3.1.67 \(\int \frac {x^3 (A+B x^2)}{(b x^2+c x^4)^2} \, dx\) [67]

Optimal. Leaf size=51 \[ -\frac {b B-A c}{2 b c \left (b+c x^2\right )}+\frac {A \log (x)}{b^2}-\frac {A \log \left (b+c x^2\right )}{2 b^2} \]

[Out]

1/2*(A*c-B*b)/b/c/(c*x^2+b)+A*ln(x)/b^2-1/2*A*ln(c*x^2+b)/b^2

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Rubi [A]
time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1598, 457, 78} \begin {gather*} -\frac {A \log \left (b+c x^2\right )}{2 b^2}+\frac {A \log (x)}{b^2}-\frac {b B-A c}{2 b c \left (b+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

-1/2*(b*B - A*c)/(b*c*(b + c*x^2)) + (A*Log[x])/b^2 - (A*Log[b + c*x^2])/(2*b^2)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1598

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]

Rubi steps

\begin {align*} \int \frac {x^3 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {A+B x^2}{x \left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x (b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{b^2 x}+\frac {b B-A c}{b (b+c x)^2}-\frac {A c}{b^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b B-A c}{2 b c \left (b+c x^2\right )}+\frac {A \log (x)}{b^2}-\frac {A \log \left (b+c x^2\right )}{2 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.90 \begin {gather*} \frac {\frac {b (-b B+A c)}{c \left (b+c x^2\right )}+2 A \log (x)-A \log \left (b+c x^2\right )}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

((b*(-(b*B) + A*c))/(c*(b + c*x^2)) + 2*A*Log[x] - A*Log[b + c*x^2])/(2*b^2)

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Maple [A]
time = 0.37, size = 48, normalized size = 0.94

method	result	size
default	\(-\frac {A \ln \left (c \,x^{2}+b \right )-\frac {b \left (A c -B b \right )}{c \left (c \,x^{2}+b \right )}}{2 b^{2}}+\frac {A \ln \left (x \right )}{b^{2}}\)	\(48\)
norman	\(-\frac {\left (A c -B b \right ) x^{2}}{2 b^{2} \left (c \,x^{2}+b \right )}+\frac {A \ln \left (x \right )}{b^{2}}-\frac {A \ln \left (c \,x^{2}+b \right )}{2 b^{2}}\)	\(48\)
risch	\(\frac {A}{2 b \left (c \,x^{2}+b \right )}-\frac {B}{2 c \left (c \,x^{2}+b \right )}+\frac {A \ln \left (x \right )}{b^{2}}-\frac {A \ln \left (c \,x^{2}+b \right )}{2 b^{2}}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(B*x^2+A)/(c*x^4+b*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

-1/2/b^2*(A*ln(c*x^2+b)-b*(A*c-B*b)/c/(c*x^2+b))+A*ln(x)/b^2

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Maxima [A]
time = 0.27, size = 51, normalized size = 1.00 \begin {gather*} -\frac {B b - A c}{2 \, {\left (b c^{2} x^{2} + b^{2} c\right )}} - \frac {A \log \left (c x^{2} + b\right )}{2 \, b^{2}} + \frac {A \log \left (x^{2}\right )}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x^2+A)/(c*x^4+b*x^2)^2,x, algorithm="maxima")

[Out]

-1/2*(B*b - A*c)/(b*c^2*x^2 + b^2*c) - 1/2*A*log(c*x^2 + b)/b^2 + 1/2*A*log(x^2)/b^2

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Fricas [A]
time = 1.48, size = 70, normalized size = 1.37 \begin {gather*} -\frac {B b^{2} - A b c + {\left (A c^{2} x^{2} + A b c\right )} \log \left (c x^{2} + b\right ) - 2 \, {\left (A c^{2} x^{2} + A b c\right )} \log \left (x\right )}{2 \, {\left (b^{2} c^{2} x^{2} + b^{3} c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x^2+A)/(c*x^4+b*x^2)^2,x, algorithm="fricas")

[Out]

-1/2*(B*b^2 - A*b*c + (A*c^2*x^2 + A*b*c)*log(c*x^2 + b) - 2*(A*c^2*x^2 + A*b*c)*log(x))/(b^2*c^2*x^2 + b^3*c)

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Sympy [A]
time = 0.21, size = 46, normalized size = 0.90 \begin {gather*} \frac {A \log {\left (x \right )}}{b^{2}} - \frac {A \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{2}} + \frac {A c - B b}{2 b^{2} c + 2 b c^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

A*log(x)/b**2 - A*log(b/c + x**2)/(2*b**2) + (A*c - B*b)/(2*b**2*c + 2*b*c**2*x**2)

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Giac [A]
time = 1.20, size = 52, normalized size = 1.02 \begin {gather*} -\frac {A \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac {A \log \left ({\left | x \right |}\right )}{b^{2}} - \frac {B b^{2} - A b c}{2 \, {\left (c x^{2} + b\right )} b^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x^2+A)/(c*x^4+b*x^2)^2,x, algorithm="giac")

[Out]

-1/2*A*log(abs(c*x^2 + b))/b^2 + A*log(abs(x))/b^2 - 1/2*(B*b^2 - A*b*c)/((c*x^2 + b)*b^2*c)

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Mupad [B]
time = 0.16, size = 47, normalized size = 0.92 \begin {gather*} \frac {A\,\ln \left (x\right )}{b^2}-\frac {A\,\ln \left (c\,x^2+b\right )}{2\,b^2}+\frac {A\,c-B\,b}{2\,b\,c\,\left (c\,x^2+b\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(A + B*x^2))/(b*x^2 + c*x^4)^2,x)

[Out]

(A*log(x))/b^2 - (A*log(b + c*x^2))/(2*b^2) + (A*c - B*b)/(2*b*c*(b + c*x^2))

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