3.1137 \(\int \frac{c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=181 \[ -\frac{24 \sqrt{b} \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{7/2} e^4 \sqrt [4]{a+b x^2}}+\frac{12 (2 b c-a d)}{5 a^3 e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}-\frac{2 (2 b c-a d)}{5 a^2 e^3 \sqrt{e x} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{5/4}} \]

[Out]

(-2*c)/(5*a*e*(e*x)^(5/2)*(a + b*x^2)^(5/4)) - (2*(2*b*c - a*d))/(5*a^2*e^3*Sqrt
[e*x]*(a + b*x^2)^(5/4)) + (12*(2*b*c - a*d))/(5*a^3*e^3*Sqrt[e*x]*(a + b*x^2)^(
1/4)) - (24*Sqrt[b]*(2*b*c - a*d)*(1 + a/(b*x^2))^(1/4)*Sqrt[e*x]*EllipticE[ArcC
ot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*a^(7/2)*e^4*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.30389, antiderivative size = 181, 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 \[ -\frac{24 \sqrt{b} \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{7/2} e^4 \sqrt [4]{a+b x^2}}+\frac{12 (2 b c-a d)}{5 a^3 e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}-\frac{2 (2 b c-a d)}{5 a^2 e^3 \sqrt{e x} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(9/4)),x]

[Out]

(-2*c)/(5*a*e*(e*x)^(5/2)*(a + b*x^2)^(5/4)) - (2*(2*b*c - a*d))/(5*a^2*e^3*Sqrt
[e*x]*(a + b*x^2)^(5/4)) + (12*(2*b*c - a*d))/(5*a^3*e^3*Sqrt[e*x]*(a + b*x^2)^(
1/4)) - (24*Sqrt[b]*(2*b*c - a*d)*(1 + a/(b*x^2))^(1/4)*Sqrt[e*x]*EllipticE[ArcC
ot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*a^(7/2)*e^4*(a + b*x^2)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 c}{5 a e \left (e x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{4 \left (\frac{a d}{2} - b c\right )}{5 a^{2} e^{3} \sqrt{e x} \left (a + b x^{2}\right )^{\frac{5}{4}}} - \frac{24 \left (\frac{a d}{2} - b c\right )}{5 a^{3} e^{3} \sqrt{e x} \sqrt [4]{a + b x^{2}}} + \frac{24 \sqrt{e x} \left (\frac{a d}{2} - b c\right ) \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{5 a^{3} e^{4} \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(7/2)/(b*x**2+a)**(9/4),x)

[Out]

-2*c/(5*a*e*(e*x)**(5/2)*(a + b*x**2)**(5/4)) + 4*(a*d/2 - b*c)/(5*a**2*e**3*sqr
t(e*x)*(a + b*x**2)**(5/4)) - 24*(a*d/2 - b*c)/(5*a**3*e**3*sqrt(e*x)*(a + b*x**
2)**(1/4)) + 24*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))/(5*a**3*e**4*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 0.244187, size = 140, normalized size = 0.77 \[ \frac{x \left (-2 a^3 \left (c+5 d x^2\right )-4 a^2 b x^2 \left (9 d x^2-5 c\right )-24 a b^2 x^4 \left (d x^2-3 c\right )+16 b x^4 \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} (a d-2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+48 b^3 c x^6\right )}{5 a^4 (e x)^{7/2} \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(9/4)),x]

[Out]

(x*(48*b^3*c*x^6 - 24*a*b^2*x^4*(-3*c + d*x^2) - 2*a^3*(c + 5*d*x^2) - 4*a^2*b*x
^2*(-5*c + 9*d*x^2) + 16*b*(-2*b*c + a*d)*x^4*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*
Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^2)/a)]))/(5*a^4*(e*x)^(7/2)*(a + b*x^2)^
(5/4))

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Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{7}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(7/2)/(b*x^2+a)^(9/4),x)

[Out]

int((d*x^2+c)/(e*x)^(7/2)/(b*x^2+a)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{9}{4}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(7/2)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{2} + c}{{\left (b^{2} e^{3} x^{7} + 2 \, a b e^{3} x^{5} + a^{2} e^{3} x^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(7/2)),x, algorithm="fricas")

[Out]

integral((d*x^2 + c)/((b^2*e^3*x^7 + 2*a*b*e^3*x^5 + a^2*e^3*x^3)*(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)**(7/2)/(b*x**2+a)**(9/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{9}{4}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(7/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(7/2)), x)