3.732 \(\int \frac{x^3 \sqrt{c+d x^2}}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=136 \[ -\frac{(2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (2 b c-3 a d)}{2 b^2 (b c-a d)}+\frac{a \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right ) (b c-a d)} \]

[Out]

((2*b*c - 3*a*d)*Sqrt[c + d*x^2])/(2*b^2*(b*c - a*d)) + (a*(c + d*x^2)^(3/2))/(2
*b*(b*c - a*d)*(a + b*x^2)) - ((2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])
/Sqrt[b*c - a*d]])/(2*b^(5/2)*Sqrt[b*c - a*d])

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Rubi [A]  time = 0.296521, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{(2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (2 b c-3 a d)}{2 b^2 (b c-a d)}+\frac{a \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

((2*b*c - 3*a*d)*Sqrt[c + d*x^2])/(2*b^2*(b*c - a*d)) + (a*(c + d*x^2)^(3/2))/(2
*b*(b*c - a*d)*(a + b*x^2)) - ((2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])
/Sqrt[b*c - a*d]])/(2*b^(5/2)*Sqrt[b*c - a*d])

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Rubi in Sympy [A]  time = 35.1264, size = 110, normalized size = 0.81 \[ - \frac{a \left (c + d x^{2}\right )^{\frac{3}{2}}}{2 b \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{\sqrt{c + d x^{2}} \left (\frac{3 a d}{2} - b c\right )}{b^{2} \left (a d - b c\right )} - \frac{\left (\frac{3 a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{5}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(d*x**2+c)**(1/2)/(b*x**2+a)**2,x)

[Out]

-a*(c + d*x**2)**(3/2)/(2*b*(a + b*x**2)*(a*d - b*c)) + sqrt(c + d*x**2)*(3*a*d/
2 - b*c)/(b**2*(a*d - b*c)) - (3*a*d/2 - b*c)*atan(sqrt(b)*sqrt(c + d*x**2)/sqrt
(a*d - b*c))/(b**(5/2)*sqrt(a*d - b*c))

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Mathematica [A]  time = 0.157149, size = 91, normalized size = 0.67 \[ \frac{1}{2} \left (\frac{\left (\frac{a}{a+b x^2}+2\right ) \sqrt{c+d x^2}}{b^2}-\frac{(2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2} \sqrt{b c-a d}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[c + d*x^2]*(2 + a/(a + b*x^2)))/b^2 - ((2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*S
qrt[c + d*x^2])/Sqrt[b*c - a*d]])/(b^(5/2)*Sqrt[b*c - a*d]))/2

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Maple [B]  time = 0.021, size = 2543, normalized size = 18.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(d*x^2+c)^(1/2)/(b*x^2+a)^2,x)

[Out]

1/2/b^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)+1/2/b^3*d^(1/2)*(-a*b)^(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))+1/2/b^3/(-(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-1/2/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)))*c+1/2/b^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/
2/b^3*d^(1/2)*(-a*b)^(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))+1/2/b^3/(-(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-1/2/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)))*c+1/4/b^2*(-a*b)^(1/2)/(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*a*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)+1/4/b^3*(-a*b)^(1/2)*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*a^2*d^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)))-1/4/b^2*
a*d/(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^2*(-a*b
)^(1/2)*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^2*(-a*b)^(1/2)*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^2*(-a*b)^(1/2)/(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*a*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)-1/4/b^3*(-a*b)^(1/2)*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*
a^2*d^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)))-1/4/b^2*a*d
/(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^2*(-a*b)^(
1/2)*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^2*(-a*b)^(1/2)*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

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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(d*x^2 + c)*x^3/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274448, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{b^{2} c - a b d}{\left (2 \, b x^{2} + 3 \, a\right )} \sqrt{d x^{2} + c} -{\left (2 \, a b c - 3 \, a^{2} d +{\left (2 \, b^{2} c - 3 \, a b d\right )} x^{2}\right )} \log \left (\frac{{\left (b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d} + 4 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{b^{2} c - a b d}}, \frac{2 \, \sqrt{-b^{2} c + a b d}{\left (2 \, b x^{2} + 3 \, a\right )} \sqrt{d x^{2} + c} +{\left (2 \, a b c - 3 \, a^{2} d +{\left (2 \, b^{2} c - 3 \, a b d\right )} x^{2}\right )} \arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d}}{2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}\right )}{4 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)*x^3/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \sqrt{c + d x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(d*x**2+c)**(1/2)/(b*x**2+a)**2,x)

[Out]

Integral(x**3*sqrt(c + d*x**2)/(a + b*x**2)**2, x)

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GIAC/XCAS [A]  time = 0.230687, size = 150, normalized size = 1.1 \[ \frac{\frac{\sqrt{d x^{2} + c} a d^{2}}{{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} b^{2}} + \frac{2 \, \sqrt{d x^{2} + c} d}{b^{2}} + \frac{{\left (2 \, b c d - 3 \, a d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)*x^3/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

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