3.1113 \(\int \frac{c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=103 \[ \frac{2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt{b} e^2 \sqrt [4]{a+b x^2}}-\frac{2 c}{a e \sqrt{e x} \sqrt [4]{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.179151, antiderivative size = 103, 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 \[ \frac{2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt{b} e^2 \sqrt [4]{a+b x^2}}-\frac{2 c}{a e \sqrt{e x} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/4)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(3/2)/(b*x**2+a)**(5/4),x)

[Out]

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

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Mathematica [C]  time = 0.120509, size = 93, normalized size = 0.9 \[ \frac{x \left (-4 x^2 \sqrt [4]{\frac{b x^2}{a}+1} (a d-2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-6 \left (a \left (c-d x^2\right )+2 b c x^2\right )\right )}{3 a^2 (e x)^{3/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/4)),x]

[Out]

(x*(-6*(2*b*c*x^2 + a*(c - d*x^2)) - 4*(-2*b*c + a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*
Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^2)/a)]))/(3*a^2*(e*x)^(3/2)*(a + b*x^2)^
(1/4))

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Maple [F]  time = 0.094, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(3/2)/(b*x^2+a)^(5/4),x)

[Out]

int((d*x^2+c)/(e*x)^(3/2)/(b*x^2+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(3/2)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{2} + c}{{\left (b e x^{3} + a e x\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(3/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)/(e*x)**(3/2)/(b*x**2+a)**(5/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(3/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(3/2)), x)