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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]