\(\int \frac {(c+d x^2)^3}{x (a+b x^2)^2} \, dx\) [285]

Optimal result

Integrand size = 22, antiderivative size = 88 \[ \int \frac {\left (c+d x^2\right )^3}{x \left (a+b x^2\right )^2} \, dx=\frac {d^3 x^2}{2 b^2}+\frac {(b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac {c^3 \log (x)}{a^2}-\frac {(b c-a d)^2 (b c+2 a d) \log \left (a+b x^2\right )}{2 a^2 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 90} \[ \int \frac {\left (c+d x^2\right )^3}{x \left (a+b x^2\right )^2} \, dx=-\frac {(b c-a d)^2 (2 a d+b c) \log \left (a+b x^2\right )}{2 a^2 b^3}+\frac {c^3 \log (x)}{a^2}+\frac {(b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac {d^3 x^2}{2 b^2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.26 \[ \int \frac {\left (c+d x^2\right )^3}{x \left (a+b x^2\right )^2} \, dx=\frac {2 c^3 \log (x)+\frac {\frac {a \left (b^3 c^3-a^3 d^3+a^2 b d^2 \left (3 c+d x^2\right )+a b^2 \left (-3 c^2 d+d^3 x^4\right )\right )}{a+b x^2}-(b c-a d)^2 (b c+2 a d) \log \left (a+b x^2\right )}{b^3}}{2 a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.62 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {d^{3} x^{2}}{2 b^{2}}+\frac {c^{3} \ln \left (x \right )}{a^{2}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (2 a d +b c \right ) \ln \left (b \,x^{2}+a \right )}{b}+\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{2} b^{2}}\)	\(94\)
norman	\(\frac {\frac {d^{3} x^{4}}{2 b}-\frac {2 a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{2 a \,b^{3}}}{b \,x^{2}+a}+\frac {c^{3} \ln \left (x \right )}{a^{2}}-\frac {\left (2 a^{3} d^{3}-3 a^{2} b c \,d^{2}+b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2} b^{3}}\)	\(120\)
risch	\(\frac {d^{3} x^{2}}{2 b^{2}}-\frac {a^{2} d^{3}}{2 b^{3} \left (b \,x^{2}+a \right )}+\frac {3 a c \,d^{2}}{2 b^{2} \left (b \,x^{2}+a \right )}-\frac {3 c^{2} d}{2 b \left (b \,x^{2}+a \right )}+\frac {c^{3}}{2 a \left (b \,x^{2}+a \right )}+\frac {c^{3} \ln \left (x \right )}{a^{2}}-\frac {a \ln \left (b \,x^{2}+a \right ) d^{3}}{b^{3}}+\frac {3 \ln \left (b \,x^{2}+a \right ) c \,d^{2}}{2 b^{2}}-\frac {\ln \left (b \,x^{2}+a \right ) c^{3}}{2 a^{2}}\)	\(146\)
parallelrisch	\(\frac {x^{4} a^{2} b^{2} d^{3}+2 \ln \left (x \right ) x^{2} b^{4} c^{3}-2 \ln \left (b \,x^{2}+a \right ) x^{2} a^{3} b \,d^{3}+3 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b^{2} c \,d^{2}-\ln \left (b \,x^{2}+a \right ) x^{2} b^{4} c^{3}+2 a \,b^{3} c^{3} \ln \left (x \right )-2 \ln \left (b \,x^{2}+a \right ) a^{4} d^{3}+3 \ln \left (b \,x^{2}+a \right ) a^{3} b c \,d^{2}-\ln \left (b \,x^{2}+a \right ) a \,b^{3} c^{3}-2 a^{4} d^{3}+3 a^{3} b c \,d^{2}-3 a^{2} b^{2} c^{2} d +a \,b^{3} c^{3}}{2 a^{2} b^{3} \left (b \,x^{2}+a \right )}\)	\(207\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (82) = 164\).

Time = 0.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.02 \[ \int \frac {\left (c+d x^2\right )^3}{x \left (a+b x^2\right )^2} \, dx=\frac {a^{2} b^{2} d^{3} x^{4} + a^{3} b d^{3} x^{2} + a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} - {\left (a b^{3} c^{3} - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left (b^{4} c^{3} x^{2} + a b^{3} c^{3}\right )} \log \left (x\right )}{2 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.46 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.25 \[ \int \frac {\left (c+d x^2\right )^3}{x \left (a+b x^2\right )^2} \, dx=\frac {- a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}}{2 a^{2} b^{3} + 2 a b^{4} x^{2}} + \frac {d^{3} x^{2}}{2 b^{2}} + \frac {c^{3} \log {\left (x \right )}}{a^{2}} - \frac {\left (a d - b c\right )^{2} \cdot \left (2 a d + b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2} b^{3}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.39 \[ \int \frac {\left (c+d x^2\right )^3}{x \left (a+b x^2\right )^2} \, dx=\frac {d^{3} x^{2}}{2 \, b^{2}} + \frac {c^{3} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} - \frac {{\left (b^{3} c^{3} - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2} b^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.70 \[ \int \frac {\left (c+d x^2\right )^3}{x \left (a+b x^2\right )^2} \, dx=\frac {d^{3} x^{2}}{2 \, b^{2}} + \frac {c^{3} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b^{3}} + \frac {b^{4} c^{3} x^{2} - 3 \, a^{2} b^{2} c d^{2} x^{2} + 2 \, a^{3} b d^{3} x^{2} + 2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}}{2 \, {\left (b x^{2} + a\right )} a^{2} b^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.95 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.39 \[ \int \frac {\left (c+d x^2\right )^3}{x \left (a+b x^2\right )^2} \, dx=\frac {d^3\,x^2}{2\,b^2}+\frac {c^3\,\ln \left (x\right )}{a^2}-\frac {\ln \left (b\,x^2+a\right )\,\left (2\,a^3\,d^3-3\,a^2\,b\,c\,d^2+b^3\,c^3\right )}{2\,a^2\,b^3}-\frac {a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}{2\,a\,b\,\left (b^3\,x^2+a\,b^2\right )} \]

[In]

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

[Out]

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