\(\int \frac {x^3 (a+b x^2)^2}{(c+d x^2)^3} \, dx\) [190]

Optimal result

Integrand size = 22, antiderivative size = 99 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {b^2 x^2}{2 d^3}+\frac {c (b c-a d)^2}{4 d^4 \left (c+d x^2\right )^2}-\frac {(b c-a d) (3 b c-a d)}{2 d^4 \left (c+d x^2\right )}-\frac {b (3 b c-2 a d) \log \left (c+d x^2\right )}{2 d^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78} \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {c (b c-a d)^2}{4 d^4 \left (c+d x^2\right )^2}-\frac {(3 b c-a d) (b c-a d)}{2 d^4 \left (c+d x^2\right )}-\frac {b (3 b c-2 a d) \log \left (c+d x^2\right )}{2 d^4}+\frac {b^2 x^2}{2 d^3} \]

[In]

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

[Out]

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

Rule 78

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]))))

Rule 457

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]]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.15 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {-a^2 d^2 \left (c+2 d x^2\right )+2 a b c d \left (3 c+4 d x^2\right )+b^2 \left (-5 c^3-4 c^2 d x^2+4 c d^2 x^4+2 d^3 x^6\right )-2 b (3 b c-2 a d) \left (c+d x^2\right )^2 \log \left (c+d x^2\right )}{4 d^4 \left (c+d x^2\right )^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.66 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.06

method	result	size
norman	\(\frac {\frac {b^{2} x^{6}}{2 d}-\frac {c \left (a^{2} d^{2}-6 a b c d +9 b^{2} c^{2}\right )}{4 d^{4}}-\frac {\left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) x^{2}}{2 d^{3}}}{\left (d \,x^{2}+c \right )^{2}}+\frac {b \left (2 a d -3 b c \right ) \ln \left (d \,x^{2}+c \right )}{2 d^{4}}\)	\(105\)
default	\(\frac {b^{2} x^{2}}{2 d^{3}}+\frac {\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d \left (d \,x^{2}+c \right )^{2}}+\frac {b \left (2 a d -3 b c \right ) \ln \left (d \,x^{2}+c \right )}{d}-\frac {a^{2} d^{2}-4 a b c d +3 b^{2} c^{2}}{d \left (d \,x^{2}+c \right )}}{2 d^{3}}\)	\(113\)
risch	\(\frac {b^{2} x^{2}}{2 d^{3}}+\frac {\left (-\frac {1}{2} a^{2} d^{2}+2 a b c d -\frac {3}{2} b^{2} c^{2}\right ) x^{2}-\frac {c \left (a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right )}{4 d}}{d^{3} \left (d \,x^{2}+c \right )^{2}}+\frac {b \ln \left (d \,x^{2}+c \right ) a}{d^{3}}-\frac {3 b^{2} \ln \left (d \,x^{2}+c \right ) c}{2 d^{4}}\)	\(113\)
parallelrisch	\(\frac {2 b^{2} d^{3} x^{6}+4 \ln \left (d \,x^{2}+c \right ) x^{4} a b \,d^{3}-6 \ln \left (d \,x^{2}+c \right ) x^{4} b^{2} c \,d^{2}+8 \ln \left (d \,x^{2}+c \right ) x^{2} a b c \,d^{2}-12 \ln \left (d \,x^{2}+c \right ) x^{2} b^{2} c^{2} d -2 x^{2} a^{2} d^{3}+8 x^{2} a b c \,d^{2}-12 x^{2} b^{2} c^{2} d +4 \ln \left (d \,x^{2}+c \right ) a b \,c^{2} d -6 \ln \left (d \,x^{2}+c \right ) b^{2} c^{3}-c \,a^{2} d^{2}+6 a b \,c^{2} d -9 b^{2} c^{3}}{4 d^{4} \left (d \,x^{2}+c \right )^{2}}\)	\(195\)

[In]

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

[Out]

(1/2*b^2*x^6/d-1/4*c*(a^2*d^2-6*a*b*c*d+9*b^2*c^2)/d^4-1/2*(a^2*d^2-4*a*b*c*d+6*b^2*c^2)/d^3*x^2)/(d*x^2+c)^2+
1/2*b*(2*a*d-3*b*c)/d^4*ln(d*x^2+c)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.80 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {2 \, b^{2} d^{3} x^{6} + 4 \, b^{2} c d^{2} x^{4} - 5 \, b^{2} c^{3} + 6 \, a b c^{2} d - a^{2} c d^{2} - 2 \, {\left (2 \, b^{2} c^{2} d - 4 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2} - 2 \, {\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d + {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, {\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 3.57 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.23 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {b^{2} x^{2}}{2 d^{3}} + \frac {b \left (2 a d - 3 b c\right ) \log {\left (c + d x^{2} \right )}}{2 d^{4}} + \frac {- a^{2} c d^{2} + 6 a b c^{2} d - 5 b^{2} c^{3} + x^{2} \left (- 2 a^{2} d^{3} + 8 a b c d^{2} - 6 b^{2} c^{2} d\right )}{4 c^{2} d^{4} + 8 c d^{5} x^{2} + 4 d^{6} x^{4}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.21 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {b^{2} x^{2}}{2 \, d^{3}} - \frac {5 \, b^{2} c^{3} - 6 \, a b c^{2} d + a^{2} c d^{2} + 2 \, {\left (3 \, b^{2} c^{2} d - 4 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{4 \, {\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )}} - \frac {{\left (3 \, b^{2} c - 2 \, a b d\right )} \log \left (d x^{2} + c\right )}{2 \, d^{4}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {b^{2} x^{2}}{2 \, d^{3}} - \frac {{\left (3 \, b^{2} c - 2 \, a b d\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, d^{4}} - \frac {5 \, b^{2} c^{3} - 6 \, a b c^{2} d + a^{2} c d^{2} + 2 \, {\left (3 \, b^{2} c^{2} d - 4 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{4 \, {\left (d x^{2} + c\right )}^{2} d^{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.24 \[ \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {b^2\,x^2}{2\,d^3}-\frac {\ln \left (d\,x^2+c\right )\,\left (3\,b^2\,c-2\,a\,b\,d\right )}{2\,d^4}-\frac {x^2\,\left (\frac {a^2\,d^2}{2}-2\,a\,b\,c\,d+\frac {3\,b^2\,c^2}{2}\right )+\frac {a^2\,c\,d^2-6\,a\,b\,c^2\,d+5\,b^2\,c^3}{4\,d}}{c^2\,d^3+2\,c\,d^4\,x^2+d^5\,x^4} \]

[In]

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

[Out]

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