3.79 \(\int \frac{x \sqrt{a+b x+c x^2}}{d-f x^2} \, dx\)

Optimal. Leaf size=282 \[ -\frac{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 f^{3/2}}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 f^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{f}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} f} \]

[Out]

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

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Rubi [A]  time = 0.734932, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 f^{3/2}}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 f^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{f}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} f} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 138.653, size = 257, normalized size = 0.91 \[ - \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 \sqrt{c} f} - \frac{\sqrt{a + b x + c x^{2}}}{f} - \frac{\sqrt{a f - b \sqrt{d} \sqrt{f} + c d} \operatorname{atanh}{\left (\frac{- 2 a \sqrt{f} + b \sqrt{d} + x \left (- b \sqrt{f} + 2 c \sqrt{d}\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a f - b \sqrt{d} \sqrt{f} + c d}} \right )}}{2 f^{\frac{3}{2}}} - \frac{\sqrt{a f + b \sqrt{d} \sqrt{f} + c d} \operatorname{atanh}{\left (\frac{- 2 a \sqrt{f} - b \sqrt{d} + x \left (- b \sqrt{f} - 2 c \sqrt{d}\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a f + b \sqrt{d} \sqrt{f} + c d}} \right )}}{2 f^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(2*sqrt(c)*f) - sqrt(a
+ b*x + c*x**2)/f - sqrt(a*f - b*sqrt(d)*sqrt(f) + c*d)*atanh((-2*a*sqrt(f) + b*
sqrt(d) + x*(-b*sqrt(f) + 2*c*sqrt(d)))/(2*sqrt(a + b*x + c*x**2)*sqrt(a*f - b*s
qrt(d)*sqrt(f) + c*d)))/(2*f**(3/2)) - sqrt(a*f + b*sqrt(d)*sqrt(f) + c*d)*atanh
((-2*a*sqrt(f) - b*sqrt(d) + x*(-b*sqrt(f) - 2*c*sqrt(d)))/(2*sqrt(a + b*x + c*x
**2)*sqrt(a*f + b*sqrt(d)*sqrt(f) + c*d)))/(2*f**(3/2))

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Mathematica [A]  time = 2.17196, size = 448, normalized size = 1.59 \[ -\frac{\frac{\log \left (\sqrt{d} \sqrt{f}-f x\right ) \left (a f^{3/2}+b \sqrt{d} f+c d \sqrt{f}\right )}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}+\frac{\log \left (\sqrt{d} \sqrt{f}+f x\right ) \left (a f^{3/2}-b \sqrt{d} f+c d \sqrt{f}\right )}{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}-\frac{\left (a f^{3/2}-b \sqrt{d} f+c d \sqrt{f}\right ) \log \left (\sqrt{d} \left (2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}+2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x\right )\right )}{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}-\frac{\left (a f^{3/2}+b \sqrt{d} f+c d \sqrt{f}\right ) \log \left (\sqrt{d} \left (2 \left (\sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+a \sqrt{f}+c \sqrt{d} x\right )+b \left (\sqrt{d}+\sqrt{f} x\right )\right )\right )}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}+2 f \sqrt{a+x (b+c x)}+\frac{b f \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}}{2 f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 1667, normalized size = 5.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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)-1/2/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)-1/4/f*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^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)+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+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))*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)+1/2/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
)-1/4/f*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^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)+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+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))*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(-sqrt(c*x^2 + b*x + a)*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(-sqrt(c*x^2 + b*x + a)*x/(f*x^2 - d),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x \sqrt{a + b x + c x^{2}}}{- d + f x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: TypeError