3.645 \(\int \frac{1}{x^7 \left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=115 \[ \frac{b^2 \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^4} (2 a d+3 b c)}{6 a^2 c^2 x^2}-\frac{\sqrt{c+d x^4}}{6 a c x^6} \]

[Out]

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

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Rubi [A]  time = 0.471375, antiderivative size = 115, 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 \[ \frac{b^2 \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^4} (2 a d+3 b c)}{6 a^2 c^2 x^2}-\frac{\sqrt{c+d x^4}}{6 a c x^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 63.3789, size = 100, normalized size = 0.87 \[ - \frac{\sqrt{c + d x^{4}}}{6 a c x^{6}} + \frac{\sqrt{c + d x^{4}} \left (2 a d + 3 b c\right )}{6 a^{2} c^{2} x^{2}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{2 a^{\frac{5}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.28554, size = 149, normalized size = 1.3 \[ \frac{\sqrt{c+d x^4} \left (-a^2 c+\frac{3 b^2 c x^8 \sin ^{-1}\left (\frac{\sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^4}{a}+1}}\right )}{\sqrt{\frac{b x^4}{a}+1} \sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )} \sqrt{\frac{a \left (c+d x^4\right )}{c \left (a+b x^4\right )}}}+a x^4 (2 a d+3 b c)\right )}{6 a^3 c^2 x^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 383, normalized size = 3.3 \[ -{\frac{-2\,d{x}^{4}+c}{6\,a{x}^{6}{c}^{2}}\sqrt{d{x}^{4}+c}}-{\frac{{b}^{2}}{4\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{{b}^{2}}{4\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{b}{2\,{a}^{2}c{x}^{2}}\sqrt{d{x}^{4}+c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^7),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^7), x)

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Fricas [A]  time = 0.329006, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} c^{2} x^{6} \log \left (\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{6} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2}\right )} \sqrt{d x^{4} + c} +{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \,{\left ({\left (3 \, b c + 2 \, a d\right )} x^{4} - a c\right )} \sqrt{d x^{4} + c} \sqrt{-a b c + a^{2} d}}{24 \, \sqrt{-a b c + a^{2} d} a^{2} c^{2} x^{6}}, \frac{3 \, b^{2} c^{2} x^{6} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} x^{2}}\right ) + 2 \,{\left ({\left (3 \, b c + 2 \, a d\right )} x^{4} - a c\right )} \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d}}{12 \, \sqrt{a b c - a^{2} d} a^{2} c^{2} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^7),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{7} \left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**7*(a + b*x**4)*sqrt(c + d*x**4)), x)

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GIAC/XCAS [A]  time = 0.223181, size = 144, normalized size = 1.25 \[ -\frac{b^{2} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{2 \, \sqrt{a b c - a^{2} d} a^{2}} + \frac{3 \, a b c^{5} \sqrt{d + \frac{c}{x^{4}}} - a^{2} c^{4}{\left (d + \frac{c}{x^{4}}\right )}^{\frac{3}{2}} + 3 \, a^{2} c^{4} \sqrt{d + \frac{c}{x^{4}}} d}{6 \, a^{3} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^7),x, algorithm="giac")

[Out]

-1/2*b^2*arctan(a*sqrt(d + c/x^4)/sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*a^2)
+ 1/6*(3*a*b*c^5*sqrt(d + c/x^4) - a^2*c^4*(d + c/x^4)^(3/2) + 3*a^2*c^4*sqrt(d
+ c/x^4)*d)/(a^3*c^6)