3.640 \(\int \frac{1}{x^5 \left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{4 a^2 c^{3/2}}-\frac{\sqrt{c+d x^4}}{4 a c x^4} \]

[Out]

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

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Rubi [A]  time = 0.354658, antiderivative size = 117, 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 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{4 a^2 c^{3/2}}-\frac{\sqrt{c+d x^4}}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 43.99, size = 100, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{4}}}{4 a c x^{4}} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 a^{2} \sqrt{a d - b c}} + \frac{\left (\frac{a d}{2} + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{c}} \right )}}{2 a^{2} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 0.548571, size = 409, normalized size = 3.5 \[ \frac{\frac{6 b d x^8 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{x^4 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}+\frac{5 b d x^4 \left (3 a c+2 a d x^4+b c x^4+3 b d x^8\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )-3 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}{a c \left (-5 b d x^4 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}}{12 x^4 \left (a+b x^4\right ) \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^5*(a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

((6*b*d*x^8*AppellF1[1, 1/2, 1, 2, -((d*x^4)/c), -((b*x^4)/a)])/(-4*a*c*AppellF1
[1, 1/2, 1, 2, -((d*x^4)/c), -((b*x^4)/a)] + x^4*(2*b*c*AppellF1[2, 1/2, 2, 3, -
((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[2, 3/2, 1, 3, -((d*x^4)/c), -((b*x^4)/
a)])) + (5*b*d*x^4*(3*a*c + b*c*x^4 + 2*a*d*x^4 + 3*b*d*x^8)*AppellF1[3/2, 1/2,
1, 5/2, -(c/(d*x^4)), -(a/(b*x^4))] - 3*(a + b*x^4)*(c + d*x^4)*(2*a*d*AppellF1[
5/2, 1/2, 2, 7/2, -(c/(d*x^4)), -(a/(b*x^4))] + b*c*AppellF1[5/2, 3/2, 1, 7/2, -
(c/(d*x^4)), -(a/(b*x^4))]))/(a*c*(-5*b*d*x^4*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*
x^4)), -(a/(b*x^4))] + 2*a*d*AppellF1[5/2, 1/2, 2, 7/2, -(c/(d*x^4)), -(a/(b*x^4
))] + b*c*AppellF1[5/2, 3/2, 1, 7/2, -(c/(d*x^4)), -(a/(b*x^4))])))/(12*x^4*(a +
b*x^4)*Sqrt[c + d*x^4])

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Maple [B]  time = 0.023, size = 402, normalized size = 3.4 \[ -{\frac{1}{4\,ac{x}^{4}}\sqrt{d{x}^{4}+c}}+{\frac{d}{4\,a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{4}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-{\frac{b}{4\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{b}{4\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{b}{2\,{a}^{2}}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{4}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^4+a)/(d*x^4+c)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^5),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^5), x)

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Fricas [A]  time = 0.286413, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, b c^{\frac{3}{2}} x^{4} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{4} + 2 \, b c - a d - 2 \, \sqrt{d x^{4} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{4} + a}\right ) +{\left (2 \, b c + a d\right )} x^{4} \log \left (\frac{{\left (d x^{4} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{4} + c} c}{x^{4}}\right ) - 2 \, \sqrt{d x^{4} + c} a \sqrt{c}}{8 \, a^{2} c^{\frac{3}{2}} x^{4}}, -\frac{4 \, b c^{\frac{3}{2}} x^{4} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{4} + c} b}\right ) -{\left (2 \, b c + a d\right )} x^{4} \log \left (\frac{{\left (d x^{4} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{4} + c} c}{x^{4}}\right ) + 2 \, \sqrt{d x^{4} + c} a \sqrt{c}}{8 \, a^{2} c^{\frac{3}{2}} x^{4}}, \frac{b \sqrt{-c} c x^{4} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{4} + 2 \, b c - a d - 2 \, \sqrt{d x^{4} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{4} + a}\right ) -{\left (2 \, b c + a d\right )} x^{4} \arctan \left (\frac{c}{\sqrt{d x^{4} + c} \sqrt{-c}}\right ) - \sqrt{d x^{4} + c} a \sqrt{-c}}{4 \, a^{2} \sqrt{-c} c x^{4}}, -\frac{2 \, b \sqrt{-c} c x^{4} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{4} + c} b}\right ) +{\left (2 \, b c + a d\right )} x^{4} \arctan \left (\frac{c}{\sqrt{d x^{4} + c} \sqrt{-c}}\right ) + \sqrt{d x^{4} + c} a \sqrt{-c}}{4 \, a^{2} \sqrt{-c} c x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^5),x, algorithm="fricas")

[Out]

[1/8*(2*b*c^(3/2)*x^4*sqrt(b/(b*c - a*d))*log((b*d*x^4 + 2*b*c - a*d - 2*sqrt(d*
x^4 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^4 + a)) + (2*b*c + a*d)*x^4*log((
(d*x^4 + 2*c)*sqrt(c) + 2*sqrt(d*x^4 + c)*c)/x^4) - 2*sqrt(d*x^4 + c)*a*sqrt(c))
/(a^2*c^(3/2)*x^4), -1/8*(4*b*c^(3/2)*x^4*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*
d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x^4 + c)*b)) - (2*b*c + a*d)*x^4*log(((d*x^4 + 2
*c)*sqrt(c) + 2*sqrt(d*x^4 + c)*c)/x^4) + 2*sqrt(d*x^4 + c)*a*sqrt(c))/(a^2*c^(3
/2)*x^4), 1/4*(b*sqrt(-c)*c*x^4*sqrt(b/(b*c - a*d))*log((b*d*x^4 + 2*b*c - a*d -
2*sqrt(d*x^4 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^4 + a)) - (2*b*c + a*d)
*x^4*arctan(c/(sqrt(d*x^4 + c)*sqrt(-c))) - sqrt(d*x^4 + c)*a*sqrt(-c))/(a^2*sqr
t(-c)*c*x^4), -1/4*(2*b*sqrt(-c)*c*x^4*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*
sqrt(-b/(b*c - a*d))/(sqrt(d*x^4 + c)*b)) + (2*b*c + a*d)*x^4*arctan(c/(sqrt(d*x
^4 + c)*sqrt(-c))) + sqrt(d*x^4 + c)*a*sqrt(-c))/(a^2*sqrt(-c)*c*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} \left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

Integral(1/(x**5*(a + b*x**4)*sqrt(c + d*x**4)), x)

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GIAC/XCAS [A]  time = 0.219547, size = 159, normalized size = 1.36 \[ \frac{1}{4} \, d^{2}{\left (\frac{2 \, b^{2} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c d^{2}} - \frac{\sqrt{d x^{4} + c}}{a c d^{2} x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^5),x, algorithm="giac")

[Out]

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