Optimal. Leaf size=196 \[ \frac{x \left (c+d x^2\right )^{3/2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{192 d^2}+\frac{c x \sqrt{c+d x^2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{128 d^2}+\frac{c^2 \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{5/2}}-\frac{b x \left (c+d x^2\right )^{5/2} (3 b c-10 a d)}{48 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d} \]
[Out]
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Rubi [A] time = 0.270342, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x \left (c+d x^2\right )^{3/2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{192 d^2}+\frac{c x \sqrt{c+d x^2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{128 d^2}+\frac{c^2 \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{5/2}}-\frac{b x \left (c+d x^2\right )^{5/2} (3 b c-10 a d)}{48 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2*(c + d*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 27.3776, size = 192, normalized size = 0.98 \[ \frac{b x \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{5}{2}}}{8 d} + \frac{b x \left (c + d x^{2}\right )^{\frac{5}{2}} \left (10 a d - 3 b c\right )}{48 d^{2}} + \frac{c^{2} \left (48 a^{2} d^{2} - 16 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{128 d^{\frac{5}{2}}} + \frac{c x \sqrt{c + d x^{2}} \left (48 a^{2} d^{2} - 16 a b c d + 3 b^{2} c^{2}\right )}{128 d^{2}} + \frac{x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (48 a^{2} d^{2} - 16 a b c d + 3 b^{2} c^{2}\right )}{192 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.163647, size = 159, normalized size = 0.81 \[ \frac{3 c^2 \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+\sqrt{d} x \sqrt{c+d x^2} \left (48 a^2 d^2 \left (5 c+2 d x^2\right )+16 a b d \left (3 c^2+14 c d x^2+8 d^2 x^4\right )+b^2 \left (-9 c^3+6 c^2 d x^2+72 c d^2 x^4+48 d^3 x^6\right )\right )}{384 d^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2*(c + d*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.011, size = 249, normalized size = 1.3 \[{\frac{{a}^{2}x}{4} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}cx}{8}\sqrt{d{x}^{2}+c}}+{\frac{3\,{a}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}+{\frac{{b}^{2}{x}^{3}}{8\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}cx}{16\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}{c}^{2}x}{64\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,x{b}^{2}{c}^{3}}{128\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{b}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{abx}{3\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{abcx}{12\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{ab{c}^{2}x}{8\,d}\sqrt{d{x}^{2}+c}}-{\frac{ab{c}^{3}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.327499, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, b^{2} d^{3} x^{7} + 8 \,{\left (9 \, b^{2} c d^{2} + 16 \, a b d^{3}\right )} x^{5} + 2 \,{\left (3 \, b^{2} c^{2} d + 112 \, a b c d^{2} + 48 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (3 \, b^{2} c^{3} - 16 \, a b c^{2} d - 80 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 3 \,{\left (3 \, b^{2} c^{4} - 16 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{768 \, d^{\frac{5}{2}}}, \frac{{\left (48 \, b^{2} d^{3} x^{7} + 8 \,{\left (9 \, b^{2} c d^{2} + 16 \, a b d^{3}\right )} x^{5} + 2 \,{\left (3 \, b^{2} c^{2} d + 112 \, a b c d^{2} + 48 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (3 \, b^{2} c^{3} - 16 \, a b c^{2} d - 80 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} + 3 \,{\left (3 \, b^{2} c^{4} - 16 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{384 \, \sqrt{-d} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 88.1589, size = 440, normalized size = 2.24 \[ \frac{a^{2} c^{\frac{3}{2}} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} + \frac{a^{2} c^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a^{2} \sqrt{c} d x^{3}}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a^{2} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 \sqrt{d}} + \frac{a^{2} d^{2} x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{a b c^{\frac{5}{2}} x}{8 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 a b c^{\frac{3}{2}} x^{3}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{11 a b \sqrt{c} d x^{5}}{12 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 d^{\frac{3}{2}}} + \frac{a b d^{2} x^{7}}{3 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{3 b^{2} c^{\frac{7}{2}} x}{128 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{5}{2}} x^{3}}{128 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{13 b^{2} c^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} \sqrt{c} d x^{7}}{16 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{128 d^{\frac{5}{2}}} + \frac{b^{2} d^{2} x^{9}}{8 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.235794, size = 236, normalized size = 1.2 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, b^{2} d x^{2} + \frac{9 \, b^{2} c d^{6} + 16 \, a b d^{7}}{d^{6}}\right )} x^{2} + \frac{3 \, b^{2} c^{2} d^{5} + 112 \, a b c d^{6} + 48 \, a^{2} d^{7}}{d^{6}}\right )} x^{2} - \frac{3 \,{\left (3 \, b^{2} c^{3} d^{4} - 16 \, a b c^{2} d^{5} - 80 \, a^{2} c d^{6}\right )}}{d^{6}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (3 \, b^{2} c^{4} - 16 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{128 \, d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2),x, algorithm="giac")
[Out]