3.52 \(\int \frac{x^2 \sqrt{a+c x^2}}{d+e x+f x^2} \, dx\)

Optimal. Leaf size=452 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (a f^2+2 c \left (e^2-d f\right )\right )}{2 \sqrt{c} f^3}-\frac{\left (e \left (e-\sqrt{e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (e \left (\sqrt{e^2-4 d f}+e\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{a+c x^2} (2 e-f x)}{2 f^2} \]

[Out]

-((2*e - f*x)*Sqrt[a + c*x^2])/(2*f^2) + ((a*f^2 + 2*c*(e^2 - d*f))*ArcTanh[(Sqr
t[c]*x)/Sqrt[a + c*x^2]])/(2*Sqrt[c]*f^3) - ((e*(e - Sqrt[e^2 - 4*d*f])*(a*f^2 +
c*(e^2 - 2*d*f)) - 2*d*f*(a*f^2 + c*(e^2 - 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]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f
- e*Sqrt[e^2 - 4*d*f])]) + ((e*(e + Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - 2*d*f))
- 2*d*f*(a*f^2 + c*(e^2 - 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]*f^3*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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Rubi [A]  time = 4.09631, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (a f^2+2 c \left (e^2-d f\right )\right )}{2 \sqrt{c} f^3}-\frac{\left (e \left (e-\sqrt{e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (e \left (\sqrt{e^2-4 d f}+e\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{a+c x^2} (2 e-f x)}{2 f^2} \]

Antiderivative was successfully verified.

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

[Out]

-((2*e - f*x)*Sqrt[a + c*x^2])/(2*f^2) + ((a*f^2 + 2*c*(e^2 - d*f))*ArcTanh[(Sqr
t[c]*x)/Sqrt[a + c*x^2]])/(2*Sqrt[c]*f^3) - ((e*(e - Sqrt[e^2 - 4*d*f])*(a*f^2 +
c*(e^2 - 2*d*f)) - 2*d*f*(a*f^2 + c*(e^2 - 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]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f
- e*Sqrt[e^2 - 4*d*f])]) + ((e*(e + Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - 2*d*f))
- 2*d*f*(a*f^2 + c*(e^2 - 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]*f^3*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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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(x**2*(c*x**2+a)**(1/2)/(f*x**2+e*x+d),x)

[Out]

Timed out

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Mathematica [A]  time = 3.32588, size = 882, normalized size = 1.95 \[ \frac{f \sqrt{c x^2+a} (f x-2 e)-\frac{\sqrt{2} \left (a \left (-e^2+\sqrt{e^2-4 d f} e+2 d f\right ) f^2+c \left (-e^4+\sqrt{e^2-4 d f} e^3+4 d f e^2-2 d f \sqrt{e^2-4 d f} e-2 d^2 f^2\right )\right ) \log \left (-e-2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}-\frac{\sqrt{2} \left (a \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) f^2+c \left (e^4+\sqrt{e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right )\right ) \log \left (e+2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}}+\frac{\left (a f^2+2 c \left (e^2-d f\right )\right ) \log \left (c x+\sqrt{c} \sqrt{c x^2+a}\right )}{\sqrt{c}}+\frac{\sqrt{2} \left (a \left (-e^2+\sqrt{e^2-4 d f} e+2 d f\right ) f^2+c \left (-e^4+\sqrt{e^2-4 d f} e^3+4 d f e^2-2 d f \sqrt{e^2-4 d f} e-2 d^2 f^2\right )\right ) \log \left (2 a \sqrt{e^2-4 d f} f+c \left (e^2-\sqrt{e^2-4 d f} e-4 d f\right ) x+\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}+\frac{\sqrt{2} \left (a \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) f^2+c \left (e^4+\sqrt{e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right )\right ) \log \left (2 a \sqrt{e^2-4 d f} f-c \left (e^2+\sqrt{e^2-4 d f} e-4 d f\right ) x+\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}}}{2 f^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.077, size = 7739, normalized size = 17.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^2+a)^(1/2)/(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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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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(sqrt(c*x^2 + a)*x^2/(f*x^2 + e*x + d),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(c*x**2+a)**(1/2)/(f*x**2+e*x+d),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.667244, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done