3.86 \(\int \frac{x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx\)
Optimal. Leaf size=349 \[ -\frac{\sqrt{a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c f^2}-\frac{b \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} f^2}-\frac{\left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 f^{5/2}}+\frac{\left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 f^{5/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 f} \]
[Out]
-((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^2) - (a
+ b*x + c*x^2)^(3/2)/(3*f) - (b*(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^2) - ((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^(5/2)) + ((
c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqr
t[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^(5/2))
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Rubi [A] time = 1.34543, antiderivative size = 349, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269
\[ -\frac{\sqrt{a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c f^2}-\frac{b \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} f^2}-\frac{\left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 f^{5/2}}+\frac{\left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 f^{5/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 f} \]
Antiderivative was successfully verified.
[In] Int[(x*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x]
[Out]
-((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^2) - (a
+ b*x + c*x^2)^(3/2)/(3*f) - (b*(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^2) - ((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^(5/2)) + ((
c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqr
t[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^(5/2))
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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*(c*x**2+b*x+a)**(3/2)/(-f*x**2+d),x)
[Out]
Timed out
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Mathematica [A] time = 0.969773, size = 416, normalized size = 1.19 \[ \frac{-\frac{2 \sqrt{f} \sqrt{a+x (b+c x)} \left (2 c f (16 a+7 b x)+3 b^2 f+8 c^2 \left (3 d+f x^2\right )\right )}{c}+\frac{3 b \sqrt{f} \left (-12 a c f+b^2 f-24 c^2 d\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{3/2}}-24 \log \left (\sqrt{d} \sqrt{f}+f x\right ) \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}+24 \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2} \log \left (\sqrt{d} \left (2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}+2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x\right )\right )-24 \log \left (\sqrt{d} \sqrt{f}-f x\right ) \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}+24 \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2} \log \left (\sqrt{d} \left (2 \left (\sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+a \sqrt{f}+c \sqrt{d} x\right )+b \left (\sqrt{d}+\sqrt{f} x\right )\right )\right )}{48 f^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x]
[Out]
((-2*Sqrt[f]*Sqrt[a + x*(b + c*x)]*(3*b^2*f + 2*c*f*(16*a + 7*b*x) + 8*c^2*(3*d
+ f*x^2)))/c - 24*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*Log[Sqrt[d]*Sqrt[f] - f*
x] - 24*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*Log[Sqrt[d]*Sqrt[f] + f*x] + (3*b*
Sqrt[f]*(-24*c^2*d + b^2*f - 12*a*c*f)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b +
c*x)]])/c^(3/2) + 24*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*Log[Sqrt[d]*(-(b*Sqr
t[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)])] + 24*(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*Sq
rt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)]))])/(48*f^(5/2))
_______________________________________________________________________________________
Maple [B] time = 0.022, size = 4567, normalized size = 13.1 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)
[Out]
1/32/f/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^3-1/2/f^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*d+1/2/f/(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^2-1/16/f/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^2-1/8/f*((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^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*(d*f)^(1/2)-1/2/f^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*d+1/2/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))*a^2-5/8/f^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*(d*
f)^(1/2)-1/16/f/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/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^3-1/8/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)*x*b-1/6/f*((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*((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)-3/16/f^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^
2*(d*f)^(1/2)-3/4/f^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)*d*b-1/2/f^3*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)*d+1/2/f^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^2*d+1/2/f^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*d^2+1/4/f^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*(d*f)^(1/2)+3/4/f^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)*(d*f)^(1/2)*a-3/8/f/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+3/16/f^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^2*(d*f)^(1/2)-3/4/f^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)*d*b+1/2/f
^3*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)*d+1/2/f^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*d+1/2/f^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*d^2+1/f^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*d+1/f^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)*a-1/f^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*(d*f)^(1/2)*a+1/f^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))*
a*c*d-1/2/f*((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*((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-3/8/f/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-1/4/f^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*(d*f)^(1/2)-3/4/f^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)*(d*f)^(1/2)*a-1/f^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))*b*(d*f)^(1/2)*c*d+1/f^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))*b*(d*f)^(1/2)*c*d
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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/(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/(f*x^2 - d),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{a x \sqrt{a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac{b x^{2} \sqrt{a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac{c x^{3} \sqrt{a + b x + c x^{2}}}{- d + f x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c*x**2+b*x+a)**(3/2)/(-f*x**2+d),x)
[Out]
-Integral(a*x*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x) - Integral(b*x**2*sqrt(a
+ b*x + c*x**2)/(-d + f*x**2), x) - Integral(c*x**3*sqrt(a + b*x + c*x**2)/(-d +
f*x**2), x)
_______________________________________________________________________________________
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/(f*x^2 - d),x, algorithm="giac")
[Out]
Exception raised: TypeError