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

Optimal. Leaf size=158 \[ \frac{a^{3/4} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{1}{3} d x \sqrt{a+c x^4}+\frac{1}{4} e x^2 \sqrt{a+c x^4}+\frac{a e \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}} \]

[Out]

(d*x*Sqrt[a + c*x^4])/3 + (e*x^2*Sqrt[a + c*x^4])/4 + (a*e*ArcTanh[(Sqrt[c]*x^2)
/Sqrt[a + c*x^4]])/(4*Sqrt[c]) + (a^(3/4)*d*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*
x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(
3*c^(1/4)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.181423, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{a^{3/4} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{1}{3} d x \sqrt{a+c x^4}+\frac{1}{4} e x^2 \sqrt{a+c x^4}+\frac{a e \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

(d*x*Sqrt[a + c*x^4])/3 + (e*x^2*Sqrt[a + c*x^4])/4 + (a*e*ArcTanh[(Sqrt[c]*x^2)
/Sqrt[a + c*x^4]])/(4*Sqrt[c]) + (a^(3/4)*d*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*
x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(
3*c^(1/4)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 16.5768, size = 143, normalized size = 0.91 \[ \frac{a^{\frac{3}{4}} d \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{c} \sqrt{a + c x^{4}}} + \frac{a e \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{4 \sqrt{c}} + \frac{d x \sqrt{a + c x^{4}}}{3} + \frac{e x^{2} \sqrt{a + c x^{4}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/4)*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)/(3*c**(1/4)*sqrt(a + c*x**4)) +
a*e*atanh(sqrt(c)*x**2/sqrt(a + c*x**4))/(4*sqrt(c)) + d*x*sqrt(a + c*x**4)/3 +
e*x**2*sqrt(a + c*x**4)/4

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Mathematica [C]  time = 0.553328, size = 132, normalized size = 0.84 \[ \frac{1}{12} \left (x \sqrt{a+c x^4} (4 d+3 e x)-\frac{8 i a d \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}}+\frac{3 a e \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(4*d + 3*e*x)*Sqrt[a + c*x^4] + (3*a*e*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]]
)/Sqrt[c] - ((8*I)*a*d*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/
Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*Sqrt[a + c*x^4]))/12

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Maple [C]  time = 0.006, size = 127, normalized size = 0.8 \[{\frac{dx}{3}\sqrt{c{x}^{4}+a}}+{\frac{2\,ad}{3}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{e{x}^{2}}{4}\sqrt{c{x}^{4}+a}}+{\frac{ae}{4}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^4+a)^(1/2),x)

[Out]

1/3*d*x*(c*x^4+a)^(1/2)+2/3*d*a/(I/a^(1/2)*c^(1/2))^(1/2)*(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)*EllipticF(x*(I/a^(1/2)
*c^(1/2))^(1/2),I)+1/4*e*x^2*(c*x^4+a)^(1/2)+1/4*e*a/c^(1/2)*ln(x^2*c^(1/2)+(c*x
^4+a)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{4} + a}{\left (e x + d\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x^4 + a)*(e*x + d), x)

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Sympy [A]  time = 8.5021, size = 88, normalized size = 0.56 \[ \frac{\sqrt{a} d x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{a} e x^{2} \sqrt{1 + \frac{c x^{4}}{a}}}{4} + \frac{a e \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*d*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(I*pi)/a)/(4*g
amma(5/4)) + sqrt(a)*e*x**2*sqrt(1 + c*x**4/a)/4 + a*e*asinh(sqrt(c)*x**2/sqrt(a
))/(4*sqrt(c))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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