Optimal. Leaf size=281 \[ \frac{c^4 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{9/2}}-\frac{c^3 x \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1024 d^4}+\frac{c^2 x^3 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1536 d^3}+\frac{x^5 \left (c+d x^2\right )^{3/2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{192 d^2}+\frac{c x^5 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{384 d^2}-\frac{b x^5 \left (c+d x^2\right )^{5/2} (7 b c-24 a d)}{120 d^2}+\frac{b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d} \]
[Out]
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Rubi [A] time = 0.669416, antiderivative size = 278, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{c^4 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{9/2}}-\frac{c^3 x \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1024 d^4}+\frac{c^2 x^3 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1536 d^3}+\frac{1}{192} x^5 \left (c+d x^2\right )^{3/2} \left (24 a^2+\frac{b c (7 b c-24 a d)}{d^2}\right )+\frac{c x^5 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{384 d^2}-\frac{b x^5 \left (c+d x^2\right )^{5/2} (7 b c-24 a d)}{120 d^2}+\frac{b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d} \]
Antiderivative was successfully verified.
[In] Int[x^4*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 49.2156, size = 270, normalized size = 0.96 \[ \frac{b^{2} x^{7} \left (c + d x^{2}\right )^{\frac{5}{2}}}{12 d} + \frac{b x^{5} \left (c + d x^{2}\right )^{\frac{5}{2}} \left (24 a d - 7 b c\right )}{120 d^{2}} + \frac{c^{4} \left (24 a^{2} d^{2} - b c \left (24 a d - 7 b c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{1024 d^{\frac{9}{2}}} - \frac{c^{3} x \sqrt{c + d x^{2}} \left (24 a^{2} d^{2} - b c \left (24 a d - 7 b c\right )\right )}{1024 d^{4}} + \frac{c^{2} x^{3} \sqrt{c + d x^{2}} \left (24 a^{2} d^{2} - b c \left (24 a d - 7 b c\right )\right )}{1536 d^{3}} + \frac{c x^{5} \sqrt{c + d x^{2}} \left (24 a^{2} d^{2} - b c \left (24 a d - 7 b c\right )\right )}{384 d^{2}} + \frac{x^{5} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (24 a^{2} d^{2} - b c \left (24 a d - 7 b c\right )\right )}{192 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(b*x**2+a)**2*(d*x**2+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.244492, size = 225, normalized size = 0.8 \[ \frac{15 c^4 \left (24 a^2 d^2-24 a b c d+7 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+\sqrt{d} x \sqrt{c+d x^2} \left (120 a^2 d^2 \left (-3 c^3+2 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right )+24 a b d \left (15 c^4-10 c^3 d x^2+8 c^2 d^2 x^4+176 c d^3 x^6+128 d^4 x^8\right )+b^2 \left (-105 c^5+70 c^4 d x^2-56 c^3 d^2 x^4+48 c^2 d^3 x^6+1664 c d^4 x^8+1280 d^5 x^{10}\right )\right )}{15360 d^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.03, size = 389, normalized size = 1.4 \[{\frac{{a}^{2}{x}^{3}}{8\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}cx}{16\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}{c}^{2}x}{64\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}{c}^{3}x}{128\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{a}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{{b}^{2}{x}^{7}}{12\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{b}^{2}c{x}^{5}}{120\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}{c}^{2}{x}^{3}}{192\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{7\,x{b}^{2}{c}^{3}}{384\,{d}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}{c}^{4}x}{1536\,{d}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}{c}^{5}x}{1024\,{d}^{4}}\sqrt{d{x}^{2}+c}}+{\frac{7\,{b}^{2}{c}^{6}}{1024}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{9}{2}}}}+{\frac{ab{x}^{5}}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{abc{x}^{3}}{8\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{ab{c}^{2}x}{16\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{ab{c}^{3}x}{64\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,ab{c}^{4}x}{128\,{d}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{3\,ab{c}^{5}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(b*x^2+a)^2*(d*x^2+c)^(3/2),x)
[Out]
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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)^2*(d*x^2 + c)^(3/2)*x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.83707, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, b^{2} d^{5} x^{11} + 128 \,{\left (13 \, b^{2} c d^{4} + 24 \, a b d^{5}\right )} x^{9} + 48 \,{\left (b^{2} c^{2} d^{3} + 88 \, a b c d^{4} + 40 \, a^{2} d^{5}\right )} x^{7} - 8 \,{\left (7 \, b^{2} c^{3} d^{2} - 24 \, a b c^{2} d^{3} - 360 \, a^{2} c d^{4}\right )} x^{5} + 10 \,{\left (7 \, b^{2} c^{4} d - 24 \, a b c^{3} d^{2} + 24 \, a^{2} c^{2} d^{3}\right )} x^{3} - 15 \,{\left (7 \, b^{2} c^{5} - 24 \, a b c^{4} d + 24 \, a^{2} c^{3} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 15 \,{\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{30720 \, d^{\frac{9}{2}}}, \frac{{\left (1280 \, b^{2} d^{5} x^{11} + 128 \,{\left (13 \, b^{2} c d^{4} + 24 \, a b d^{5}\right )} x^{9} + 48 \,{\left (b^{2} c^{2} d^{3} + 88 \, a b c d^{4} + 40 \, a^{2} d^{5}\right )} x^{7} - 8 \,{\left (7 \, b^{2} c^{3} d^{2} - 24 \, a b c^{2} d^{3} - 360 \, a^{2} c d^{4}\right )} x^{5} + 10 \,{\left (7 \, b^{2} c^{4} d - 24 \, a b c^{3} d^{2} + 24 \, a^{2} c^{2} d^{3}\right )} x^{3} - 15 \,{\left (7 \, b^{2} c^{5} - 24 \, a b c^{4} d + 24 \, a^{2} c^{3} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} + 15 \,{\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{15360 \, \sqrt{-d} d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)*x^4,x, algorithm="fricas")
[Out]
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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(x**4*(b*x**2+a)**2*(d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.246887, size = 355, normalized size = 1.26 \[ \frac{1}{15360} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, b^{2} d x^{2} + \frac{13 \, b^{2} c d^{10} + 24 \, a b d^{11}}{d^{10}}\right )} x^{2} + \frac{3 \,{\left (b^{2} c^{2} d^{9} + 88 \, a b c d^{10} + 40 \, a^{2} d^{11}\right )}}{d^{10}}\right )} x^{2} - \frac{7 \, b^{2} c^{3} d^{8} - 24 \, a b c^{2} d^{9} - 360 \, a^{2} c d^{10}}{d^{10}}\right )} x^{2} + \frac{5 \,{\left (7 \, b^{2} c^{4} d^{7} - 24 \, a b c^{3} d^{8} + 24 \, a^{2} c^{2} d^{9}\right )}}{d^{10}}\right )} x^{2} - \frac{15 \,{\left (7 \, b^{2} c^{5} d^{6} - 24 \, a b c^{4} d^{7} + 24 \, a^{2} c^{3} d^{8}\right )}}{d^{10}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{1024 \, d^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)*x^4,x, algorithm="giac")
[Out]