Optimal. Leaf size=131 \[ -\frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 \sqrt{a} c^{5/2}}-\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)}{8 c^2 x^2}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 c x^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.349063, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 \sqrt{a} c^{5/2}}-\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)}{8 c^2 x^2}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 c x^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(3/2)/(x^5*Sqrt[c + d*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 29.0505, size = 117, normalized size = 0.89 \[ - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{4 c x^{4}} + \frac{3 \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d - b c\right )}{8 c^{2} x^{2}} - \frac{3 \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{8 \sqrt{a} c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(3/2)/x**5/(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.42053, size = 208, normalized size = 1.59 \[ \frac{\frac{6 b d x^6 (b c-a d)^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{-4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}+\left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 a c+3 a d x^2-5 b c x^2\right )}{8 c^2 x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^2)^(3/2)/(x^5*Sqrt[c + d*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.029, size = 352, normalized size = 2.7 \[ -{\frac{1}{16\,{c}^{2}{x}^{4}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 3\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}{a}^{2}{d}^{2}-6\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}abcd+3\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}{b}^{2}{c}^{2}-6\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}da{x}^{2}\sqrt{ac}+10\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}bc{x}^{2}\sqrt{ac}+4\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}ca\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(3/2)/x^5/(d*x^2+c)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/(sqrt(d*x^2 + c)*x^5),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.339389, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} -{\left ({\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{a c}}{x^{4}}\right ) - 4 \,{\left ({\left (5 \, b c - 3 \, a d\right )} x^{2} + 2 \, a c\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{a c}}{32 \, \sqrt{a c} c^{2} x^{4}}, -\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} \arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt{-a c}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} a c}\right ) + 2 \,{\left ({\left (5 \, b c - 3 \, a d\right )} x^{2} + 2 \, a c\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-a c}}{16 \, \sqrt{-a c} c^{2} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/(sqrt(d*x^2 + c)*x^5),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{x^{5} \sqrt{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(3/2)/x**5/(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/(sqrt(d*x^2 + c)*x^5),x, algorithm="giac")
[Out]