\(\int \frac {x (c+d x^2)^3}{(a+b x^2)^2} \, dx\) [283]

Optimal result

Integrand size = 20, antiderivative size = 88 \[ \int \frac {x \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {d^2 (3 b c-2 a d) x^2}{2 b^3}+\frac {d^3 x^4}{4 b^2}-\frac {(b c-a d)^3}{2 b^4 \left (a+b x^2\right )}+\frac {3 d (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {455, 45} \[ \int \frac {x \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {(b c-a d)^3}{2 b^4 \left (a+b x^2\right )}+\frac {3 d (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}+\frac {d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac {d^3 x^4}{4 b^2} \]

[In]

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

[Out]

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

Rule 45

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])

Rule 455

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]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.44 \[ \int \frac {x \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {d^2 (3 b c-2 a d) x^2}{2 b^3}+\frac {d^3 x^4}{4 b^2}+\frac {-b^3 c^3+3 a b^2 c^2 d-3 a^2 b c d^2+a^3 d^3}{2 b^4 \left (a+b x^2\right )}+\frac {3 \left (b^2 c^2 d-2 a b c d^2+a^2 d^3\right ) \log \left (a+b x^2\right )}{2 b^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.65 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {d \left (-b d \,x^{2}+2 a d -3 b c \right )^{2}}{4 b^{4}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {3 d \ln \left (b \,x^{2}+a \right )}{b}-\frac {-a d +b c}{b \left (b \,x^{2}+a \right )}\right )}{2 b^{3}}\)	\(89\)
norman	\(\frac {\frac {3 a^{3} d^{3}-6 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{2 b^{4}}+\frac {d^{3} x^{6}}{4 b}-\frac {3 d^{2} \left (a d -2 b c \right ) x^{4}}{4 b^{2}}}{b \,x^{2}+a}+\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\)	\(120\)
risch	\(\frac {d^{3} x^{4}}{4 b^{2}}-\frac {d^{3} x^{2} a}{b^{3}}+\frac {3 d^{2} x^{2} c}{2 b^{2}}+\frac {d^{3} a^{2}}{b^{4}}-\frac {3 d^{2} a c}{b^{3}}+\frac {9 d \,c^{2}}{4 b^{2}}+\frac {a^{3} d^{3}}{2 b^{4} \left (b \,x^{2}+a \right )}-\frac {3 a^{2} c \,d^{2}}{2 b^{3} \left (b \,x^{2}+a \right )}+\frac {3 a \,c^{2} d}{2 b^{2} \left (b \,x^{2}+a \right )}-\frac {c^{3}}{2 b \left (b \,x^{2}+a \right )}+\frac {3 d^{3} \ln \left (b \,x^{2}+a \right ) a^{2}}{2 b^{4}}-\frac {3 d^{2} \ln \left (b \,x^{2}+a \right ) a c}{b^{3}}+\frac {3 d \ln \left (b \,x^{2}+a \right ) c^{2}}{2 b^{2}}\)	\(197\)
parallelrisch	\(\frac {b^{3} d^{3} x^{6}-3 a \,b^{2} d^{3} x^{4}+6 x^{4} b^{3} c \,d^{2}+6 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b \,d^{3}-12 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{2} c \,d^{2}+6 \ln \left (b \,x^{2}+a \right ) x^{2} b^{3} c^{2} d +6 \ln \left (b \,x^{2}+a \right ) a^{3} d^{3}-12 \ln \left (b \,x^{2}+a \right ) a^{2} b c \,d^{2}+6 \ln \left (b \,x^{2}+a \right ) a \,b^{2} c^{2} d +6 a^{3} d^{3}-12 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}{4 b^{4} \left (b \,x^{2}+a \right )}\)	\(199\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (80) = 160\).

Time = 0.24 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.06 \[ \int \frac {x \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {b^{3} d^{3} x^{6} - 2 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} + 3 \, {\left (2 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 2 \, {\left (3 \, a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (b^{5} x^{2} + a b^{4}\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.27 \[ \int \frac {x \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=x^{2} \left (- \frac {a d^{3}}{b^{3}} + \frac {3 c d^{2}}{2 b^{2}}\right ) + \frac {a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {d^{3} x^{4}}{4 b^{2}} + \frac {3 d \left (a d - b c\right )^{2} \log {\left (a + b x^{2} \right )}}{2 b^{4}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.41 \[ \int \frac {x \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {b d^{3} x^{4} + 2 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x^{2}}{4 \, b^{3}} + \frac {3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (80) = 160\).

Time = 0.30 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.08 \[ \int \frac {x \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (d^{3} + \frac {6 \, {\left (b^{2} c d^{2} - a b d^{3}\right )}}{{\left (b x^{2} + a\right )} b}\right )} {\left (b x^{2} + a\right )}^{2}}{4 \, b^{4}} - \frac {3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{2 \, b^{4}} - \frac {\frac {b^{5} c^{3}}{b x^{2} + a} - \frac {3 \, a b^{4} c^{2} d}{b x^{2} + a} + \frac {3 \, a^{2} b^{3} c d^{2}}{b x^{2} + a} - \frac {a^{3} b^{2} d^{3}}{b x^{2} + a}}{2 \, b^{6}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.85 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.48 \[ \int \frac {x \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (3\,a^2\,d^3-6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{2\,b^4}-x^2\,\left (\frac {a\,d^3}{b^3}-\frac {3\,c\,d^2}{2\,b^2}\right )+\frac {d^3\,x^4}{4\,b^2}+\frac {a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}{2\,b\,\left (b^4\,x^2+a\,b^3\right )} \]

[In]

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

[Out]

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