Optimal. Leaf size=292 \[ -\frac{d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c \left (a+b x^2\right )^{3/4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2} \]
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Rubi [A] time = 0.646343, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c \left (a+b x^2\right )^{3/4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)^(3/4)*(c + d*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 113.161, size = 250, normalized size = 0.86 \[ - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - \frac{5 b c}{2}\right ) \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 c x \left (a d - b c\right )^{2}} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - \frac{5 b c}{2}\right ) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 c x \left (a d - b c\right )^{2}} + \frac{\sqrt{a} \sqrt{b} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 c \left (a + b x^{2}\right )^{\frac{3}{4}} \left (a d - b c\right )} + \frac{d x \sqrt [4]{a + b x^{2}}}{2 c \left (c + d x^{2}\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)**(3/4)/(d*x**2+c)**2,x)
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Mathematica [C] time = 0.811733, size = 340, normalized size = 1.16 \[ \frac{x \left (\frac{5 a b d x^2 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-10 a c F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}+\frac{18 a (a d-2 b c) F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}-\frac{3 d \left (a+b x^2\right )}{c}\right )}{6 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) (b c-a d)} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((a + b*x^2)^(3/4)*(c + d*x^2)^2),x]
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Maple [F] time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)^2),x, algorithm="maxima")
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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^2 + a)^(3/4)*(d*x^2 + c)^2),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)**(3/4)/(d*x**2+c)**2,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)^2),x, algorithm="giac")
[Out]