Optimal. Leaf size=272 \[ \frac{3 a^2 (4 b c-a d) \left (a^2 d^2-2 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{7/2}}+\frac{d x \left (a+b x^2\right )^{5/2} \left (5 a^2 d^2-20 a b c d+36 b^2 c^2\right )}{160 b^3}+\frac{x \left (a+b x^2\right )^{3/2} (4 b c-a d) \left (a^2 d^2-2 a b c d+8 b^2 c^2\right )}{128 b^3}+\frac{3 a x \sqrt{a+b x^2} (4 b c-a d) \left (a^2 d^2-2 a b c d+8 b^2 c^2\right )}{256 b^3}+\frac{d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) (14 b c-5 a d)}{80 b^2}+\frac{d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b} \]
[Out]
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Rubi [A] time = 0.495243, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{3 a^2 (4 b c-a d) \left (a^2 d^2-2 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{7/2}}+\frac{d x \left (a+b x^2\right )^{5/2} \left (5 a^2 d^2-20 a b c d+36 b^2 c^2\right )}{160 b^3}+\frac{x \left (a+b x^2\right )^{3/2} (4 b c-a d) \left (a^2 d^2-2 a b c d+8 b^2 c^2\right )}{128 b^3}+\frac{3 a x \sqrt{a+b x^2} (4 b c-a d) \left (a^2 d^2-2 a b c d+8 b^2 c^2\right )}{256 b^3}+\frac{d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) (14 b c-5 a d)}{80 b^2}+\frac{d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(3/2)*(c + d*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 53.8301, size = 269, normalized size = 0.99 \[ - \frac{3 a^{2} \left (a d - 4 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{256 b^{\frac{7}{2}}} - \frac{3 a x \sqrt{a + b x^{2}} \left (a d - 4 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 8 b^{2} c^{2}\right )}{256 b^{3}} + \frac{d x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (c + d x^{2}\right )^{2}}{10 b} - \frac{d x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (c + d x^{2}\right ) \left (5 a d - 14 b c\right )}{80 b^{2}} + \frac{d x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (5 a^{2} d^{2} - 20 a b c d + 36 b^{2} c^{2}\right )}{160 b^{3}} - \frac{x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - 4 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 8 b^{2} c^{2}\right )}{128 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(3/2)*(d*x**2+c)**3,x)
[Out]
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Mathematica [A] time = 0.207767, size = 220, normalized size = 0.81 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (15 a^4 d^3-10 a^3 b d^2 \left (9 c+d x^2\right )+4 a^2 b^2 d \left (60 c^2+15 c d x^2+2 d^2 x^4\right )+16 a b^3 \left (50 c^3+70 c^2 d x^2+45 c d^2 x^4+11 d^3 x^6\right )+32 b^4 x^2 \left (10 c^3+20 c^2 d x^2+15 c d^2 x^4+4 d^3 x^6\right )\right )-15 a^2 (a d-4 b c) \left (a^2 d^2-2 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{1280 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(3/2)*(c + d*x^2)^3,x]
[Out]
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Maple [A] time = 0.018, size = 393, normalized size = 1.4 \[{\frac{{c}^{3}x}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,a{c}^{3}x}{8}\sqrt{b{x}^{2}+a}}+{\frac{3\,{c}^{3}{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{{d}^{3}{x}^{5}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{a{d}^{3}{x}^{3}}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{3}{a}^{2}x}{32\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{3}{d}^{3}x}{128\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{3}{a}^{4}x}{256\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{3\,{d}^{3}{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{3\,c{d}^{2}{x}^{3}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{3\,ac{d}^{2}x}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{a}^{2}c{d}^{2}x}{64\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{9\,{a}^{3}c{d}^{2}x}{128\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{9\,c{d}^{2}{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{{c}^{2}dx}{2\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{a{c}^{2}dx}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{c}^{2}d{a}^{2}x}{16\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,{a}^{3}{c}^{2}d}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(3/2)*(d*x^2+c)^3,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)^(3/2)*(d*x^2 + c)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.690348, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (128 \, b^{4} d^{3} x^{9} + 16 \,{\left (30 \, b^{4} c d^{2} + 11 \, a b^{3} d^{3}\right )} x^{7} + 8 \,{\left (80 \, b^{4} c^{2} d + 90 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 10 \,{\left (32 \, b^{4} c^{3} + 112 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} + 5 \,{\left (160 \, a b^{3} c^{3} + 48 \, a^{2} b^{2} c^{2} d - 18 \, a^{3} b c d^{2} + 3 \, a^{4} d^{3}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (32 \, a^{2} b^{3} c^{3} - 16 \, a^{3} b^{2} c^{2} d + 6 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{2560 \, b^{\frac{7}{2}}}, \frac{{\left (128 \, b^{4} d^{3} x^{9} + 16 \,{\left (30 \, b^{4} c d^{2} + 11 \, a b^{3} d^{3}\right )} x^{7} + 8 \,{\left (80 \, b^{4} c^{2} d + 90 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 10 \,{\left (32 \, b^{4} c^{3} + 112 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} + 5 \,{\left (160 \, a b^{3} c^{3} + 48 \, a^{2} b^{2} c^{2} d - 18 \, a^{3} b c d^{2} + 3 \, a^{4} d^{3}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 15 \,{\left (32 \, a^{2} b^{3} c^{3} - 16 \, a^{3} b^{2} c^{2} d + 6 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{1280 \, \sqrt{-b} b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 151.033, size = 665, normalized size = 2.44 \[ \frac{3 a^{\frac{9}{2}} d^{3} x}{256 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{9 a^{\frac{7}{2}} c d^{2} x}{128 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{\frac{7}{2}} d^{3} x^{3}}{256 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{\frac{5}{2}} c^{2} d x}{16 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{5}{2}} c d^{2} x^{3}}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{\frac{5}{2}} d^{3} x^{5}}{640 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{\frac{3}{2}} c^{3} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{a^{\frac{3}{2}} c^{3} x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 a^{\frac{3}{2}} c^{2} d x^{3}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{39 a^{\frac{3}{2}} c d^{2} x^{5}}{64 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 a^{\frac{3}{2}} d^{3} x^{7}}{160 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 \sqrt{a} b c^{3} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{11 \sqrt{a} b c^{2} d x^{5}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 \sqrt{a} b c d^{2} x^{7}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{19 \sqrt{a} b d^{3} x^{9}}{80 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{5} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{7}{2}}} + \frac{9 a^{4} c d^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{5}{2}}} - \frac{3 a^{3} c^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + \frac{3 a^{2} c^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 \sqrt{b}} + \frac{b^{2} c^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{2} c^{2} d x^{7}}{2 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 b^{2} c d^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{2} d^{3} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(3/2)*(d*x**2+c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.284076, size = 351, normalized size = 1.29 \[ \frac{1}{1280} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, b d^{3} x^{2} + \frac{30 \, b^{9} c d^{2} + 11 \, a b^{8} d^{3}}{b^{8}}\right )} x^{2} + \frac{80 \, b^{9} c^{2} d + 90 \, a b^{8} c d^{2} + a^{2} b^{7} d^{3}}{b^{8}}\right )} x^{2} + \frac{5 \,{\left (32 \, b^{9} c^{3} + 112 \, a b^{8} c^{2} d + 6 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )}}{b^{8}}\right )} x^{2} + \frac{5 \,{\left (160 \, a b^{8} c^{3} + 48 \, a^{2} b^{7} c^{2} d - 18 \, a^{3} b^{6} c d^{2} + 3 \, a^{4} b^{5} d^{3}\right )}}{b^{8}}\right )} \sqrt{b x^{2} + a} x - \frac{3 \,{\left (32 \, a^{2} b^{3} c^{3} - 16 \, a^{3} b^{2} c^{2} d + 6 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)^3,x, algorithm="giac")
[Out]