Optimal. Leaf size=278 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\sqrt{b} \sqrt [4]{a^2 d+b^2 c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\sqrt{b} \sqrt [4]{a^2 d+b^2 c}}-\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{b x \sqrt{a^2 d+b^2 c}}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{b x \sqrt{a^2 d+b^2 c}} \]
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Rubi [A] time = 0.769595, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\sqrt{b} \sqrt [4]{a^2 d+b^2 c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\sqrt{b} \sqrt [4]{a^2 d+b^2 c}}-\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{b x \sqrt{a^2 d+b^2 c}}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{b x \sqrt{a^2 d+b^2 c}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)*(c + d*x^2)^(1/4)),x]
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Rubi in Sympy [A] time = 110.108, size = 248, normalized size = 0.89 \[ - \frac{a \sqrt [4]{c} \sqrt{- \frac{d x^{2}}{c}} \Pi \left (- \frac{b \sqrt{c}}{\sqrt{a^{2} d + b^{2} c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{c + d x^{2}}}{\sqrt [4]{c}} \right )}\middle | -1\right )}{b x \sqrt{a^{2} d + b^{2} c}} + \frac{a \sqrt [4]{c} \sqrt{- \frac{d x^{2}}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{a^{2} d + b^{2} c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{c + d x^{2}}}{\sqrt [4]{c}} \right )}\middle | -1\right )}{b x \sqrt{a^{2} d + b^{2} c}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt [4]{c + d x^{2}}}{\sqrt [4]{a^{2} d + b^{2} c}} \right )}}{\sqrt{b} \sqrt [4]{a^{2} d + b^{2} c}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [4]{c + d x^{2}}}{\sqrt [4]{a^{2} d + b^{2} c}} \right )}}{\sqrt{b} \sqrt [4]{a^{2} d + b^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)/(d*x**2+c)**(1/4),x)
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Mathematica [C] time = 0.144068, size = 126, normalized size = 0.45 \[ -\frac{2 \sqrt [4]{\frac{b \left (x-\sqrt{-\frac{c}{d}}\right )}{a+b x}} \sqrt [4]{\frac{b \left (\sqrt{-\frac{c}{d}}+x\right )}{a+b x}} F_1\left (\frac{1}{2};\frac{1}{4},\frac{1}{4};\frac{3}{2};\frac{a-b \sqrt{-\frac{c}{d}}}{a+b x},\frac{a+b \sqrt{-\frac{c}{d}}}{a+b x}\right )}{b \sqrt [4]{c+d x^2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((a + b*x)*(c + d*x^2)^(1/4)),x]
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Maple [F] time = 0.061, size = 0, normalized size = 0. \[ \int{\frac{1}{bx+a}{\frac{1}{\sqrt [4]{d{x}^{2}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)/(d*x^2+c)^(1/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{2} + c\right )}^{\frac{1}{4}}{\left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^2 + c)^(1/4)*(b*x + a)),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/((d*x^2 + c)^(1/4)*(b*x + a)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right ) \sqrt [4]{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)/(d*x**2+c)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{2} + c\right )}^{\frac{1}{4}}{\left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^2 + c)^(1/4)*(b*x + a)),x, algorithm="giac")
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