\(\int \frac {1}{(a+b x^2)^{5/2} \sqrt {a^2-b^2 x^4}} \, dx\) [204]

Optimal result

Integrand size = 28, antiderivative size = 168 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {a^2-b^2 x^4}} \, dx=\frac {x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a-b x^2\right )}{32 a^3 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {19 \sqrt {a-b x^2} \sqrt {a+b x^2} \arctan \left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{32 \sqrt {2} a^3 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]

[Out]

1/8*x*(-b*x^2+a)/a^2/(b*x^2+a)^(3/2)/(-b^2*x^4+a^2)^(1/2)+9/32*x*(-b*x^2+a)/a^3/(b*x^2+a)^(1/2)/(-b^2*x^4+a^2)
^(1/2)+19/64*arctan(x*2^(1/2)*b^(1/2)/(-b*x^2+a)^(1/2))*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/a^3*2^(1/2)/b^(1/2)/(
-b^2*x^4+a^2)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1166, 425, 541, 12, 385, 211} \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {a^2-b^2 x^4}} \, dx=\frac {x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {19 \sqrt {a+b x^2} \sqrt {a-b x^2} \arctan \left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{32 \sqrt {2} a^3 \sqrt {b} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a-b x^2\right )}{32 a^3 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}} \]

[In]

Int[1/((a + b*x^2)^(5/2)*Sqrt[a^2 - b^2*x^4]),x]

[Out]

(x*(a - b*x^2))/(8*a^2*(a + b*x^2)^(3/2)*Sqrt[a^2 - b^2*x^4]) + (9*x*(a - b*x^2))/(32*a^3*Sqrt[a + b*x^2]*Sqrt
[a^2 - b^2*x^4]) + (19*Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a - b*x^2]])/(32*Sqrt[2
]*a^3*Sqrt[b]*Sqrt[a^2 - b^2*x^4])

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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + c*x^4)^FracPart[p]/((d + e*x
^2)^FracPart[p]*(a/d + c*(x^2/e))^FracPart[p]), Int[(d + e*x^2)^(p + q)*(a/d + (c/e)*x^2)^p, x], x] /; FreeQ[{
a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a-b x^2} \left (a+b x^2\right )^3} \, dx}{\sqrt {a^2-b^2 x^4}} \\ & = \frac {x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}-\frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {-7 a b+2 b^2 x^2}{\sqrt {a-b x^2} \left (a+b x^2\right )^2} \, dx}{8 a^2 b \sqrt {a^2-b^2 x^4}} \\ & = \frac {x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a-b x^2\right )}{32 a^3 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {19 a^2 b^2}{\sqrt {a-b x^2} \left (a+b x^2\right )} \, dx}{32 a^4 b^2 \sqrt {a^2-b^2 x^4}} \\ & = \frac {x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a-b x^2\right )}{32 a^3 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {\left (19 \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a-b x^2} \left (a+b x^2\right )} \, dx}{32 a^2 \sqrt {a^2-b^2 x^4}} \\ & = \frac {x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a-b x^2\right )}{32 a^3 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {\left (19 \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{a+2 a b x^2} \, dx,x,\frac {x}{\sqrt {a-b x^2}}\right )}{32 a^2 \sqrt {a^2-b^2 x^4}} \\ & = \frac {x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a-b x^2\right )}{32 a^3 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {19 \sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{32 \sqrt {2} a^3 \sqrt {b} \sqrt {a^2-b^2 x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.69 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {a^2-b^2 x^4}} \, dx=\frac {\sqrt {a^2-b^2 x^4} \left (2 \sqrt {b} x \sqrt {a-b x^2} \left (13 a+9 b x^2\right )+19 \sqrt {2} \left (a+b x^2\right )^2 \arctan \left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )\right )}{64 a^3 \sqrt {b} \sqrt {a-b x^2} \left (a+b x^2\right )^{5/2}} \]

[In]

Integrate[1/((a + b*x^2)^(5/2)*Sqrt[a^2 - b^2*x^4]),x]

[Out]

(Sqrt[a^2 - b^2*x^4]*(2*Sqrt[b]*x*Sqrt[a - b*x^2]*(13*a + 9*b*x^2) + 19*Sqrt[2]*(a + b*x^2)^2*ArcTan[(Sqrt[2]*
Sqrt[b]*x)/Sqrt[a - b*x^2]]))/(64*a^3*Sqrt[b]*Sqrt[a - b*x^2]*(a + b*x^2)^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(710\) vs. \(2(138)=276\).

Time = 0.23 (sec) , antiderivative size = 711, normalized size of antiderivative = 4.23

method	result	size
default	\(-\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, b^{\frac {9}{2}} \left (19 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {-b \,x^{2}+a}-\sqrt {-a b}\, x +a \right )}{b x -\sqrt {-a b}}\right ) b^{\frac {5}{2}} x^{4} \sqrt {a}-19 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {-b \,x^{2}+a}+\sqrt {-a b}\, x +a \right )}{b x +\sqrt {-a b}}\right ) b^{\frac {5}{2}} x^{4} \sqrt {a}+38 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {-b \,x^{2}+a}-\sqrt {-a b}\, x +a \right )}{b x -\sqrt {-a b}}\right ) a^{\frac {3}{2}} b^{\frac {3}{2}} x^{2}-38 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {-b \,x^{2}+a}+\sqrt {-a b}\, x +a \right )}{b x +\sqrt {-a b}}\right ) a^{\frac {3}{2}} b^{\frac {3}{2}} x^{2}+16 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right ) b^{2} x^{4} \sqrt {-a b}-16 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {\frac {\left (-b x +\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right )}{b}}}\right ) b^{2} x^{4} \sqrt {-a b}-36 b^{\frac {3}{2}} \sqrt {-a b}\, \sqrt {-b \,x^{2}+a}\, x^{3}+19 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {-b \,x^{2}+a}-\sqrt {-a b}\, x +a \right )}{b x -\sqrt {-a b}}\right ) a^{\frac {5}{2}} \sqrt {b}-19 \sqrt {2}\, \ln \left (\frac {2 b \left (\sqrt {2}\, \sqrt {a}\, \sqrt {-b \,x^{2}+a}+\sqrt {-a b}\, x +a \right )}{b x +\sqrt {-a b}}\right ) a^{\frac {5}{2}} \sqrt {b}+32 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right ) a b \,x^{2} \sqrt {-a b}-32 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {\frac {\left (-b x +\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right )}{b}}}\right ) a b \,x^{2} \sqrt {-a b}-52 \sqrt {b}\, \sqrt {-a b}\, a \sqrt {-b \,x^{2}+a}\, x +16 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right ) a^{2} \sqrt {-a b}-16 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {\frac {\left (-b x +\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right )}{b}}}\right ) a^{2} \sqrt {-a b}\right )}{16 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, \left (-\sqrt {-a b}+\sqrt {a b}\right )^{3} \left (\sqrt {-a b}+\sqrt {a b}\right )^{3} \left (b x +\sqrt {-a b}\right )^{2} \left (b x -\sqrt {-a b}\right )^{2} \sqrt {-a b}}\)	\(711\)

[In]

int(1/(b*x^2+a)^(5/2)/(-b^2*x^4+a^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/16*(-b^2*x^4+a^2)^(1/2)*b^(9/2)*(19*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1/2)-(-a*b)^(1/2)*x+a)/(b*x
-(-a*b)^(1/2)))*b^(5/2)*x^4*a^(1/2)-19*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1/2)+(-a*b)^(1/2)*x+a)/(b*x
+(-a*b)^(1/2)))*b^(5/2)*x^4*a^(1/2)+38*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1/2)-(-a*b)^(1/2)*x+a)/(b*x
-(-a*b)^(1/2)))*a^(3/2)*b^(3/2)*x^2-38*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1/2)+(-a*b)^(1/2)*x+a)/(b*x
+(-a*b)^(1/2)))*a^(3/2)*b^(3/2)*x^2+16*arctan(b^(1/2)*x/(-b*x^2+a)^(1/2))*b^2*x^4*(-a*b)^(1/2)-16*arctan(b^(1/
2)*x/(1/b*(-b*x+(a*b)^(1/2))*(b*x+(a*b)^(1/2)))^(1/2))*b^2*x^4*(-a*b)^(1/2)-36*b^(3/2)*(-a*b)^(1/2)*(-b*x^2+a)
^(1/2)*x^3+19*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1/2)-(-a*b)^(1/2)*x+a)/(b*x-(-a*b)^(1/2)))*a^(5/2)*b
^(1/2)-19*2^(1/2)*ln(2*b*(2^(1/2)*a^(1/2)*(-b*x^2+a)^(1/2)+(-a*b)^(1/2)*x+a)/(b*x+(-a*b)^(1/2)))*a^(5/2)*b^(1/
2)+32*arctan(b^(1/2)*x/(-b*x^2+a)^(1/2))*a*b*x^2*(-a*b)^(1/2)-32*arctan(b^(1/2)*x/(1/b*(-b*x+(a*b)^(1/2))*(b*x
+(a*b)^(1/2)))^(1/2))*a*b*x^2*(-a*b)^(1/2)-52*b^(1/2)*(-a*b)^(1/2)*a*(-b*x^2+a)^(1/2)*x+16*arctan(b^(1/2)*x/(-
b*x^2+a)^(1/2))*a^2*(-a*b)^(1/2)-16*arctan(b^(1/2)*x/(1/b*(-b*x+(a*b)^(1/2))*(b*x+(a*b)^(1/2)))^(1/2))*a^2*(-a
*b)^(1/2))/(b*x^2+a)^(1/2)/(-b*x^2+a)^(1/2)/(-(-a*b)^(1/2)+(a*b)^(1/2))^3/((-a*b)^(1/2)+(a*b)^(1/2))^3/(b*x+(-
a*b)^(1/2))^2/(b*x-(-a*b)^(1/2))^2/(-a*b)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.17 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {a^2-b^2 x^4}} \, dx=\left [-\frac {19 \, \sqrt {2} {\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {-b} \log \left (-\frac {3 \, b^{2} x^{4} + 2 \, a b x^{2} - 2 \, \sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {-b} x - a^{2}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, \sqrt {-b^{2} x^{4} + a^{2}} {\left (9 \, b^{2} x^{3} + 13 \, a b x\right )} \sqrt {b x^{2} + a}}{128 \, {\left (a^{3} b^{4} x^{6} + 3 \, a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{2} + a^{6} b\right )}}, -\frac {19 \, \sqrt {2} {\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {b}}{2 \, {\left (b^{2} x^{3} + a b x\right )}}\right ) - 2 \, \sqrt {-b^{2} x^{4} + a^{2}} {\left (9 \, b^{2} x^{3} + 13 \, a b x\right )} \sqrt {b x^{2} + a}}{64 \, {\left (a^{3} b^{4} x^{6} + 3 \, a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{2} + a^{6} b\right )}}\right ] \]

[In]

integrate(1/(b*x^2+a)^(5/2)/(-b^2*x^4+a^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/128*(19*sqrt(2)*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(-b)*log(-(3*b^2*x^4 + 2*a*b*x^2 - 2*sqrt(
2)*sqrt(-b^2*x^4 + a^2)*sqrt(b*x^2 + a)*sqrt(-b)*x - a^2)/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*sqrt(-b^2*x^4 + a^2
)*(9*b^2*x^3 + 13*a*b*x)*sqrt(b*x^2 + a))/(a^3*b^4*x^6 + 3*a^4*b^3*x^4 + 3*a^5*b^2*x^2 + a^6*b), -1/64*(19*sqr
t(2)*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(b)*arctan(1/2*sqrt(2)*sqrt(-b^2*x^4 + a^2)*sqrt(b*x^2 +
a)*sqrt(b)/(b^2*x^3 + a*b*x)) - 2*sqrt(-b^2*x^4 + a^2)*(9*b^2*x^3 + 13*a*b*x)*sqrt(b*x^2 + a))/(a^3*b^4*x^6 +
3*a^4*b^3*x^4 + 3*a^5*b^2*x^2 + a^6*b)]

Sympy [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {a^2-b^2 x^4}} \, dx=\int \frac {1}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(b*x**2+a)**(5/2)/(-b**2*x**4+a**2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(-a + b*x**2)*(a + b*x**2))*(a + b*x**2)**(5/2)), x)

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(b*x^2+a)^(5/2)/(-b^2*x^4+a^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-b^2*x^4 + a^2)*(b*x^2 + a)^(5/2)), x)

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(b*x^2+a)^(5/2)/(-b^2*x^4+a^2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-b^2*x^4 + a^2)*(b*x^2 + a)^(5/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {a^2-b^2 x^4}} \, dx=\int \frac {1}{\sqrt {a^2-b^2\,x^4}\,{\left (b\,x^2+a\right )}^{5/2}} \,d x \]

[In]

int(1/((a^2 - b^2*x^4)^(1/2)*(a + b*x^2)^(5/2)),x)

[Out]

int(1/((a^2 - b^2*x^4)^(1/2)*(a + b*x^2)^(5/2)), x)