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

Optimal. Leaf size=501 \[ \frac{3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2} f}-\frac{3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^3 f}-\frac{d \sqrt{a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c f^3}-\frac{b d \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} f^3}+\frac{b (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2 f}-\frac{d \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 f^{7/2}}+\frac{d \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 f^{7/2}}-\frac{d \left (a+b x+c x^2\right )^{3/2}}{3 f^2}-\frac{\left (a+b x+c x^2\right )^{5/2}}{5 c f} \]

[Out]

(-3*b*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^3*f) - (d*(8*c^2*d
+ b^2*f + 8*a*c*f + 2*b*c*f*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^3) - (d*(a + b*x +
c*x^2)^(3/2))/(3*f^2) + (b*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(16*c^2*f) - (a
+ b*x + c*x^2)^(5/2)/(5*c*f) + (3*b*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt
[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(7/2)*f) - (b*d*(24*c^2*d - b^2*f + 12*a*c*f
)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*f^3) - (d*
(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*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*f^(7/2)) + (d*(c*d + b*Sqrt[d]*Sqrt[f] + a*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*f^(7/2))

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Rubi [A]  time = 2.85712, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2} f}-\frac{3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^3 f}-\frac{d \sqrt{a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c f^3}-\frac{b d \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} f^3}+\frac{b (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2 f}-\frac{d \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 f^{7/2}}+\frac{d \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 f^{7/2}}-\frac{d \left (a+b x+c x^2\right )^{3/2}}{3 f^2}-\frac{\left (a+b x+c x^2\right )^{5/2}}{5 c f} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^3*f) - (d*(8*c^2*d
+ b^2*f + 8*a*c*f + 2*b*c*f*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^3) - (d*(a + b*x +
c*x^2)^(3/2))/(3*f^2) + (b*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(16*c^2*f) - (a
+ b*x + c*x^2)^(5/2)/(5*c*f) + (3*b*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt
[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(7/2)*f) - (b*d*(24*c^2*d - b^2*f + 12*a*c*f
)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*f^3) - (d*
(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*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*f^(7/2)) + (d*(c*d + b*Sqrt[d]*Sqrt[f] + a*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*f^(7/2))

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

[Out]

Timed out

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Mathematica [A]  time = 1.57451, size = 536, normalized size = 1.07 \[ \frac{-\frac{2 \sqrt{f} \sqrt{a+x (b+c x)} \left (24 c^2 f \left (16 a^2 f+7 a b f x+b^2 \left (10 d+f x^2\right )\right )-30 b^2 c f^2 (10 a+b x)+16 c^3 f \left (160 a d+48 a f x^2+70 b d x+33 b f x^3\right )+45 b^4 f^2+128 c^4 \left (15 d^2+5 d f x^2+3 f^2 x^4\right )\right )}{c^3}+\frac{15 b \sqrt{f} \left (16 c^2 f \left (3 a^2 f+b^2 d\right )-24 a b^2 c f^2-192 a c^3 d f+3 b^4 f^2-384 c^4 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{7/2}}-1920 d \log \left (\sqrt{d} \sqrt{f}+f x\right ) \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}+1920 d \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2} \log \left (\sqrt{d} \left (2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}+2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x\right )\right )-1920 d \log \left (\sqrt{d} \sqrt{f}-f x\right ) \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}+1920 d \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2} \log \left (\sqrt{d} \left (2 \left (\sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+a \sqrt{f}+c \sqrt{d} x\right )+b \left (\sqrt{d}+\sqrt{f} x\right )\right )\right )}{3840 f^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[f]*Sqrt[a + x*(b + c*x)]*(45*b^4*f^2 - 30*b^2*c*f^2*(10*a + b*x) + 16*
c^3*f*(160*a*d + 70*b*d*x + 48*a*f*x^2 + 33*b*f*x^3) + 128*c^4*(15*d^2 + 5*d*f*x
^2 + 3*f^2*x^4) + 24*c^2*f*(16*a^2*f + 7*a*b*f*x + b^2*(10*d + f*x^2))))/c^3 - 1
920*d*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*Log[Sqrt[d]*Sqrt[f] - f*x] - 1920*d*
(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*Log[Sqrt[d]*Sqrt[f] + f*x] + (15*b*Sqrt[f]
*(-384*c^4*d^2 - 192*a*c^3*d*f + 3*b^4*f^2 - 24*a*b^2*c*f^2 + 16*c^2*f*(b^2*d +
3*a^2*f))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(7/2) + 1920*d*(c*
d - b*Sqrt[d]*Sqrt[f] + a*f)^(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)])] + 1920*d*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(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)]))])/(3840*f^(7/2))

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Maple [B]  time = 0.04, size = 4884, normalized size = 9.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/f^2*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)*a-1/2/f^3*d^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)*c-1/2/f
^2*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)*a-1/2/f^3*d^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)*c+1/16/f*b^2/c^2*(c*x^2+b
*x+a)^(3/2)-3/128/f*b^4/c^3*(c*x^2+b*x+a)^(1/2)+3/256/f*b^5/c^(7/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/8/f*b/c*(c*x^2+b*x+a)^(3/2)*x+3/16/f*b/c*(c*x^
2+b*x+a)^(1/2)*x*a-1/4/f^3*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)*x*c*(d*f)^(1/2)-3/4/f^3*d*ln((1/2
*(2*c*(d*f)^(1/2)+b*f)/f+c*(x-(d*f)^(1/2)/f))/c^(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))*c^(1/2)
*(d*f)^(1/2)*a-3/8/f^2*d/c^(1/2)*ln((1/2*(2*c*(d*f)^(1/2)+b*f)/f+c*(x-(d*f)^(1/2
)/f))/c^(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))*a*b-3/16/f^3*d*ln((1/2*(2*c*(d*f)^(1/2)+b*f)/f+
c*(x-(d*f)^(1/2)/f))/c^(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))/c^(1/2)*b^2*(d*f)^(1/2)+1/f^3*d^
2/((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))*a*c+3/16/f^3*d*ln((1/2/f*(-2*c*(d*f)^(1/2)+b*f)+c*(x+(d*
f)^(1/2)/f))/c^(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))/c^(1/2)*b^2*(d*f)^(1/2)+1/f^3*d^2/
(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))*a*c+1/4/f^3*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)*x*c*
(d*f)^(1/2)+3/4/f^3*d*ln((1/2/f*(-2*c*(d*f)^(1/2)+b*f)+c*(x+(d*f)^(1/2)/f))/c^(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))*c^(1/2)*(d*f)^(1/2)*a-3/8/f^2*d/c^(1/2)*ln((1/2/f*(
-2*c*(d*f)^(1/2)+b*f)+c*(x+(d*f)^(1/2)/f))/c^(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))*a*b+
3/32/f*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a+3/16/f*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))*a^2-3/32/f*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))*a-1/8/f^2*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)*x*b-5/8/f^3*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)*b*(d*f)^(1/2)
-1/16/f^2*d/c*((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+1/32/f^2*d/c^(3/2)*ln((1/2*(2*c*(d*f)^(1/2)+
b*f)/f+c*(x-(d*f)^(1/2)/f))/c^(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))*b^3-3/4/f^3*d^2*ln((1/2*(
2*c*(d*f)^(1/2)+b*f)/f+c*(x-(d*f)^(1/2)/f))/c^(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))*c^(1/2)*b
-1/2/f^4*d^2*ln((1/2*(2*c*(d*f)^(1/2)+b*f)/f+c*(x-(d*f)^(1/2)/f))/c^(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))*c^(3/2)*(d*f)^(1/2)+1/2/f^3*d^2/((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*l
n((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))*b^2+1/2/f^2*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))*a^2+1/2/f^4*d^3/((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))*c^2-1/8/f^2*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)*x*b+5/8/f^3*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)*b*(d*f)^(1/2)-1/16/f^2*
d/c*((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+1/32/f^2*d/c^(3/2)*ln((1/2/f*(-2*c*(d*f)^(1/2)+b
*f)+c*(x+(d*f)^(1/2)/f))/c^(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))*b^3-3/4/f^3*d^2*ln((1/
2/f*(-2*c*(d*f)^(1/2)+b*f)+c*(x+(d*f)^(1/2)/f))/c^(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))
*c^(1/2)*b+1/2/f^4*d^2*ln((1/2/f*(-2*c*(d*f)^(1/2)+b*f)+c*(x+(d*f)^(1/2)/f))/c^(
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))*c^(3/2)*(d*f)^(1/2)-1/6/f^2*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))^(3
/2)-1/6/f^2*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)^(3/2)-1/f^4*d^2/(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*l
n((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))*
b*(d*f)^(1/2)*c-1/f^3*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))*b*(d*f)^(1/2)*a+1/
f^3*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))*b*(d*f)^(1/2)*a+1/f^4*d^2/((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))*b*(d
*f)^(1/2)*c-3/64/f*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x+1/2/f^3*d^2/(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))*b^2+1/2/f^2*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))*a^2+1/2/f
^4*d^3/(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))*c^2-1/5*(c*x^2+b*x+a)^(5/2)/c/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(-(c*x^2 + b*x + a)^(3/2)*x^3/(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(-(c*x^2 + b*x + a)^(3/2)*x^3/(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**3*(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(-(c*x^2 + b*x + a)^(3/2)*x^3/(f*x^2 - d),x, algorithm="giac")

[Out]

Exception raised: TypeError