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

Optimal. Leaf size=411 \[ \frac{f \left (2 c d e-\left (e-\sqrt{e^2-4 d f}\right ) (c d-a f)\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} \sqrt{e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{f \left (2 c d e-\left (\sqrt{e^2-4 d f}+e\right ) (c d-a f)\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} \sqrt{e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{-a f+c d-c e x}{\sqrt{a+c x^2} \left ((c d-a f)^2+a c e^2\right )} \]

[Out]

-((c*d - a*f - c*e*x)/((a*c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2])) + (f*(2*c*d*e
- (c*d - a*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]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e
^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - (f*(2*c*d*e - (c*d - a*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]*Sqrt[e^2 - 4*d
*f]*(a*c*e^2 + (c*d - a*f)^2)*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 = 1.96015, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{f \left (2 c d e-\left (e-\sqrt{e^2-4 d f}\right ) (c d-a f)\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} \sqrt{e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{f \left (2 c d e-\left (\sqrt{e^2-4 d f}+e\right ) (c d-a f)\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} \sqrt{e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{-a f+c d-c e x}{\sqrt{a+c x^2} \left ((c d-a f)^2+a c e^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((c*d - a*f - c*e*x)/((a*c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2])) + (f*(2*c*d*e
- (c*d - a*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]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e
^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - (f*(2*c*d*e - (c*d - a*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]*Sqrt[e^2 - 4*d
*f]*(a*c*e^2 + (c*d - a*f)^2)*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 [A]  time = 158.234, size = 400, normalized size = 0.97 \[ \frac{\sqrt{2} f \left (2 c d e + \left (e - \sqrt{- 4 d f + e^{2}}\right ) \left (a f - c d\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (2 a f - c x \left (e - \sqrt{- 4 d f + e^{2}}\right )\right )}{2 \sqrt{a + c x^{2}} \sqrt{2 a f^{2} - 2 c d f + c e^{2} - c e \sqrt{- 4 d f + e^{2}}}} \right )}}{2 \sqrt{- 4 d f + e^{2}} \left (a c e^{2} + \left (a f - c d\right )^{2}\right ) \sqrt{2 a f^{2} - 2 c d f + c e^{2} - c e \sqrt{- 4 d f + e^{2}}}} - \frac{\sqrt{2} f \left (2 c d e + \left (e + \sqrt{- 4 d f + e^{2}}\right ) \left (a f - c d\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (2 a f - c x \left (e + \sqrt{- 4 d f + e^{2}}\right )\right )}{2 \sqrt{a + c x^{2}} \sqrt{2 a f^{2} - 2 c d f + c e^{2} + c e \sqrt{- 4 d f + e^{2}}}} \right )}}{2 \sqrt{- 4 d f + e^{2}} \left (a c e^{2} + \left (a f - c d\right )^{2}\right ) \sqrt{2 a f^{2} - 2 c d f + c e^{2} + c e \sqrt{- 4 d f + e^{2}}}} + \frac{2 a f - 2 c d + 2 c e x}{2 \sqrt{a + c x^{2}} \left (a c e^{2} + \left (a f - c d\right )^{2}\right )} \]

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]

sqrt(2)*f*(2*c*d*e + (e - sqrt(-4*d*f + e**2))*(a*f - c*d))*atanh(sqrt(2)*(2*a*f
- c*x*(e - sqrt(-4*d*f + e**2)))/(2*sqrt(a + c*x**2)*sqrt(2*a*f**2 - 2*c*d*f +
c*e**2 - c*e*sqrt(-4*d*f + e**2))))/(2*sqrt(-4*d*f + e**2)*(a*c*e**2 + (a*f - c*
d)**2)*sqrt(2*a*f**2 - 2*c*d*f + c*e**2 - c*e*sqrt(-4*d*f + e**2))) - sqrt(2)*f*
(2*c*d*e + (e + sqrt(-4*d*f + e**2))*(a*f - c*d))*atanh(sqrt(2)*(2*a*f - c*x*(e
+ sqrt(-4*d*f + e**2)))/(2*sqrt(a + c*x**2)*sqrt(2*a*f**2 - 2*c*d*f + c*e**2 + c
*e*sqrt(-4*d*f + e**2))))/(2*sqrt(-4*d*f + e**2)*(a*c*e**2 + (a*f - c*d)**2)*sqr
t(2*a*f**2 - 2*c*d*f + c*e**2 + c*e*sqrt(-4*d*f + e**2))) + (2*a*f - 2*c*d + 2*c
*e*x)/(2*sqrt(a + c*x**2)*(a*c*e**2 + (a*f - c*d)**2))

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Mathematica [A]  time = 1.75164, size = 617, normalized size = 1.5 \[ \frac{-\frac{\sqrt{2} f \left (a f \left (\sqrt{e^2-4 d f}-e\right )-c d \left (\sqrt{e^2-4 d f}+e\right )\right ) \log \left (\sqrt{a+c x^2} \sqrt{4 a f^2-2 c e \sqrt{e^2-4 d f}-4 c d f+2 c e^2}+2 a f+c x \left (\sqrt{e^2-4 d f}-e\right )\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{2} f \left (a f \left (\sqrt{e^2-4 d f}+e\right )+c d \left (e-\sqrt{e^2-4 d f}\right )\right ) \log \left (-\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}-2 a f+c x \sqrt{e^2-4 d f}+c e x\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\sqrt{2} f \left (a f \left (\sqrt{e^2-4 d f}-e\right )-c d \left (\sqrt{e^2-4 d f}+e\right )\right ) \log \left (\sqrt{e^2-4 d f}-e-2 f x\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\sqrt{2} f \left (a f \left (\sqrt{e^2-4 d f}+e\right )+c d \left (e-\sqrt{e^2-4 d f}\right )\right ) \log \left (\sqrt{e^2-4 d f}+e+2 f x\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{2 (a f-c d+c e x)}{\sqrt{a+c x^2}}}{2 \left (a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 3000, normalized size = 7.3 \[ \text{output too large to display} \]

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]

-1/(-4*d*f+e^2)^(1/2)*f*e/(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/((x-1/
2*(-e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2*(-e+(-4*d*f+e
^2)^(1/2))/f)+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)+4*e
*c^2/(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/(4*a*c-4*c^2/f*d+c^2/f^2*e^
2-1/f^2*(-4*d*f+e^2)*c^2)/((x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e-(-4*d*f+e^
2)^(1/2))/f*(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f
^2-2*c*d*f+e^2*c)/f^2)^(1/2)*x-2/(-4*d*f+e^2)^(1/2)*c^2/(-(-4*d*f+e^2)^(1/2)*c*e
+2*a*f^2-2*c*d*f+e^2*c)/(4*a*c-4*c^2/f*d+c^2/f^2*e^2-1/f^2*(-4*d*f+e^2)*c^2)/((x
-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2*(-e+(-4*d*
f+e^2)^(1/2))/f)+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*
x*e^2+1/(-4*d*f+e^2)^(1/2)*f*e/(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)*2
^(1/2)/((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*ln(((-(-4*d*f
+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2*(-e
+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^
2*c)/f^2)^(1/2)*(4*(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)^2*c-4*c*(e-(-4*d*f+e^2)^(1/
2))/f*(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)+2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d
*f+e^2*c)/f^2)^(1/2))/(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f))+f/(-(-4*d*f+e^2)^(1/2)*
c*e+2*a*f^2-2*c*d*f+e^2*c)/((x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e-(-4*d*f+e
^2)^(1/2))/f*(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*
f^2-2*c*d*f+e^2*c)/f^2)^(1/2)-2*(-4*d*f+e^2)^(1/2)*c^2/(-(-4*d*f+e^2)^(1/2)*c*e+
2*a*f^2-2*c*d*f+e^2*c)/(4*a*c-4*c^2/f*d+c^2/f^2*e^2-1/f^2*(-4*d*f+e^2)*c^2)/((x-
1/2*(-e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2*(-e+(-4*d*f
+e^2)^(1/2))/f)+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*x
-f/(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)*2^(1/2)/((-(-4*d*f+e^2)^(1/2)
*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*ln(((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*
d*f+e^2*c)/f^2-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)+1/2*
2^(1/2)*((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*(4*(x-1/2*(-
e+(-4*d*f+e^2)^(1/2))/f)^2*c-4*c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2*(-e+(-4*d*f+e^2
)^(1/2))/f)+2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2))/(x-1/2
*(-e+(-4*d*f+e^2)^(1/2))/f))+1/(-4*d*f+e^2)^(1/2)*f*e/((-4*d*f+e^2)^(1/2)*c*e+2*
a*f^2-2*c*d*f+e^2*c)/((x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e+(-4*d*f+e^2)^(1/
2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d
*f+e^2*c)/f^2)^(1/2)+4*e*c^2/((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/(4*a
*c-4*c^2/f*d+c^2/f^2*e^2-1/f^2*(-4*d*f+e^2)*c^2)/((x+1/2*(e+(-4*d*f+e^2)^(1/2))/
f)^2*c-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*((-4*d*f+
e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*x+2/(-4*d*f+e^2)^(1/2)*c^2/((-4
*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/(4*a*c-4*c^2/f*d+c^2/f^2*e^2-1/f^2*(-
4*d*f+e^2)*c^2)/((x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e+(-4*d*f+e^2)^(1/2))/f
*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^
2*c)/f^2)^(1/2)*x*e^2-1/(-4*d*f+e^2)^(1/2)*f*e/((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2
*c*d*f+e^2*c)*2^(1/2)/(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)
*ln((((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2-c*(e+(-4*d*f+e^2)^(1/2))
/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2
-2*c*d*f+e^2*c)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-4*c*(e+(-4*d*
f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f
^2-2*c*d*f+e^2*c)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f))+f/((-4*d*f+e^2)^
(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/((x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e+(-4*
d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*((-4*d*f+e^2)^(1/2)*c*e+2
*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)+2*(-4*d*f+e^2)^(1/2)*c^2/((-4*d*f+e^2)^(1/2)*c*
e+2*a*f^2-2*c*d*f+e^2*c)/(4*a*c-4*c^2/f*d+c^2/f^2*e^2-1/f^2*(-4*d*f+e^2)*c^2)/((
x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f
+e^2)^(1/2))/f)+1/2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*x-
f/((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)*2^(1/2)/(((-4*d*f+e^2)^(1/2)*c*
e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*ln((((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+
e^2*c)/f^2-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/
2)*(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d
*f+e^2)^(1/2))/f)^2*c-4*c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))
/f)+2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2))/(x+1/2*(e+(-4*d
*f+e^2)^(1/2))/f))

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

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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, algorithm="giac")

[Out]

Exception raised: TypeError