3.198 \(\int \frac{x^5 (c+d x^2)}{a+b x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0852302, 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} \[ \frac{a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4}+\frac{x^4 (b c-a d)}{4 b^2}-\frac{a x^2 (b c-a d)}{2 b^3}+\frac{d x^6}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(c + d*x^2))/(a + b*x^2),x]

[Out]

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

Mathematica [A]  time = 0.03034, size = 71, normalized size = 0.95 \[ \frac{b x^2 \left (6 a^2 d-3 a b \left (2 c+d x^2\right )+b^2 x^2 \left (3 c+2 d x^2\right )\right )+6 a^2 (b c-a d) \log \left (a+b x^2\right )}{12 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(c + d*x^2))/(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 86, normalized size = 1.2 \begin{align*}{\frac{d{x}^{6}}{6\,b}}-{\frac{{x}^{4}ad}{4\,{b}^{2}}}+{\frac{c{x}^{4}}{4\,b}}+{\frac{{x}^{2}{a}^{2}d}{2\,{b}^{3}}}-{\frac{a{x}^{2}c}{2\,{b}^{2}}}-{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) d}{2\,{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) c}{2\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(d*x^2+c)/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 0.971569, size = 100, normalized size = 1.33 \begin{align*} \frac{2 \, b^{2} d x^{6} + 3 \,{\left (b^{2} c - a b d\right )} x^{4} - 6 \,{\left (a b c - a^{2} d\right )} x^{2}}{12 \, b^{3}} + \frac{{\left (a^{2} b c - a^{3} d\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*(d*x^2+c)/(b*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.44851, size = 155, normalized size = 2.07 \begin{align*} \frac{2 \, b^{3} d x^{6} + 3 \,{\left (b^{3} c - a b^{2} d\right )} x^{4} - 6 \,{\left (a b^{2} c - a^{2} b d\right )} x^{2} + 6 \,{\left (a^{2} b c - a^{3} d\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*(d*x^2+c)/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.441804, size = 65, normalized size = 0.87 \begin{align*} - \frac{a^{2} \left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{d x^{6}}{6 b} - \frac{x^{4} \left (a d - b c\right )}{4 b^{2}} + \frac{x^{2} \left (a^{2} d - a b c\right )}{2 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(d*x**2+c)/(b*x**2+a),x)

[Out]

-a**2*(a*d - b*c)*log(a + b*x**2)/(2*b**4) + d*x**6/(6*b) - x**4*(a*d - b*c)/(4*b**2) + x**2*(a**2*d - a*b*c)/
(2*b**3)

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Giac [A]  time = 1.16312, size = 104, normalized size = 1.39 \begin{align*} \frac{2 \, b^{2} d x^{6} + 3 \, b^{2} c x^{4} - 3 \, a b d x^{4} - 6 \, a b c x^{2} + 6 \, a^{2} d x^{2}}{12 \, b^{3}} + \frac{{\left (a^{2} b c - a^{3} d\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*(d*x^2+c)/(b*x^2+a),x, algorithm="giac")

[Out]

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