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

Optimal. Leaf size=110 \[ \frac{b \sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{5/2}}+\frac{\sqrt{c+d x^4} (3 b c-a d)}{6 a^2 c x^2}-\frac{\sqrt{c+d x^4}}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.481663, antiderivative size = 110, 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 \[ \frac{b \sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{5/2}}+\frac{\sqrt{c+d x^4} (3 b c-a d)}{6 a^2 c x^2}-\frac{\sqrt{c+d x^4}}{6 a x^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 59.7717, size = 95, normalized size = 0.86 \[ - \frac{\sqrt{c + d x^{4}}}{6 a x^{6}} - \frac{\sqrt{c + d x^{4}} \left (a d - 3 b c\right )}{6 a^{2} c x^{2}} - \frac{b \sqrt{a d - b c} \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.828968, size = 156, normalized size = 1.42 \[ \frac{\sqrt{c+d x^4} \left (-a^2+\frac{a x^4 (3 b c-a d)}{c}+\frac{3 b x^8 (b c-a d) \sin ^{-1}\left (\frac{\sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^4}{a}+1}}\right )}{c \sqrt{\frac{b x^4}{a}+1} \sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )} \sqrt{\frac{a \left (c+d x^4\right )}{c \left (a+b x^4\right )}}}\right )}{6 a^3 x^6} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^4]*(-a^2 + (a*(3*b*c - a*d)*x^4)/c + (3*b*(b*c - a*d)*x^8*ArcSin[S
qrt[(b/a - d/c)*x^4]/Sqrt[1 + (b*x^4)/a]])/(c*Sqrt[(b/a - d/c)*x^4]*Sqrt[1 + (b*
x^4)/a]*Sqrt[(a*(c + d*x^4))/(c*(a + b*x^4))])))/(6*a^3*x^6)

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Maple [B]  time = 0.023, size = 1116, normalized size = 10.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.279906, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b c x^{6} \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \,{\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \,{\left ({\left (3 \, b c - a d\right )} x^{4} - a c\right )} \sqrt{d x^{4} + c}}{24 \, a^{2} c x^{6}}, -\frac{3 \, b c x^{6} \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} a x^{2} \sqrt{\frac{b c - a d}{a}}}\right ) - 2 \,{\left ({\left (3 \, b c - a d\right )} x^{4} - a c\right )} \sqrt{d x^{4} + c}}{12 \, a^{2} c x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/24*(3*b*c*x^6*sqrt(-(b*c - a*d)/a)*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*((a*b*c - 2*a^2*d)*x^6 - a^2*c*x^
2)*sqrt(d*x^4 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^8 + 2*a*b*x^4 + a^2)) + 4*((3*b*
c - a*d)*x^4 - a*c)*sqrt(d*x^4 + c))/(a^2*c*x^6), -1/12*(3*b*c*x^6*sqrt((b*c - a
*d)/a)*arctan(-1/2*((b*c - 2*a*d)*x^4 - a*c)/(sqrt(d*x^4 + c)*a*x^2*sqrt((b*c -
a*d)/a))) - 2*((3*b*c - a*d)*x^4 - a*c)*sqrt(d*x^4 + c))/(a^2*c*x^6)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217342, size = 131, normalized size = 1.19 \[ -\frac{\frac{3 \,{\left (b^{2} c^{2} - a b c d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d} a^{2}} - \frac{3 \, a b c \sqrt{d + \frac{c}{x^{4}}} - a^{2}{\left (d + \frac{c}{x^{4}}\right )}^{\frac{3}{2}}}{a^{3}}}{6 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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