Optimal. Leaf size=307 \[ \frac{2 \sqrt{d} \sqrt{a+b x^2} (a d+b c) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{2 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{3 a^2 c^2 x}-\frac{2 d x \sqrt{a+b x^2} (a d+b c)}{3 a^2 c^2 \sqrt{c+d x^2}}-\frac{b \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a c x^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.757315, 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 \[ \frac{2 \sqrt{d} \sqrt{a+b x^2} (a d+b c) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{2 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{3 a^2 c^2 x}-\frac{2 d x \sqrt{a+b x^2} (a d+b c)}{3 a^2 c^2 \sqrt{c+d x^2}}-\frac{b \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a c x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 105.076, size = 274, normalized size = 0.89 \[ - \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{3 a c x^{3}} - \frac{2 b x \sqrt{c + d x^{2}} \left (a d + b c\right )}{3 a^{2} c^{2} \sqrt{a + b x^{2}}} + \frac{2 \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d + b c\right )}{3 a^{2} c^{2} x} - \frac{\sqrt{b} d \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 \sqrt{a} c^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{2 \sqrt{b} \sqrt{c + d x^{2}} \left (a d + b c\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 a^{\frac{3}{2}} c^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.595675, size = 229, normalized size = 0.75 \[ \frac{\sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c+2 a d x^2+2 b c x^2\right )-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+2 b c) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+2 i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{3 a^2 c^2 x^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.034, size = 435, normalized size = 1.4 \[{\frac{1}{3\,{a}^{2}{c}^{2}{x}^{3} \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) } \left ( 2\,\sqrt{-{\frac{b}{a}}}{x}^{6}ab{d}^{2}+2\,\sqrt{-{\frac{b}{a}}}{x}^{6}{b}^{2}cd+bd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}ac+2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{2}{c}^{2}-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}abcd-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{2}{c}^{2}+2\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{2}{d}^{2}+3\,\sqrt{-{\frac{b}{a}}}{x}^{4}abcd+2\,\sqrt{-{\frac{b}{a}}}{x}^{4}{b}^{2}{c}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{2}cd+\sqrt{-{\frac{b}{a}}}{x}^{2}ab{c}^{2}-\sqrt{-{\frac{b}{a}}}{a}^{2}{c}^{2} \right ) \sqrt{d{x}^{2}+c}\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^4),x, algorithm="giac")
[Out]