\(\int \frac {c+d x}{\sqrt {-a+b x^4}} \, dx\) [212]

Optimal result

Integrand size = 19, antiderivative size = 89 \[ \int \frac {c+d x}{\sqrt {-a+b x^4}} \, dx=\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {-a+b x^4}}\right )}{2 \sqrt {b}}+\frac {\sqrt [4]{a} c \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {-a+b x^4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1899, 230, 227, 281, 223, 212} \[ \int \frac {c+d x}{\sqrt {-a+b x^4}} \, dx=\frac {\sqrt [4]{a} c \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {b x^4-a}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {b x^4-a}}\right )}{2 \sqrt {b}} \]

[In]

Int[(c + d*x)/Sqrt[-a + b*x^4],x]

[Out]

(d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[-a + b*x^4]])/(2*Sqrt[b]) + (a^(1/4)*c*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(
1/4)*x)/a^(1/4)], -1])/(b^(1/4)*Sqrt[-a + b*x^4])

Rule 212

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

Rule 223

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 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + b*(x^4/a)]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + b*(x^4/
a)], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 281

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

Rule 1899

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[P
q, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b
, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c}{\sqrt {-a+b x^4}}+\frac {d x}{\sqrt {-a+b x^4}}\right ) \, dx \\ & = c \int \frac {1}{\sqrt {-a+b x^4}} \, dx+d \int \frac {x}{\sqrt {-a+b x^4}} \, dx \\ & = \frac {1}{2} d \text {Subst}\left (\int \frac {1}{\sqrt {-a+b x^2}} \, dx,x,x^2\right )+\frac {\left (c \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{\sqrt {-a+b x^4}} \\ & = \frac {\sqrt [4]{a} c \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt {-a+b x^4}}+\frac {1}{2} d \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {-a+b x^4}}\right ) \\ & = \frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {-a+b x^4}}\right )}{2 \sqrt {b}}+\frac {\sqrt [4]{a} c \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt {-a+b x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x}{\sqrt {-a+b x^4}} \, dx=\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {-a+b x^4}}\right )}{2 \sqrt {b}}+\frac {c x \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^4}{a}\right )}{\sqrt {-a+b x^4}} \]

[In]

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

[Out]

(d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[-a + b*x^4]])/(2*Sqrt[b]) + (c*x*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2,
5/4, (b*x^4)/a])/Sqrt[-a + b*x^4]

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {c \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}-a}}+\frac {d \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}-a}\right )}{2 \sqrt {b}}\)	\(95\)
elliptic	\(\frac {c \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}-a}}+\frac {d \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}-a}\right )}{2 \sqrt {b}}\)	\(98\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82 \[ \int \frac {c+d x}{\sqrt {-a+b x^4}} \, dx=-\frac {4 \, b^{\frac {3}{2}} c \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - a \sqrt {b} d \log \left (2 \, b x^{4} + 2 \, \sqrt {b x^{4} - a} \sqrt {b} x^{2} - a\right )}{4 \, a b} \]

[In]

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

[Out]

-1/4*(4*b^(3/2)*c*(a/b)^(3/4)*elliptic_f(arcsin((a/b)^(1/4)/x), -1) - a*sqrt(b)*d*log(2*b*x^4 + 2*sqrt(b*x^4 -
a)*sqrt(b)*x^2 - a))/(a*b)

Sympy [A] (verification not implemented)

Time = 1.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x}{\sqrt {-a+b x^4}} \, dx=d \left (\begin {cases} \frac {\operatorname {acosh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2 \sqrt {b}} & \text {for}\: \left |{\frac {b x^{4}}{a}}\right | > 1 \\- \frac {i \operatorname {asin}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2 \sqrt {b}} & \text {otherwise} \end {cases}\right ) - \frac {i c x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4}}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} \]

[In]

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

[Out]

d*Piecewise((acosh(sqrt(b)*x**2/sqrt(a))/(2*sqrt(b)), Abs(b*x**4/a) > 1), (-I*asin(sqrt(b)*x**2/sqrt(a))/(2*sq
rt(b)), True)) - I*c*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b*x**4/a)/(4*sqrt(a)*gamma(5/4))

Maxima [F]

\[ \int \frac {c+d x}{\sqrt {-a+b x^4}} \, dx=\int { \frac {d x + c}{\sqrt {b x^{4} - a}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {c+d x}{\sqrt {-a+b x^4}} \, dx=\int { \frac {d x + c}{\sqrt {b x^{4} - a}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x}{\sqrt {-a+b x^4}} \, dx=\int \frac {c+d\,x}{\sqrt {b\,x^4-a}} \,d x \]

[In]

int((c + d*x)/(b*x^4 - a)^(1/2),x)

[Out]

int((c + d*x)/(b*x^4 - a)^(1/2), x)