Optimal. Leaf size=241 \[ \frac{x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{480 b^2}+\frac{a x \left (a+b x^2\right )^{3/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{384 b^2}+\frac{a^2 x \sqrt{a+b x^2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^2}+\frac{a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{3 d x \left (a+b x^2\right )^{7/2} (4 b c-a d)}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b} \]
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Rubi [A] time = 0.323662, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{480 b^2}+\frac{a x \left (a+b x^2\right )^{3/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{384 b^2}+\frac{a^2 x \sqrt{a+b x^2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^2}+\frac{a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{3 d x \left (a+b x^2\right )^{7/2} (4 b c-a d)}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(5/2)*(c + d*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 32.758, size = 238, normalized size = 0.99 \[ \frac{a^{3} \left (3 a^{2} d^{2} - 20 a b c d + 80 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{256 b^{\frac{5}{2}}} + \frac{a^{2} x \sqrt{a + b x^{2}} \left (3 a^{2} d^{2} - 20 a b c d + 80 b^{2} c^{2}\right )}{256 b^{2}} + \frac{a x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (3 a^{2} d^{2} - 20 a b c d + 80 b^{2} c^{2}\right )}{384 b^{2}} + \frac{d x \left (a + b x^{2}\right )^{\frac{7}{2}} \left (c + d x^{2}\right )}{10 b} - \frac{3 d x \left (a + b x^{2}\right )^{\frac{7}{2}} \left (a d - 4 b c\right )}{80 b^{2}} + \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (3 a^{2} d^{2} - 20 a b c d + 80 b^{2} c^{2}\right )}{480 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(5/2)*(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.211399, size = 191, normalized size = 0.79 \[ \frac{15 a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{b} x \sqrt{a+b x^2} \left (-45 a^4 d^2+30 a^3 b d \left (10 c+d x^2\right )+8 a^2 b^2 \left (330 c^2+295 c d x^2+93 d^2 x^4\right )+16 a b^3 x^2 \left (130 c^2+170 c d x^2+63 d^2 x^4\right )+64 b^4 x^4 \left (10 c^2+15 c d x^2+6 d^2 x^4\right )\right )}{3840 b^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(5/2)*(c + d*x^2)^2,x]
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Maple [A] time = 0.011, size = 308, normalized size = 1.3 \[{\frac{{c}^{2}x}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a{c}^{2}x}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{c}^{2}x}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,{c}^{2}{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{{d}^{2}{x}^{3}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,a{d}^{2}x}{80\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}{d}^{2}x}{160\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{2}{a}^{3}x}{128\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{2}{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{d}^{2}{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{cdx}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{acdx}{24\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,cd{a}^{2}x}{96\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,cd{a}^{3}x}{64\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,cd{a}^{4}}{64}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(5/2)*(d*x^2+c)^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)^(5/2)*(d*x^2 + c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.556019, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (384 \, b^{4} d^{2} x^{9} + 48 \,{\left (20 \, b^{4} c d + 21 \, a b^{3} d^{2}\right )} x^{7} + 8 \,{\left (80 \, b^{4} c^{2} + 340 \, a b^{3} c d + 93 \, a^{2} b^{2} d^{2}\right )} x^{5} + 10 \,{\left (208 \, a b^{3} c^{2} + 236 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + 15 \,{\left (176 \, a^{2} b^{2} c^{2} + 20 \, a^{3} b c d - 3 \, a^{4} d^{2}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} + 15 \,{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{7680 \, b^{\frac{5}{2}}}, \frac{{\left (384 \, b^{4} d^{2} x^{9} + 48 \,{\left (20 \, b^{4} c d + 21 \, a b^{3} d^{2}\right )} x^{7} + 8 \,{\left (80 \, b^{4} c^{2} + 340 \, a b^{3} c d + 93 \, a^{2} b^{2} d^{2}\right )} x^{5} + 10 \,{\left (208 \, a b^{3} c^{2} + 236 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + 15 \,{\left (176 \, a^{2} b^{2} c^{2} + 20 \, a^{3} b c d - 3 \, a^{4} d^{2}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 15 \,{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{3840 \, \sqrt{-b} b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)*(d*x^2 + c)^2,x, algorithm="fricas")
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Sympy [A] time = 170.682, size = 537, normalized size = 2.23 \[ - \frac{3 a^{\frac{9}{2}} d^{2} x}{256 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{7}{2}} c d x}{64 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{\frac{7}{2}} d^{2} x^{3}}{256 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{\frac{5}{2}} c^{2} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 a^{\frac{5}{2}} c^{2} x}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 a^{\frac{5}{2}} c d x^{3}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{129 a^{\frac{5}{2}} d^{2} x^{5}}{640 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 a^{\frac{3}{2}} b c^{2} x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 a^{\frac{3}{2}} b c d x^{5}}{96 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{73 a^{\frac{3}{2}} b d^{2} x^{7}}{160 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 \sqrt{a} b^{2} c^{2} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 \sqrt{a} b^{2} c d x^{7}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{29 \sqrt{a} b^{2} d^{2} x^{9}}{80 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{5} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{5}{2}}} - \frac{5 a^{4} c d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{64 b^{\frac{3}{2}}} + \frac{5 a^{3} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{b^{3} c^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{3} c d x^{9}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{3} d^{2} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(5/2)*(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.366916, size = 298, normalized size = 1.24 \[ \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, b^{2} d^{2} x^{2} + \frac{20 \, b^{10} c d + 21 \, a b^{9} d^{2}}{b^{8}}\right )} x^{2} + \frac{80 \, b^{10} c^{2} + 340 \, a b^{9} c d + 93 \, a^{2} b^{8} d^{2}}{b^{8}}\right )} x^{2} + \frac{5 \,{\left (208 \, a b^{9} c^{2} + 236 \, a^{2} b^{8} c d + 3 \, a^{3} b^{7} d^{2}\right )}}{b^{8}}\right )} x^{2} + \frac{15 \,{\left (176 \, a^{2} b^{8} c^{2} + 20 \, a^{3} b^{7} c d - 3 \, a^{4} b^{6} d^{2}\right )}}{b^{8}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)*(d*x^2 + c)^2,x, algorithm="giac")
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