\(\int \frac {x^2 (c+d x^2)^3}{a+b x^2} \, dx\) [221]

Optimal result

Integrand size = 22, antiderivative size = 119 \[ \int \frac {x^2 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {(b c-a d)^3 x}{b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^7}{7 b}-\frac {\sqrt {a} (b c-a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \]

[Out]

(-a*d+b*c)^3*x/b^4+1/3*d*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)*x^3/b^3+1/5*d^2*(-a*d+3*b*c)*x^5/b^2+1/7*d^3*x^7/b-(-a*
d+b*c)^3*arctan(x*b^(1/2)/a^(1/2))*a^(1/2)/b^(9/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 119, 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 = {472, 211} \[ \int \frac {x^2 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^3}{b^{9/2}}+\frac {x (b c-a d)^3}{b^4}+\frac {d^2 x^5 (3 b c-a d)}{5 b^2}+\frac {d^3 x^7}{7 b} \]

[In]

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

[Out]

((b*c - a*d)^3*x)/b^4 + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^3)/(3*b^3) + (d^2*(3*b*c - a*d)*x^5)/(5*b^2) +
(d^3*x^7)/(7*b) - (Sqrt[a]*(b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^3}{b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^2}{b^3}+\frac {d^2 (3 b c-a d) x^4}{b^2}+\frac {d^3 x^6}{b}+\frac {-a b^3 c^3+3 a^2 b^2 c^2 d-3 a^3 b c d^2+a^4 d^3}{b^4 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {(b c-a d)^3 x}{b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^7}{7 b}-\frac {\left (a (b c-a d)^3\right ) \int \frac {1}{a+b x^2} \, dx}{b^4} \\ & = \frac {(b c-a d)^3 x}{b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^7}{7 b}-\frac {\sqrt {a} (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {(b c-a d)^3 x}{b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^7}{7 b}+\frac {\sqrt {a} (-b c+a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \]

[In]

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

[Out]

((b*c - a*d)^3*x)/b^4 + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^3)/(3*b^3) + (d^2*(3*b*c - a*d)*x^5)/(5*b^2) +
(d^3*x^7)/(7*b) + (Sqrt[a]*(-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)

Maple [A] (verified)

Time = 2.69 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.45

method	result	size
default	\(-\frac {-\frac {1}{7} b^{3} d^{3} x^{7}+\frac {1}{5} a \,b^{2} d^{3} x^{5}-\frac {3}{5} b^{3} c \,d^{2} x^{5}-\frac {1}{3} a^{2} b \,d^{3} x^{3}+a \,b^{2} c \,d^{2} x^{3}-b^{3} c^{2} d \,x^{3}+a^{3} d^{3} x -3 a^{2} b c \,d^{2} x +3 a \,b^{2} c^{2} d x -b^{3} c^{3} x}{b^{4}}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{4} \sqrt {a b}}\)	\(173\)
risch	\(\frac {d^{3} x^{7}}{7 b}-\frac {a \,d^{3} x^{5}}{5 b^{2}}+\frac {3 c \,d^{2} x^{5}}{5 b}+\frac {a^{2} d^{3} x^{3}}{3 b^{3}}-\frac {a c \,d^{2} x^{3}}{b^{2}}+\frac {c^{2} d \,x^{3}}{b}-\frac {a^{3} d^{3} x}{b^{4}}+\frac {3 a^{2} c \,d^{2} x}{b^{3}}-\frac {3 a \,c^{2} d x}{b^{2}}+\frac {c^{3} x}{b}+\frac {\sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a^{3} d^{3}}{2 b^{5}}-\frac {3 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a^{2} c \,d^{2}}{2 b^{4}}+\frac {3 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a \,c^{2} d}{2 b^{3}}-\frac {\sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) c^{3}}{2 b^{2}}-\frac {\sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a^{3} d^{3}}{2 b^{5}}+\frac {3 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a^{2} c \,d^{2}}{2 b^{4}}-\frac {3 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a \,c^{2} d}{2 b^{3}}+\frac {\sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) c^{3}}{2 b^{2}}\)	\(341\)

[In]

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

[Out]

-1/b^4*(-1/7*b^3*d^3*x^7+1/5*a*b^2*d^3*x^5-3/5*b^3*c*d^2*x^5-1/3*a^2*b*d^3*x^3+a*b^2*c*d^2*x^3-b^3*c^2*d*x^3+a
^3*d^3*x-3*a^2*b*c*d^2*x+3*a*b^2*c^2*d*x-b^3*c^3*x)+a*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^4/(a*b)^
(1/2)*arctan(b*x/(a*b)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.06 \[ \int \frac {x^2 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\left [\frac {30 \, b^{3} d^{3} x^{7} + 42 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{5} + 70 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} - 105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 210 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{210 \, b^{4}}, \frac {15 \, b^{3} d^{3} x^{7} + 21 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{5} + 35 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} - 105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{105 \, b^{4}}\right ] \]

[In]

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

[Out]

[1/210*(30*b^3*d^3*x^7 + 42*(3*b^3*c*d^2 - a*b^2*d^3)*x^5 + 70*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3)*x^3 -
105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2
+ a)) + 210*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x)/b^4, 1/105*(15*b^3*d^3*x^7 + 21*(3*b^3*c*d^
2 - a*b^2*d^3)*x^5 + 35*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3)*x^3 - 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b
*c*d^2 - a^3*d^3)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*
x)/b^4]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (109) = 218\).

Time = 0.37 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.30 \[ \int \frac {x^2 \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^{5} \left (- \frac {a d^{3}}{5 b^{2}} + \frac {3 c d^{2}}{5 b}\right ) + x^{3} \left (\frac {a^{2} d^{3}}{3 b^{3}} - \frac {a c d^{2}}{b^{2}} + \frac {c^{2} d}{b}\right ) + x \left (- \frac {a^{3} d^{3}}{b^{4}} + \frac {3 a^{2} c d^{2}}{b^{3}} - \frac {3 a c^{2} d}{b^{2}} + \frac {c^{3}}{b}\right ) - \frac {\sqrt {- \frac {a}{b^{9}}} \left (a d - b c\right )^{3} \log {\left (- \frac {b^{4} \sqrt {- \frac {a}{b^{9}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {a}{b^{9}}} \left (a d - b c\right )^{3} \log {\left (\frac {b^{4} \sqrt {- \frac {a}{b^{9}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {d^{3} x^{7}}{7 b} \]

[In]

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

[Out]

x**5*(-a*d**3/(5*b**2) + 3*c*d**2/(5*b)) + x**3*(a**2*d**3/(3*b**3) - a*c*d**2/b**2 + c**2*d/b) + x*(-a**3*d**
3/b**4 + 3*a**2*c*d**2/b**3 - 3*a*c**2*d/b**2 + c**3/b) - sqrt(-a/b**9)*(a*d - b*c)**3*log(-b**4*sqrt(-a/b**9)
*(a*d - b*c)**3/(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3) + x)/2 + sqrt(-a/b**9)*(a*d - b*c)
**3*log(b**4*sqrt(-a/b**9)*(a*d - b*c)**3/(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3) + x)/2 +
d**3*x**7/(7*b)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.45 \[ \int \frac {x^2 \left (c+d x^2\right )^3}{a+b x^2} \, dx=-\frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{3} d^{3} x^{7} + 21 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{5} + 35 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{105 \, b^{4}} \]

[In]

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

[Out]

-(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/105*(15*b^3
*d^3*x^7 + 21*(3*b^3*c*d^2 - a*b^2*d^3)*x^5 + 35*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3)*x^3 + 105*(b^3*c^3
- 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x)/b^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.55 \[ \int \frac {x^2 \left (c+d x^2\right )^3}{a+b x^2} \, dx=-\frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{6} d^{3} x^{7} + 63 \, b^{6} c d^{2} x^{5} - 21 \, a b^{5} d^{3} x^{5} + 105 \, b^{6} c^{2} d x^{3} - 105 \, a b^{5} c d^{2} x^{3} + 35 \, a^{2} b^{4} d^{3} x^{3} + 105 \, b^{6} c^{3} x - 315 \, a b^{5} c^{2} d x + 315 \, a^{2} b^{4} c d^{2} x - 105 \, a^{3} b^{3} d^{3} x}{105 \, b^{7}} \]

[In]

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

[Out]

-(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/105*(15*b^6
*d^3*x^7 + 63*b^6*c*d^2*x^5 - 21*a*b^5*d^3*x^5 + 105*b^6*c^2*d*x^3 - 105*a*b^5*c*d^2*x^3 + 35*a^2*b^4*d^3*x^3
+ 105*b^6*c^3*x - 315*a*b^5*c^2*d*x + 315*a^2*b^4*c*d^2*x - 105*a^3*b^3*d^3*x)/b^7

Mupad [B] (verification not implemented)

Time = 5.31 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.67 \[ \int \frac {x^2 \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^3\,\left (\frac {c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{3\,b}\right )-x^5\,\left (\frac {a\,d^3}{5\,b^2}-\frac {3\,c\,d^2}{5\,b}\right )+x\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )+\frac {d^3\,x^7}{7\,b}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3}{a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^3}{b^{9/2}} \]

[In]

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

[Out]

x^3*((c^2*d)/b + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/(3*b)) - x^5*((a*d^3)/(5*b^2) - (3*c*d^2)/(5*b)) + x*(c^3/b -
(a*((3*c^2*d)/b + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/b))/b) + (d^3*x^7)/(7*b) + (a^(1/2)*atan((a^(1/2)*b^(1/2)*x
*(a*d - b*c)^3)/(a^4*d^3 - a*b^3*c^3 + 3*a^2*b^2*c^2*d - 3*a^3*b*c*d^2))*(a*d - b*c)^3)/b^(9/2)