\(\int \frac {x^2}{(a+b x^2)^2 \sqrt {c+d x^2}} \, dx\) [760]

Optimal result

Integrand size = 24, antiderivative size = 89 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {x \sqrt {c+d x^2}}{2 (b c-a d) \left (a+b x^2\right )}+\frac {c \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {482, 12, 385, 211} \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {c \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{3/2}}-\frac {x \sqrt {c+d x^2}}{2 \left (a+b x^2\right ) (b c-a d)} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 482

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(608\) vs. \(2(89)=178\).

Time = 3.00 (sec) , antiderivative size = 608, normalized size of antiderivative = 6.83 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {1}{2} \left (\frac {2 c^{3/2} x+2 \sqrt {c} d x^3-2 c x \sqrt {c+d x^2}-d x^3 \sqrt {c+d x^2}}{(b c-a d) \left (a+b x^2\right ) \left (2 c+d x^2-2 \sqrt {c} \sqrt {c+d x^2}\right )}+\frac {\sqrt {b} c^{3/2} \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} (b c-a d)^{3/2} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {c \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} (b c-a d) \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {\sqrt {b} c^{3/2} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} (b c-a d)^{3/2} \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {c \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} (b c-a d) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}\right ) \]

[In]

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

[Out]

((2*c^(3/2)*x + 2*Sqrt[c]*d*x^3 - 2*c*x*Sqrt[c + d*x^2] - d*x^3*Sqrt[c + d*x^2])/((b*c - a*d)*(a + b*x^2)*(2*c
+ d*x^2 - 2*Sqrt[c]*Sqrt[c + d*x^2])) + (Sqrt[b]*c^(3/2)*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*
c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/(Sqrt[a]*(b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*S
qrt[c]*Sqrt[b*c - a*d]]) + (c*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt
[c] + Sqrt[c + d*x^2]))])/(Sqrt[a]*(b*c - a*d)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (Sqrt[
b]*c^(3/2)*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^
2]))])/(Sqrt[a]*(b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (c*ArcTan[(Sqrt[2*b
*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/(Sqrt[a]*(b*c - a*d)
*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]))/2

Maple [A] (verified)

Time = 2.99 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(-\frac {c \left (-\frac {\sqrt {d \,x^{2}+c}\, x}{c \left (b \,x^{2}+a \right )}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 \left (a d -b c \right )}\)	\(79\)
default	\(\frac {\frac {b \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{4 b^{2}}+\frac {\frac {b \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{4 b^{2}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}\)	\(816\)

[In]

int(x^2/(b*x^2+a)^2/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (73) = 146\).

Time = 0.32 (sec) , antiderivative size = 418, normalized size of antiderivative = 4.70 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\left [-\frac {4 \, {\left (a b c - a^{2} d\right )} \sqrt {d x^{2} + c} x - {\left (b c x^{2} + a c\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}, -\frac {2 \, {\left (a b c - a^{2} d\right )} \sqrt {d x^{2} + c} x - {\left (b c x^{2} + a c\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{4 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}\right ] \]

[In]

integrate(x^2/(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

[-1/8*(4*(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x - (b*c*x^2 + a*c)*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d +
8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*
sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (a*b^3*c^2 - 2*a^2*b^2*c
*d + a^3*b*d^2)*x^2), -1/4*(2*(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x - (b*c*x^2 + a*c)*sqrt(a*b*c - a^2*d)*arctan(1
/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d
)*x)))/(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^2)]

Sympy [F]

\[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {x^{2}}{\left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}}}\, dx \]

[In]

integrate(x**2/(b*x**2+a)**2/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**2/((a + b*x**2)**2*sqrt(c + d*x**2)), x)

Maxima [F]

\[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}} \,d x } \]

[In]

integrate(x^2/(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (73) = 146\).

Time = 0.84 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.60 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {c \sqrt {d} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} {\left (b c - a d\right )}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d^{\frac {3}{2}} - b c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} {\left (b^{2} c - a b d\right )}} \]

[In]

integrate(x^2/(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {x^2}{{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}} \,d x \]

[In]

int(x^2/((a + b*x^2)^2*(c + d*x^2)^(1/2)),x)

[Out]

int(x^2/((a + b*x^2)^2*(c + d*x^2)^(1/2)), x)