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

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]

((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])

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Rubi [A]  time = 0.60376, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} (e f-d g) \Pi \left (\frac{\sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt{c x^4-a}}+\frac{(e f-d g) \tanh ^{-1}\left (\frac{a e^2-c d^2 x^2}{\sqrt{c x^4-a} \sqrt{c d^4-a e^4}}\right )}{2 \sqrt{c d^4-a e^4}}+\frac{\sqrt [4]{a} g \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt{c x^4-a}} \]

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])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)/(e*x+d)/(c*x**4-a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 2.00761, size = 719, normalized size = 3.3 \[ \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)*EllipticF[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*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)*(-(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 = 0.039, size = 247, normalized size = 1.1 \[{\frac{g}{e}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}}+{\frac{-dg+ef}{{e}^{2}} \left ( -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}-2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}-a}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}-a}}}}+{\frac{e}{d}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},-{\frac{{e}^{2}}{{d}^{2}}\sqrt{a}{\frac{1}{\sqrt{c}}}},{1\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}} \right ) } \]

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]

1/e*g/(-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., size = 0, normalized size = 0. \[ \int \frac{g x + f}{\sqrt{c x^{4} - a}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/(sqrt(c*x^4 - a)*(e*x + d)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/(sqrt(c*x^4 - a)*(e*x + d)),x, algorithm="fricas")

[Out]

Timed out

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/(sqrt(c*x^4 - a)*(e*x + d)),x, algorithm="giac")

[Out]

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