3.407 \(\int \frac {f+g x}{(d+e x) \sqrt {-a+c x^4}} \, dx\)

Optimal. Leaf size=218 \[ \frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} (e f-d g) \Pi \left (\frac {\sqrt {a} e^2}{\sqrt {c} d^2};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt {c x^4-a}}+\frac {(e f-d g) \tanh ^{-1}\left (\frac {a e^2-c d^2 x^2}{\sqrt {c x^4-a} \sqrt {c d^4-a e^4}}\right )}{2 \sqrt {c d^4-a e^4}}+\frac {\sqrt [4]{a} g \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt {c x^4-a}} \]

[Out]

1/2*(-d*g+e*f)*arctanh((-c*d^2*x^2+a*e^2)/(-a*e^4+c*d^4)^(1/2)/(c*x^4-a)^(1/2))/(-a*e^4+c*d^4)^(1/2)+a^(1/4)*g
*EllipticF(c^(1/4)*x/a^(1/4),I)*(1-c*x^4/a)^(1/2)/c^(1/4)/e/(c*x^4-a)^(1/2)+a^(1/4)*(-d*g+e*f)*EllipticPi(c^(1
/4)*x/a^(1/4),e^2*a^(1/2)/d^2/c^(1/2),I)*(1-c*x^4/a)^(1/2)/c^(1/4)/d/e/(c*x^4-a)^(1/2)

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Rubi [A]  time = 0.29, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1742, 12, 1248, 725, 206, 1711, 224, 221, 1219, 1218} \[ \frac {(e f-d g) \tanh ^{-1}\left (\frac {a e^2-c d^2 x^2}{\sqrt {c x^4-a} \sqrt {c d^4-a e^4}}\right )}{2 \sqrt {c d^4-a e^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} (e f-d g) \Pi \left (\frac {\sqrt {a} e^2}{\sqrt {c} d^2};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt {c x^4-a}}+\frac {\sqrt [4]{a} g \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt {c x^4-a}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)/((d + e*x)*Sqrt[-a + c*x^4]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

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]

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 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1711

Int[((A_.) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[B/e, Int[1/Sqr
t[a + c*x^4], x], x] + Dist[(e*A - d*B)/e, Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e, A
, B}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]

Rule 1742

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[P
x, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)*Sq
rt[a + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x]] /; FreeQ[{a,
c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + a*e^4, 0]

Rubi steps

\begin {align*} \int \frac {f+g x}{(d+e x) \sqrt {-a+c x^4}} \, dx &=\int \frac {(-e f+d g) x}{\left (d^2-e^2 x^2\right ) \sqrt {-a+c x^4}} \, dx+\int \frac {d f-e g x^2}{\left (d^2-e^2 x^2\right ) \sqrt {-a+c x^4}} \, dx\\ &=\frac {g \int \frac {1}{\sqrt {-a+c x^4}} \, dx}{e}+\frac {(d (e f-d g)) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {-a+c x^4}} \, dx}{e}+(-e f+d g) \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {-a+c x^4}} \, dx\\ &=\frac {1}{2} (-e f+d g) \operatorname {Subst}\left (\int \frac {1}{\left (d^2-e^2 x\right ) \sqrt {-a+c x^2}} \, dx,x,x^2\right )+\frac {\left (g \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{e \sqrt {-a+c x^4}}+\frac {\left (d (e f-d g) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {1-\frac {c x^4}{a}}} \, dx}{e \sqrt {-a+c x^4}}\\ &=\frac {\sqrt [4]{a} g \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt {-a+c x^4}}+\frac {\sqrt [4]{a} (e f-d g) \sqrt {1-\frac {c x^4}{a}} \Pi \left (\frac {\sqrt {a} e^2}{\sqrt {c} d^2};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt {-a+c x^4}}+\frac {1}{2} (e f-d g) \operatorname {Subst}\left (\int \frac {1}{c d^4-a e^4-x^2} \, dx,x,\frac {a e^2-c d^2 x^2}{\sqrt {-a+c x^4}}\right )\\ &=\frac {(e f-d g) \tanh ^{-1}\left (\frac {a e^2-c d^2 x^2}{\sqrt {c d^4-a e^4} \sqrt {-a+c x^4}}\right )}{2 \sqrt {c d^4-a e^4}}+\frac {\sqrt [4]{a} g \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt {-a+c x^4}}+\frac {\sqrt [4]{a} (e f-d g) \sqrt {1-\frac {c x^4}{a}} \Pi \left (\frac {\sqrt {a} e^2}{\sqrt {c} d^2};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt {-a+c x^4}}\\ \end {align*}

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Mathematica [C]  time = 1.26, size = 719, normalized size = 3.30 \[ \frac {\frac {i f \sqrt {-\frac {(1-i) \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [4]{c} x+i \sqrt [4]{a}}} \sqrt {\frac {(1+i) \left (\sqrt [4]{a}+i \sqrt [4]{c} x\right ) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2}} \left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2 \left (\left (\sqrt [4]{a} e-\sqrt [4]{c} d\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {(1+i) \left (\sqrt [4]{c} x+\sqrt [4]{a}\right )}{2 \sqrt [4]{c} x+2 i \sqrt [4]{a}}}\right )\right |2\right )-(1-i) \sqrt [4]{a} e \Pi \left (\frac {(1-i) \left (\sqrt [4]{c} d-i \sqrt [4]{a} e\right )}{\sqrt [4]{c} d-\sqrt [4]{a} e};\left .\sin ^{-1}\left (\sqrt {\frac {(1+i) \left (\sqrt [4]{c} x+\sqrt [4]{a}\right )}{2 \sqrt [4]{c} x+2 i \sqrt [4]{a}}}\right )\right |2\right )\right )}{\sqrt [4]{a} \left (\sqrt [4]{a} e-\sqrt [4]{c} d\right ) \left (\sqrt [4]{a} e+i \sqrt [4]{c} d\right )}+\frac {d g \sqrt {-\frac {(1-i) \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [4]{c} x+i \sqrt [4]{a}}} \sqrt {\frac {(1+i) \left (\sqrt [4]{a}+i \sqrt [4]{c} x\right ) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2}} \left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2 \left (i \left (\sqrt [4]{c} d-\sqrt [4]{a} e\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {(1+i) \left (\sqrt [4]{c} x+\sqrt [4]{a}\right )}{2 \sqrt [4]{c} x+2 i \sqrt [4]{a}}}\right )\right |2\right )+(1+i) \sqrt [4]{a} e \Pi \left (\frac {(1-i) \left (\sqrt [4]{c} d-i \sqrt [4]{a} e\right )}{\sqrt [4]{c} d-\sqrt [4]{a} e};\left .\sin ^{-1}\left (\sqrt {\frac {(1+i) \left (\sqrt [4]{c} x+\sqrt [4]{a}\right )}{2 \sqrt [4]{c} x+2 i \sqrt [4]{a}}}\right )\right |2\right )\right )}{\sqrt [4]{a} e \left (\sqrt [4]{a} e-\sqrt [4]{c} d\right ) \left (\sqrt [4]{a} e+i \sqrt [4]{c} d\right )}-\frac {i g \sqrt {1-\frac {c x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )}{e \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}}{\sqrt {c x^4-a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-I)*g*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(Sqrt[-(Sqrt[c]/Sqrt[a])]*e
) + (I*f*(a^(1/4) - I*c^(1/4)*x)^2*Sqrt[((-1 + I)*(a^(1/4) - c^(1/4)*x))/(I*a^(1/4) + c^(1/4)*x)]*Sqrt[((1 + I
)*(a^(1/4) + I*c^(1/4)*x)*(a^(1/4) + c^(1/4)*x))/(a^(1/4) - I*c^(1/4)*x)^2]*((-(c^(1/4)*d) + a^(1/4)*e)*Ellipt
icF[ArcSin[Sqrt[((1 + I)*(a^(1/4) + c^(1/4)*x))/((2*I)*a^(1/4) + 2*c^(1/4)*x)]], 2] - (1 - I)*a^(1/4)*e*Ellipt
icPi[((1 - I)*(c^(1/4)*d - I*a^(1/4)*e))/(c^(1/4)*d - a^(1/4)*e), ArcSin[Sqrt[((1 + I)*(a^(1/4) + c^(1/4)*x))/
((2*I)*a^(1/4) + 2*c^(1/4)*x)]], 2]))/(a^(1/4)*(-(c^(1/4)*d) + a^(1/4)*e)*(I*c^(1/4)*d + a^(1/4)*e)) + (d*g*(a
^(1/4) - I*c^(1/4)*x)^2*Sqrt[((-1 + I)*(a^(1/4) - c^(1/4)*x))/(I*a^(1/4) + c^(1/4)*x)]*Sqrt[((1 + I)*(a^(1/4)
+ I*c^(1/4)*x)*(a^(1/4) + c^(1/4)*x))/(a^(1/4) - I*c^(1/4)*x)^2]*(I*(c^(1/4)*d - a^(1/4)*e)*EllipticF[ArcSin[S
qrt[((1 + I)*(a^(1/4) + c^(1/4)*x))/((2*I)*a^(1/4) + 2*c^(1/4)*x)]], 2] + (1 + I)*a^(1/4)*e*EllipticPi[((1 - I
)*(c^(1/4)*d - I*a^(1/4)*e))/(c^(1/4)*d - a^(1/4)*e), ArcSin[Sqrt[((1 + I)*(a^(1/4) + c^(1/4)*x))/((2*I)*a^(1/
4) + 2*c^(1/4)*x)]], 2]))/(a^(1/4)*e*(-(c^(1/4)*d) + a^(1/4)*e)*(I*c^(1/4)*d + a^(1/4)*e)))/Sqrt[-a + c*x^4]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {g x + f}{\sqrt {c x^{4} - a} {\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*x + f)/(sqrt(c*x^4 - a)*(e*x + d)), x)

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maple [A]  time = 0.02, size = 247, normalized size = 1.13 \[ \frac {\sqrt {\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, g \EllipticF \left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, x , i\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}\, e}+\frac {\left (-d g +e f \right ) \left (\frac {\sqrt {\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, e \EllipticPi \left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, x , -\frac {\sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}\, d}-\frac {\arctanh \left (\frac {\frac {2 c \,d^{2} x^{2}}{e^{2}}-2 a}{2 \sqrt {-a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}-a}}\right )}{2 \sqrt {-a +\frac {c \,d^{4}}{e^{4}}}}\right )}{e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g/e/(-1/a^(1/2)*c^(1/2))^(1/2)*(1/a^(1/2)*c^(1/2)*x^2+1)^(1/2)*(-1/a^(1/2)*c^(1/2)*x^2+1)^(1/2)/(c*x^4-a)^(1/2
)*EllipticF((-1/a^(1/2)*c^(1/2))^(1/2)*x,I)+(-d*g+e*f)/e^2*(-1/2/(c*d^4/e^4-a)^(1/2)*arctanh(1/2*(2*c*d^2/e^2*
x^2-2*a)/(c*d^4/e^4-a)^(1/2)/(c*x^4-a)^(1/2))+1/(-1/a^(1/2)*c^(1/2))^(1/2)/d*e*(1/a^(1/2)*c^(1/2)*x^2+1)^(1/2)
*(-1/a^(1/2)*c^(1/2)*x^2+1)^(1/2)/(c*x^4-a)^(1/2)*EllipticPi((-1/a^(1/2)*c^(1/2))^(1/2)*x,-e^2*a^(1/2)/d^2/c^(
1/2),(1/a^(1/2)*c^(1/2))^(1/2)/(-1/a^(1/2)*c^(1/2))^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {g x + f}{\sqrt {c x^{4} - a} {\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*x + f)/(sqrt(c*x^4 - a)*(e*x + d)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {f+g\,x}{\sqrt {c\,x^4-a}\,\left (d+e\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f + g x}{\sqrt {- a + c x^{4}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)/(c*x**4-a)**(1/2),x)

[Out]

Integral((f + g*x)/(sqrt(-a + c*x**4)*(d + e*x)), x)

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