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

Optimal. Leaf size=139 \[ -\frac{(2 c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{c^2 \sqrt{e} \sqrt{c d-b e}}+\frac{(5 c d-2 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c^2 \sqrt{e}}+\frac{x \sqrt{d+e x^2}}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.551695, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{(2 c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{c^2 \sqrt{e} \sqrt{c d-b e}}+\frac{(5 c d-2 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c^2 \sqrt{e}}+\frac{x \sqrt{d+e x^2}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 102.221, size = 124, normalized size = 0.89 \[ \frac{x \sqrt{d + e x^{2}}}{2 c} + \frac{\left (b e - 2 c d\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} x \sqrt{b e - 2 c d}}{\sqrt{d + e x^{2}} \sqrt{b e - c d}} \right )}}{c^{2} \sqrt{e} \sqrt{b e - c d}} - \frac{\left (2 b e - 5 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{2 c^{2} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(d + e*x**2)/(2*c) + (b*e - 2*c*d)**(3/2)*atanh(sqrt(e)*x*sqrt(b*e - 2*c*d
)/(sqrt(d + e*x**2)*sqrt(b*e - c*d)))/(c**2*sqrt(e)*sqrt(b*e - c*d)) - (2*b*e -
5*c*d)*atanh(sqrt(e)*x/sqrt(d + e*x**2))/(2*c**2*sqrt(e))

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Mathematica [A]  time = 0.392921, size = 134, normalized size = 0.96 \[ -\frac{\frac{(2 b e-5 c d) \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{\sqrt{e}}-\frac{2 (b e-2 c d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{b e-2 c d}}{\sqrt{d+e x^2} \sqrt{b e-c d}}\right )}{\sqrt{e} \sqrt{b e-c d}}-c x \sqrt{d+e x^2}}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.069, size = 7043, normalized size = 50.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.783323, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*sqrt(e*x^2 + d)*c*sqrt(e)*x - (2*c*d - b*e)*sqrt(e)*sqrt((2*c*d - b*e)/(
c*d*e - b*e^2))*log((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (17*c^2*d^2*e^2 - 24*
b*c*d*e^3 + 8*b^2*e^4)*x^4 + 2*(7*c^2*d^3*e - 11*b*c*d^2*e^2 + 4*b^2*d*e^3)*x^2
+ 4*((3*c^2*d^2*e^2 - 5*b*c*d*e^3 + 2*b^2*e^4)*x^3 + (c^2*d^3*e - 2*b*c*d^2*e^2
+ b^2*d*e^3)*x)*sqrt(e*x^2 + d)*sqrt((2*c*d - b*e)/(c*d*e - b*e^2)))/(c^2*e^2*x^
4 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d*e - b*c*e^2)*x^2)) - (5*c*d - 2*b*e
)*log(2*sqrt(e*x^2 + d)*e*x - (2*e*x^2 + d)*sqrt(e)))/(c^2*sqrt(e)), 1/4*(2*sqrt
(e*x^2 + d)*c*sqrt(-e)*x - (2*c*d - b*e)*sqrt(-e)*sqrt((2*c*d - b*e)/(c*d*e - b*
e^2))*log((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (17*c^2*d^2*e^2 - 24*b*c*d*e^3
+ 8*b^2*e^4)*x^4 + 2*(7*c^2*d^3*e - 11*b*c*d^2*e^2 + 4*b^2*d*e^3)*x^2 + 4*((3*c^
2*d^2*e^2 - 5*b*c*d*e^3 + 2*b^2*e^4)*x^3 + (c^2*d^3*e - 2*b*c*d^2*e^2 + b^2*d*e^
3)*x)*sqrt(e*x^2 + d)*sqrt((2*c*d - b*e)/(c*d*e - b*e^2)))/(c^2*e^2*x^4 + c^2*d^
2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d*e - b*c*e^2)*x^2)) + 2*(5*c*d - 2*b*e)*arctan
(sqrt(-e)*x/sqrt(e*x^2 + d)))/(c^2*sqrt(-e)), 1/4*(2*sqrt(e*x^2 + d)*c*sqrt(e)*x
- 2*(2*c*d - b*e)*sqrt(e)*sqrt(-(2*c*d - b*e)/(c*d*e - b*e^2))*arctan(1/2*(c*d^
2 - b*d*e + (3*c*d*e - 2*b*e^2)*x^2)/((c*d*e - b*e^2)*sqrt(e*x^2 + d)*x*sqrt(-(2
*c*d - b*e)/(c*d*e - b*e^2)))) - (5*c*d - 2*b*e)*log(2*sqrt(e*x^2 + d)*e*x - (2*
e*x^2 + d)*sqrt(e)))/(c^2*sqrt(e)), 1/2*(sqrt(e*x^2 + d)*c*sqrt(-e)*x - (2*c*d -
b*e)*sqrt(-e)*sqrt(-(2*c*d - b*e)/(c*d*e - b*e^2))*arctan(1/2*(c*d^2 - b*d*e +
(3*c*d*e - 2*b*e^2)*x^2)/((c*d*e - b*e^2)*sqrt(e*x^2 + d)*x*sqrt(-(2*c*d - b*e)/
(c*d*e - b*e^2)))) + (5*c*d - 2*b*e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)))/(c^2*sq
rt(-e))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x**2)**(3/2)/(b*e - c*d + c*e*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out