Optimal. Leaf size=130 \[ -\frac{3 (b c-a d) \left ((a d+b c)^2+4 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{7/2}}+\frac{3 x (b c-a d)^2 (a d+3 b c)}{8 c^2 d^3 \left (c+d x^2\right )}-\frac{x (b c-a d)^3}{4 c d^3 \left (c+d x^2\right )^2}+\frac{b^3 x}{d^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.365222, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{3 (b c-a d) \left ((a d+b c)^2+4 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{7/2}}+\frac{3 x (b c-a d)^2 (a d+3 b c)}{8 c^2 d^3 \left (c+d x^2\right )}-\frac{x (b c-a d)^3}{4 c d^3 \left (c+d x^2\right )^2}+\frac{b^3 x}{d^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^3/(c + d*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int b^{3}\, dx}{d^{3}} + \frac{x \left (a d - b c\right )^{3}}{4 c d^{3} \left (c + d x^{2}\right )^{2}} + \frac{3 x \left (a d - b c\right )^{2} \left (a d + 3 b c\right )}{8 c^{2} d^{3} \left (c + d x^{2}\right )} + \frac{3 \left (a d - b c\right ) \left (4 b^{2} c^{2} + \left (a d + b c\right )^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{5}{2}} d^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**3/(d*x**2+c)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.127599, size = 141, normalized size = 1.08 \[ -\frac{3 \left (-a^3 d^3-a^2 b c d^2-3 a b^2 c^2 d+5 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{7/2}}+\frac{3 x (b c-a d)^2 (a d+3 b c)}{8 c^2 d^3 \left (c+d x^2\right )}-\frac{x (b c-a d)^3}{4 c d^3 \left (c+d x^2\right )^2}+\frac{b^3 x}{d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^3/(c + d*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.015, size = 266, normalized size = 2.1 \[{\frac{{b}^{3}x}{{d}^{3}}}+{\frac{3\,d{x}^{3}{a}^{3}}{8\, \left ( d{x}^{2}+c \right ) ^{2}{c}^{2}}}+{\frac{3\,{x}^{3}{a}^{2}b}{8\, \left ( d{x}^{2}+c \right ) ^{2}c}}-{\frac{15\,a{b}^{2}{x}^{3}}{8\,d \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{9\,c{x}^{3}{b}^{3}}{8\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,x{a}^{3}}{8\, \left ( d{x}^{2}+c \right ) ^{2}c}}-{\frac{3\,{a}^{2}bx}{8\,d \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{9\,acx{b}^{2}}{8\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{7\,{c}^{2}x{b}^{3}}{8\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{a}^{3}}{8\,{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{a}^{2}b}{8\,cd}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{9\,a{b}^{2}}{8\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,{b}^{3}c}{8\,{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^3/(d*x^2+c)^3,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/(d*x^2 + c)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.247622, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, b^{3} c^{5} - 3 \, a b^{2} c^{4} d - a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} +{\left (5 \, b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} - a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{4} + 2 \,{\left (5 \, b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} - a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) - 2 \,{\left (8 \, b^{3} c^{2} d^{2} x^{5} +{\left (25 \, b^{3} c^{3} d - 15 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + 3 \, a^{3} d^{4}\right )} x^{3} +{\left (15 \, b^{3} c^{4} - 9 \, a b^{2} c^{3} d - 3 \, a^{2} b c^{2} d^{2} + 5 \, a^{3} c d^{3}\right )} x\right )} \sqrt{-c d}}{16 \,{\left (c^{2} d^{5} x^{4} + 2 \, c^{3} d^{4} x^{2} + c^{4} d^{3}\right )} \sqrt{-c d}}, -\frac{3 \,{\left (5 \, b^{3} c^{5} - 3 \, a b^{2} c^{4} d - a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} +{\left (5 \, b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} - a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{4} + 2 \,{\left (5 \, b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} - a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (8 \, b^{3} c^{2} d^{2} x^{5} +{\left (25 \, b^{3} c^{3} d - 15 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + 3 \, a^{3} d^{4}\right )} x^{3} +{\left (15 \, b^{3} c^{4} - 9 \, a b^{2} c^{3} d - 3 \, a^{2} b c^{2} d^{2} + 5 \, a^{3} c d^{3}\right )} x\right )} \sqrt{c d}}{8 \,{\left (c^{2} d^{5} x^{4} + 2 \, c^{3} d^{4} x^{2} + c^{4} d^{3}\right )} \sqrt{c d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^3/(d*x^2 + c)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 8.27276, size = 422, normalized size = 3.25 \[ \frac{b^{3} x}{d^{3}} - \frac{3 \sqrt{- \frac{1}{c^{5} d^{7}}} \left (a d - b c\right ) \left (a^{2} d^{2} + 2 a b c d + 5 b^{2} c^{2}\right ) \log{\left (- \frac{3 c^{3} d^{3} \sqrt{- \frac{1}{c^{5} d^{7}}} \left (a d - b c\right ) \left (a^{2} d^{2} + 2 a b c d + 5 b^{2} c^{2}\right )}{3 a^{3} d^{3} + 3 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - 15 b^{3} c^{3}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{c^{5} d^{7}}} \left (a d - b c\right ) \left (a^{2} d^{2} + 2 a b c d + 5 b^{2} c^{2}\right ) \log{\left (\frac{3 c^{3} d^{3} \sqrt{- \frac{1}{c^{5} d^{7}}} \left (a d - b c\right ) \left (a^{2} d^{2} + 2 a b c d + 5 b^{2} c^{2}\right )}{3 a^{3} d^{3} + 3 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - 15 b^{3} c^{3}} + x \right )}}{16} + \frac{x^{3} \left (3 a^{3} d^{4} + 3 a^{2} b c d^{3} - 15 a b^{2} c^{2} d^{2} + 9 b^{3} c^{3} d\right ) + x \left (5 a^{3} c d^{3} - 3 a^{2} b c^{2} d^{2} - 9 a b^{2} c^{3} d + 7 b^{3} c^{4}\right )}{8 c^{4} d^{3} + 16 c^{3} d^{4} x^{2} + 8 c^{2} d^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**3/(d*x**2+c)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.229855, size = 243, normalized size = 1.87 \[ \frac{b^{3} x}{d^{3}} - \frac{3 \,{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} c^{2} d^{3}} + \frac{9 \, b^{3} c^{3} d x^{3} - 15 \, a b^{2} c^{2} d^{2} x^{3} + 3 \, a^{2} b c d^{3} x^{3} + 3 \, a^{3} d^{4} x^{3} + 7 \, b^{3} c^{4} x - 9 \, a b^{2} c^{3} d x - 3 \, a^{2} b c^{2} d^{2} x + 5 \, a^{3} c d^{3} x}{8 \,{\left (d x^{2} + c\right )}^{2} c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^3/(d*x^2 + c)^3,x, algorithm="giac")
[Out]