3.157 \(\int \frac{x^5}{a+b x} \, dx\)

Optimal. Leaf size=70 \[ -\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^3}{3 b^3}+\frac{a^4 x}{b^5}-\frac{a^5 \log (a+b x)}{b^6}-\frac{a x^4}{4 b^2}+\frac{x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.0344801, antiderivative size = 70, normalized size of antiderivative = 1., 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} \[ -\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^3}{3 b^3}+\frac{a^4 x}{b^5}-\frac{a^5 \log (a+b x)}{b^6}-\frac{a x^4}{4 b^2}+\frac{x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0053479, size = 70, normalized size = 1. \[ -\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^3}{3 b^3}+\frac{a^4 x}{b^5}-\frac{a^5 \log (a+b x)}{b^6}-\frac{a x^4}{4 b^2}+\frac{x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x),x]

[Out]

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

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Maple [A]  time = 0.002, size = 63, normalized size = 0.9 \begin{align*}{\frac{{a}^{4}x}{{b}^{5}}}-{\frac{{a}^{3}{x}^{2}}{2\,{b}^{4}}}+{\frac{{a}^{2}{x}^{3}}{3\,{b}^{3}}}-{\frac{a{x}^{4}}{4\,{b}^{2}}}+{\frac{{x}^{5}}{5\,b}}-{\frac{{a}^{5}\ln \left ( bx+a \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.06738, size = 86, normalized size = 1.23 \begin{align*} -\frac{a^{5} \log \left (b x + a\right )}{b^{6}} + \frac{12 \, b^{4} x^{5} - 15 \, a b^{3} x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a^{3} b x^{2} + 60 \, a^{4} x}{60 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a),x, algorithm="maxima")

[Out]

-a^5*log(b*x + a)/b^6 + 1/60*(12*b^4*x^5 - 15*a*b^3*x^4 + 20*a^2*b^2*x^3 - 30*a^3*b*x^2 + 60*a^4*x)/b^5

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Fricas [A]  time = 1.49602, size = 144, normalized size = 2.06 \begin{align*} \frac{12 \, b^{5} x^{5} - 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} - 30 \, a^{3} b^{2} x^{2} + 60 \, a^{4} b x - 60 \, a^{5} \log \left (b x + a\right )}{60 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a),x, algorithm="fricas")

[Out]

1/60*(12*b^5*x^5 - 15*a*b^4*x^4 + 20*a^2*b^3*x^3 - 30*a^3*b^2*x^2 + 60*a^4*b*x - 60*a^5*log(b*x + a))/b^6

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Sympy [A]  time = 0.373252, size = 61, normalized size = 0.87 \begin{align*} - \frac{a^{5} \log{\left (a + b x \right )}}{b^{6}} + \frac{a^{4} x}{b^{5}} - \frac{a^{3} x^{2}}{2 b^{4}} + \frac{a^{2} x^{3}}{3 b^{3}} - \frac{a x^{4}}{4 b^{2}} + \frac{x^{5}}{5 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x+a),x)

[Out]

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

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Giac [A]  time = 1.15778, size = 88, normalized size = 1.26 \begin{align*} -\frac{a^{5} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{12 \, b^{4} x^{5} - 15 \, a b^{3} x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a^{3} b x^{2} + 60 \, a^{4} x}{60 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a),x, algorithm="giac")

[Out]

-a^5*log(abs(b*x + a))/b^6 + 1/60*(12*b^4*x^5 - 15*a*b^3*x^4 + 20*a^2*b^2*x^3 - 30*a^3*b*x^2 + 60*a^4*x)/b^5