3.158 \(\int \frac{\left (d+e x^2\right )^3}{\sqrt{a+c x^4}} \, dx\)

Optimal. Leaf size=326 \[ \frac{3 e x \sqrt{a+c x^4} \left (5 c d^2-a e^2\right )}{5 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{3 \sqrt [4]{a} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (5 c d^2-a e^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{7/4} \sqrt{a+c x^4}}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{5 \sqrt{c} d \left (c d^2-a e^2\right )}{\sqrt{a}}-3 a e^3+15 c d^2 e\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 c^{7/4} \sqrt{a+c x^4}}+\frac{d e^2 x \sqrt{a+c x^4}}{c}+\frac{e^3 x^3 \sqrt{a+c x^4}}{5 c} \]

[Out]

(d*e^2*x*Sqrt[a + c*x^4])/c + (e^3*x^3*Sqrt[a + c*x^4])/(5*c) + (3*e*(5*c*d^2 -
a*e^2)*x*Sqrt[a + c*x^4])/(5*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (3*a^(1/4)*e*(5*
c*d^2 - a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^
2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(7/4)*Sqrt[a + c*x^4]) +
(a^(1/4)*(15*c*d^2*e - 3*a*e^3 + (5*Sqrt[c]*d*(c*d^2 - a*e^2))/Sqrt[a])*(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])/(10*c^(7/4)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.545271, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{3 e x \sqrt{a+c x^4} \left (5 c d^2-a e^2\right )}{5 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{3 \sqrt [4]{a} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (5 c d^2-a e^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{7/4} \sqrt{a+c x^4}}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{5 \sqrt{c} d \left (c d^2-a e^2\right )}{\sqrt{a}}-3 a e^3+15 c d^2 e\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 c^{7/4} \sqrt{a+c x^4}}+\frac{d e^2 x \sqrt{a+c x^4}}{c}+\frac{e^3 x^3 \sqrt{a+c x^4}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2)^3/Sqrt[a + c*x^4],x]

[Out]

(d*e^2*x*Sqrt[a + c*x^4])/c + (e^3*x^3*Sqrt[a + c*x^4])/(5*c) + (3*e*(5*c*d^2 -
a*e^2)*x*Sqrt[a + c*x^4])/(5*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (3*a^(1/4)*e*(5*
c*d^2 - a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^
2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(7/4)*Sqrt[a + c*x^4]) +
(a^(1/4)*(15*c*d^2*e - 3*a*e^3 + (5*Sqrt[c]*d*(c*d^2 - a*e^2))/Sqrt[a])*(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])/(10*c^(7/4)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 69.7671, size = 299, normalized size = 0.92 \[ \frac{3 \sqrt [4]{a} e \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (a e^{2} - 5 c d^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{7}{4}} \sqrt{a + c x^{4}}} + \frac{d e^{2} x \sqrt{a + c x^{4}}}{c} + \frac{e^{3} x^{3} \sqrt{a + c x^{4}}}{5 c} - \frac{3 e x \sqrt{a + c x^{4}} \left (a e^{2} - 5 c d^{2}\right )}{5 c^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} - \frac{\sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (3 \sqrt{a} e \left (a e^{2} - 5 c d^{2}\right ) + 5 \sqrt{c} d \left (a e^{2} - c d^{2}\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{10 \sqrt [4]{a} c^{\frac{7}{4}} \sqrt{a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d)**3/(c*x**4+a)**(1/2),x)

[Out]

3*a**(1/4)*e*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x
**2)*(a*e**2 - 5*c*d**2)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(5*c**(7/4
)*sqrt(a + c*x**4)) + d*e**2*x*sqrt(a + c*x**4)/c + e**3*x**3*sqrt(a + c*x**4)/(
5*c) - 3*e*x*sqrt(a + c*x**4)*(a*e**2 - 5*c*d**2)/(5*c**(3/2)*(sqrt(a) + sqrt(c)
*x**2)) - sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2
)*(3*sqrt(a)*e*(a*e**2 - 5*c*d**2) + 5*sqrt(c)*d*(a*e**2 - c*d**2))*elliptic_f(2
*atan(c**(1/4)*x/a**(1/4)), 1/2)/(10*a**(1/4)*c**(7/4)*sqrt(a + c*x**4))

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Mathematica [C]  time = 0.55683, size = 235, normalized size = 0.72 \[ \frac{\sqrt{\frac{c x^4}{a}+1} \left (3 a^{3/2} e^3-15 \sqrt{a} c d^2 e+5 i a \sqrt{c} d e^2-5 i c^{3/2} d^3\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-3 \sqrt{a} e \sqrt{\frac{c x^4}{a}+1} \left (a e^2-5 c d^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{c} e^2 x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (a+c x^4\right ) \left (5 d+e x^2\right )}{5 c^{3/2} \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2)^3/Sqrt[a + c*x^4],x]

[Out]

(Sqrt[(I*Sqrt[c])/Sqrt[a]]*Sqrt[c]*e^2*x*(5*d + e*x^2)*(a + c*x^4) - 3*Sqrt[a]*e
*(-5*c*d^2 + a*e^2)*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqr
t[a]]*x], -1] + ((-5*I)*c^(3/2)*d^3 - 15*Sqrt[a]*c*d^2*e + (5*I)*a*Sqrt[c]*d*e^2
+ 3*a^(3/2)*e^3)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[
a]]*x], -1])/(5*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^(3/2)*Sqrt[a + c*x^4])

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Maple [C]  time = 0.017, size = 388, normalized size = 1.2 \[{{d}^{3}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{e}^{3} \left ({\frac{{x}^{3}}{5\,c}\sqrt{c{x}^{4}+a}}-{{\frac{3\,i}{5}}{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){c}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ) +{3\,i{d}^{2}e\sqrt{a}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}{\frac{1}{\sqrt{c}}}}+3\,{e}^{2}d \left ( 1/3\,{\frac{x\sqrt{c{x}^{4}+a}}{c}}-1/3\,{\frac{a}{c\sqrt{c{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}},i \right ){\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)^3/(c*x^4+a)^(1/2),x)

[Out]

d^3/(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)+e^3*(1/5/
c*x^3*(c*x^4+a)^(1/2)-3/5*I/c^(3/2)*a^(3/2)/(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)-EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)))+3*I*d
^2*e*a^(1/2)/(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)/c^(1/2)*(EllipticF(x*(I/a^(1/2)*c^(1/2))^
(1/2),I)-EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I))+3*e^2*d*(1/3*x*(c*x^4+a)^(1/2
)/c-1/3/c*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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{3}}{\sqrt{c x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^3/sqrt(c*x^4 + a),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^3/sqrt(c*x^4 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}{\sqrt{c x^{4} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^3/sqrt(c*x^4 + a),x, algorithm="fricas")

[Out]

integral((e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3)/sqrt(c*x^4 + a), x)

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Sympy [A]  time = 7.89018, size = 173, normalized size = 0.53 \[ \frac{d^{3} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} + \frac{3 d^{2} e x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} + \frac{3 d e^{2} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} + \frac{e^{3} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d)**3/(c*x**4+a)**(1/2),x)

[Out]

d**3*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)
*gamma(5/4)) + 3*d**2*e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x**4*exp_pol
ar(I*pi)/a)/(4*sqrt(a)*gamma(7/4)) + 3*d*e**2*x**5*gamma(5/4)*hyper((1/2, 5/4),
(9/4,), c*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(9/4)) + e**3*x**7*gamma(7/4)*
hyper((1/2, 7/4), (11/4,), c*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{3}}{\sqrt{c x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^3/sqrt(c*x^4 + a),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^3/sqrt(c*x^4 + a), x)