3.102 \(\int \frac{x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx\)

Optimal. Leaf size=466 \[ -\frac{2 d^2 \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{f^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac{2 d (b+2 c x)}{f^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 b \sqrt{a+b x+c x^2}}{c f \left (b^2-4 a c\right )}-\frac{2 x (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2} f}+\frac{d^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 f \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{d^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 f \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 2.90387, antiderivative size = 466, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ -\frac{2 d^2 \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{f^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac{2 d (b+2 c x)}{f^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 b \sqrt{a+b x+c x^2}}{c f \left (b^2-4 a c\right )}-\frac{2 x (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2} f}+\frac{d^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 f \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{d^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 f \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 5.48617, size = 480, normalized size = 1.03 \[ \frac{1}{2} \left (\frac{4 \left (-a^3 f (b-2 c x)-a^2 \left (b^2 f x+3 b c d-2 c^2 d x\right )+a b^2 d (b-4 c x)+b^4 d x\right )}{c \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)} \left (f \left (a^2 f-b^2 d\right )+2 a c d f+c^2 d^2\right )}-\frac{2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{3/2} f}-\frac{d^{3/2} \log \left (\sqrt{d} \sqrt{f}-f x\right )}{f \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}}+\frac{d^{3/2} \log \left (\sqrt{d} \sqrt{f}+f x\right )}{f \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}-\frac{d^{3/2} \log \left (\sqrt{d} \left (2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}+2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x\right )\right )}{f \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{d^{3/2} \log \left (\sqrt{d} \left (2 \left (\sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+a \sqrt{f}+c \sqrt{d} x\right )+b \left (\sqrt{d}+\sqrt{f} x\right )\right )\right )}{f \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]

[Out]

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

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Maple [B]  time = 0.025, size = 1648, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)

[Out]

-4/f^2*d/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x-2/f^2*d/(4*a*c-b^2)/(c*x^2+b*x+a)^(
1/2)*b+1/f*x/c/(c*x^2+b*x+a)^(1/2)-1/2/f*b/c^2/(c*x^2+b*x+a)^(1/2)-1/f*b^2/c/(4*
a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-1/2/f*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/f
/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/2/f*d^2/(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^2*d^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*c^2+1/f*d^2/(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+1/f^2*d^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*c+1/2/f*d^2/(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*d^2/(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*d^2/(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^2*
d^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*c^2-1/f*d^2/(
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+1/
f^2*d^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*c-1/2/f*d
^2/(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*d^2/(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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^4/((c*x^2 + b*x + a)^(3/2)*(f*x^2 - 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^4/((c*x^2 + b*x + a)^(3/2)*(f*x^2 - d)),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(x**4/(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(-x^4/((c*x^2 + b*x + a)^(3/2)*(f*x^2 - d)),x, algorithm="giac")

[Out]

Exception raised: TypeError