3.213 \(\int \frac{x^6}{a x^2+b x^3} \, dx\)

Optimal. Leaf size=57 \[ \frac{a^2 x^2}{2 b^3}-\frac{a^3 x}{b^4}+\frac{a^4 \log (a+b x)}{b^5}-\frac{a x^3}{3 b^2}+\frac{x^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0355117, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 43} \[ \frac{a^2 x^2}{2 b^3}-\frac{a^3 x}{b^4}+\frac{a^4 \log (a+b x)}{b^5}-\frac{a x^3}{3 b^2}+\frac{x^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a*x^2 + b*x^3),x]

[Out]

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

Rule 1584

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

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

Mathematica [A]  time = 0.0045484, size = 57, normalized size = 1. \[ \frac{a^2 x^2}{2 b^3}-\frac{a^3 x}{b^4}+\frac{a^4 \log (a+b x)}{b^5}-\frac{a x^3}{3 b^2}+\frac{x^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a*x^2 + b*x^3),x]

[Out]

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

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Maple [A]  time = 0.003, size = 52, normalized size = 0.9 \begin{align*} -{\frac{x{a}^{3}}{{b}^{4}}}+{\frac{{a}^{2}{x}^{2}}{2\,{b}^{3}}}-{\frac{a{x}^{3}}{3\,{b}^{2}}}+{\frac{{x}^{4}}{4\,b}}+{\frac{{a}^{4}\ln \left ( bx+a \right ) }{{b}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^3+a*x^2),x)

[Out]

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

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Maxima [A]  time = 0.961211, size = 70, normalized size = 1.23 \begin{align*} \frac{a^{4} \log \left (b x + a\right )}{b^{5}} + \frac{3 \, b^{3} x^{4} - 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} - 12 \, a^{3} x}{12 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^3+a*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.734538, size = 117, normalized size = 2.05 \begin{align*} \frac{3 \, b^{4} x^{4} - 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a^{3} b x + 12 \, a^{4} \log \left (b x + a\right )}{12 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^3+a*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.721893, size = 49, normalized size = 0.86 \begin{align*} \frac{a^{4} \log{\left (a + b x \right )}}{b^{5}} - \frac{a^{3} x}{b^{4}} + \frac{a^{2} x^{2}}{2 b^{3}} - \frac{a x^{3}}{3 b^{2}} + \frac{x^{4}}{4 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x**3+a*x**2),x)

[Out]

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

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Giac [A]  time = 1.14571, size = 72, normalized size = 1.26 \begin{align*} \frac{a^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac{3 \, b^{3} x^{4} - 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} - 12 \, a^{3} x}{12 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^3+a*x^2),x, algorithm="giac")

[Out]

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