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

Optimal. Leaf size=591 \[ -\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt{e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (\sqrt{e^2-4 d f}+e\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )}{f^4}+\frac{\sqrt{a+c x^2} \left (a f^2+c \left (e^2-d f\right )\right )}{f^3}-\frac{c e x \sqrt{a+c x^2}}{2 f^2}-\frac{a \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 f^2}+\frac{\left (a+c x^2\right )^{3/2}}{3 f} \]

[Out]

((a*f^2 + c*(e^2 - d*f))*Sqrt[a + c*x^2])/f^3 - (c*e*x*Sqrt[a + c*x^2])/(2*f^2)
+ (a + c*x^2)^(3/2)/(3*f) - (a*Sqrt[c]*e*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(
2*f^2) - (Sqrt[c]*e*(a*f^2 + c*(e^2 - 2*d*f))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2
]])/f^4 - ((2*c*d*e*f*(2*a*f^2 + c*(e^2 - 2*d*f)) - (e - Sqrt[e^2 - 4*d*f])*(a^2
*f^4 + 2*a*c*f^2*(e^2 - d*f) + c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*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^4*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*c*d*e*f*(2*a*f^2 + c*(e^2 - 2*d*f))
- (e + Sqrt[e^2 - 4*d*f])*(a^2*f^4 + 2*a*c*f^2*(e^2 - d*f) + c^2*(e^4 - 3*d*e^2
*f + d^2*f^2)))*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^4*Sqr
t[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 = 7.02553, antiderivative size = 591, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ -\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt{e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (\sqrt{e^2-4 d f}+e\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )}{f^4}+\frac{\sqrt{a+c x^2} \left (a f^2+c \left (e^2-d f\right )\right )}{f^3}-\frac{c e x \sqrt{a+c x^2}}{2 f^2}-\frac{a \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 f^2}+\frac{\left (a+c x^2\right )^{3/2}}{3 f} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 4.08487, size = 1176, normalized size = 1.99 \[ \frac{f \sqrt{c x^2+a} \left (8 a f^2+c \left (6 e^2-3 f x e+2 f \left (f x^2-3 d\right )\right )\right )+\frac{3 \sqrt{2} \left (a^2 \left (\sqrt{e^2-4 d f}-e\right ) f^4-2 a c \left (e^3-\sqrt{e^2-4 d f} e^2-3 d f e+d f \sqrt{e^2-4 d f}\right ) f^2+c^2 \left (-e^5+\sqrt{e^2-4 d f} e^4+5 d f e^3-3 d f \sqrt{e^2-4 d f} e^2-5 d^2 f^2 e+d^2 f^2 \sqrt{e^2-4 d f}\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{3 \sqrt{2} \left (a^2 \left (e+\sqrt{e^2-4 d f}\right ) f^4+2 a c \left (e^3+\sqrt{e^2-4 d f} e^2-3 d f e-d f \sqrt{e^2-4 d f}\right ) f^2+c^2 \left (e^5+\sqrt{e^2-4 d f} e^4-5 d f e^3-3 d f \sqrt{e^2-4 d f} e^2+5 d^2 f^2 e+d^2 f^2 \sqrt{e^2-4 d f}\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 )}}-3 \sqrt{c} e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right ) \log \left (c x+\sqrt{c} \sqrt{c x^2+a}\right )-\frac{3 \sqrt{2} \left (a^2 \left (\sqrt{e^2-4 d f}-e\right ) f^4-2 a c \left (e^3-\sqrt{e^2-4 d f} e^2-3 d f e+d f \sqrt{e^2-4 d f}\right ) f^2+c^2 \left (-e^5+\sqrt{e^2-4 d f} e^4+5 d f e^3-3 d f \sqrt{e^2-4 d f} e^2-5 d^2 f^2 e+d^2 f^2 \sqrt{e^2-4 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{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{3 \sqrt{2} \left (a^2 \left (e+\sqrt{e^2-4 d f}\right ) f^4+2 a c \left (e^3+\sqrt{e^2-4 d f} e^2-3 d f e-d f \sqrt{e^2-4 d f}\right ) f^2+c^2 \left (e^5+\sqrt{e^2-4 d f} e^4-5 d f e^3-3 d f \sqrt{e^2-4 d f} e^2+5 d^2 f^2 e+d^2 f^2 \sqrt{e^2-4 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{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 )}}}{6 f^4} \]

Antiderivative was successfully verified.

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

[Out]

(f*Sqrt[a + c*x^2]*(8*a*f^2 + c*(6*e^2 - 3*e*f*x + 2*f*(-3*d + f*x^2))) + (3*Sqr
t[2]*(a^2*f^4*(-e + Sqrt[e^2 - 4*d*f]) - 2*a*c*f^2*(e^3 - 3*d*e*f - e^2*Sqrt[e^2
- 4*d*f] + d*f*Sqrt[e^2 - 4*d*f]) + c^2*(-e^5 + 5*d*e^3*f - 5*d^2*e*f^2 + e^4*S
qrt[e^2 - 4*d*f] - 3*d*e^2*f*Sqrt[e^2 - 4*d*f] + d^2*f^2*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])]) + (3*Sqrt[2]*(a^2*f^4*(e + Sqrt[e^2 - 4*d*f]) + 2*a*
c*f^2*(e^3 - 3*d*e*f + e^2*Sqrt[e^2 - 4*d*f] - d*f*Sqrt[e^2 - 4*d*f]) + c^2*(e^5
- 5*d*e^3*f + 5*d^2*e*f^2 + e^4*Sqrt[e^2 - 4*d*f] - 3*d*e^2*f*Sqrt[e^2 - 4*d*f]
+ d^2*f^2*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])]) - 3*Sqrt[c]*e*(3*a*
f^2 + 2*c*(e^2 - 2*d*f))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - (3*Sqrt[2]*(a^2*f^
4*(-e + Sqrt[e^2 - 4*d*f]) - 2*a*c*f^2*(e^3 - 3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] +
d*f*Sqrt[e^2 - 4*d*f]) + c^2*(-e^5 + 5*d*e^3*f - 5*d^2*e*f^2 + e^4*Sqrt[e^2 - 4*
d*f] - 3*d*e^2*f*Sqrt[e^2 - 4*d*f] + d^2*f^2*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])]) - (3*Sqrt[2
]*(a^2*f^4*(e + Sqrt[e^2 - 4*d*f]) + 2*a*c*f^2*(e^3 - 3*d*e*f + e^2*Sqrt[e^2 - 4
*d*f] - d*f*Sqrt[e^2 - 4*d*f]) + c^2*(e^5 - 5*d*e^3*f + 5*d^2*e*f^2 + e^4*Sqrt[e
^2 - 4*d*f] - 3*d*e^2*f*Sqrt[e^2 - 4*d*f] + d^2*f^2*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])]))/(6*
f^4)

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Maple [B]  time = 0.024, size = 14709, normalized size = 24.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^2+a)^(3/2)/(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((c*x^2 + a)^(3/2)*x/(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((c*x^2 + a)^(3/2)*x/(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. \[ \int \frac{x \left (a + c x^{2}\right )^{\frac{3}{2}}}{d + e x + f x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.825786, 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)*x/(f*x^2 + e*x + d),x, algorithm="giac")

[Out]

Done