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

Optimal. Leaf size=454 \[ \frac{3 b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{5/2} d}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x+c x^2}}{a^2 d x \left (b^2-4 a c\right )}-\frac{2 f \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a d x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f^2 \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^{3/2} \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{f^2 \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^{3/2} \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]

[Out]

(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*x*Sqrt[a + b*x + c*x^2]) - (2*f*(b*
(b^2*f - c*(c*d + 3*a*f)) - c*(2*c^2*d - b^2*f + 2*a*c*f)*x))/((b^2 - 4*a*c)*d*(
b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) - ((3*b^2 - 8*a*c)*Sqrt[a + b*x
+ c*x^2])/(a^2*(b^2 - 4*a*c)*d*x) + (3*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a +
b*x + c*x^2])])/(2*a^(5/2)*d) + (f^2*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sq
rt[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^(3/2)*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (f^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^(3/2)*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))

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Rubi [A]  time = 2.75034, antiderivative size = 454, 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 \[ \frac{3 b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{5/2} d}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x+c x^2}}{a^2 d x \left (b^2-4 a c\right )}-\frac{2 f \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a d x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f^2 \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^{3/2} \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{f^2 \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^{3/2} \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(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*x*Sqrt[a + b*x + c*x^2]) - (2*f*(b*
(b^2*f - c*(c*d + 3*a*f)) - c*(2*c^2*d - b^2*f + 2*a*c*f)*x))/((b^2 - 4*a*c)*d*(
b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) - ((3*b^2 - 8*a*c)*Sqrt[a + b*x
+ c*x^2])/(a^2*(b^2 - 4*a*c)*d*x) + (3*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a +
b*x + c*x^2])])/(2*a^(5/2)*d) + (f^2*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sq
rt[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^(3/2)*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (f^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^(3/2)*(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**2/(c*x**2+b*x+a)**(3/2)/(-f*x**2+d),x)

[Out]

Timed out

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Mathematica [A]  time = 3.39167, size = 537, normalized size = 1.18 \[ \frac{1}{2} \left (\frac{3 b \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{a^{5/2} d}-\frac{3 b \log (x)}{a^{5/2} d}-\frac{4 \left (-b^3 c (5 a f+c d)-b^2 c^2 x (4 a f+c d)+a b c^2 (5 a f+3 c d)+2 a c^3 x (a f+c d)+b^5 f+b^4 c f x\right )}{a^2 \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)} \left (f \left (a^2 f-b^2 d\right )+2 a c d f+c^2 d^2\right )}-\frac{2 \sqrt{a+x (b+c x)}}{a^2 d x}-\frac{f^2 \log \left (\sqrt{d} \sqrt{f}-f x\right )}{d^{3/2} \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}}+\frac{f^2 \log \left (\sqrt{d} \sqrt{f}+f x\right )}{d^{3/2} \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}-\frac{f^2 \log \left (\sqrt{d} \left (2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}+2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x\right )\right )}{d^{3/2} \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{f^2 \log \left (\sqrt{d} \left (2 \left (\sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+a \sqrt{f}+c \sqrt{d} x\right )+b \left (\sqrt{d}+\sqrt{f} x\right )\right )\right )}{d^{3/2} \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-4*(b^5*f - b^3*c*(c*d + 5*a*f) + a*b*c^2*(3*c*d + 5*a*f) + b^4*c*f*x + 2*a*c^
3*(c*d + a*f)*x - b^2*c^2*(c*d + 4*a*f)*x))/(a^2*(-b^2 + 4*a*c)*(c^2*d^2 + 2*a*c
*d*f + f*(-(b^2*d) + a^2*f))*Sqrt[a + x*(b + c*x)]) - (2*Sqrt[a + x*(b + c*x)])/
(a^2*d*x) - (3*b*Log[x])/(a^(5/2)*d) - (f^2*Log[Sqrt[d]*Sqrt[f] - f*x])/(d^(3/2)
*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (f^2*Log[Sqrt[d]*Sqrt[f] + f*x])/(d^(3
/2)*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (3*b*Log[2*a + b*x + 2*Sqrt[a]*Sqrt
[a + x*(b + c*x)]])/(a^(5/2)*d) - (f^2*Log[Sqrt[d]*(-(b*Sqrt[d]) + 2*a*Sqrt[f] -
2*c*Sqrt[d]*x + b*Sqrt[f]*x + 2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*
(b + c*x)])])/(d^(3/2)*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (f^2*Log[Sqrt[d]
*(b*(Sqrt[d] + Sqrt[f]*x) + 2*(a*Sqrt[f] + c*Sqrt[d]*x + Sqrt[c*d + b*Sqrt[d]*Sq
rt[f] + a*f]*Sqrt[a + x*(b + c*x)]))])/(d^(3/2)*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^
(3/2)))/2

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Maple [B]  time = 0.025, size = 1656, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/d/a/x/(c*x^2+b*x+a)^(1/2)-3/2/d*b/a^2/(c*x^2+b*x+a)^(1/2)+3/d*b^2/a^2/(4*a*c-
b^2)/(c*x^2+b*x+a)^(1/2)*c*x+3/2/d*b^3/a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+3/2/d
*b/a^(5/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-8/d*c^2/a/(4*a*c-b^2)/(
c*x^2+b*x+a)^(1/2)*x-4/d*c/a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*b-1/2*f^2/d/(d*f)^(
1/2)/(b*(d*f)^(1/2)+f*a+c*d)/((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*f/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*c^2+f^2/d/(d*f)^(1/2)/(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/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*c+1/2*f^2/d/(d*f)^(1/2)/(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+1/2*f^2/d/(d*f)^(1/2)/(b*(d*f)^(1/2)+f*a+c*d)/((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*f^2/d/(d*f)^(1/2)/(-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*f/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*c^2-f^2/d/(d*f)^(1/
2)/(-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/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*c-1/2*f^2/d/(d*f)^(1/2)
/(-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^2-1/2*f^2/d/(d*f
)^(1/2)/(-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. \[ -\int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (f x^{2} - d\right )} x^{2}}\,{d x} \]

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^2),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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

[Out]

Exception raised: TypeError