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

Optimal. Leaf size=35 \[ \frac{a c}{2 b^2 \left (a+b x^2\right )}+\frac{c \log \left (a+b x^2\right )}{2 b^2} \]

[Out]

(a*c)/(2*b^2*(a + b*x^2)) + (c*Log[a + b*x^2])/(2*b^2)

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Rubi [A]  time = 0.0268245, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {21, 266, 43} \[ \frac{a c}{2 b^2 \left (a+b x^2\right )}+\frac{c \log \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(a*c + b*c*x^2))/(a + b*x^2)^3,x]

[Out]

(a*c)/(2*b^2*(a + b*x^2)) + (c*Log[a + b*x^2])/(2*b^2)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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

Mathematica [A]  time = 0.0089612, size = 28, normalized size = 0.8 \[ \frac{c \left (\frac{a}{a+b x^2}+\log \left (a+b x^2\right )\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a*c + b*c*x^2))/(a + b*x^2)^3,x]

[Out]

(c*(a/(a + b*x^2) + Log[a + b*x^2]))/(2*b^2)

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Maple [A]  time = 0.007, size = 32, normalized size = 0.9 \begin{align*}{\frac{ac}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*c/b^2/(b*x^2+a)+1/2*c*ln(b*x^2+a)/b^2

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Maxima [A]  time = 0.997979, size = 46, normalized size = 1.31 \begin{align*} \frac{a c}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} + \frac{c \log \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*c/(b^3*x^2 + a*b^2) + 1/2*c*log(b*x^2 + a)/b^2

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Fricas [A]  time = 1.21433, size = 84, normalized size = 2.4 \begin{align*} \frac{a c +{\left (b c x^{2} + a c\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*c + (b*c*x^2 + a*c)*log(b*x^2 + a))/(b^3*x^2 + a*b^2)

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Sympy [A]  time = 0.340099, size = 31, normalized size = 0.89 \begin{align*} c \left (\frac{a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(a/(2*a*b**2 + 2*b**3*x**2) + log(a + b*x**2)/(2*b**2))

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Giac [A]  time = 1.23018, size = 43, normalized size = 1.23 \begin{align*} \frac{c \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} + \frac{a c}{2 \,{\left (b x^{2} + a\right )} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*log(abs(b*x^2 + a))/b^2 + 1/2*a*c/((b*x^2 + a)*b^2)