3.59 \(\int \frac {x^3 (A+B x^2)}{a+b x^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac {a (A b-a B) \log \left (a+b x^2\right )}{2 b^3}+\frac {x^2 (A b-a B)}{2 b^2}+\frac {B x^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 77} \[ \frac {x^2 (A b-a B)}{2 b^2}-\frac {a (A b-a B) \log \left (a+b x^2\right )}{2 b^3}+\frac {B x^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(A + B*x^2))/(a + b*x^2),x]

[Out]

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

Rule 77

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 446

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]]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 47, normalized size = 0.87 \[ \frac {b x^2 \left (-2 a B+2 A b+b B x^2\right )+2 a (a B-A b) \log \left (a+b x^2\right )}{4 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 51, normalized size = 0.94 \[ \frac {B b^{2} x^{4} - 2 \, {\left (B a b - A b^{2}\right )} x^{2} + 2 \, {\left (B a^{2} - A a b\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.27, size = 52, normalized size = 0.96 \[ \frac {B b x^{4} - 2 \, B a x^{2} + 2 \, A b x^{2}}{4 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 62, normalized size = 1.15 \[ \frac {B \,x^{4}}{4 b}+\frac {A \,x^{2}}{2 b}-\frac {B a \,x^{2}}{2 b^{2}}-\frac {A a \ln \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {B \,a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 50, normalized size = 0.93 \[ \frac {B b x^{4} - 2 \, {\left (B a - A b\right )} x^{2}}{4 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.10, size = 52, normalized size = 0.96 \[ x^2\,\left (\frac {A}{2\,b}-\frac {B\,a}{2\,b^2}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (B\,a^2-A\,a\,b\right )}{2\,b^3}+\frac {B\,x^4}{4\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(A + B*x^2))/(a + b*x^2),x)

[Out]

x^2*(A/(2*b) - (B*a)/(2*b^2)) + (log(a + b*x^2)*(B*a^2 - A*a*b))/(2*b^3) + (B*x^4)/(4*b)

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sympy [A]  time = 0.29, size = 46, normalized size = 0.85 \[ \frac {B x^{4}}{4 b} + \frac {a \left (- A b + B a\right ) \log {\left (a + b x^{2} \right )}}{2 b^{3}} + x^{2} \left (\frac {A}{2 b} - \frac {B a}{2 b^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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