3.703 \(\int \frac{x^5}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2} (a d+b c)}{b^2 d^2}+\frac{\left (c+d x^2\right )^{3/2}}{3 b d^2} \]

[Out]

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

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Rubi [A]  time = 0.285799, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^2} (a d+b c)}{b^2 d^2}+\frac{\left (c+d x^2\right )^{3/2}}{3 b d^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((a + b*x^2)*Sqrt[c + d*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 34.0468, size = 85, normalized size = 0.85 \[ \frac{a^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{5}{2}} \sqrt{a d - b c}} + \frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 b d^{2}} - \frac{\sqrt{c + d x^{2}} \left (a d + b c\right )}{b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**2+a)/(d*x**2+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.159616, size = 89, normalized size = 0.89 \[ \frac{\sqrt{c+d x^2} \left (-3 a d-2 b c+b d x^2\right )}{3 b^2 d^2}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((a + b*x^2)*Sqrt[c + d*x^2]),x]

[Out]

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

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Maple [B]  time = 0.02, size = 362, normalized size = 3.6 \[{\frac{{x}^{2}}{3\,bd}\sqrt{d{x}^{2}+c}}-{\frac{2\,c}{3\,b{d}^{2}}\sqrt{d{x}^{2}+c}}-{\frac{a}{{b}^{2}d}\sqrt{d{x}^{2}+c}}-{\frac{{a}^{2}}{2\,{b}^{3}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{{a}^{2}}{2\,{b}^{3}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a)/(d*x^2+c)^(1/2),x)

[Out]

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

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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^5/((b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256928, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} d^{2} \log \left (\frac{{\left (b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d} - 4 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left (b d x^{2} - 2 \, b c - 3 \, a d\right )} \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c}}{12 \, \sqrt{b^{2} c - a b d} b^{2} d^{2}}, \frac{3 \, a^{2} d^{2} \arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d}}{2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}\right ) + 2 \,{\left (b d x^{2} - 2 \, b c - 3 \, a d\right )} \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c}}{6 \, \sqrt{-b^{2} c + a b d} b^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

[1/12*(3*a^2*d^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)) +
4*(b*d*x^2 - 2*b*c - 3*a*d)*sqrt(b^2*c - a*b*d)*sqrt(d*x^2 + c))/(sqrt(b^2*c - a
*b*d)*b^2*d^2), 1/6*(3*a^2*d^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))) + 2*(b*d*x^2 - 2*b*c - 3*a*d)*sqrt(-b
^2*c + a*b*d)*sqrt(d*x^2 + c))/(sqrt(-b^2*c + a*b*d)*b^2*d^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (a + b x^{2}\right ) \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**2+a)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**5/((a + b*x**2)*sqrt(c + d*x**2)), x)

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GIAC/XCAS [A]  time = 0.239251, size = 142, normalized size = 1.42 \[ \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}} + \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} d^{4} - 3 \, \sqrt{d x^{2} + c} b^{2} c d^{4} - 3 \, \sqrt{d x^{2} + c} a b d^{5}}{3 \, b^{3} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

a^2*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2) +
1/3*((d*x^2 + c)^(3/2)*b^2*d^4 - 3*sqrt(d*x^2 + c)*b^2*c*d^4 - 3*sqrt(d*x^2 + c)
*a*b*d^5)/(b^3*d^6)