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

Optimal. Leaf size=668 \[ -\frac{a^{3/2} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^3}+\frac{\left (a^2 f \left (e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (\sqrt{e^2-4 d f}+e\right )+c^2 d^3 \left (e-\sqrt{e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\left (a^2 f \left (-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (e-\sqrt{e^2-4 d f}\right )+c^2 d^3 \left (\sqrt{e^2-4 d f}+e\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{a \sqrt{a+c x^2} \left (e^2-d f\right )}{d^3}+\frac{e \left (a+c x^2\right )^{3/2}}{d^2 x}-\frac{3 c e x \sqrt{a+c x^2}}{2 d^2}-\frac{\sqrt{a+c x^2} \left (2 \left (a \left (e^2-d f\right )+c d^2\right )-c d e x\right )}{2 d^3}-\frac{\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac{3 c \sqrt{a+c x^2}}{2 d}-\frac{3 \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 d} \]

[Out]

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

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Rubi [A]  time = 6.97277, antiderivative size = 668, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 15, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556 \[ -\frac{a^{3/2} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^3}+\frac{\left (a^2 f \left (e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (\sqrt{e^2-4 d f}+e\right )+c^2 d^3 \left (e-\sqrt{e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\left (a^2 f \left (-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (e-\sqrt{e^2-4 d f}\right )+c^2 d^3 \left (\sqrt{e^2-4 d f}+e\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{a \sqrt{a+c x^2} \left (e^2-d f\right )}{d^3}+\frac{e \left (a+c x^2\right )^{3/2}}{d^2 x}-\frac{3 c e x \sqrt{a+c x^2}}{2 d^2}-\frac{\sqrt{a+c x^2} \left (2 \left (a \left (e^2-d f\right )+c d^2\right )-c d e x\right )}{2 d^3}-\frac{\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac{3 c \sqrt{a+c x^2}}{2 d}-\frac{3 \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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

[Out]

Timed out

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Mathematica [A]  time = 4.53441, size = 949, normalized size = 1.42 \[ \frac{\frac{a d \sqrt{c x^2+a} (2 e x-d)}{x^2}-\sqrt{a} \left (-3 c d^2+2 a f d-2 a e^2\right ) \log (x)-\frac{\sqrt{2} \left (c^2 \left (e-\sqrt{e^2-4 d f}\right ) d^3+2 a c f \left (e+\sqrt{e^2-4 d f}\right ) d^2+a^2 f \left (e^3+\sqrt{e^2-4 d f} e^2-3 d f e-d f \sqrt{e^2-4 d f}\right )\right ) \log \left (-e-2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}-\frac{\sqrt{2} \left (-c^2 \left (e+\sqrt{e^2-4 d f}\right ) d^3+2 a c f \left (\sqrt{e^2-4 d f}-e\right ) d^2-a^2 f \left (e^3-\sqrt{e^2-4 d f} e^2-3 d f e+d f \sqrt{e^2-4 d f}\right )\right ) \log \left (e+2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}}+\sqrt{a} \left (-3 c d^2+2 a f d-2 a e^2\right ) \log \left (a+\sqrt{c x^2+a} \sqrt{a}\right )+\frac{\sqrt{2} \left (c^2 \left (e-\sqrt{e^2-4 d f}\right ) d^3+2 a c f \left (e+\sqrt{e^2-4 d f}\right ) d^2+a^2 f \left (e^3+\sqrt{e^2-4 d f} e^2-3 d f e-d f \sqrt{e^2-4 d f}\right )\right ) \log \left (2 a \sqrt{e^2-4 d f} f+c \left (e^2-\sqrt{e^2-4 d f} e-4 d f\right ) x+\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}+\frac{\sqrt{2} \left (-c^2 \left (e+\sqrt{e^2-4 d f}\right ) d^3+2 a c f \left (\sqrt{e^2-4 d f}-e\right ) d^2-a^2 f \left (e^3-\sqrt{e^2-4 d f} e^2-3 d f e+d f \sqrt{e^2-4 d f}\right )\right ) \log \left (2 a \sqrt{e^2-4 d f} f-c \left (e^2+\sqrt{e^2-4 d f} e-4 d f\right ) x+\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}}}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.03, size = 10298, normalized size = 15.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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^3),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.61857, 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^3),x, algorithm="giac")

[Out]

Done