3.188 \(\int \frac{1}{\sqrt{a-b x^4} (c-d x^4)^2} \, dx\)

Optimal. Leaf size=310 \[ -\frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{4 c \sqrt{a-b x^4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (5 b c-3 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (5 b c-3 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)}-\frac{d x \sqrt{a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)} \]

[Out]

-(d*x*Sqrt[a - b*x^4])/(4*c*(b*c - a*d)*(c - d*x^4)) - (a^(1/4)*b^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(
b^(1/4)*x)/a^(1/4)], -1])/(4*c*(b*c - a*d)*Sqrt[a - b*x^4]) + (a^(1/4)*(5*b*c - 3*a*d)*Sqrt[1 - (b*x^4)/a]*Ell
ipticPi[-((Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)*S
qrt[a - b*x^4]) + (a^(1/4)*(5*b*c - 3*a*d)*Sqrt[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]),
ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)*Sqrt[a - b*x^4])

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Rubi [A]  time = 0.247204, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {414, 523, 224, 221, 409, 1219, 1218} \[ -\frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt{a-b x^4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (5 b c-3 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (5 b c-3 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)}-\frac{d x \sqrt{a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*x*Sqrt[a - b*x^4])/(4*c*(b*c - a*d)*(c - d*x^4)) - (a^(1/4)*b^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(
b^(1/4)*x)/a^(1/4)], -1])/(4*c*(b*c - a*d)*Sqrt[a - b*x^4]) + (a^(1/4)*(5*b*c - 3*a*d)*Sqrt[1 - (b*x^4)/a]*Ell
ipticPi[-((Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)*S
qrt[a - b*x^4]) + (a^(1/4)*(5*b*c - 3*a*d)*Sqrt[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]),
ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)*Sqrt[a - b*x^4])

Rule 414

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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

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 409

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

Rule 1219

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + (c*x^4)/a]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a-b x^4} \left (c-d x^4\right )^2} \, dx &=-\frac{d x \sqrt{a-b x^4}}{4 c (b c-a d) \left (c-d x^4\right )}-\frac{\int \frac{-4 b c+3 a d-b d x^4}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=-\frac{d x \sqrt{a-b x^4}}{4 c (b c-a d) \left (c-d x^4\right )}-\frac{b \int \frac{1}{\sqrt{a-b x^4}} \, dx}{4 c (b c-a d)}+\frac{(5 b c-3 a d) \int \frac{1}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=-\frac{d x \sqrt{a-b x^4}}{4 c (b c-a d) \left (c-d x^4\right )}+\frac{(5 b c-3 a d) \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{8 c^2 (b c-a d)}+\frac{(5 b c-3 a d) \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{8 c^2 (b c-a d)}-\frac{\left (b \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{4 c (b c-a d) \sqrt{a-b x^4}}\\ &=-\frac{d x \sqrt{a-b x^4}}{4 c (b c-a d) \left (c-d x^4\right )}-\frac{\sqrt [4]{a} b^{3/4} \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 )}{4 c (b c-a d) \sqrt{a-b x^4}}+\frac{\left ((5 b c-3 a d) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{1-\frac{b x^4}{a}}} \, dx}{8 c^2 (b c-a d) \sqrt{a-b x^4}}+\frac{\left ((5 b c-3 a d) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{1-\frac{b x^4}{a}}} \, dx}{8 c^2 (b c-a d) \sqrt{a-b x^4}}\\ &=-\frac{d x \sqrt{a-b x^4}}{4 c (b c-a d) \left (c-d x^4\right )}-\frac{\sqrt [4]{a} b^{3/4} \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 )}{4 c (b c-a d) \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} (5 b c-3 a d) \sqrt{1-\frac{b x^4}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d) \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} (5 b c-3 a d) \sqrt{1-\frac{b x^4}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d) \sqrt{a-b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.25887, size = 386, normalized size = 1.25 \[ \frac{2 d x^5 \left (b x^4 \sqrt{1-\frac{b x^4}{a}} \left (d x^4-c\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+5 c \left (a-b x^4\right )\right ) \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c x F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right ) \left (b d x^4 \sqrt{1-\frac{b x^4}{a}} \left (d x^4-c\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )-5 c \left (-4 a d+4 b c+b d x^4\right )\right )}{20 c^2 \sqrt{a-b x^4} \left (d x^4-c\right ) (b c-a d) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c]*(-5*c*(4*b*c - 4*a*d + b*d*x^4) + b*d*x^4*Sqrt[1 - (
b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]) + 2*d*x^5*(5*c*(a - b*x^4) + b*x^4*Sq
rt[1 - (b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])*(2*a*d*AppellF1[5/4, 1/2, 2,
9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))/(20*c^2*(b*c - a*d)*Sqrt[a
- b*x^4]*(-c + d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/
2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))

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Maple [C]  time = 0.024, size = 322, normalized size = 1. \begin{align*} -{\frac{dx}{ \left ( 4\,ad-4\,bc \right ) c \left ( d{x}^{4}-c \right ) }\sqrt{-b{x}^{4}+a}}+{\frac{b}{ \left ( 4\,ad-4\,bc \right ) 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}}}}-{\frac{1}{32\,cd}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{3\,ad-5\,bc}{ \left ( ad-bc \right ){{\it \_alpha}}^{3}} \left ( -{{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{{x}^{2}\sqrt{b}}{\sqrt{a}}}}\sqrt{1+{\frac{{x}^{2}\sqrt{b}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*d/(a*d-b*c)/c*x*(-b*x^4+a)^(1/2)/(d*x^4-c)+1/4*b/(a*d-b*c)/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)-1/32/c/d*s
um((3*a*d-5*b*c)/(a*d-b*c)/_alpha^3*(-1/((a*d-b*c)/d)^(1/2)*arctanh(1/2*(-2*_alpha^2*b*x^2+2*a)/((a*d-b*c)/d)^
(1/2)/(-b*x^4+a)^(1/2))-2/(1/a^(1/2)*b^(1/2))^(1/2)*_alpha^3*d/c*(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)*EllipticPi(x*(1/a^(1/2)*b^(1/2))^(1/2),a^(1/2)/b^(1/2)*_alpha^2/c*d,(-1/a^(1/2
)*b^(1/2))^(1/2)/(1/a^(1/2)*b^(1/2))^(1/2))),_alpha=RootOf(_Z^4*d-c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-b x^{4} + a}{\left (d x^{4} - c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-b x^{4} + a}{\left (d x^{4} - c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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