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

Optimal. Leaf size=120 \[ \frac{(b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} b^2 \sqrt{b c-a d}}-\frac{x \sqrt{c+d x^2}}{2 b \left (a+b x^2\right )}+\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.256267, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} b^2 \sqrt{b c-a d}}-\frac{x \sqrt{c+d x^2}}{2 b \left (a+b x^2\right )}+\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.6842, size = 105, normalized size = 0.88 \[ - \frac{x \sqrt{c + d x^{2}}}{2 b \left (a + b x^{2}\right )} + \frac{\sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{b^{2}} - \frac{\left (2 a d - b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 \sqrt{a} b^{2} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.133627, size = 118, normalized size = 0.98 \[ \frac{-\frac{b x \sqrt{c+d x^2}}{a+b x^2}+\frac{(b c-2 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} \sqrt{b c-a d}}+2 \sqrt{d} \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 2547, normalized size = 21.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/b/(a*d-b*c)/(x-1/b*(-a*b)^(1/2))*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/
b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-1/4/b^2*d*(-a*b)^(1/2)/(a*d-b*c)*((x-1
/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+
1/4/b^2*d^(3/2)*a/(a*d-b*c)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)
+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^
(1/2))-1/4/b^3*d^2*(-a*b)^(1/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/
b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^
(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-
a*b)^(1/2)))*a+1/4/b^2*d*(-a*b)^(1/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d
-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(
-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-
1/b*(-a*b)^(1/2)))*c-1/4/b*d/(a*d-b*c)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2
)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-1/4/b*d^(1/2)/(a*d-b*c)*ln((d*(-a*
b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^
(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*c+1/4/b/(a*d-b*c)/(x+1/b*(-a*b)
^(1/2))*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b
*c)/b)^(3/2)+1/4/b^2*d*(-a*b)^(1/2)/(a*d-b*c)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*
b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/4/b^2*d^(3/2)*a/(a*d-b*c)*l
n((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2
*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+1/4/b^3*d^2*(-a*b)^(1
/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*
(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/
b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*a-1/4/b^2*d*(-a
*b)^(1/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(
x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^
(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*c-1/4/b*d
/(a*d-b*c)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*
d-b*c)/b)^(1/2)*x-1/4/b*d^(1/2)/(a*d-b*c)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1
/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2)
)-(a*d-b*c)/b)^(1/2))*c+1/4/(-a*b)^(1/2)/b*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^
(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/4/b^2*d^(1/2)*ln((d*(-a*b)^(1/
2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/
b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+1/4/(-a*b)^(1/2)/b^2/(-(a*d-b*c)/b)^(
1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)
^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*
c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*a*d-1/4/(-a*b)^(1/2)/b/(-(a*d-b*c)/b)^(1/2)*l
n((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)
*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^
(1/2))/(x-1/b*(-a*b)^(1/2)))*c-1/4/(-a*b)^(1/2)/b*((x+1/b*(-a*b)^(1/2))^2*d-2*d*
(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/4/b^2*d^(1/2)*ln((-d*(-
a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b
)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/4/(-a*b)^(1/2)/b^2/(-(a*d-b
*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d
-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))
-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*a*d+1/4/(-a*b)^(1/2)/b/(-(a*d-b*c)/b)
^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/
b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-
b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*c

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(d*x^2 + c)*x^2/(b*x^2 + a)^2, x)

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Fricas [A]  time = 0.342458, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(4*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c)*b*x - 4*sqrt(-a*b*c + a^2*d)*(b*x^
2 + a)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + (a*b*c - 2*a^2*
d + (b^2*c - 2*a*b*d)*x^2)*log((((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2
- 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2)*sqrt(-a*b*c + a^2*d) - 4*((a*b^2*c^2 - 3*a^2*b
*c*d + 2*a^3*d^2)*x^3 - (a^2*b*c^2 - a^3*c*d)*x)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a
*b*x^2 + a^2)))/((b^3*x^2 + a*b^2)*sqrt(-a*b*c + a^2*d)), -1/8*(4*sqrt(-a*b*c +
a^2*d)*sqrt(d*x^2 + c)*b*x - 8*sqrt(-a*b*c + a^2*d)*(b*x^2 + a)*sqrt(-d)*arctan(
d*x/(sqrt(d*x^2 + c)*sqrt(-d))) + (a*b*c - 2*a^2*d + (b^2*c - 2*a*b*d)*x^2)*log(
(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^
2)*sqrt(-a*b*c + a^2*d) - 4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^3 - (a^2*b*
c^2 - a^3*c*d)*x)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/((b^3*x^2 + a*b
^2)*sqrt(-a*b*c + a^2*d)), -1/4*(2*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*b*x - 2*s
qrt(a*b*c - a^2*d)*(b*x^2 + a)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*
x - c) - (a*b*c - 2*a^2*d + (b^2*c - 2*a*b*d)*x^2)*arctan(1/2*((b*c - 2*a*d)*x^2
- a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)))/((b^3*x^2 + a*b^2)*sqrt(a*b*c
- a^2*d)), -1/4*(2*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*b*x - 4*sqrt(a*b*c - a^2*
d)*(b*x^2 + a)*sqrt(-d)*arctan(d*x/(sqrt(d*x^2 + c)*sqrt(-d))) - (a*b*c - 2*a^2*
d + (b^2*c - 2*a*b*d)*x^2)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(a*b*c - a^
2*d)*sqrt(d*x^2 + c)*x)))/((b^3*x^2 + a*b^2)*sqrt(a*b*c - a^2*d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.568549, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x