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

Optimal. Leaf size=270 \[ \frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} c^{3/4} \sqrt{a+c x^4}}+\frac{e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt{a+c x^4}}+\frac{x (d+e x)^2}{2 a \sqrt{a+c x^4}}-\frac{e^2 x \sqrt{a+c x^4}}{2 a \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )} \]

[Out]

(x*(d + e*x)^2)/(2*a*Sqrt[a + c*x^4]) - (e^2*x*Sqrt[a + c*x^4])/(2*a*Sqrt[c]*(Sq
rt[a] + Sqrt[c]*x^2)) + (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])/(2*a^(3/4)*c^(3/
4)*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]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)],
1/2])/(4*a^(5/4)*c^(3/4)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.313343, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} c^{3/4} \sqrt{a+c x^4}}+\frac{e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt{a+c x^4}}+\frac{x (d+e x)^2}{2 a \sqrt{a+c x^4}}-\frac{e^2 x \sqrt{a+c x^4}}{2 a \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d + e*x)^2)/(2*a*Sqrt[a + c*x^4]) - (e^2*x*Sqrt[a + c*x^4])/(2*a*Sqrt[c]*(Sq
rt[a] + Sqrt[c]*x^2)) + (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])/(2*a^(3/4)*c^(3/
4)*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]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)],
1/2])/(4*a^(5/4)*c^(3/4)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 34.6986, size = 238, normalized size = 0.88 \[ \frac{x \left (d + e x\right )^{2}}{2 a \sqrt{a + c x^{4}}} - \frac{e^{2} x \sqrt{a + c x^{4}}}{2 a \sqrt{c} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} + \frac{e^{2} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} c^{\frac{3}{4}} \sqrt{a + c x^{4}}} - \frac{\sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{5}{4}} c^{\frac{3}{4}} \sqrt{a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.364497, size = 188, normalized size = 0.7 \[ \frac{i \left (\sqrt{\frac{c x^4}{a}+1} \left (\sqrt{a} e^2-i \sqrt{c} d^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{c} x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} (d+e x)^2-\sqrt{a} e^2 \sqrt{\frac{c x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{2 a^{3/2} \left (\frac{i \sqrt{c}}{\sqrt{a}}\right )^{3/2} \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.011, size = 239, normalized size = 0.9 \[{d}^{2} \left ({\frac{x}{2\,a}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{c}} \right ) c}}}}+{\frac{1}{2\,a}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ) +{e}^{2} \left ({\frac{{x}^{3}}{2\,a}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{c}} \right ) c}}}}-{{\frac{i}{2}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}{\frac{1}{\sqrt{c}}}} \right ) +{\frac{de{x}^{2}}{a}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^2*(1/2/a*x/((x^4+1/c*a)*c)^(1/2)+1/2/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))+e^2*(1/2/a*x^3/((x^4+1/c*a)*c)^(1/2)-1/2*I/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)-Elli
pticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)))+d*e*x^2/a/(c*x^4+a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**2/(a + c*x**4)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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