3.111 \(\int \frac{1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx\)

Optimal. Leaf size=266 \[ \frac{b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{20 a^2 c \sqrt [4]{a+b x^4} (b c-a d)^3}+\frac{3 d^2 (4 b c-a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}+\frac{3 d^2 (4 b c-a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}-\frac{d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}+\frac{b x (5 a d+4 b c)}{20 a c \left (a+b x^4\right )^{5/4} (b c-a d)^2} \]

[Out]

(b*(4*b*c + 5*a*d)*x)/(20*a*c*(b*c - a*d)^2*(a + b*x^4)^(5/4)) + (b*(16*b^2*c^2
- 56*a*b*c*d - 5*a^2*d^2)*x)/(20*a^2*c*(b*c - a*d)^3*(a + b*x^4)^(1/4)) - (d*x)/
(4*c*(b*c - a*d)*(a + b*x^4)^(5/4)*(c + d*x^4)) + (3*d^2*(4*b*c - a*d)*ArcTan[((
b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*(b*c - a*d)^(13/4))
+ (3*d^2*(4*b*c - a*d)*ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4)
)])/(8*c^(7/4)*(b*c - a*d)^(13/4))

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Rubi [A]  time = 0.884767, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{20 a^2 c \sqrt [4]{a+b x^4} (b c-a d)^3}+\frac{3 d^2 (4 b c-a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}+\frac{3 d^2 (4 b c-a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}-\frac{d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}+\frac{b x (5 a d+4 b c)}{20 a c \left (a+b x^4\right )^{5/4} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)^(9/4)*(c + d*x^4)^2),x]

[Out]

(b*(4*b*c + 5*a*d)*x)/(20*a*c*(b*c - a*d)^2*(a + b*x^4)^(5/4)) + (b*(16*b^2*c^2
- 56*a*b*c*d - 5*a^2*d^2)*x)/(20*a^2*c*(b*c - a*d)^3*(a + b*x^4)^(1/4)) - (d*x)/
(4*c*(b*c - a*d)*(a + b*x^4)^(5/4)*(c + d*x^4)) + (3*d^2*(4*b*c - a*d)*ArcTan[((
b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*(b*c - a*d)^(13/4))
+ (3*d^2*(4*b*c - a*d)*ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4)
)])/(8*c^(7/4)*(b*c - a*d)^(13/4))

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Rubi in Sympy [A]  time = 144.937, size = 240, normalized size = 0.9 \[ \frac{d x}{4 c \left (a + b x^{4}\right )^{\frac{5}{4}} \left (c + d x^{4}\right ) \left (a d - b c\right )} - \frac{3 d^{2} \left (a d - 4 b c\right ) \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} \left (- a d + b c\right )^{\frac{13}{4}}} - \frac{3 d^{2} \left (a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} \left (- a d + b c\right )^{\frac{13}{4}}} + \frac{b x \left (5 a d + 4 b c\right )}{20 a c \left (a + b x^{4}\right )^{\frac{5}{4}} \left (a d - b c\right )^{2}} + \frac{b x \left (5 a^{2} d^{2} + 56 a b c d - 16 b^{2} c^{2}\right )}{20 a^{2} c \sqrt [4]{a + b x^{4}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(9/4)/(d*x**4+c)**2,x)

[Out]

d*x/(4*c*(a + b*x**4)**(5/4)*(c + d*x**4)*(a*d - b*c)) - 3*d**2*(a*d - 4*b*c)*at
an(x*(-a*d + b*c)**(1/4)/(c**(1/4)*(a + b*x**4)**(1/4)))/(8*c**(7/4)*(-a*d + b*c
)**(13/4)) - 3*d**2*(a*d - 4*b*c)*atanh(x*(-a*d + b*c)**(1/4)/(c**(1/4)*(a + b*x
**4)**(1/4)))/(8*c**(7/4)*(-a*d + b*c)**(13/4)) + b*x*(5*a*d + 4*b*c)/(20*a*c*(a
+ b*x**4)**(5/4)*(a*d - b*c)**2) + b*x*(5*a**2*d**2 + 56*a*b*c*d - 16*b**2*c**2
)/(20*a**2*c*(a + b*x**4)**(1/4)*(a*d - b*c)**3)

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Mathematica [A]  time = 0.822363, size = 242, normalized size = 0.91 \[ \frac{1}{20} x \left (a+b x^4\right )^{3/4} \left (\frac{8 b^2 (7 a d-2 b c)}{a^2 \left (a+b x^4\right ) (a d-b c)^3}+\frac{4 b^2}{a \left (a+b x^4\right )^2 (b c-a d)^2}-\frac{5 d^3}{c \left (c+d x^4\right ) (b c-a d)^3}\right )+\frac{3 d^2 (4 b c-a d) \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{16 c^{7/4} (b c-a d)^{13/4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^4)^(9/4)*(c + d*x^4)^2),x]

[Out]

(x*(a + b*x^4)^(3/4)*((4*b^2)/(a*(b*c - a*d)^2*(a + b*x^4)^2) + (8*b^2*(-2*b*c +
7*a*d))/(a^2*(-(b*c) + a*d)^3*(a + b*x^4)) - (5*d^3)/(c*(b*c - a*d)^3*(c + d*x^
4))))/20 + (3*d^2*(4*b*c - a*d)*(2*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*
x^4)^(1/4))] - Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + Log[c^(1
/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(16*c^(7/4)*(b*c - a*d)^(13/4))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x)

[Out]

int(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{9}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)^2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)^2), x)

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Fricas [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/((b*x^4 + a)^(9/4)*(d*x^4 + c)^2),x, algorithm="fricas")

[Out]

Timed out

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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/(b*x**4+a)**(9/4)/(d*x**4+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{9}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)^2), x)