3.193 \(\int \frac{1}{(d+e x) \sqrt{a+c x^4}} \, dx\)

Optimal. Leaf size=413 \[ \frac{\sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}+\frac{\tan ^{-1}\left (\frac{x \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}{\sqrt{a+c x^4}}\right )}{2 d \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}-\frac{e \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{2 \sqrt{a e^4+c d^4}} \]

[Out]

ArcTan[(Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]]/(2*d*Sqrt[-((c*d^
4 + a*e^4)/(d^2*e^2))]) - (e*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sq
rt[a + c*x^4])])/(2*Sqrt[c*d^4 + a*e^4]) + (c^(1/4)*d*(Sqrt[a] + Sqrt[c]*x^2)*Sq
rt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)
], 1/2])/(2*a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*Sqrt[a + c*x^4]) - ((Sqrt[c]*d^2
- Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)
^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcT
an[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*d*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*
Sqrt[a + c*x^4])

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Rubi [A]  time = 0.602332, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{\sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}+\frac{\tan ^{-1}\left (\frac{x \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}{\sqrt{a+c x^4}}\right )}{2 d \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}-\frac{e \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{2 \sqrt{a e^4+c d^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*Sqrt[a + c*x^4]),x]

[Out]

ArcTan[(Sqrt[-((c*d^4 + a*e^4)/(d^2*e^2))]*x)/Sqrt[a + c*x^4]]/(2*d*Sqrt[-((c*d^
4 + a*e^4)/(d^2*e^2))]) - (e*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sq
rt[a + c*x^4])])/(2*Sqrt[c*d^4 + a*e^4]) + (c^(1/4)*d*(Sqrt[a] + Sqrt[c]*x^2)*Sq
rt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)
], 1/2])/(2*a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*Sqrt[a + c*x^4]) - ((Sqrt[c]*d^2
- Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)
^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcT
an[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*d*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*
Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 56.3311, size = 367, normalized size = 0.89 \[ \frac{e \operatorname{atanh}{\left (\frac{- a e^{2} - c d^{2} x^{2}}{\sqrt{a + c x^{4}} \sqrt{a e^{4} + c d^{4}}} \right )}}{2 \sqrt{a e^{4} + c d^{4}}} + \frac{\operatorname{atan}{\left (\frac{x \sqrt{- \frac{a e^{2}}{d^{2}} - \frac{c d^{2}}{e^{2}}}}{\sqrt{a + c x^{4}}} \right )}}{2 d \sqrt{- \frac{a e^{2}}{d^{2}} - \frac{c d^{2}}{e^{2}}}} + \frac{\sqrt [4]{c} d \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a + c x^{4}} \left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right )} + \frac{\sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) \Pi \left (\frac{\left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right )^{2}}{4 \sqrt{a} \sqrt{c} d^{2} e^{2}}; 2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \sqrt{a + c x^{4}} \left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*atanh((-a*e**2 - c*d**2*x**2)/(sqrt(a + c*x**4)*sqrt(a*e**4 + c*d**4)))/(2*sqr
t(a*e**4 + c*d**4)) + atan(x*sqrt(-a*e**2/d**2 - c*d**2/e**2)/sqrt(a + c*x**4))/
(2*d*sqrt(-a*e**2/d**2 - c*d**2/e**2)) + c**(1/4)*d*sqrt((a + c*x**4)/(sqrt(a) +
sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4
)), 1/2)/(2*a**(1/4)*sqrt(a + c*x**4)*(sqrt(a)*e**2 + sqrt(c)*d**2)) + sqrt((a +
c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*(sqrt(a)*e**2 - s
qrt(c)*d**2)*elliptic_pi((sqrt(a)*e**2 + sqrt(c)*d**2)**2/(4*sqrt(a)*sqrt(c)*d**
2*e**2), 2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(4*a**(1/4)*c**(1/4)*d*sqrt(a + c*x**
4)*(sqrt(a)*e**2 + sqrt(c)*d**2))

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Mathematica [C]  time = 0.433786, size = 200, normalized size = 0.48 \[ \frac{\sqrt{\frac{c x^4}{a}+1} \left (\sqrt [4]{c} d \log \left (\frac{e^2 x^2-d^2}{a e^2 \left (\sqrt{\frac{c x^4}{a}+1} \sqrt{\frac{c d^4}{a e^4}+1}+1\right )+c d^2 x^2}\right )-2 \sqrt [4]{-1} \sqrt [4]{a} e \sqrt{\frac{c d^4}{a e^4}+1} \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )\right )}{2 \sqrt [4]{c} d e \sqrt{a+c x^4} \sqrt{\frac{c d^4}{a e^4}+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*Sqrt[a + c*x^4]),x]

[Out]

(Sqrt[1 + (c*x^4)/a]*(-2*(-1)^(1/4)*a^(1/4)*Sqrt[1 + (c*d^4)/(a*e^4)]*e*Elliptic
Pi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1] +
c^(1/4)*d*Log[(-d^2 + e^2*x^2)/(c*d^2*x^2 + a*e^2*(1 + Sqrt[1 + (c*d^4)/(a*e^4)]
*Sqrt[1 + (c*x^4)/a]))]))/(2*c^(1/4)*d*Sqrt[1 + (c*d^4)/(a*e^4)]*e*Sqrt[a + c*x^
4])

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Maple [C]  time = 0.008, size = 169, normalized size = 0.4 \[{\frac{1}{e} \left ( -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}+2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}}+{\frac{e}{d}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},{\frac{-i{e}^{2}}{{d}^{2}}\sqrt{a}{\frac{1}{\sqrt{c}}}},{1\sqrt{{-i\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^4+a)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + a}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + a)*(e*x + d)),x, algorithm="maxima")

[Out]

integrate(1/(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(1/(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. \[ \int \frac{1}{\sqrt{a + c x^{4}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(c*x**4+a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + c*x**4)*(d + e*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + a}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + a)*(e*x + d)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^4 + a)*(e*x + d)), x)