3.620 \(\int \frac{x^3 \sqrt{c+d x^4}}{a+b x^4} \, dx\)

Optimal. Leaf size=70 \[ \frac{\sqrt{c+d x^4}}{2 b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{3/2}} \]

[Out]

Sqrt[c + d*x^4]/(2*b) - (Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[
b*c - a*d]])/(2*b^(3/2))

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Rubi [A]  time = 0.182226, antiderivative size = 70, 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 \[ \frac{\sqrt{c+d x^4}}{2 b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

Sqrt[c + d*x^4]/(2*b) - (Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[
b*c - a*d]])/(2*b^(3/2))

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Rubi in Sympy [A]  time = 18.9067, size = 56, normalized size = 0.8 \[ \frac{\sqrt{c + d x^{4}}}{2 b} - \frac{\sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(d*x**4+c)**(1/2)/(b*x**4+a),x)

[Out]

sqrt(c + d*x**4)/(2*b) - sqrt(a*d - b*c)*atan(sqrt(b)*sqrt(c + d*x**4)/sqrt(a*d
- b*c))/(2*b**(3/2))

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Mathematica [A]  time = 0.0640862, size = 70, normalized size = 1. \[ \frac{\sqrt{c+d x^4}}{2 b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

Sqrt[c + d*x^4]/(2*b) - (Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[
b*c - a*d]])/(2*b^(3/2))

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Maple [B]  time = 0.01, size = 988, normalized size = 14.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217246, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{4} + 2 \, b c - a d - 2 \, \sqrt{d x^{4} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{4} + a}\right ) + 2 \, \sqrt{d x^{4} + c}}{4 \, b}, -\frac{\sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) - \sqrt{d x^{4} + c}}{2 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)*x^3/(b*x^4 + a),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \sqrt{c + d x^{4}}}{a + b x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(d*x**4+c)**(1/2)/(b*x**4+a),x)

[Out]

Integral(x**3*sqrt(c + d*x**4)/(a + b*x**4), x)

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GIAC/XCAS [A]  time = 0.211631, size = 89, normalized size = 1.27 \[ \frac{{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{2 \, \sqrt{-b^{2} c + a b d} b} + \frac{\sqrt{d x^{4} + c}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)*x^3/(b*x^4 + a),x, algorithm="giac")

[Out]

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