3.790 \(\int \frac {x \sqrt {c+d x^4}}{a+b x^4} \, dx\)

Optimal. Leaf size=91 \[ \frac {\sqrt {b c-a d} \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 \sqrt {a} b}+\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{2 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {465, 402, 217, 206, 377, 205} \[ \frac {\sqrt {b c-a d} \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 \sqrt {a} b}+\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

(Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(2*Sqrt[a]*b) + (Sqrt[d]*ArcTanh[(Sq
rt[d]*x^2)/Sqrt[c + d*x^4]])/(2*b)

Rule 205

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 377

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 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 465

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

Rubi steps

\begin {align*} \int \frac {x \sqrt {c+d x^4}}{a+b x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {c+d x^2}}{a+b x^2} \, dx,x,x^2\right )\\ &=\frac {d \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,x^2\right )}{2 b}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{2 b}\\ &=\frac {d \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x^2}{\sqrt {c+d x^4}}\right )}{2 b}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^2}{\sqrt {c+d x^4}}\right )}{2 b}\\ &=\frac {\sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 \sqrt {a} b}+\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{2 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 89, normalized size = 0.98 \[ \frac {\frac {\sqrt {b c-a d} \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{\sqrt {a}}+\sqrt {d} \log \left (\sqrt {d} \sqrt {c+d x^4}+d x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 612, normalized size = 6.73 \[ \left [\frac {2 \, \sqrt {d} \log \left (-2 \, d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right ) + \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \, {\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, b}, -\frac {4 \, \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{2}}{\sqrt {d x^{4} + c}}\right ) - \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \, {\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, b}, \frac {\sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{6} + {\left (b c^{2} - a c d\right )} x^{2}\right )}}\right ) + \sqrt {d} \log \left (-2 \, d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right )}{4 \, b}, -\frac {2 \, \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{2}}{\sqrt {d x^{4} + c}}\right ) - \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{6} + {\left (b c^{2} - a c d\right )} x^{2}\right )}}\right )}{4 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(2*sqrt(d)*log(-2*d*x^4 - 2*sqrt(d*x^4 + c)*sqrt(d)*x^2 - c) + sqrt(-(b*c - a*d)/a)*log(((b^2*c^2 - 8*a*b
*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 + 4*((a*b*c - 2*a^2*d)*x^6 - a^2*c*x^2)*sqrt(d
*x^4 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^8 + 2*a*b*x^4 + a^2)))/b, -1/8*(4*sqrt(-d)*arctan(sqrt(-d)*x^2/sqrt(d*x
^4 + c)) - sqrt(-(b*c - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a
^2*c^2 + 4*((a*b*c - 2*a^2*d)*x^6 - a^2*c*x^2)*sqrt(d*x^4 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^8 + 2*a*b*x^4 + a^
2)))/b, 1/4*(sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)*sqrt(d*x^4 + c)*sqrt((b*c - a*d)/a)/((b*
c*d - a*d^2)*x^6 + (b*c^2 - a*c*d)*x^2)) + sqrt(d)*log(-2*d*x^4 - 2*sqrt(d*x^4 + c)*sqrt(d)*x^2 - c))/b, -1/4*
(2*sqrt(-d)*arctan(sqrt(-d)*x^2/sqrt(d*x^4 + c)) - sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)*sq
rt(d*x^4 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^6 + (b*c^2 - a*c*d)*x^2)))/b]

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:inde
x.cc index_m i_lex_is_greater Error: Bad Argument Value

________________________________________________________________________________________

maple [B]  time = 0.20, size = 1000, normalized size = 10.99 \[ \frac {a d \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, b}-\frac {a d \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, b}-\frac {c \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {c \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {\sqrt {d}\, \ln \left (\frac {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d -\frac {\sqrt {-a b}\, d}{b}}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\right )}{4 b}+\frac {\sqrt {d}\, \ln \left (\frac {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d +\frac {\sqrt {-a b}\, d}{b}}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\right )}{4 b}-\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{4 \sqrt {-a b}}+\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{4 \sqrt {-a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{4} + c} x}{b x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x^4 + c)*x/(b*x^4 + a), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\sqrt {d\,x^4+c}}{b\,x^4+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {c + d x^{4}}}{a + b x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(c + d*x**4)/(a + b*x**4), x)

________________________________________________________________________________________