3.62 \(\int \frac{\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx\)

Optimal. Leaf size=604 \[ \frac{a^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^2}-\frac{\left (a^2 f^2 \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )+4 a c d^2 f^2+c^2 d^2 \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (a^2 f^2 \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )+4 a c d^2 f^2+c^2 d^2 \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{a e \sqrt{a+c x^2}}{d^2}+\frac{\sqrt{a+c x^2} (2 a e-c d x)}{2 d^2}+\frac{\sqrt{c} (2 c d-3 a f) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 d f}-\frac{\left (a+c x^2\right )^{3/2}}{d x}+\frac{3 c x \sqrt{a+c x^2}}{2 d}+\frac{3 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 d} \]

[Out]

-((a*e*Sqrt[a + c*x^2])/d^2) + (3*c*x*Sqrt[a + c*x^2])/(2*d) + ((2*a*e - c*d*x)*
Sqrt[a + c*x^2])/(2*d^2) - (a + c*x^2)^(3/2)/(d*x) + (3*a*Sqrt[c]*ArcTanh[(Sqrt[
c]*x)/Sqrt[a + c*x^2]])/(2*d) + (Sqrt[c]*(2*c*d - 3*a*f)*ArcTanh[(Sqrt[c]*x)/Sqr
t[a + c*x^2]])/(2*d*f) - ((4*a*c*d^2*f^2 + c^2*d^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4
*d*f]) + a^2*f^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sq
rt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]
)]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2
*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((4*a*c*d^2*f^2 + a^2*f^2*(e^2 - 2*d*f - e*Sqrt[
e^2 - 4*d*f]) + c^2*d^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*
(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 -
4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(
e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) + (a^(3/2)*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a
]])/d^2

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Rubi [A]  time = 5.93298, antiderivative size = 604, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 14, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.518 \[ \frac{a^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^2}-\frac{\left (a^2 f^2 \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )+4 a c d^2 f^2+c^2 d^2 \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (a^2 f^2 \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )+4 a c d^2 f^2+c^2 d^2 \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{a e \sqrt{a+c x^2}}{d^2}+\frac{\sqrt{a+c x^2} (2 a e-c d x)}{2 d^2}+\frac{\sqrt{c} (2 c d-3 a f) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 d f}-\frac{\left (a+c x^2\right )^{3/2}}{d x}+\frac{3 c x \sqrt{a+c x^2}}{2 d}+\frac{3 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

-((a*e*Sqrt[a + c*x^2])/d^2) + (3*c*x*Sqrt[a + c*x^2])/(2*d) + ((2*a*e - c*d*x)*
Sqrt[a + c*x^2])/(2*d^2) - (a + c*x^2)^(3/2)/(d*x) + (3*a*Sqrt[c]*ArcTanh[(Sqrt[
c]*x)/Sqrt[a + c*x^2]])/(2*d) + (Sqrt[c]*(2*c*d - 3*a*f)*ArcTanh[(Sqrt[c]*x)/Sqr
t[a + c*x^2]])/(2*d*f) - ((4*a*c*d^2*f^2 + c^2*d^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4
*d*f]) + a^2*f^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sq
rt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]
)]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2
*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((4*a*c*d^2*f^2 + a^2*f^2*(e^2 - 2*d*f - e*Sqrt[
e^2 - 4*d*f]) + c^2*d^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*
(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 -
4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(
e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) + (a^(3/2)*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a
]])/d^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.49819, size = 880, normalized size = 1.46 \[ -\frac{e \log (x) a^{3/2}}{d^2}+\frac{e \log \left (a+\sqrt{c x^2+a} \sqrt{a}\right ) a^{3/2}}{d^2}-\frac{\sqrt{c x^2+a} a}{d x}-\frac{\left (-4 a c f^2 d^2+c^2 \left (-e^2+\sqrt{e^2-4 d f} e+2 d f\right ) d^2+a^2 f^2 \left (-e^2-\sqrt{e^2-4 d f} e+2 d f\right )\right ) \log \left (-e-2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}-\frac{\left (4 a c f^2 d^2+c^2 \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) d^2+a^2 f^2 \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )\right ) \log \left (e+2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}}+\frac{c^{3/2} \log \left (c x+\sqrt{c} \sqrt{c x^2+a}\right )}{f}+\frac{\left (-4 a c f^2 d^2+c^2 \left (-e^2+\sqrt{e^2-4 d f} e+2 d f\right ) d^2+a^2 f^2 \left (-e^2-\sqrt{e^2-4 d f} e+2 d f\right )\right ) \log \left (2 a \sqrt{e^2-4 d f} f+c \left (e^2-\sqrt{e^2-4 d f} e-4 d f\right ) x+\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}\right )}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}+\frac{\left (4 a c f^2 d^2+c^2 \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) d^2+a^2 f^2 \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )\right ) \log \left (2 a \sqrt{e^2-4 d f} f-c \left (e^2+\sqrt{e^2-4 d f} e-4 d f\right ) x+\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}\right )}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}} \]

Antiderivative was successfully verified.

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

[Out]

-((a*Sqrt[a + c*x^2])/(d*x)) - (a^(3/2)*e*Log[x])/d^2 - ((-4*a*c*d^2*f^2 + a^2*f
^2*(-e^2 + 2*d*f - e*Sqrt[e^2 - 4*d*f]) + c^2*d^2*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4
*d*f]))*Log[-e + Sqrt[e^2 - 4*d*f] - 2*f*x])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sq
rt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((4*a*c*d^2*f^2 + a^2*f^2
*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + c^2*d^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f
]))*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*
a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) + (a^(3/2)*e*Log[a + Sqrt[a]*Sqr
t[a + c*x^2]])/d^2 + (c^(3/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/f + ((-4*a*c*d
^2*f^2 + a^2*f^2*(-e^2 + 2*d*f - e*Sqrt[e^2 - 4*d*f]) + c^2*d^2*(-e^2 + 2*d*f +
e*Sqrt[e^2 - 4*d*f]))*Log[2*a*f*Sqrt[e^2 - 4*d*f] + c*(e^2 - 4*d*f - e*Sqrt[e^2
- 4*d*f])*x + Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e
^2 - 4*d*f])]*Sqrt[a + c*x^2]])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 +
c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((4*a*c*d^2*f^2 + a^2*f^2*(e^2 - 2*d*f
- e*Sqrt[e^2 - 4*d*f]) + c^2*d^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*Log[2*a*f
*Sqrt[e^2 - 4*d*f] - c*(e^2 - 4*d*f + e*Sqrt[e^2 - 4*d*f])*x + Sqrt[2]*Sqrt[e^2
- 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2]])
/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4
*d*f])])

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Maple [B]  time = 0.029, size = 9912, normalized size = 16.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + a)^(3/2)/((f*x^2 + e*x + d)*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + c*x**2)**(3/2)/(x**2*(d + e*x + f*x**2)), x)

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GIAC/XCAS [A]  time = 0.664234, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done