3.619 \(\int \frac{x^5 \sqrt{c+d x^4}}{a+b x^4} \, dx\)

Optimal. Leaf size=120 \[ -\frac{\sqrt{a} \sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 b^2}+\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{4 b^2 \sqrt{d}}+\frac{x^2 \sqrt{c+d x^4}}{4 b} \]

[Out]

(x^2*Sqrt[c + d*x^4])/(4*b) - (Sqrt[a]*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x
^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(2*b^2) + ((b*c - 2*a*d)*ArcTanh[(Sqrt[d]*x^2)/S
qrt[c + d*x^4]])/(4*b^2*Sqrt[d])

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Rubi [A]  time = 0.42646, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\sqrt{a} \sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 b^2}+\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{4 b^2 \sqrt{d}}+\frac{x^2 \sqrt{c+d x^4}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

(x^2*Sqrt[c + d*x^4])/(4*b) - (Sqrt[a]*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x
^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(2*b^2) + ((b*c - 2*a*d)*ArcTanh[(Sqrt[d]*x^2)/S
qrt[c + d*x^4]])/(4*b^2*Sqrt[d])

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Rubi in Sympy [A]  time = 45.9007, size = 105, normalized size = 0.88 \[ \frac{\sqrt{a} \sqrt{a d - b c} \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{2 b^{2}} + \frac{x^{2} \sqrt{c + d x^{4}}}{4 b} - \frac{\left (2 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c + d x^{4}}} \right )}}{4 b^{2} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(d*x**4+c)**(1/2)/(b*x**4+a),x)

[Out]

sqrt(a)*sqrt(a*d - b*c)*atanh(x**2*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**4)))/(
2*b**2) + x**2*sqrt(c + d*x**4)/(4*b) - (2*a*d - b*c)*atanh(sqrt(d)*x**2/sqrt(c
+ d*x**4))/(4*b**2*sqrt(d))

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Mathematica [A]  time = 0.234717, size = 114, normalized size = 0.95 \[ \frac{\frac{(b c-2 a d) \log \left (\sqrt{d} \sqrt{c+d x^4}+d x^2\right )}{\sqrt{d}}-2 \sqrt{a} \sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )+b x^2 \sqrt{c+d x^4}}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

(b*x^2*Sqrt[c + d*x^4] - 2*Sqrt[a]*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x^2)/
(Sqrt[a]*Sqrt[c + d*x^4])] + ((b*c - 2*a*d)*Log[d*x^2 + Sqrt[d]*Sqrt[c + d*x^4]]
)/Sqrt[d])/(4*b^2)

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Maple [B]  time = 0.026, size = 1066, normalized size = 8.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(d*x^4+c)^(1/2)/(b*x^4+a),x)

[Out]

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

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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(sqrt(d*x^4 + c)*x^5/(b*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.309671, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{d x^{4} + c} b \sqrt{d} x^{2} -{\left (b c - 2 \, a d\right )} \log \left (2 \, \sqrt{d x^{4} + c} d x^{2} -{\left (2 \, d x^{4} + c\right )} \sqrt{d}\right ) + \sqrt{-a b c + a^{2} d} \sqrt{d} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \,{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, b^{2} \sqrt{d}}, \frac{2 \, \sqrt{d x^{4} + c} b \sqrt{-d} x^{2} + 2 \,{\left (b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{-d} x^{2}}{\sqrt{d x^{4} + c}}\right ) + \sqrt{-a b c + a^{2} d} \sqrt{-d} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \,{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, b^{2} \sqrt{-d}}, \frac{2 \, \sqrt{d x^{4} + c} b \sqrt{d} x^{2} + 2 \, \sqrt{a b c - a^{2} d} \sqrt{d} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} x^{2}}\right ) -{\left (b c - 2 \, a d\right )} \log \left (2 \, \sqrt{d x^{4} + c} d x^{2} -{\left (2 \, d x^{4} + c\right )} \sqrt{d}\right )}{8 \, b^{2} \sqrt{d}}, \frac{\sqrt{d x^{4} + c} b \sqrt{-d} x^{2} +{\left (b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{-d} x^{2}}{\sqrt{d x^{4} + c}}\right ) + \sqrt{a b c - a^{2} d} \sqrt{-d} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} x^{2}}\right )}{4 \, b^{2} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)*x^5/(b*x^4 + a),x, algorithm="fricas")

[Out]

[1/8*(2*sqrt(d*x^4 + c)*b*sqrt(d)*x^2 - (b*c - 2*a*d)*log(2*sqrt(d*x^4 + c)*d*x^
2 - (2*d*x^4 + c)*sqrt(d)) + sqrt(-a*b*c + a^2*d)*sqrt(d)*log(((b^2*c^2 - 8*a*b*
c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 - 4*((b*c - 2*a*d
)*x^6 - a*c*x^2)*sqrt(d*x^4 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^8 + 2*a*b*x^4 + a^
2)))/(b^2*sqrt(d)), 1/8*(2*sqrt(d*x^4 + c)*b*sqrt(-d)*x^2 + 2*(b*c - 2*a*d)*arct
an(sqrt(-d)*x^2/sqrt(d*x^4 + c)) + sqrt(-a*b*c + a^2*d)*sqrt(-d)*log(((b^2*c^2 -
8*a*b*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 - 4*((b*c
- 2*a*d)*x^6 - a*c*x^2)*sqrt(d*x^4 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^8 + 2*a*b*x
^4 + a^2)))/(b^2*sqrt(-d)), 1/8*(2*sqrt(d*x^4 + c)*b*sqrt(d)*x^2 + 2*sqrt(a*b*c
- a^2*d)*sqrt(d)*arctan(-1/2*((b*c - 2*a*d)*x^4 - a*c)/(sqrt(d*x^4 + c)*sqrt(a*b
*c - a^2*d)*x^2)) - (b*c - 2*a*d)*log(2*sqrt(d*x^4 + c)*d*x^2 - (2*d*x^4 + c)*sq
rt(d)))/(b^2*sqrt(d)), 1/4*(sqrt(d*x^4 + c)*b*sqrt(-d)*x^2 + (b*c - 2*a*d)*arcta
n(sqrt(-d)*x^2/sqrt(d*x^4 + c)) + sqrt(a*b*c - a^2*d)*sqrt(-d)*arctan(-1/2*((b*c
- 2*a*d)*x^4 - a*c)/(sqrt(d*x^4 + c)*sqrt(a*b*c - a^2*d)*x^2)))/(b^2*sqrt(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5} \sqrt{c + d x^{4}}}{a + b x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(d*x**4+c)**(1/2)/(b*x**4+a),x)

[Out]

Integral(x**5*sqrt(c + d*x**4)/(a + b*x**4), x)

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GIAC/XCAS [A]  time = 0.332472, size = 136, normalized size = 1.13 \[ \frac{\sqrt{d x^{4} + c} b^{2} x^{2}}{384 \, d^{3}} + \frac{\sqrt{a b c - a^{2} d} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{2 \, b^{2}} - \frac{{\left (b^{2} c - 2 \, a b d\right )} \arctan \left (\frac{\sqrt{d + \frac{c}{x^{4}}}}{\sqrt{-d}}\right )}{384 \, \sqrt{-d} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)*x^5/(b*x^4 + a),x, algorithm="giac")

[Out]

1/384*sqrt(d*x^4 + c)*b^2*x^2/d^3 + 1/2*sqrt(a*b*c - a^2*d)*arctan(a*sqrt(d + c/
x^4)/sqrt(a*b*c - a^2*d))/b^2 - 1/384*(b^2*c - 2*a*b*d)*arctan(sqrt(d + c/x^4)/s
qrt(-d))/(sqrt(-d)*d^3)