3.5.7 \(\int \frac {f+g x}{(d+e x) \sqrt {-a+c x^4}} \, dx\) [407]

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

[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 = 0.17, 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 = {1756, 12, 1262, 739, 212, 1725, 230, 227, 1233, 1232} \begin {gather*} \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 .\text {ArcSin}\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 .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} 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}} \end {gather*}

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 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 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 739

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 1232

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

Rule 1233

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 1262

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 1725

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 1756

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) \text {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) \text {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] Result contains complex when optimal does not.
time = 11.00, size = 719, normalized size = 3.30 \begin {gather*} \frac {-\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 )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} e}+\frac {i f \left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2 \sqrt {-\frac {(1-i) \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{i \sqrt [4]{a}+\sqrt [4]{c} x}} \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 (\left (-\sqrt [4]{c} d+\sqrt [4]{a} e\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {(1+i) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\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]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\right )\right |2\right )\right )}{\sqrt [4]{a} \left (-\sqrt [4]{c} d+\sqrt [4]{a} e\right ) \left (i \sqrt [4]{c} d+\sqrt [4]{a} e\right )}+\frac {d g \left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2 \sqrt {-\frac {(1-i) \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{i \sqrt [4]{a}+\sqrt [4]{c} x}} \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 (i \left (\sqrt [4]{c} d-\sqrt [4]{a} e\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {(1+i) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\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]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\right )\right |2\right )\right )}{\sqrt [4]{a} e \left (-\sqrt [4]{c} d+\sqrt [4]{a} e\right ) \left (i \sqrt [4]{c} d+\sqrt [4]{a} e\right )}}{\sqrt {-a+c x^4}} \end {gather*}

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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Maple [A]
time = 0.14, size = 247, normalized size = 1.13

method	result	size
default	\(\frac {g \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}+\frac {\left (-d g +e f \right ) \left (-\frac {\arctanh \left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}-2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}\, \sqrt {c \,x^{4}-a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}}+\frac {e \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e^{2} \sqrt {a}}{d^{2} \sqrt {c}}, \frac {\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}-a}}\right )}{e^{2}}\)	\(247\)
elliptic	\(\frac {g \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}-\frac {\left (d g -e f \right ) \left (-\frac {\arctanh \left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}-2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}\, \sqrt {c \,x^{4}-a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}}+\frac {e \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e^{2} \sqrt {a}}{d^{2} \sqrt {c}}, \frac {\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}-a}}\right )}{e^{2}}\)	\(248\)

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,method=_RETURNVERBOSE)

[Out]

g/e/(-1/a^(1/2)*c^(1/2))^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4-a)^(1/2)
*EllipticF(x*(-1/a^(1/2)*c^(1/2))^(1/2),I)+(-d*g+e*f)/e^2*(-1/2/(c*d^4/e^4-a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e
^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+1/a^(1/2)*c^(1/2)*x^2)^(1/2)*
(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4-a)^(1/2)*EllipticPi(x*(-1/a^(1/2)*c^(1/2))^(1/2),-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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)*(x*e + d)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

integral(sqrt(c*x^4 - a)*(g*x + f)/(c*d*x^4 - a*d + (c*x^5 - a*x)*e), x)

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)*(x*e + d)), x)

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

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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