3.272 \(\int \frac{x^3 (c+d x^2)^2}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=88 \[ \frac{d x^2 (b c-a d)}{b^3}+\frac{a (b c-a d)^2}{2 b^4 \left (a+b x^2\right )}+\frac{(b c-3 a d) (b c-a d) \log \left (a+b x^2\right )}{2 b^4}+\frac{d^2 x^4}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.105801, antiderivative size = 88, 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, 77} \[ \frac{d x^2 (b c-a d)}{b^3}+\frac{a (b c-a d)^2}{2 b^4 \left (a+b x^2\right )}+\frac{(b c-3 a d) (b c-a d) \log \left (a+b x^2\right )}{2 b^4}+\frac{d^2 x^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x^2)^2)/(a + b*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0601013, size = 87, normalized size = 0.99 \[ \frac{2 \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) \log \left (a+b x^2\right )+4 b d x^2 (b c-a d)+\frac{2 a (b c-a d)^2}{a+b x^2}+b^2 d^2 x^4}{4 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(c + d*x^2)^2)/(a + b*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.01, size = 142, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{4}}{4\,{b}^{2}}}-{\frac{a{d}^{2}{x}^{2}}{{b}^{3}}}+{\frac{d{x}^{2}c}{{b}^{2}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ){a}^{2}{d}^{2}}{2\,{b}^{4}}}-2\,{\frac{\ln \left ( b{x}^{2}+a \right ) adc}{{b}^{3}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,{b}^{2}}}+{\frac{{a}^{3}{d}^{2}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}cd}{{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{a{c}^{2}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(d*x^2+c)^2/(b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.1633, size = 144, normalized size = 1.64 \begin{align*} \frac{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}}{2 \,{\left (b^{5} x^{2} + a b^{4}\right )}} + \frac{b d^{2} x^{4} + 4 \,{\left (b c d - a d^{2}\right )} x^{2}}{4 \, b^{3}} + \frac{{\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\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^3*(d*x^2+c)^2/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.48316, size = 327, normalized size = 3.72 \begin{align*} \frac{b^{3} d^{2} x^{6} + 2 \, a b^{2} c^{2} - 4 \, a^{2} b c d + 2 \, a^{3} d^{2} +{\left (4 \, b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{4} + 4 \,{\left (a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + 2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} b c d + 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{2} + a b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.09655, size = 97, normalized size = 1.1 \begin{align*} \frac{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}}{2 a b^{4} + 2 b^{5} x^{2}} + \frac{d^{2} x^{4}}{4 b^{2}} - \frac{x^{2} \left (a d^{2} - b c d\right )}{b^{3}} + \frac{\left (a d - b c\right ) \left (3 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(d*x**2+c)**2/(b*x**2+a)**2,x)

[Out]

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

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Giac [A]  time = 1.14321, size = 220, normalized size = 2.5 \begin{align*} \frac{\frac{{\left (b x^{2} + a\right )}^{2}{\left (d^{2} + \frac{2 \,{\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )}}{{\left (b x^{2} + a\right )} b}\right )}}{b^{3}} - \frac{2 \,{\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{2 \,{\left (\frac{a b^{4} c^{2}}{b x^{2} + a} - \frac{2 \, a^{2} b^{3} c d}{b x^{2} + a} + \frac{a^{3} b^{2} d^{2}}{b x^{2} + a}\right )}}{b^{5}}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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