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

Optimal. Leaf size=87 \[ \frac{\sqrt [4]{a} c \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt{a-b x^4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

[Out]

(d*ArcTan[(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])

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Rubi [A]  time = 0.0621967, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1885, 224, 221, 275, 217, 203} \[ \frac{\sqrt [4]{a} c \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{a-b x^4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/Sqrt[a - b*x^4],x]

[Out]

(d*ArcTan[(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 1885

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)]

Rule 224

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 221

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 275

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 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 203

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

Rubi steps

\begin{align*} \int \frac{c+d x}{\sqrt{a-b x^4}} \, dx &=\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 \operatorname{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 \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{x^2}{\sqrt{a-b x^4}}\right )\\ &=\frac{d \tan ^{-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]  time = 0.0472358, size = 81, normalized size = 0.93 \[ \frac{c x \sqrt{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b x^4}{a}\right )}{\sqrt{a-b x^4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*ArcTan[(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]

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Maple [A]  time = 0.015, size = 90, normalized size = 1. \begin{align*}{\frac{d}{2}\arctan \left ({{x}^{2}\sqrt{b}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{b}}}}+{c\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x + c}{\sqrt{-b x^{4} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-b x^{4} + a}{\left (d x + c\right )}}{b x^{4} - a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.96213, size = 97, normalized size = 1.11 \begin{align*} d \left (\begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{for}\: \frac{\left |{b x^{4}}\right |}{\left |{a}\right |} > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{otherwise} \end{cases}\right ) + \frac{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} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x + c}{\sqrt{-b x^{4} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)