\(\int \frac {b+a x^2}{x^2 (-b+a x^2) \sqrt [4]{b x^2+a x^4}} \, dx\) [1657]

Optimal result

Integrand size = 37, antiderivative size = 112 \[ \int \frac {b+a x^2}{x^2 \left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {2 \left (b x^2+a x^4\right )^{3/4}}{3 b x^3}-\frac {2^{3/4} a^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{b}-\frac {2^{3/4} a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{b} \]

[Out]

2/3*(a*x^4+b*x^2)^(3/4)/b/x^3-2^(3/4)*a^(3/4)*arctan(2^(1/4)*a^(1/4)*x/(a*x^4+b*x^2)^(1/4))/b-2^(3/4)*a^(3/4)*
arctanh(2^(1/4)*a^(1/4)*x/(a*x^4+b*x^2)^(1/4))/b

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.49, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {2081, 477, 525, 524} \[ \int \frac {b+a x^2}{x^2 \left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {2 \left (a x^2+b\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},\frac {2 a x^2}{a x^2+b}\right )}{3 b x \sqrt [4]{a x^4+b x^2}} \]

[In]

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

[Out]

(2*(b + a*x^2)*Hypergeometric2F1[-3/4, 1, 1/4, (2*a*x^2)/(b + a*x^2)])/(3*b*x*(b*x^2 + a*x^4)^(1/4))

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

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

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {\left (b+a x^2\right )^{3/4}}{x^{5/2} \left (-b+a x^2\right )} \, dx}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {\left (b+a x^4\right )^{3/4}}{x^4 \left (-b+a x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \sqrt {x} \left (b+a x^2\right )\right ) \text {Subst}\left (\int \frac {\left (1+\frac {a x^4}{b}\right )^{3/4}}{x^4 \left (-b+a x^4\right )} \, dx,x,\sqrt {x}\right )}{\left (1+\frac {a x^2}{b}\right )^{3/4} \sqrt [4]{b x^2+a x^4}} \\ & = \frac {2 \left (b+a x^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},\frac {2 a x^2}{b+a x^2}\right )}{3 b x \sqrt [4]{b x^2+a x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.25 \[ \int \frac {b+a x^2}{x^2 \left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\left (x^2 \left (b+a x^2\right )\right )^{3/4} \left (2 \left (b+a x^2\right )^{3/4}-3\ 2^{3/4} a^{3/4} x^{3/2} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-3\ 2^{3/4} a^{3/4} x^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )\right )}{3 b x^3 \left (b+a x^2\right )^{3/4}} \]

[In]

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

[Out]

((x^2*(b + a*x^2))^(3/4)*(2*(b + a*x^2)^(3/4) - 3*2^(3/4)*a^(3/4)*x^(3/2)*ArcTan[(2^(1/4)*a^(1/4)*Sqrt[x])/(b
+ a*x^2)^(1/4)] - 3*2^(3/4)*a^(3/4)*x^(3/2)*ArcTanh[(2^(1/4)*a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]))/(3*b*x^3*(b
+ a*x^2)^(3/4))

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(-\frac {\left (3 \ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right ) a \,x^{3}-2 \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {3}{4}} 2^{\frac {1}{4}} a^{\frac {1}{4}}-6 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right ) a \,x^{3}\right ) 2^{\frac {3}{4}}}{6 x^{3} a^{\frac {1}{4}} b}\)	\(127\)

[In]

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

[Out]

-1/6*(3*ln((-x*2^(1/4)*a^(1/4)-(x^2*(a*x^2+b))^(1/4))/(x*2^(1/4)*a^(1/4)-(x^2*(a*x^2+b))^(1/4)))*a*x^3-2*(x^2*
(a*x^2+b))^(3/4)*2^(1/4)*a^(1/4)-6*arctan(1/2*(x^2*(a*x^2+b))^(1/4)/x*2^(3/4)/a^(1/4))*a*x^3)*2^(3/4)/x^3/a^(1
/4)/b

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 42.42 (sec) , antiderivative size = 611, normalized size of antiderivative = 5.46 \[ \int \frac {b+a x^2}{x^2 \left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {3 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} + 4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b x^{2}} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + 2 \, {\left (a x^{4} + b x^{2}\right )}^{\frac {3}{4}} a^{2} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a^{2} b x^{3} + a b^{2} x\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b x}\right ) + 3 i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} + 4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b x^{2}} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (a x^{4} + b x^{2}\right )}^{\frac {3}{4}} a^{2} - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 i \, a^{2} b x^{3} + i \, a b^{2} x\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b x}\right ) - 3 i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} - 4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b x^{2}} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (a x^{4} + b x^{2}\right )}^{\frac {3}{4}} a^{2} - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-3 i \, a^{2} b x^{3} - i \, a b^{2} x\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b x}\right ) - 3 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} - 4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b x^{2}} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + 2 \, {\left (a x^{4} + b x^{2}\right )}^{\frac {3}{4}} a^{2} - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a^{2} b x^{3} + a b^{2} x\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b x}\right ) - 4 \, {\left (a x^{4} + b x^{2}\right )}^{\frac {3}{4}}}{6 \, b x^{3}} \]

[In]

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

[Out]

-1/6*(3*(1/2)^(1/4)*b*x^3*(a^3/b^4)^(1/4)*log((4*sqrt(1/2)*(a*x^4 + b*x^2)^(1/4)*a*b^2*x^2*sqrt(a^3/b^4) + 4*(
1/2)^(3/4)*sqrt(a*x^4 + b*x^2)*b^3*x*(a^3/b^4)^(3/4) + 2*(a*x^4 + b*x^2)^(3/4)*a^2 + (1/2)^(1/4)*(3*a^2*b*x^3
+ a*b^2*x)*(a^3/b^4)^(1/4))/(a*x^3 - b*x)) + 3*I*(1/2)^(1/4)*b*x^3*(a^3/b^4)^(1/4)*log(-(4*sqrt(1/2)*(a*x^4 +
b*x^2)^(1/4)*a*b^2*x^2*sqrt(a^3/b^4) + 4*I*(1/2)^(3/4)*sqrt(a*x^4 + b*x^2)*b^3*x*(a^3/b^4)^(3/4) - 2*(a*x^4 +
b*x^2)^(3/4)*a^2 - (1/2)^(1/4)*(3*I*a^2*b*x^3 + I*a*b^2*x)*(a^3/b^4)^(1/4))/(a*x^3 - b*x)) - 3*I*(1/2)^(1/4)*b
*x^3*(a^3/b^4)^(1/4)*log(-(4*sqrt(1/2)*(a*x^4 + b*x^2)^(1/4)*a*b^2*x^2*sqrt(a^3/b^4) - 4*I*(1/2)^(3/4)*sqrt(a*
x^4 + b*x^2)*b^3*x*(a^3/b^4)^(3/4) - 2*(a*x^4 + b*x^2)^(3/4)*a^2 - (1/2)^(1/4)*(-3*I*a^2*b*x^3 - I*a*b^2*x)*(a
^3/b^4)^(1/4))/(a*x^3 - b*x)) - 3*(1/2)^(1/4)*b*x^3*(a^3/b^4)^(1/4)*log((4*sqrt(1/2)*(a*x^4 + b*x^2)^(1/4)*a*b
^2*x^2*sqrt(a^3/b^4) - 4*(1/2)^(3/4)*sqrt(a*x^4 + b*x^2)*b^3*x*(a^3/b^4)^(3/4) + 2*(a*x^4 + b*x^2)^(3/4)*a^2 -
(1/2)^(1/4)*(3*a^2*b*x^3 + a*b^2*x)*(a^3/b^4)^(1/4))/(a*x^3 - b*x)) - 4*(a*x^4 + b*x^2)^(3/4))/(b*x^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (88) = 176\).

Time = 0.30 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.86 \[ \int \frac {b+a x^2}{x^2 \left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{b} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{b} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{2 \, b} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{2 \, b} + \frac {2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{4}}}{3 \, b} \]

[In]

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

[Out]

-2^(1/4)*(-a)^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) + 2*(a + b/x^2)^(1/4))/(-a)^(1/4))/b - 2^(1/4)*(-a)
^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) - 2*(a + b/x^2)^(1/4))/(-a)^(1/4))/b + 1/2*2^(1/4)*(-a)^(3/4)*l
og(2^(3/4)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^2))/b - 1/2*2^(1/4)*(-a)^(3/4)*log(-
2^(3/4)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^2))/b + 2/3*(a + b/x^2)^(3/4)/b

Mupad [F(-1)]

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

[In]

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

[Out]

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