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

Optimal. Leaf size=115 \[ \frac{d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}+\frac{d^2 x^6 (3 b c-a d)}{6 b^2}+\frac{x^2 (b c-a d)^3}{2 b^4}-\frac{a (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^5}+\frac{d^3 x^8}{8 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.124793, antiderivative size = 115, 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} \[ \frac{d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}+\frac{d^2 x^6 (3 b c-a d)}{6 b^2}+\frac{x^2 (b c-a d)^3}{2 b^4}-\frac{a (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^5}+\frac{d^3 x^8}{8 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0561667, size = 125, normalized size = 1.09 \[ \frac{b x^2 \left (6 a^2 b d^2 \left (6 c+d x^2\right )-12 a^3 d^3-2 a b^2 d \left (18 c^2+9 c d x^2+2 d^2 x^4\right )+3 b^3 \left (6 c^2 d x^2+4 c^3+4 c d^2 x^4+d^3 x^6\right )\right )+12 a (a d-b c)^3 \log \left (a+b x^2\right )}{24 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^2*(-12*a^3*d^3 + 6*a^2*b*d^2*(6*c + d*x^2) - 2*a*b^2*d*(18*c^2 + 9*c*d*x^2 + 2*d^2*x^4) + 3*b^3*(4*c^3 +
6*c^2*d*x^2 + 4*c*d^2*x^4 + d^3*x^6)) + 12*a*(-(b*c) + a*d)^3*Log[a + b*x^2])/(24*b^5)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 205, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{8}}{8\,b}}-{\frac{{x}^{6}a{d}^{3}}{6\,{b}^{2}}}+{\frac{{x}^{6}c{d}^{2}}{2\,b}}+{\frac{{x}^{4}{a}^{2}{d}^{3}}{4\,{b}^{3}}}-{\frac{3\,{x}^{4}ac{d}^{2}}{4\,{b}^{2}}}+{\frac{3\,{x}^{4}{c}^{2}d}{4\,b}}-{\frac{{a}^{3}{d}^{3}{x}^{2}}{2\,{b}^{4}}}+{\frac{3\,{x}^{2}{a}^{2}c{d}^{2}}{2\,{b}^{3}}}-{\frac{3\,a{c}^{2}d{x}^{2}}{2\,{b}^{2}}}+{\frac{{c}^{3}{x}^{2}}{2\,b}}+{\frac{{a}^{4}\ln \left ( b{x}^{2}+a \right ){d}^{3}}{2\,{b}^{5}}}-{\frac{3\,{a}^{3}\ln \left ( b{x}^{2}+a \right ) c{d}^{2}}{2\,{b}^{4}}}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{2}+a \right ){c}^{2}d}{2\,{b}^{3}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02363, size = 227, normalized size = 1.97 \begin{align*} \frac{3 \, b^{3} d^{3} x^{8} + 4 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{6} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + 12 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{24 \, b^{4}} - \frac{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\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^3*(d*x^2+c)^3/(b*x^2+a),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.40042, size = 342, normalized size = 2.97 \begin{align*} \frac{3 \, b^{4} d^{3} x^{8} + 4 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{6} + 6 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{4} + 12 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2} - 12 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\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^3*(d*x^2+c)^3/(b*x^2+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.67253, size = 136, normalized size = 1.18 \begin{align*} \frac{a \left (a d - b c\right )^{3} \log{\left (a + b x^{2} \right )}}{2 b^{5}} + \frac{d^{3} x^{8}}{8 b} - \frac{x^{6} \left (a d^{3} - 3 b c d^{2}\right )}{6 b^{2}} + \frac{x^{4} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{4 b^{3}} - \frac{x^{2} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{2 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.14939, size = 243, normalized size = 2.11 \begin{align*} \frac{3 \, b^{3} d^{3} x^{8} + 12 \, b^{3} c d^{2} x^{6} - 4 \, a b^{2} d^{3} x^{6} + 18 \, b^{3} c^{2} d x^{4} - 18 \, a b^{2} c d^{2} x^{4} + 6 \, a^{2} b d^{3} x^{4} + 12 \, b^{3} c^{3} x^{2} - 36 \, a b^{2} c^{2} d x^{2} + 36 \, a^{2} b c d^{2} x^{2} - 12 \, a^{3} d^{3} x^{2}}{24 \, b^{4}} - \frac{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\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^3*(d*x^2+c)^3/(b*x^2+a),x, algorithm="giac")

[Out]

1/24*(3*b^3*d^3*x^8 + 12*b^3*c*d^2*x^6 - 4*a*b^2*d^3*x^6 + 18*b^3*c^2*d*x^4 - 18*a*b^2*c*d^2*x^4 + 6*a^2*b*d^3
*x^4 + 12*b^3*c^3*x^2 - 36*a*b^2*c^2*d*x^2 + 36*a^2*b*c*d^2*x^2 - 12*a^3*d^3*x^2)/b^4 - 1/2*(a*b^3*c^3 - 3*a^2
*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*log(abs(b*x^2 + a))/b^5