3.664 \(\int \frac{1}{x^7 \left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=208 \[ \frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} (5 b c-2 a d)}{12 a^2 c x^6 (b c-a d)}+\frac{\sqrt{c+d x^4} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{12 a^3 c^2 x^2 (b c-a d)}+\frac{b \sqrt{c+d x^4}}{4 a x^6 \left (a+b x^4\right ) (b c-a d)} \]

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^4])/(12*a^2*c*(b*c - a*d)*x^6) + ((15*b^2*c^2 - 8
*a*b*c*d - 4*a^2*d^2)*Sqrt[c + d*x^4])/(12*a^3*c^2*(b*c - a*d)*x^2) + (b*Sqrt[c
+ d*x^4])/(4*a*(b*c - a*d)*x^6*(a + b*x^4)) + (b^2*(5*b*c - 6*a*d)*ArcTan[(Sqrt[
b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(4*a^(7/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.869462, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} (5 b c-2 a d)}{12 a^2 c x^6 (b c-a d)}+\frac{\sqrt{c+d x^4} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{12 a^3 c^2 x^2 (b c-a d)}+\frac{b \sqrt{c+d x^4}}{4 a x^6 \left (a+b x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^4])/(12*a^2*c*(b*c - a*d)*x^6) + ((15*b^2*c^2 - 8
*a*b*c*d - 4*a^2*d^2)*Sqrt[c + d*x^4])/(12*a^3*c^2*(b*c - a*d)*x^2) + (b*Sqrt[c
+ d*x^4])/(4*a*(b*c - a*d)*x^6*(a + b*x^4)) + (b^2*(5*b*c - 6*a*d)*ArcTan[(Sqrt[
b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(4*a^(7/2)*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 136.229, size = 184, normalized size = 0.88 \[ - \frac{b \sqrt{c + d x^{4}}}{4 a x^{6} \left (a + b x^{4}\right ) \left (a d - b c\right )} - \frac{\sqrt{c + d x^{4}} \left (2 a d - 5 b c\right )}{12 a^{2} c x^{6} \left (a d - b c\right )} + \frac{\sqrt{c + d x^{4}} \left (4 a^{2} d^{2} + 8 a b c d - 15 b^{2} c^{2}\right )}{12 a^{3} c^{2} x^{2} \left (a d - b c\right )} + \frac{b^{2} \left (6 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{4 a^{\frac{7}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*sqrt(c + d*x**4)/(4*a*x**6*(a + b*x**4)*(a*d - b*c)) - sqrt(c + d*x**4)*(2*a*
d - 5*b*c)/(12*a**2*c*x**6*(a*d - b*c)) + sqrt(c + d*x**4)*(4*a**2*d**2 + 8*a*b*
c*d - 15*b**2*c**2)/(12*a**3*c**2*x**2*(a*d - b*c)) + b**2*(6*a*d - 5*b*c)*atanh
(x**2*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**4)))/(4*a**(7/2)*(a*d - b*c)**(3/2)
)

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Mathematica [A]  time = 2.13903, size = 195, normalized size = 0.94 \[ \frac{\sqrt{c+d x^4} \left (-\frac{2 a^2}{c}+\frac{3 a b^3 x^8}{\left (a+b x^4\right ) (b c-a d)}+\frac{3 b^2 x^{12} (5 b c-6 a d) \sin ^{-1}\left (\frac{\sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^4}{a}+1}}\right )}{a c^2 \sqrt{\frac{b x^4}{a}+1} \left (\frac{x^4 (b c-a d)}{a c}\right )^{3/2} \sqrt{\frac{a \left (c+d x^4\right )}{c \left (a+b x^4\right )}}}+\frac{4 a x^4 (a d+3 b c)}{c^2}\right )}{12 a^4 x^6} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^4]*((-2*a^2)/c + (4*a*(3*b*c + a*d)*x^4)/c^2 + (3*a*b^3*x^8)/((b*c
- a*d)*(a + b*x^4)) + (3*b^2*(5*b*c - 6*a*d)*x^12*ArcSin[Sqrt[(b/a - d/c)*x^4]/
Sqrt[1 + (b*x^4)/a]])/(a*c^2*(((b*c - a*d)*x^4)/(a*c))^(3/2)*Sqrt[1 + (b*x^4)/a]
*Sqrt[(a*(c + d*x^4))/(c*(a + b*x^4))])))/(12*a^4*x^6)

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Maple [B]  time = 0.024, size = 923, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/a^2*(d*x^4+c)^(1/2)*(-2*d*x^4+c)/x^6/c^2-1/8/a^3*b^2/(a*d-b*c)/(x^2-1/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)+1/8/a^3*b*d*(-a*b)^(1/2)/(a*d-b*c)/(-(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)))-1/8/a^3*b^2/(a*d-b*c)/(x^2+1/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)-1/8/a^3*b*d*(-a*b)^(1/2)/(a*d-b*c)/(-(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)))-5/8/a^3*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)))+5/8/a^3*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)))+b/a^3/c/x^2*(d*x^4+c)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^7),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^7), x)

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Fricas [A]  time = 0.60998, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left ({\left (15 \, b^{3} c^{2} - 8 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{8} - 2 \, a^{2} b c^{2} + 2 \, a^{3} c d + 2 \,{\left (5 \, a b^{2} c^{2} - 3 \, a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{4}\right )} \sqrt{d x^{4} + c} \sqrt{-a b c + a^{2} d} + 3 \,{\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{10} +{\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{6}\right )} \log \left (\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{6} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2}\right )} \sqrt{d x^{4} + c} +{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{48 \,{\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{10} +{\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{6}\right )} \sqrt{-a b c + a^{2} d}}, \frac{2 \,{\left ({\left (15 \, b^{3} c^{2} - 8 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{8} - 2 \, a^{2} b c^{2} + 2 \, a^{3} c d + 2 \,{\left (5 \, a b^{2} c^{2} - 3 \, a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{4}\right )} \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} + 3 \,{\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{10} +{\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{6}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} x^{2}}\right )}{24 \,{\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{10} +{\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{6}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^7),x, algorithm="fricas")

[Out]

[1/48*(4*((15*b^3*c^2 - 8*a*b^2*c*d - 4*a^2*b*d^2)*x^8 - 2*a^2*b*c^2 + 2*a^3*c*d
+ 2*(5*a*b^2*c^2 - 3*a^2*b*c*d - 2*a^3*d^2)*x^4)*sqrt(d*x^4 + c)*sqrt(-a*b*c +
a^2*d) + 3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^10 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*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)))/(((a^
3*b^2*c^3 - a^4*b*c^2*d)*x^10 + (a^4*b*c^3 - a^5*c^2*d)*x^6)*sqrt(-a*b*c + a^2*d
)), 1/24*(2*((15*b^3*c^2 - 8*a*b^2*c*d - 4*a^2*b*d^2)*x^8 - 2*a^2*b*c^2 + 2*a^3*
c*d + 2*(5*a*b^2*c^2 - 3*a^2*b*c*d - 2*a^3*d^2)*x^4)*sqrt(d*x^4 + c)*sqrt(a*b*c
- a^2*d) + 3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^10 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)
*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)))/(((a^3*b^2*c^3 - a^4*b*c^2*d)*x^10 + (a^4*b*c^3 - a^5*c^2*d)*x^6)*sqrt(a*
b*c - a^2*d))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226708, size = 244, normalized size = 1.17 \[ \frac{b^{3} c \sqrt{d + \frac{c}{x^{4}}}}{4 \,{\left (a^{3} b c - a^{4} d\right )}{\left (b c + a{\left (d + \frac{c}{x^{4}}\right )} - a d\right )}} - \frac{{\left (5 \, b^{3} c - 6 \, a b^{2} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{4 \,{\left (a^{3} b c - a^{4} d\right )} \sqrt{a b c - a^{2} d}} + \frac{6 \, a^{3} b c^{5} \sqrt{d + \frac{c}{x^{4}}} - a^{4} c^{4}{\left (d + \frac{c}{x^{4}}\right )}^{\frac{3}{2}} + 3 \, a^{4} c^{4} \sqrt{d + \frac{c}{x^{4}}} d}{6 \, a^{6} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^7),x, algorithm="giac")

[Out]

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