Optimal. Leaf size=377 \[ -\frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+9 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 a^{7/4} \sqrt{a+b x^4}}+\frac{3 b^{5/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} \sqrt{a+b x^4}}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{3 b^{3/2} c x \sqrt{a+b x^4}}{5 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3 b c \sqrt{a+b x^4}}{5 a^2 x}-\frac{c \sqrt{a+b x^4}}{5 a x^5}-\frac{d \sqrt{a+b x^4}}{4 a x^4}-\frac{e \sqrt{a+b x^4}}{3 a x^3}-\frac{f \sqrt{a+b x^4}}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.864639, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367 \[ -\frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (5 \sqrt{a} e+9 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 a^{7/4} \sqrt{a+b x^4}}+\frac{3 b^{5/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} \sqrt{a+b x^4}}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{3 b^{3/2} c x \sqrt{a+b x^4}}{5 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3 b c \sqrt{a+b x^4}}{5 a^2 x}-\frac{c \sqrt{a+b x^4}}{5 a x^5}-\frac{d \sqrt{a+b x^4}}{4 a x^4}-\frac{e \sqrt{a+b x^4}}{3 a x^3}-\frac{f \sqrt{a+b x^4}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3)/(x^6*Sqrt[a + b*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 98.6803, size = 342, normalized size = 0.91 \[ - \frac{c \sqrt{a + b x^{4}}}{5 a x^{5}} - \frac{d \sqrt{a + b x^{4}}}{4 a x^{4}} - \frac{e \sqrt{a + b x^{4}}}{3 a x^{3}} - \frac{f \sqrt{a + b x^{4}}}{2 a x^{2}} - \frac{3 b^{\frac{3}{2}} c x \sqrt{a + b x^{4}}}{5 a^{2} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{3 b c \sqrt{a + b x^{4}}}{5 a^{2} x} + \frac{b d \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{4}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} + \frac{3 b^{\frac{5}{4}} c \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{7}{4}} \sqrt{a + b x^{4}}} - \frac{b^{\frac{3}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) \left (5 \sqrt{a} e + 9 \sqrt{b} c\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{30 a^{\frac{7}{4}} \sqrt{a + b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**3+e*x**2+d*x+c)/x**6/(b*x**4+a)**(1/2),x)
[Out]
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Mathematica [C] time = 0.905436, size = 268, normalized size = 0.71 \[ \frac{-36 \sqrt{a} b^{3/2} c x^5 \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (15 \sqrt{a} b d x^5 \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\left (a+b x^4\right ) \left (12 a c+5 a x \left (3 d+4 e x+6 f x^2\right )-36 b c x^4\right )\right )+4 \sqrt{a} b x^5 \sqrt{\frac{b x^4}{a}+1} \left (9 \sqrt{b} c+5 i \sqrt{a} e\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{60 a^2 x^5 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3)/(x^6*Sqrt[a + b*x^4]),x]
[Out]
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Maple [C] time = 0.015, size = 354, normalized size = 0.9 \[ -{\frac{c}{5\,a{x}^{5}}\sqrt{b{x}^{4}+a}}+{\frac{3\,bc}{5\,{a}^{2}x}\sqrt{b{x}^{4}+a}}-{{\frac{3\,i}{5}}c{b}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{3\,i}{5}}c{b}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{d}{4\,a{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{bd}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{e}{3\,a{x}^{3}}\sqrt{b{x}^{4}+a}}-{\frac{be}{3\,a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{f}{2\,a{x}^{2}}\sqrt{b{x}^{4}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^3+e*x^2+d*x+c)/x^6/(b*x^4+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x^{3} + e x^{2} + d x + c}{\sqrt{b x^{4} + a} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^3 + e*x^2 + d*x + c)/(sqrt(b*x^4 + a)*x^6),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{f x^{3} + e x^{2} + d x + c}{\sqrt{b x^{4} + a} x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^3 + e*x^2 + d*x + c)/(sqrt(b*x^4 + a)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.77864, size = 163, normalized size = 0.43 \[ - \frac{\sqrt{b} d \sqrt{\frac{a}{b x^{4}} + 1}}{4 a x^{2}} - \frac{\sqrt{b} f \sqrt{\frac{a}{b x^{4}} + 1}}{2 a} + \frac{c \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x^{5} \Gamma \left (- \frac{1}{4}\right )} + \frac{e \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x^{3} \Gamma \left (\frac{1}{4}\right )} + \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**3+e*x**2+d*x+c)/x**6/(b*x**4+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x^{3} + e x^{2} + d x + c}{\sqrt{b x^{4} + a} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^3 + e*x^2 + d*x + c)/(sqrt(b*x^4 + a)*x^6),x, algorithm="giac")
[Out]