Optimal. Leaf size=149 \[ \frac{c (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}+\frac{x \sqrt{c+d x^2} (3 b c-4 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)}+\frac{b x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.229334, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{c (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}+\frac{x \sqrt{c+d x^2} (3 b c-4 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)}+\frac{b x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^2]/(a + b*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 32.7139, size = 129, normalized size = 0.87 \[ - \frac{b x \left (c + d x^{2}\right )^{\frac{3}{2}}}{4 a \left (a + b x^{2}\right )^{2} \left (a d - b c\right )} + \frac{x \sqrt{c + d x^{2}} \left (4 a d - 3 b c\right )}{8 a^{2} \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{c \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{8 a^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**(1/2)/(b*x**2+a)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.262205, size = 130, normalized size = 0.87 \[ \frac{\frac{\sqrt{a} x \sqrt{c+d x^2} \left (-4 a^2 d+a b \left (5 c-2 d x^2\right )+3 b^2 c x^2\right )}{\left (a+b x^2\right )^2 (b c-a d)}+\frac{c (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{3/2}}}{8 a^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x^2]/(a + b*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.05, size = 5177, normalized size = 34.7 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^(1/2)/(b*x^2+a)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.404444, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{3} +{\left (5 \, a b c - 4 \, a^{2} d\right )} x\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} +{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d +{\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{4} + 2 \,{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} + 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{32 \,{\left (a^{4} b c - a^{5} d +{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{4} + 2 \,{\left (a^{3} b^{2} c - a^{4} b d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d}}, \frac{2 \,{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{3} +{\left (5 \, a b c - 4 \, a^{2} d\right )} x\right )} \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} +{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d +{\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{4} + 2 \,{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{16 \,{\left (a^{4} b c - a^{5} d +{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{4} + 2 \,{\left (a^{3} b^{2} c - a^{4} b d\right )} x^{2}\right )} \sqrt{a b c - a^{2} d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**(1/2)/(b*x**2+a)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 1.24272, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^3,x, algorithm="giac")
[Out]