5 + (-13*a**4 - 30*a**3*b*x - 18*a**2*b**2*x**2)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*
x**2 + 3*b**8*x**3) + x/b**4

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