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

Optimal. Leaf size=68 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2}}{b d} \]

[Out]

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

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Rubi [A]  time = 0.193159, antiderivative size = 68, 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 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2}}{b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.6863, size = 56, normalized size = 0.82 \[ - \frac{a \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{3}{2}} \sqrt{a d - b c}} + \frac{\sqrt{c + d x^{2}}}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0708919, size = 68, normalized size = 1. \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2}}{b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 318, normalized size = 4.7 \[{\frac{1}{bd}\sqrt{d{x}^{2}+c}}+{\frac{a}{2\,{b}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{a}{2\,{b}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x^2+c)^(1/2)/b/d+1/2*a/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*a/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)))

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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)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267235, size = 1, normalized size = 0.01 \[ \left [\frac{a d \log \left (\frac{{\left (b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d} + 4 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c}}{4 \, \sqrt{b^{2} c - a b d} b d}, -\frac{a d \arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d}}{2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}\right ) - 2 \, \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c}}{2 \, \sqrt{-b^{2} c + a b d} b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23375, size = 86, normalized size = 1.26 \[ -\frac{\frac{a d \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b} - \frac{\sqrt{d x^{2} + c}}{b}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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