\(\int \frac {(c+d x^2)^3}{x^3 (a+b x^2)^2} \, dx\) [287]

Optimal result

Integrand size = 22, antiderivative size = 98 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {c^3}{2 a^2 x^2}-\frac {(b c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}-\frac {c^2 (2 b c-3 a d) \log (x)}{a^3}+\frac {(b c-a d)^2 (2 b c+a d) \log \left (a+b x^2\right )}{2 a^3 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 98, 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, 90} \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {(b c-a d)^2 (a d+2 b c) \log \left (a+b x^2\right )}{2 a^3 b^2}-\frac {c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac {(b c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}-\frac {c^3}{2 a^2 x^2} \]

[In]

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

[Out]

-1/2*c^3/(a^2*x^2) - (b*c - a*d)^3/(2*a^2*b^2*(a + b*x^2)) - (c^2*(2*b*c - 3*a*d)*Log[x])/a^3 + ((b*c - a*d)^2
*(2*b*c + a*d)*Log[a + b*x^2])/(2*a^3*b^2)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {a c^3}{x^2}+\frac {a (-b c+a d)^3}{b^2 \left (a+b x^2\right )}+2 c^2 (-2 b c+3 a d) \log (x)+\frac {(b c-a d)^2 (2 b c+a d) \log \left (a+b x^2\right )}{b^2}}{2 a^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.73 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02

method	result	size
default	\(-\frac {c^{3}}{2 a^{2} x^{2}}+\frac {c^{2} \left (3 a d -2 b c \right ) \ln \left (x \right )}{a^{3}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (a d +2 b c \right ) \ln \left (b \,x^{2}+a \right )}{b^{2}}+\frac {\left (a d -b c \right ) a}{b^{2} \left (b \,x^{2}+a \right )}\right )}{2 a^{3}}\)	\(100\)
norman	\(\frac {-\frac {c^{3}}{2 a}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{2}}{2 a^{2} b^{2}}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {c^{2} \left (3 a d -2 b c \right ) \ln \left (x \right )}{a^{3}}+\frac {\left (a^{3} d^{3}-3 a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{3} b^{2}}\)	\(131\)
risch	\(\frac {-\frac {c^{3}}{2 a}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{2}}{2 a^{2} b^{2}}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {3 c^{2} \ln \left (x \right ) d}{a^{2}}-\frac {2 c^{3} \ln \left (x \right ) b}{a^{3}}+\frac {\ln \left (-b \,x^{2}-a \right ) d^{3}}{2 b^{2}}-\frac {3 \ln \left (-b \,x^{2}-a \right ) c^{2} d}{2 a^{2}}+\frac {b \ln \left (-b \,x^{2}-a \right ) c^{3}}{a^{3}}\)	\(151\)
parallelrisch	\(\frac {6 \ln \left (x \right ) x^{4} a \,b^{3} c^{2} d -4 \ln \left (x \right ) x^{4} b^{4} c^{3}+\ln \left (b \,x^{2}+a \right ) x^{4} a^{3} b \,d^{3}-3 \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{3} c^{2} d +2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{4} c^{3}+6 \ln \left (x \right ) x^{2} a^{2} b^{2} c^{2} d -4 \ln \left (x \right ) x^{2} a \,b^{3} c^{3}+\ln \left (b \,x^{2}+a \right ) x^{2} a^{4} d^{3}-3 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b^{2} c^{2} d +2 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{3} c^{3}+a^{4} d^{3} x^{2}-3 a^{3} b c \,d^{2} x^{2}+3 a^{2} b^{2} c^{2} d \,x^{2}-2 b^{3} c^{3} a \,x^{2}-a^{2} b^{2} c^{3}}{2 a^{3} b^{2} x^{2} \left (b \,x^{2}+a \right )}\)	\(262\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (92) = 184\).

Time = 0.25 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.13 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {a^{2} b^{2} c^{3} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2} - {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{4} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d\right )} x^{4} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{3} x^{4} + a^{4} b^{2} x^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 4.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {- a b^{2} c^{3} + x^{2} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{2 a^{3} b^{2} x^{2} + 2 a^{2} b^{3} x^{4}} + \frac {c^{2} \cdot \left (3 a d - 2 b c\right ) \log {\left (x \right )}}{a^{3}} + \frac {\left (a d - b c\right )^{2} \left (a d + 2 b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3} b^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.44 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {a b^{2} c^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{2 \, {\left (a^{2} b^{3} x^{4} + a^{3} b^{2} x^{2}\right )}} - \frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3} b^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.60 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b^{2}} - \frac {a^{2} b d^{3} x^{4} + 4 \, b^{3} c^{3} x^{2} - 6 \, a b^{2} c^{2} d x^{2} + 6 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 2 \, a b^{2} c^{3}}{4 \, {\left (b x^{4} + a x^{2}\right )} a^{2} b^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.38 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d^3-3\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )}{2\,a^3\,b^2}-\frac {\ln \left (x\right )\,\left (2\,b\,c^3-3\,a\,c^2\,d\right )}{a^3}-\frac {\frac {c^3}{2\,a}-\frac {x^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^2\,b^2}}{b\,x^4+a\,x^2} \]

[In]

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

[Out]

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