3.223 \(\int \frac{x^7}{(a x^2+b x^3)^2} \, dx\)

Optimal. Leaf size=46 \[ \frac{a^3}{b^4 (a+b x)}+\frac{3 a^2 \log (a+b x)}{b^4}-\frac{2 a x}{b^3}+\frac{x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0301855, antiderivative size = 46, 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} \[ \frac{a^3}{b^4 (a+b x)}+\frac{3 a^2 \log (a+b x)}{b^4}-\frac{2 a x}{b^3}+\frac{x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a*x^2 + b*x^3)^2,x]

[Out]

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

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^7}{\left (a x^2+b x^3\right )^2} \, dx &=\int \frac{x^3}{(a+b x)^2} \, dx\\ &=\int \left (-\frac{2 a}{b^3}+\frac{x}{b^2}-\frac{a^3}{b^3 (a+b x)^2}+\frac{3 a^2}{b^3 (a+b x)}\right ) \, dx\\ &=-\frac{2 a x}{b^3}+\frac{x^2}{2 b^2}+\frac{a^3}{b^4 (a+b x)}+\frac{3 a^2 \log (a+b x)}{b^4}\\ \end{align*}

Mathematica [A]  time = 0.0136239, size = 43, normalized size = 0.93 \[ \frac{\frac{2 a^3}{a+b x}+6 a^2 \log (a+b x)-4 a b x+b^2 x^2}{2 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a*x^2 + b*x^3)^2,x]

[Out]

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

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Maple [A]  time = 0.006, size = 45, normalized size = 1. \begin{align*} -2\,{\frac{ax}{{b}^{3}}}+{\frac{{x}^{2}}{2\,{b}^{2}}}+{\frac{{a}^{3}}{{b}^{4} \left ( bx+a \right ) }}+3\,{\frac{{a}^{2}\ln \left ( bx+a \right ) }{{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(b*x^3+a*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.993047, size = 63, normalized size = 1.37 \begin{align*} \frac{a^{3}}{b^{5} x + a b^{4}} + \frac{3 \, a^{2} \log \left (b x + a\right )}{b^{4}} + \frac{b x^{2} - 4 \, a x}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^3+a*x^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.7575, size = 132, normalized size = 2.87 \begin{align*} \frac{b^{3} x^{3} - 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 2 \, a^{3} + 6 \,{\left (a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^3+a*x^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.969929, size = 44, normalized size = 0.96 \begin{align*} \frac{a^{3}}{a b^{4} + b^{5} x} + \frac{3 a^{2} \log{\left (a + b x \right )}}{b^{4}} - \frac{2 a x}{b^{3}} + \frac{x^{2}}{2 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(b*x**3+a*x**2)**2,x)

[Out]

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

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Giac [A]  time = 1.23639, size = 65, normalized size = 1.41 \begin{align*} \frac{3 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac{a^{3}}{{\left (b x + a\right )} b^{4}} + \frac{b^{2} x^{2} - 4 \, a b x}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^3+a*x^2)^2,x, algorithm="giac")

[Out]

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