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

Optimal. Leaf size=103 \[ \frac{a^2 (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}+\frac{d x^6 (2 b c-a d)}{6 b^2}+\frac{x^4 (b c-a d)^2}{4 b^3}-\frac{a x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^8}{8 b} \]

[Out]

-(a*(b*c - a*d)^2*x^2)/(2*b^4) + ((b*c - a*d)^2*x^4)/(4*b^3) + (d*(2*b*c - a*d)*x^6)/(6*b^2) + (d^2*x^8)/(8*b)
+ (a^2*(b*c - a*d)^2*Log[a + b*x^2])/(2*b^5)

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Rubi [A]  time = 0.12215, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 88} \[ \frac{a^2 (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}+\frac{d x^6 (2 b c-a d)}{6 b^2}+\frac{x^4 (b c-a d)^2}{4 b^3}-\frac{a x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^8}{8 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(b*c - a*d)^2*x^2)/(2*b^4) + ((b*c - a*d)^2*x^4)/(4*b^3) + (d*(2*b*c - a*d)*x^6)/(6*b^2) + (d^2*x^8)/(8*b)
+ (a^2*(b*c - a*d)^2*Log[a + b*x^2])/(2*b^5)

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 (c+d x)^2}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a (-b c+a d)^2}{b^4}+\frac{(b c-a d)^2 x}{b^3}+\frac{d (2 b c-a d) x^2}{b^2}+\frac{d^2 x^3}{b}+\frac{a^2 (-b c+a d)^2}{b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{a (b c-a d)^2 x^2}{2 b^4}+\frac{(b c-a d)^2 x^4}{4 b^3}+\frac{d (2 b c-a d) x^6}{6 b^2}+\frac{d^2 x^8}{8 b}+\frac{a^2 (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0521949, size = 116, normalized size = 1.13 \[ \frac{\left (a^2 b^2 c^2-2 a^3 b c d+a^4 d^2\right ) \log \left (a+b x^2\right )}{2 b^5}+\frac{d x^6 (2 b c-a d)}{6 b^2}+\frac{x^4 (b c-a d)^2}{4 b^3}-\frac{a x^2 (a d-b c)^2}{2 b^4}+\frac{d^2 x^8}{8 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(-(b*c) + a*d)^2*x^2)/(2*b^4) + ((b*c - a*d)^2*x^4)/(4*b^3) + (d*(2*b*c - a*d)*x^6)/(6*b^2) + (d^2*x^8)/(8
*b) + ((a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*Log[a + b*x^2])/(2*b^5)

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Maple [A]  time = 0.003, size = 165, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{8}}{8\,b}}-{\frac{{x}^{6}a{d}^{2}}{6\,{b}^{2}}}+{\frac{{x}^{6}cd}{3\,b}}+{\frac{{x}^{4}{a}^{2}{d}^{2}}{4\,{b}^{3}}}-{\frac{{x}^{4}acd}{2\,{b}^{2}}}+{\frac{{x}^{4}{c}^{2}}{4\,b}}-{\frac{{a}^{3}{d}^{2}{x}^{2}}{2\,{b}^{4}}}+{\frac{{x}^{2}{a}^{2}cd}{{b}^{3}}}-{\frac{a{c}^{2}{x}^{2}}{2\,{b}^{2}}}+{\frac{{a}^{4}\ln \left ( b{x}^{2}+a \right ){d}^{2}}{2\,{b}^{5}}}-{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) cd}{{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d^2*x^8/b-1/6/b^2*x^6*a*d^2+1/3/b*x^6*c*d+1/4/b^3*x^4*a^2*d^2-1/2/b^2*x^4*a*c*d+1/4/b*x^4*c^2-1/2/b^4*a^3*
d^2*x^2+1/b^3*a^2*c*d*x^2-1/2/b^2*a*c^2*x^2+1/2*a^4/b^5*ln(b*x^2+a)*d^2-a^3/b^4*ln(b*x^2+a)*c*d+1/2*a^2/b^3*ln
(b*x^2+a)*c^2

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Maxima [A]  time = 0.998989, size = 185, normalized size = 1.8 \begin{align*} \frac{3 \, b^{3} d^{2} x^{8} + 4 \,{\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{6} + 6 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{4} - 12 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}}{24 \, b^{4}} + \frac{{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*b^3*d^2*x^8 + 4*(2*b^3*c*d - a*b^2*d^2)*x^6 + 6*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^4 - 12*(a*b^2*c^
2 - 2*a^2*b*c*d + a^3*d^2)*x^2)/b^4 + 1/2*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(b*x^2 + a)/b^5

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Fricas [A]  time = 1.38708, size = 285, normalized size = 2.77 \begin{align*} \frac{3 \, b^{4} d^{2} x^{8} + 4 \,{\left (2 \, b^{4} c d - a b^{3} d^{2}\right )} x^{6} + 6 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} - 12 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} + 12 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x^{2} + a\right )}{24 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*b^4*d^2*x^8 + 4*(2*b^4*c*d - a*b^3*d^2)*x^6 + 6*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^4 - 12*(a*b^3*
c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^2 + 12*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(b*x^2 + a))/b^5

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Sympy [A]  time = 0.570223, size = 119, normalized size = 1.16 \begin{align*} \frac{a^{2} \left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{5}} + \frac{d^{2} x^{8}}{8 b} - \frac{x^{6} \left (a d^{2} - 2 b c d\right )}{6 b^{2}} + \frac{x^{4} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{4 b^{3}} - \frac{x^{2} \left (a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}\right )}{2 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(a*d - b*c)**2*log(a + b*x**2)/(2*b**5) + d**2*x**8/(8*b) - x**6*(a*d**2 - 2*b*c*d)/(6*b**2) + x**4*(a**2
*d**2 - 2*a*b*c*d + b**2*c**2)/(4*b**3) - x**2*(a**3*d**2 - 2*a**2*b*c*d + a*b**2*c**2)/(2*b**4)

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Giac [A]  time = 1.20576, size = 200, normalized size = 1.94 \begin{align*} \frac{3 \, b^{3} d^{2} x^{8} + 8 \, b^{3} c d x^{6} - 4 \, a b^{2} d^{2} x^{6} + 6 \, b^{3} c^{2} x^{4} - 12 \, a b^{2} c d x^{4} + 6 \, a^{2} b d^{2} x^{4} - 12 \, a b^{2} c^{2} x^{2} + 24 \, a^{2} b c d x^{2} - 12 \, a^{3} d^{2} x^{2}}{24 \, b^{4}} + \frac{{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*b^3*d^2*x^8 + 8*b^3*c*d*x^6 - 4*a*b^2*d^2*x^6 + 6*b^3*c^2*x^4 - 12*a*b^2*c*d*x^4 + 6*a^2*b*d^2*x^4 - 1
2*a*b^2*c^2*x^2 + 24*a^2*b*c*d*x^2 - 12*a^3*d^2*x^2)/b^4 + 1/2*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(abs(b
*x^2 + a))/b^5