3.1101 \(\int \frac{c+d x^2}{\sqrt{e x} \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=102 \[ \frac{d \sqrt{e x} \sqrt [4]{a+b x^2}}{b e}-\frac{(e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} e^2 \left (a+b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.263758, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{d \sqrt{e x} \sqrt [4]{a+b x^2}}{b e}-\frac{(e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} e^2 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.3887, size = 90, normalized size = 0.88 \[ \frac{d \sqrt{e x} \sqrt [4]{a + b x^{2}}}{b e} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (\frac{a d}{2} - b c\right ) \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{\sqrt{a} \sqrt{b} e^{2} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*sqrt(e*x)*(a + b*x**2)**(1/4)/(b*e) + 2*(e*x)**(3/2)*(a*d/2 - b*c)*(a/(b*x**2)
+ 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x))/2, 2)/(sqrt(a)*sqrt(b)*e**2*(a
+ b*x**2)**(3/4))

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Mathematica [C]  time = 0.0939675, size = 77, normalized size = 0.75 \[ \frac{x \left (\frac{b x^2}{a}+1\right )^{3/4} (2 b c-a d) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+d x \left (a+b x^2\right )}{b \sqrt{e x} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(d*x*(a + b*x^2) + (2*b*c - a*d)*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4,
3/4, 5/4, -((b*x^2)/a)])/(b*Sqrt[e*x]*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c){\frac{1}{\sqrt{ex}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.5639, size = 78, normalized size = 0.76 \[ - \frac{c{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac{3}{4}} \sqrt{e} x} + \frac{d x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} \sqrt{e} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*hyper((1/2, 3/4), (3/2,), a*exp_polar(I*pi)/(b*x**2))/(b**(3/4)*sqrt(e)*x) +
d*x**(5/2)*gamma(5/4)*hyper((3/4, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**
(3/4)*sqrt(e)*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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