3.2.83 \(\int \frac {x^4}{(a+b x)^3} \, dx\)

Optimal. Leaf size=64 \[ -\frac {a^4}{2 b^5 (a+b x)^2}+\frac {4 a^3}{b^5 (a+b x)}+\frac {6 a^2 \log (a+b x)}{b^5}-\frac {3 a x}{b^4}+\frac {x^2}{2 b^3} \]

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Rubi [A]  time = 0.04, antiderivative size = 64, 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}{2 b^5 (a+b x)^2}+\frac {4 a^3}{b^5 (a+b x)}+\frac {6 a^2 \log (a+b x)}{b^5}-\frac {3 a x}{b^4}+\frac {x^2}{2 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x)^3,x]

[Out]

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

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Mathematica [A]  time = 0.02, size = 55, normalized size = 0.86 \begin {gather*} \frac {-\frac {a^4}{(a+b x)^2}+\frac {8 a^3}{a+b x}+12 a^2 \log (a+b x)-6 a b x+b^2 x^2}{2 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x)^3,x]

[Out]

(-6*a*b*x + b^2*x^2 - a^4/(a + b*x)^2 + (8*a^3)/(a + b*x) + 12*a^2*Log[a + b*x])/(2*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)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4/(a + b*x)^3,x]

[Out]

IntegrateAlgebraic[x^4/(a + b*x)^3, x]

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fricas [A]  time = 0.88, size = 95, normalized size = 1.48 \begin {gather*} \frac {b^{4} x^{4} - 4 \, a b^{3} x^{3} - 11 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b x + 7 \, a^{4} + 12 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.95, size = 61, normalized size = 0.95 \begin {gather*} \frac {6 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{3} x^{2} - 6 \, a b^{2} x}{2 \, b^{6}} + \frac {8 \, a^{3} b x + 7 \, a^{4}}{2 \, {\left (b x + a\right )}^{2} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 61, normalized size = 0.95 \begin {gather*} -\frac {a^{4}}{2 \left (b x +a \right )^{2} b^{5}}+\frac {x^{2}}{2 b^{3}}+\frac {4 a^{3}}{\left (b x +a \right ) b^{5}}+\frac {6 a^{2} \ln \left (b x +a \right )}{b^{5}}-\frac {3 a x}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x+a)^3,x)

[Out]

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

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maxima [A]  time = 1.33, size = 69, normalized size = 1.08 \begin {gather*} \frac {8 \, a^{3} b x + 7 \, a^{4}}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {6 \, a^{2} \log \left (b x + a\right )}{b^{5}} + \frac {b x^{2} - 6 \, a x}{2 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.08, size = 54, normalized size = 0.84 \begin {gather*} \frac {\frac {{\left (a+b\,x\right )}^2}{2}+\frac {4\,a^3}{a+b\,x}-\frac {a^4}{2\,{\left (a+b\,x\right )}^2}+6\,a^2\,\ln \left (a+b\,x\right )-4\,a\,b\,x}{b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b*x)^3,x)

[Out]

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

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sympy [A]  time = 0.34, size = 70, normalized size = 1.09 \begin {gather*} \frac {6 a^{2} \log {\left (a + b x \right )}}{b^{5}} - \frac {3 a x}{b^{4}} + \frac {7 a^{4} + 8 a^{3} b x}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {x^{2}}{2 b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a)**3,x)

[Out]

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

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