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

Optimal. Leaf size=114 \[ \frac{d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{x (d+e x)}{2 a \sqrt{a+c x^4}} \]

[Out]

(x*(d + e*x))/(2*a*Sqrt[a + c*x^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])/(4*a
^(5/4)*c^(1/4)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.110956, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{x (d+e x)}{2 a \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d + e*x))/(2*a*Sqrt[a + c*x^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])/(4*a
^(5/4)*c^(1/4)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 12.695, size = 100, normalized size = 0.88 \[ \frac{x \left (d + e x\right )}{2 a \sqrt{a + c x^{4}}} + \frac{d \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{5}{4}} \sqrt [4]{c} \sqrt{a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(d + e*x)/(2*a*sqrt(a + c*x**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)/(4*a
**(5/4)*c**(1/4)*sqrt(a + c*x**4))

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Mathematica [C]  time = 0.182713, size = 90, normalized size = 0.79 \[ \frac{x (d+e x)-\frac{i d \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}}{2 a \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.006, size = 115, normalized size = 1. \[ d \left ({\frac{x}{2\,a}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{c}} \right ) c}}}}+{\frac{1}{2\,a}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ) +{\frac{e{x}^{2}}{2\,a}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^4+a)^(3/2),x)

[Out]

d*(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))+1/2*e*x^2/a/(c*x^4+a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 24.2276, size = 61, normalized size = 0.54 \[ \frac{d x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{5}{4}\right )} + \frac{e x^{2}}{2 a^{\frac{3}{2}} \sqrt{1 + \frac{c x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), c*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*g
amma(5/4)) + e*x**2/(2*a**(3/2)*sqrt(1 + c*x**4/a))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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