3.1112 \(\int \frac{\sqrt{e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=99 \[ \frac{d (e x)^{3/2}}{b e \sqrt [4]{a+b x^2}}-\frac{\sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-3 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} b^{3/2} \sqrt [4]{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.166838, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{d (e x)^{3/2}}{b e \sqrt [4]{a+b x^2}}-\frac{\sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-3 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} b^{3/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(5/4),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d \left (e x\right )^{\frac{3}{2}}}{b e \sqrt [4]{a + b x^{2}}} + \frac{\sqrt{e x} \left (\frac{3 a d}{2} - b c\right ) \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{b^{2} \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(1/2)*(d*x**2+c)/(b*x**2+a)**(5/4),x)

[Out]

d*(e*x)**(3/2)/(b*e*(a + b*x**2)**(1/4)) + sqrt(e*x)*(3*a*d/2 - b*c)*(a/(b*x**2)
+ 1)**(1/4)*Integral((a*x**2/b + 1)**(-5/4), (x, 1/x))/(b**2*(a + b*x**2)**(1/4
))

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Mathematica [C]  time = 0.0964026, size = 81, normalized size = 0.82 \[ \frac{2 x \sqrt{e x} \left (\sqrt [4]{\frac{b x^2}{a}+1} (3 a d-2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-3 a d+3 b c\right )}{3 a b \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(5/4),x]

[Out]

(2*x*Sqrt[e*x]*(3*b*c - 3*a*d + (-2*b*c + 3*a*d)*(1 + (b*x^2)/a)^(1/4)*Hypergeom
etric2F1[1/4, 3/4, 7/4, -((b*x^2)/a)]))/(3*a*b*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c)\sqrt{ex} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(5/4),x)

[Out]

int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(5/4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(5/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x^{2} + c\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(5/4),x, algorithm="fricas")

[Out]

integral((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(5/4), x)

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Sympy [A]  time = 32.6515, size = 94, normalized size = 0.95 \[ \frac{c \sqrt{e} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} \Gamma \left (\frac{7}{4}\right )} + \frac{d \sqrt{e} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**(1/2)*(d*x**2+c)/(b*x**2+a)**(5/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(5/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(5/4), x)