3.708 \(\int \frac{x^4}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=114 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^2 d^{3/2}}+\frac{x \sqrt{c+d x^2}}{2 b d} \]

[Out]

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

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Rubi [A]  time = 0.279249, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^2 d^{3/2}}+\frac{x \sqrt{c+d x^2}}{2 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x^2)*Sqrt[c + d*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 39.6118, size = 100, normalized size = 0.88 \[ \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{b^{2} \sqrt{a d - b c}} + \frac{x \sqrt{c + d x^{2}}}{2 b d} - \frac{\left (2 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 b^{2} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)/(d*x**2+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.201706, size = 112, normalized size = 0.98 \[ \frac{\frac{2 a^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{b c-a d}}-\frac{(2 a d+b c) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{d^{3/2}}+\frac{b x \sqrt{c+d x^2}}{d}}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((a + b*x^2)*Sqrt[c + d*x^2]),x]

[Out]

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

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Maple [B]  time = 0.019, size = 386, normalized size = 3.4 \[{\frac{x}{2\,bd}\sqrt{d{x}^{2}+c}}-{\frac{c}{2\,b}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}-{\frac{a}{{b}^{2}}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}-{\frac{{a}^{2}}{2\,{b}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{{a}^{2}}{2\,{b}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*(d*x^2+c)^(1/2)/b/d-1/2/b*c/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-1/b^2*a*
ln(x*d^(1/2)+(d*x^2+c)^(1/2))/d^(1/2)-1/2/b^2*a^2/(-a*b)^(1/2)/(-(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^2/(-a*b)^(1/2)/(-(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^4/((b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.384246, size = 1, normalized size = 0.01 \[ \left [\frac{a d^{\frac{3}{2}} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, \sqrt{d x^{2} + c} b \sqrt{d} x +{\left (b c + 2 \, a d\right )} \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{4 \, b^{2} d^{\frac{3}{2}}}, \frac{a \sqrt{-d} d \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, \sqrt{d x^{2} + c} b \sqrt{-d} x - 2 \,{\left (b c + 2 \, a d\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{4 \, b^{2} \sqrt{-d} d}, \frac{2 \, a d^{\frac{3}{2}} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} x \sqrt{\frac{a}{b c - a d}}}\right ) + 2 \, \sqrt{d x^{2} + c} b \sqrt{d} x +{\left (b c + 2 \, a d\right )} \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{4 \, b^{2} d^{\frac{3}{2}}}, \frac{a \sqrt{-d} d \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} x \sqrt{\frac{a}{b c - a d}}}\right ) + \sqrt{d x^{2} + c} b \sqrt{-d} x -{\left (b c + 2 \, a d\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{2 \, b^{2} \sqrt{-d} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (a + b x^{2}\right ) \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**4/((a + b*x**2)*sqrt(c + d*x**2)), x)

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GIAC/XCAS [A]  time = 0.250187, size = 184, normalized size = 1.61 \[ -\frac{a^{2} \sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} b^{2}} + \frac{\sqrt{d x^{2} + c} x}{2 \, b d} + \frac{{\left (b c \sqrt{d} + 2 \, a d^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{4 \, b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

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