3.107 \(\int \frac{1}{x \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx\)

Optimal. Leaf size=394 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{a^{3/2} d}-\frac{2 f \left (a \left (2 a c f+b^2 (-f)+2 c^2 d\right )+b c x (c d-a f)\right )}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{f^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{f^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]

[Out]

(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) - (2*f*(a*(2
*c^2*d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*d*(b^2*d*f - (c*d
+ a*f)^2)*Sqrt[a + b*x + c*x^2]) - ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x
+ c*x^2])]/(a^(3/2)*d) - (f^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d
] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])]
)/(2*d*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (f^(3/2)*ArcTanh[(b*Sqrt[d] + 2*
a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*
Sqrt[a + b*x + c*x^2])])/(2*d*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))

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Rubi [A]  time = 2.33411, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{a^{3/2} d}-\frac{2 f \left (a \left (2 a c f+b^2 (-f)+2 c^2 d\right )+b c x (c d-a f)\right )}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{f^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{f^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]

[Out]

(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) - (2*f*(a*(2
*c^2*d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*d*(b^2*d*f - (c*d
+ a*f)^2)*Sqrt[a + b*x + c*x^2]) - ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x
+ c*x^2])]/(a^(3/2)*d) - (f^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d
] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])]
)/(2*d*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (f^(3/2)*ArcTanh[(b*Sqrt[d] + 2*
a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*
Sqrt[a + b*x + c*x^2])])/(2*d*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))

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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(1/x/(c*x**2+b*x+a)**(3/2)/(-f*x**2+d),x)

[Out]

Timed out

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Mathematica [A]  time = 6.4807, size = 742, normalized size = 1.88 \[ -\frac{\log \left (2 \sqrt{a} \sqrt{a+b x+c x^2}+2 a+b x\right )}{a^{3/2} d}+\frac{\log (x)}{a^{3/2} d}+\frac{f \left (a \sqrt{d} f^{3/2}+b d f+c d^{3/2} \sqrt{f}\right ) \log \left (2 \sqrt{d} \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}+2 a \sqrt{d} \sqrt{f}+b \sqrt{d} \sqrt{f} x-b d-2 c d x\right )}{2 d^{3/2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d} \left (a^2 f^2+2 a c d f+b^2 (-d) f+c^2 d^2\right )}+\frac{f \left (a \sqrt{d} f^{3/2}-b d f+c d^{3/2} \sqrt{f}\right ) \log \left (2 \sqrt{d} \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+2 a \sqrt{d} \sqrt{f}+b \sqrt{d} \sqrt{f} x+b d+2 c d x\right )}{2 d^{3/2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \left (a^2 f^2+2 a c d f+b^2 (-d) f+c^2 d^2\right )}-\frac{f \log \left (\sqrt{d} \sqrt{f}-f x\right ) \left (a \sqrt{d} f^{3/2}-b d f+c d^{3/2} \sqrt{f}\right )}{2 d^{3/2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \left (a^2 f^2+2 a c d f+b^2 (-d) f+c^2 d^2\right )}-\frac{f \log \left (\sqrt{d} \sqrt{f}+f x\right ) \left (a \sqrt{d} f^{3/2}+b d f+c d^{3/2} \sqrt{f}\right )}{2 d^{3/2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d} \left (a^2 f^2+2 a c d f+b^2 (-d) f+c^2 d^2\right )}+\frac{2 \left (2 a^2 c^2 f-4 a b^2 c f-3 a b c^2 f x+2 a c^3 d+b^4 f+b^3 c f x-b^2 c^2 d-b c^3 d x\right )}{a \left (4 a c-b^2\right ) \sqrt{a+b x+c x^2} \left (a^2 f^2+2 a c d f+b^2 (-d) f+c^2 d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]

[Out]

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

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Maple [B]  time = 0.024, size = 1518, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)

[Out]

1/d/a/(c*x^2+b*x+a)^(1/2)-2/d*b/a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x-1/d*b^2/a/
(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/d/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^
(1/2))/x)-1/2/d/(b*(d*f)^(1/2)+f*a+c*d)*f/((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2
)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)+2/d/(b*(d*f)^(1/2)+f
*a+c*d)/(4*a*c-b^2)/((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2
)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*(d*f)^(1/2)*x*c^2+1/d/(b*(d*f)^(1/2)+f*a+c
*d)/(4*a*c-b^2)/((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)
+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*x*b*c*f+1/d/(b*(d*f)^(1/2)+f*a+c*d)/(4*a*c-b^2
)/((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2
)+f*a+c*d)/f)^(1/2)*(d*f)^(1/2)*b*c+1/2/d/(b*(d*f)^(1/2)+f*a+c*d)/(4*a*c-b^2)/((
x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*
a+c*d)/f)^(1/2)*b^2*f+1/2/d/(b*(d*f)^(1/2)+f*a+c*d)*f/((b*(d*f)^(1/2)+f*a+c*d)/f
)^(1/2)*ln((2*(b*(d*f)^(1/2)+f*a+c*d)/f+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f
)+2*((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*
f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2))/(x-(d*f)^(1/2)/f))-1/2/
d*f/(-b*(d*f)^(1/2)+f*a+c*d)/((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(
x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)-2/d/(-b*(d*f)^(1/2)+f*a+c*d
)/(4*a*c-b^2)/((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f
)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*(d*f)^(1/2)*x*c^2+1/d/(-b*(d*f)^(1/2)+f*a+
c*d)/(4*a*c-b^2)/((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2
)/f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*x*b*c*f-1/d/(-b*(d*f)^(1/2)+f*a+c*d)/(4
*a*c-b^2)/((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+1/
f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*(d*f)^(1/2)*b*c+1/2/d/(-b*(d*f)^(1/2)+f*a+c*d)
/(4*a*c-b^2)/((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)
+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*b^2*f+1/2/d*f/(-b*(d*f)^(1/2)+f*a+c*d)/(1/f
*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*ln((2/f*(-b*(d*f)^(1/2)+f*a+c*d)+1/f*(-2*c*(d*f
)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*((x+(d*f)^
(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+f*
a+c*d))^(1/2))/(x+(d*f)^(1/2)/f))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (f x^{2} - d\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((c*x^2 + b*x + a)^(3/2)*(f*x^2 - d)*x),x, algorithm="maxima")

[Out]

-integrate(1/((c*x^2 + b*x + a)^(3/2)*(f*x^2 - d)*x), 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(-1/((c*x^2 + b*x + a)^(3/2)*(f*x^2 - d)*x),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(1/x/(c*x**2+b*x+a)**(3/2)/(-f*x**2+d),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.294934, size = 1, normalized size = 0. \[ \mathit{sage}_{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((c*x^2 + b*x + a)^(3/2)*(f*x^2 - d)*x),x, algorithm="giac")

[Out]

sage2