3.1124 \(\int \frac{c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{7/4}} \, dx\)

Optimal. Leaf size=144 \[ \frac{4 \sqrt{b} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{5/2} e^4 \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt{e x} (2 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.328439, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{4 \sqrt{b} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{5/2} e^4 \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt{e x} (2 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(7/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 36.2747, size = 131, normalized size = 0.91 \[ - \frac{2 c}{3 a e \left (e x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{4 \sqrt{e x} \left (\frac{a d}{2} - b c\right )}{3 a^{2} e^{3} \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{8 \sqrt{b} \left (e x\right )^{\frac{3}{2}} \left (\frac{a d}{2} - b c\right ) \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{3 a^{\frac{5}{2}} e^{4} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(5/2)/(b*x**2+a)**(7/4),x)

[Out]

-2*c/(3*a*e*(e*x)**(3/2)*(a + b*x**2)**(3/4)) + 4*sqrt(e*x)*(a*d/2 - b*c)/(3*a**
2*e**3*(a + b*x**2)**(3/4)) - 8*sqrt(b)*(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)/(3*a**(5/2)*e**4*(a + b*x*
*2)**(3/4))

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Mathematica [C]  time = 0.138028, size = 91, normalized size = 0.63 \[ \frac{x \left (4 x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} (a d-2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-2 a c+2 a d x^2-4 b c x^2\right )}{3 a^2 (e x)^{5/2} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(7/4)),x]

[Out]

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

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(5/2)/(b*x^2+a)^(7/4),x)

[Out]

int((d*x^2+c)/(e*x)^(5/2)/(b*x^2+a)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(5/2)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(5/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(5/2)),x, algorithm="fricas")

[Out]

integral((d*x^2 + c)/((b*e^2*x^4 + a*e^2*x^2)*(b*x^2 + a)^(3/4)*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)**(5/2)/(b*x**2+a)**(7/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(5/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(5/2)), x)