3.57 \(\int \frac{\left (a+b x^2\right ) \sqrt{c+d x^2}}{e+f x^2} \, dx\)

Optimal. Leaf size=128 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right ) (-2 a d f-b c f+2 b d e)}{2 \sqrt{d} f^2}+\frac{(b e-a f) \sqrt{d e-c f} \tanh ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{e} \sqrt{c+d x^2}}\right )}{\sqrt{e} f^2}+\frac{b x \sqrt{c+d x^2}}{2 f} \]

[Out]

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

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Rubi [A]  time = 0.406057, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right ) (-2 a d f-b c f+2 b d e)}{2 \sqrt{d} f^2}+\frac{(b e-a f) \sqrt{d e-c f} \tanh ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{e} \sqrt{c+d x^2}}\right )}{\sqrt{e} f^2}+\frac{b x \sqrt{c+d x^2}}{2 f} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.2829, size = 117, normalized size = 0.91 \[ \frac{b x \sqrt{c + d x^{2}}}{2 f} + \frac{\left (a f - b e\right ) \sqrt{c f - d e} \operatorname{atan}{\left (\frac{x \sqrt{c f - d e}}{\sqrt{e} \sqrt{c + d x^{2}}} \right )}}{\sqrt{e} f^{2}} + \frac{\left (2 a d f + b c f - 2 b d e\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 \sqrt{d} f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.41897, size = 124, normalized size = 0.97 \[ \frac{\frac{\log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right ) (2 a d f+b c f-2 b d e)}{\sqrt{d}}-\frac{2 (b e-a f) \sqrt{c f-d e} \tan ^{-1}\left (\frac{x \sqrt{c f-d e}}{\sqrt{e} \sqrt{c+d x^2}}\right )}{\sqrt{e}}+b f x \sqrt{c+d x^2}}{2 f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.056, size = 1942, normalized size = 15.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b*x*(d*x^2+c)^(1/2)/f+1/2*b/f*c/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-1/2/(-
e*f)^(1/2)*((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+(c*f-d*
e)/f)^(1/2)*a+1/2/(-e*f)^(1/2)/f*((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(
-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2)*b*e+1/2/f*d^(1/2)*ln((-d*(-e*f)^(1/2)/f+(x+(-e
*f)^(1/2)/f)*d)/d^(1/2)+((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/
2)/f)+(c*f-d*e)/f)^(1/2))*a-1/2/f^2*d^(1/2)*ln((-d*(-e*f)^(1/2)/f+(x+(-e*f)^(1/2
)/f)*d)/d^(1/2)+((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+(c
*f-d*e)/f)^(1/2))*b*e+1/2/(-e*f)^(1/2)/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f-2*d
*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+2*((c*f-d*e)/f)^(1/2)*((x+(-e*f)^(1/2)/f)^2*d
-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x+(-e*f)^(1/2)/f))*c
*a-1/2/(-e*f)^(1/2)/f/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f-2*d*(-e*f)^(1/2)/f*(
x+(-e*f)^(1/2)/f)+2*((c*f-d*e)/f)^(1/2)*((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)
/f*(x+(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x+(-e*f)^(1/2)/f))*c*b*e-1/2/(-e*f)^(
1/2)/f/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/
f)+2*((c*f-d*e)/f)^(1/2)*((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1
/2)/f)+(c*f-d*e)/f)^(1/2))/(x+(-e*f)^(1/2)/f))*d*e*a+1/2/(-e*f)^(1/2)/f^2/((c*f-
d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+2*((c*f-d*
e)/f)^(1/2)*((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+(c*f-d
*e)/f)^(1/2))/(x+(-e*f)^(1/2)/f))*d*e^2*b+1/2/(-e*f)^(1/2)*((x-(-e*f)^(1/2)/f)^2
*d+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2)*a-1/2/(-e*f)^(1/2)/f
*((x-(-e*f)^(1/2)/f)^2*d+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2
)*b*e+1/2/f*d^(1/2)*ln((d*(-e*f)^(1/2)/f+(x-(-e*f)^(1/2)/f)*d)/d^(1/2)+((x-(-e*f
)^(1/2)/f)^2*d+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))*a-1/2/f
^2*d^(1/2)*ln((d*(-e*f)^(1/2)/f+(x-(-e*f)^(1/2)/f)*d)/d^(1/2)+((x-(-e*f)^(1/2)/f
)^2*d+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))*b*e-1/2/(-e*f)^(
1/2)/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)
+2*((c*f-d*e)/f)^(1/2)*((x-(-e*f)^(1/2)/f)^2*d+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2
)/f)+(c*f-d*e)/f)^(1/2))/(x-(-e*f)^(1/2)/f))*c*a+1/2/(-e*f)^(1/2)/f/((c*f-d*e)/f
)^(1/2)*ln((2*(c*f-d*e)/f+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+2*((c*f-d*e)/f)^
(1/2)*((x-(-e*f)^(1/2)/f)^2*d+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)
^(1/2))/(x-(-e*f)^(1/2)/f))*c*b*e+1/2/(-e*f)^(1/2)/f/((c*f-d*e)/f)^(1/2)*ln((2*(
c*f-d*e)/f+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+2*((c*f-d*e)/f)^(1/2)*((x-(-e*f
)^(1/2)/f)^2*d+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x-(-e*
f)^(1/2)/f))*d*e*a-1/2/(-e*f)^(1/2)/f^2/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f+2*
d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+2*((c*f-d*e)/f)^(1/2)*((x-(-e*f)^(1/2)/f)^2*
d+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x-(-e*f)^(1/2)/f))*
d*e^2*b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.32636, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{d x^{2} + c} b \sqrt{d} f x -{\left (b e - a f\right )} \sqrt{d} \sqrt{\frac{d e - c f}{e}} \log \left (\frac{{\left (8 \, d^{2} e^{2} - 8 \, c d e f + c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} + 2 \,{\left (4 \, c d e^{2} - 3 \, c^{2} e f\right )} x^{2} - 4 \,{\left (c e^{2} x +{\left (2 \, d e^{2} - c e f\right )} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{\frac{d e - c f}{e}}}{f^{2} x^{4} + 2 \, e f x^{2} + e^{2}}\right ) -{\left (2 \, b d e -{\left (b c + 2 \, a d\right )} f\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{4 \, \sqrt{d} f^{2}}, \frac{2 \, \sqrt{d x^{2} + c} b \sqrt{-d} f x -{\left (b e - a f\right )} \sqrt{-d} \sqrt{\frac{d e - c f}{e}} \log \left (\frac{{\left (8 \, d^{2} e^{2} - 8 \, c d e f + c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} + 2 \,{\left (4 \, c d e^{2} - 3 \, c^{2} e f\right )} x^{2} - 4 \,{\left (c e^{2} x +{\left (2 \, d e^{2} - c e f\right )} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{\frac{d e - c f}{e}}}{f^{2} x^{4} + 2 \, e f x^{2} + e^{2}}\right ) - 2 \,{\left (2 \, b d e -{\left (b c + 2 \, a d\right )} f\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{4 \, \sqrt{-d} f^{2}}, \frac{2 \, \sqrt{d x^{2} + c} b \sqrt{d} f x - 2 \,{\left (b e - a f\right )} \sqrt{d} \sqrt{-\frac{d e - c f}{e}} \arctan \left (-\frac{{\left (2 \, d e - c f\right )} x^{2} + c e}{2 \, \sqrt{d x^{2} + c} e x \sqrt{-\frac{d e - c f}{e}}}\right ) -{\left (2 \, b d e -{\left (b c + 2 \, a d\right )} f\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{4 \, \sqrt{d} f^{2}}, \frac{\sqrt{d x^{2} + c} b \sqrt{-d} f x -{\left (b e - a f\right )} \sqrt{-d} \sqrt{-\frac{d e - c f}{e}} \arctan \left (-\frac{{\left (2 \, d e - c f\right )} x^{2} + c e}{2 \, \sqrt{d x^{2} + c} e x \sqrt{-\frac{d e - c f}{e}}}\right ) -{\left (2 \, b d e -{\left (b c + 2 \, a d\right )} f\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{2 \, \sqrt{-d} f^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*sqrt(d*x^2 + c)/(f*x^2 + e),x, algorithm="fricas")

[Out]

[1/4*(2*sqrt(d*x^2 + c)*b*sqrt(d)*f*x - (b*e - a*f)*sqrt(d)*sqrt((d*e - c*f)/e)*
log(((8*d^2*e^2 - 8*c*d*e*f + c^2*f^2)*x^4 + c^2*e^2 + 2*(4*c*d*e^2 - 3*c^2*e*f)
*x^2 - 4*(c*e^2*x + (2*d*e^2 - c*e*f)*x^3)*sqrt(d*x^2 + c)*sqrt((d*e - c*f)/e))/
(f^2*x^4 + 2*e*f*x^2 + e^2)) - (2*b*d*e - (b*c + 2*a*d)*f)*log(-2*sqrt(d*x^2 + c
)*d*x - (2*d*x^2 + c)*sqrt(d)))/(sqrt(d)*f^2), 1/4*(2*sqrt(d*x^2 + c)*b*sqrt(-d)
*f*x - (b*e - a*f)*sqrt(-d)*sqrt((d*e - c*f)/e)*log(((8*d^2*e^2 - 8*c*d*e*f + c^
2*f^2)*x^4 + c^2*e^2 + 2*(4*c*d*e^2 - 3*c^2*e*f)*x^2 - 4*(c*e^2*x + (2*d*e^2 - c
*e*f)*x^3)*sqrt(d*x^2 + c)*sqrt((d*e - c*f)/e))/(f^2*x^4 + 2*e*f*x^2 + e^2)) - 2
*(2*b*d*e - (b*c + 2*a*d)*f)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/(sqrt(-d)*f^2),
1/4*(2*sqrt(d*x^2 + c)*b*sqrt(d)*f*x - 2*(b*e - a*f)*sqrt(d)*sqrt(-(d*e - c*f)/
e)*arctan(-1/2*((2*d*e - c*f)*x^2 + c*e)/(sqrt(d*x^2 + c)*e*x*sqrt(-(d*e - c*f)/
e))) - (2*b*d*e - (b*c + 2*a*d)*f)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sq
rt(d)))/(sqrt(d)*f^2), 1/2*(sqrt(d*x^2 + c)*b*sqrt(-d)*f*x - (b*e - a*f)*sqrt(-d
)*sqrt(-(d*e - c*f)/e)*arctan(-1/2*((2*d*e - c*f)*x^2 + c*e)/(sqrt(d*x^2 + c)*e*
x*sqrt(-(d*e - c*f)/e))) - (2*b*d*e - (b*c + 2*a*d)*f)*arctan(sqrt(-d)*x/sqrt(d*
x^2 + c)))/(sqrt(-d)*f^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28336, size = 227, normalized size = 1.77 \[ \frac{\sqrt{d x^{2} + c} b x}{2 \, f} - \frac{{\left (a c \sqrt{d} f^{2} - b c \sqrt{d} f e - a d^{\frac{3}{2}} f e + b d^{\frac{3}{2}} e^{2}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} f - c f + 2 \, d e}{2 \, \sqrt{c d f e - d^{2} e^{2}}}\right )}{\sqrt{c d f e - d^{2} e^{2}} f^{2}} - \frac{{\left (b c f + 2 \, a d f - 2 \, b d e\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{4 \, \sqrt{d} f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(d*x^2 + c)*b*x/f - (a*c*sqrt(d)*f^2 - b*c*sqrt(d)*f*e - a*d^(3/2)*f*e +
b*d^(3/2)*e^2)*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*f - c*f + 2*d*e)/sqr
t(c*d*f*e - d^2*e^2))/(sqrt(c*d*f*e - d^2*e^2)*f^2) - 1/4*(b*c*f + 2*a*d*f - 2*b
*d*e)*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2)/(sqrt(d)*f^2)