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

Optimal. Leaf size=99 \[ \frac{a \sqrt{c+d x^2}}{2 b \left (a+b x^2\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} (b c-a d)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.245129, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{a \sqrt{c+d x^2}}{2 b \left (a+b x^2\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.5652, size = 78, normalized size = 0.79 \[ - \frac{a \sqrt{c + d x^{2}}}{2 b \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{\left (\frac{a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.126695, size = 99, normalized size = 1. \[ \frac{a \sqrt{c+d x^2}}{2 b \left (a+b x^2\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 807, normalized size = 8.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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)))-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)))+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)^(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)))-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)^(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)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268286, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c} a +{\left (2 \, a b c - a^{2} d +{\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \log \left (\frac{{\left (b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d} - 4 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \,{\left (a b^{2} c - a^{2} b d +{\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt{b^{2} c - a b d}}, \frac{2 \, \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c} a +{\left (2 \, a b c - a^{2} d +{\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d}}{2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}\right )}{4 \,{\left (a b^{2} c - a^{2} b d +{\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.236052, size = 157, normalized size = 1.59 \[ \frac{\frac{\sqrt{d x^{2} + c} a d^{2}}{{\left (b^{2} c - a b d\right )}{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt{-b^{2} c + a b d}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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