3.57 \(\int \frac{x^5 (A+B x^2)}{a+b x^2} \, dx\)

Optimal. Leaf size=75 \[ \frac{a^2 (A b-a B) \log \left (a+b x^2\right )}{2 b^4}+\frac{x^4 (A b-a B)}{4 b^2}-\frac{a x^2 (A b-a B)}{2 b^3}+\frac{B x^6}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.0869095, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ \frac{a^2 (A b-a B) \log \left (a+b x^2\right )}{2 b^4}+\frac{x^4 (A b-a B)}{4 b^2}-\frac{a x^2 (A b-a B)}{2 b^3}+\frac{B x^6}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(A + B*x^2))/(a + b*x^2),x]

[Out]

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

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]]

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]))))

Rubi steps

\begin{align*} \int \frac{x^5 \left (A+B x^2\right )}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 (A+B x)}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a (-A b+a B)}{b^3}+\frac{(A b-a B) x}{b^2}+\frac{B x^2}{b}-\frac{a^2 (-A b+a B)}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{a (A b-a B) x^2}{2 b^3}+\frac{(A b-a B) x^4}{4 b^2}+\frac{B x^6}{6 b}+\frac{a^2 (A b-a B) \log \left (a+b x^2\right )}{2 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0315902, size = 71, normalized size = 0.95 \[ \frac{b x^2 \left (6 a^2 B-3 a b \left (2 A+B x^2\right )+b^2 x^2 \left (3 A+2 B x^2\right )\right )+6 a^2 (A b-a B) \log \left (a+b x^2\right )}{12 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(A + B*x^2))/(a + b*x^2),x]

[Out]

(b*x^2*(6*a^2*B - 3*a*b*(2*A + B*x^2) + b^2*x^2*(3*A + 2*B*x^2)) + 6*a^2*(A*b - a*B)*Log[a + b*x^2])/(12*b^4)

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Maple [A]  time = 0.003, size = 86, normalized size = 1.2 \begin{align*}{\frac{B{x}^{6}}{6\,b}}+{\frac{A{x}^{4}}{4\,b}}-{\frac{B{x}^{4}a}{4\,{b}^{2}}}-{\frac{aA{x}^{2}}{2\,{b}^{2}}}+{\frac{B{x}^{2}{a}^{2}}{2\,{b}^{3}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) A}{2\,{b}^{3}}}-{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(B*x^2+A)/(b*x^2+a),x)

[Out]

1/6*B*x^6/b+1/4/b*A*x^4-1/4/b^2*B*x^4*a-1/2/b^2*A*x^2*a+1/2/b^3*B*x^2*a^2+1/2*a^2/b^3*ln(b*x^2+a)*A-1/2*a^3/b^
4*ln(b*x^2+a)*B

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Maxima [A]  time = 0.993539, size = 100, normalized size = 1.33 \begin{align*} \frac{2 \, B b^{2} x^{6} - 3 \,{\left (B a b - A b^{2}\right )} x^{4} + 6 \,{\left (B a^{2} - A a b\right )} x^{2}}{12 \, b^{3}} - \frac{{\left (B a^{3} - A a^{2} b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^2+A)/(b*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.1942, size = 155, normalized size = 2.07 \begin{align*} \frac{2 \, B b^{3} x^{6} - 3 \,{\left (B a b^{2} - A b^{3}\right )} x^{4} + 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} - 6 \,{\left (B a^{3} - A a^{2} b\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^2+A)/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.438359, size = 65, normalized size = 0.87 \begin{align*} \frac{B x^{6}}{6 b} - \frac{a^{2} \left (- A b + B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} - \frac{x^{4} \left (- A b + B a\right )}{4 b^{2}} + \frac{x^{2} \left (- A a b + B a^{2}\right )}{2 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(B*x**2+A)/(b*x**2+a),x)

[Out]

B*x**6/(6*b) - a**2*(-A*b + B*a)*log(a + b*x**2)/(2*b**4) - x**4*(-A*b + B*a)/(4*b**2) + x**2*(-A*a*b + B*a**2
)/(2*b**3)

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Giac [A]  time = 1.13074, size = 104, normalized size = 1.39 \begin{align*} \frac{2 \, B b^{2} x^{6} - 3 \, B a b x^{4} + 3 \, A b^{2} x^{4} + 6 \, B a^{2} x^{2} - 6 \, A a b x^{2}}{12 \, b^{3}} - \frac{{\left (B a^{3} - A a^{2} b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^2+A)/(b*x^2+a),x, algorithm="giac")

[Out]

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