3.618 \(\int \frac{x^7 \sqrt{c+d x^4}}{a+b x^4} \, dx\)

Optimal. Leaf size=93 \[ \frac{a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{5/2}}-\frac{a \sqrt{c+d x^4}}{2 b^2}+\frac{\left (c+d x^4\right )^{3/2}}{6 b d} \]

[Out]

-(a*Sqrt[c + d*x^4])/(2*b^2) + (c + d*x^4)^(3/2)/(6*b*d) + (a*Sqrt[b*c - a*d]*Ar
cTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(2*b^(5/2))

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Rubi [A]  time = 0.260944, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{5/2}}-\frac{a \sqrt{c+d x^4}}{2 b^2}+\frac{\left (c+d x^4\right )^{3/2}}{6 b d} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

-(a*Sqrt[c + d*x^4])/(2*b^2) + (c + d*x^4)^(3/2)/(6*b*d) + (a*Sqrt[b*c - a*d]*Ar
cTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(2*b^(5/2))

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Rubi in Sympy [A]  time = 24.019, size = 76, normalized size = 0.82 \[ - \frac{a \sqrt{c + d x^{4}}}{2 b^{2}} + \frac{a \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 b^{\frac{5}{2}}} + \frac{\left (c + d x^{4}\right )^{\frac{3}{2}}}{6 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(d*x**4+c)**(1/2)/(b*x**4+a),x)

[Out]

-a*sqrt(c + d*x**4)/(2*b**2) + a*sqrt(a*d - b*c)*atan(sqrt(b)*sqrt(c + d*x**4)/s
qrt(a*d - b*c))/(2*b**(5/2)) + (c + d*x**4)**(3/2)/(6*b*d)

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Mathematica [A]  time = 0.347101, size = 88, normalized size = 0.95 \[ \frac{a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{5/2}}+\frac{\sqrt{c+d x^4} \left (b \left (c+d x^4\right )-3 a d\right )}{6 b^2 d} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

(Sqrt[c + d*x^4]*(-3*a*d + b*(c + d*x^4)))/(6*b^2*d) + (a*Sqrt[b*c - a*d]*ArcTan
h[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(2*b^(5/2))

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Maple [B]  time = 0.028, size = 1015, normalized size = 10.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),x)

[Out]

1/6*(d*x^4+c)^(3/2)/b/d-1/4*a/b^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^3*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*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^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^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^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^3*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*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^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^
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^7/(b*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217318, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a d \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{4} + a}\right ) + 2 \,{\left (b d x^{4} + b c - 3 \, a d\right )} \sqrt{d x^{4} + c}}{12 \, b^{2} d}, \frac{3 \, a d \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (b d x^{4} + b c - 3 \, a d\right )} \sqrt{d x^{4} + c}}{6 \, b^{2} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)*x^7/(b*x^4 + a),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7} \sqrt{c + d x^{4}}}{a + b x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(d*x**4+c)**(1/2)/(b*x**4+a),x)

[Out]

Integral(x**7*sqrt(c + d*x**4)/(a + b*x**4), x)

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GIAC/XCAS [A]  time = 0.211739, size = 130, normalized size = 1.4 \[ -\frac{\frac{3 \,{\left (a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}} - \frac{{\left (d x^{4} + c\right )}^{\frac{3}{2}} b^{2} - 3 \, \sqrt{d x^{4} + c} a b d}{b^{3}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)*x^7/(b*x^4 + a),x, algorithm="giac")

[Out]

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