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

Optimal. Leaf size=87 \[ \frac{d x^2 (b c-a d)^2}{2 b^3}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac{\left (c+d x^2\right )^3}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.0811352, antiderivative size = 87, 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 = {444, 43} \[ \frac{d x^2 (b c-a d)^2}{2 b^3}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac{\left (c+d x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0303811, size = 82, normalized size = 0.94 \[ \frac{b d x^2 \left (6 a^2 d^2-3 a b d \left (6 c+d x^2\right )+b^2 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )\right )+6 (b c-a d)^3 \log \left (a+b x^2\right )}{12 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 149, normalized size = 1.7 \begin{align*}{\frac{{d}^{3}{x}^{6}}{6\,b}}-{\frac{{d}^{3}{x}^{4}a}{4\,{b}^{2}}}+{\frac{3\,{d}^{2}{x}^{4}c}{4\,b}}+{\frac{{d}^{3}{x}^{2}{a}^{2}}{2\,{b}^{3}}}-{\frac{3\,{d}^{2}{x}^{2}ac}{2\,{b}^{2}}}+{\frac{3\,d{x}^{2}{c}^{2}}{2\,b}}-{\frac{\ln \left ( b{x}^{2}+a \right ){a}^{3}{d}^{3}}{2\,{b}^{4}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ){a}^{2}c{d}^{2}}{2\,{b}^{3}}}-{\frac{3\,\ln \left ( b{x}^{2}+a \right ) a{c}^{2}d}{2\,{b}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.9848, size = 161, normalized size = 1.85 \begin{align*} \frac{2 \, b^{2} d^{3} x^{6} + 3 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{4} + 6 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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*(d*x^2+c)^3/(b*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.44448, size = 244, normalized size = 2.8 \begin{align*} \frac{2 \, b^{3} d^{3} x^{6} + 3 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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*(d*x^2+c)^3/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.625289, size = 88, normalized size = 1.01 \begin{align*} \frac{d^{3} x^{6}}{6 b} - \frac{x^{4} \left (a d^{3} - 3 b c d^{2}\right )}{4 b^{2}} + \frac{x^{2} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{2 b^{3}} - \frac{\left (a d - b c\right )^{3} \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*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.16644, size = 167, normalized size = 1.92 \begin{align*} \frac{2 \, b^{2} d^{3} x^{6} + 9 \, b^{2} c d^{2} x^{4} - 3 \, a b d^{3} x^{4} + 18 \, b^{2} c^{2} d x^{2} - 18 \, a b c d^{2} x^{2} + 6 \, a^{2} d^{3} x^{2}}{12 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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*(d*x^2+c)^3/(b*x^2+a),x, algorithm="giac")

[Out]

1/12*(2*b^2*d^3*x^6 + 9*b^2*c*d^2*x^4 - 3*a*b*d^3*x^4 + 18*b^2*c^2*d*x^2 - 18*a*b*c*d^2*x^2 + 6*a^2*d^3*x^2)/b
^3 + 1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(abs(b*x^2 + a))/b^4