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

Optimal. Leaf size=77 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{3/2}}-\frac{c}{d \sqrt{c+d x^2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.20655, antiderivative size = 77, 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 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{3/2}}-\frac{c}{d \sqrt{c+d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.3372, size = 63, normalized size = 0.82 \[ \frac{a \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{\sqrt{b} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{c}{d \sqrt{c + d x^{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.128109, size = 78, normalized size = 1.01 \[ \frac{\frac{c}{d \sqrt{c+d x^2}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}}}{a d-b c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.018, size = 653, normalized size = 8.5 \[ -{\frac{1}{bd}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{a}{2\, \left ( ad-bc \right ) b}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{axd}{2\,{b}^{2} \left ( ad-bc \right ) c}\sqrt{-ab}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{a}{2\, \left ( ad-bc \right ) b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{a}{2\, \left ( ad-bc \right ) b}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}+{\frac{axd}{2\,{b}^{2} \left ( ad-bc \right ) c}\sqrt{-ab}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{a}{2\, \left ( ad-bc \right ) b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/b/d/(d*x^2+c)^(1/2)+1/2*a/b/(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)-1/2*a/b^2*(-a*b)^(1/2)/(a*d-b*c)/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*d-1/2*a/b/(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
/2*a/b/(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)+1/2*a/b^2*(-a*b)^(1/2)/(a*d-b*c)/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*d-1/2*a/b/(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)*(d*x^2 + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254401, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c} c +{\left (a d^{2} x^{2} + a c d\right )} \log \left (\frac{{\left (b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d} - 4 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \,{\left (b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d}}, -\frac{2 \, \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c} c +{\left (a d^{2} x^{2} + a c d\right )} \arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d}}{2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}\right )}{2 \,{\left (b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.228596, size = 105, normalized size = 1.36 \[ -\frac{\frac{a d \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} + \frac{c}{\sqrt{d x^{2} + c}{\left (b c - a d\right )}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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