Optimal. Leaf size=307 \[ -\frac{\sqrt{d} \sqrt{a+b x^2} (b c-2 a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-2 a d)}{3 a c^2 x}+\frac{d x \sqrt{a+b x^2} (b c-2 a d)}{3 a c^2 \sqrt{c+d x^2}}-\frac{b \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3} \]
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Rubi [A] time = 0.771586, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{d} \sqrt{a+b x^2} (b c-2 a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-2 a d)}{3 a c^2 x}+\frac{d x \sqrt{a+b x^2} (b c-2 a d)}{3 a c^2 \sqrt{c+d x^2}}-\frac{b \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^2]/(x^4*Sqrt[c + d*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 98.6877, size = 269, normalized size = 0.88 \[ - \frac{\sqrt{a} \sqrt{b} d \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 c^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} - \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{3 c x^{3}} - \frac{b x \sqrt{c + d x^{2}} \left (2 a d - b c\right )}{3 a c^{2} \sqrt{a + b x^{2}}} + \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (2 a d - b c\right )}{3 a c^{2} x} + \frac{\sqrt{b} \sqrt{c + d x^{2}} \left (2 a d - b c\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 \sqrt{a} c^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(1/2)/x**4/(d*x**2+c)**(1/2),x)
[Out]
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Mathematica [C] time = 1.88978, size = 228, normalized size = 0.74 \[ \frac{-\frac{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (a c-2 a d x^2+b c x^2\right )}{a}+i c x^3 \sqrt{\frac{b}{a}} \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (b c-a d) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i c x^3 \sqrt{\frac{b}{a}} \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{3 c^2 x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x^2]/(x^4*Sqrt[c + d*x^2]),x]
[Out]
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Maple [A] time = 0.038, size = 418, normalized size = 1.4 \[{\frac{1}{ \left ( 3\,bd{x}^{4}+3\,ad{x}^{2}+3\,c{x}^{2}b+3\,ac \right ){c}^{2}{x}^{3}a}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 2\,\sqrt{-{\frac{b}{a}}}{x}^{6}ab{d}^{2}-\sqrt{-{\frac{b}{a}}}{x}^{6}{b}^{2}cd+bd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}ac-\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{2}{c}^{2}-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}abcd+\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{2}{c}^{2}+2\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{2}{d}^{2}-\sqrt{-{\frac{b}{a}}}{x}^{4}{b}^{2}{c}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{2}cd-2\,\sqrt{-{\frac{b}{a}}}{x}^{2}ab{c}^{2}-\sqrt{-{\frac{b}{a}}}{a}^{2}{c}^{2} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(1/2)/x^4/(d*x^2+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)/(sqrt(d*x^2 + c)*x^4),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)/(sqrt(d*x^2 + c)*x^4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{2}}}{x^{4} \sqrt{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(1/2)/x**4/(d*x**2+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)/(sqrt(d*x^2 + c)*x^4),x, algorithm="giac")
[Out]