Optimal. Leaf size=240 \[ \frac{x \left (c+d x^2\right )^{5/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{480 d^2}+\frac{c x \left (c+d x^2\right )^{3/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{384 d^2}+\frac{c^2 x \sqrt{c+d x^2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{256 d^2}+\frac{c^3 \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{256 d^{5/2}}-\frac{3 b x \left (c+d x^2\right )^{7/2} (b c-4 a d)}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d} \]
[Out]
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Rubi [A] time = 0.328643, antiderivative size = 240, 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 \[ \frac{x \left (c+d x^2\right )^{5/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{480 d^2}+\frac{c x \left (c+d x^2\right )^{3/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{384 d^2}+\frac{c^2 x \sqrt{c+d x^2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{256 d^2}+\frac{c^3 \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{256 d^{5/2}}-\frac{3 b x \left (c+d x^2\right )^{7/2} (b c-4 a d)}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2*(c + d*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 32.1846, size = 238, normalized size = 0.99 \[ \frac{b x \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{7}{2}}}{10 d} + \frac{3 b x \left (c + d x^{2}\right )^{\frac{7}{2}} \left (4 a d - b c\right )}{80 d^{2}} + \frac{c^{3} \left (80 a^{2} d^{2} - 20 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{256 d^{\frac{5}{2}}} + \frac{c^{2} x \sqrt{c + d x^{2}} \left (80 a^{2} d^{2} - 20 a b c d + 3 b^{2} c^{2}\right )}{256 d^{2}} + \frac{c x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (80 a^{2} d^{2} - 20 a b c d + 3 b^{2} c^{2}\right )}{384 d^{2}} + \frac{x \left (c + d x^{2}\right )^{\frac{5}{2}} \left (80 a^{2} d^{2} - 20 a b c d + 3 b^{2} c^{2}\right )}{480 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(5/2),x)
[Out]
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Mathematica [A] time = 0.205084, size = 192, normalized size = 0.8 \[ \frac{15 c^3 \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+\sqrt{d} x \sqrt{c+d x^2} \left (80 a^2 d^2 \left (33 c^2+26 c d x^2+8 d^2 x^4\right )+20 a b d \left (15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right )+b^2 \left (-45 c^4+30 c^3 d x^2+744 c^2 d^2 x^4+1008 c d^3 x^6+384 d^4 x^8\right )\right )}{3840 d^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2*(c + d*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.012, size = 308, normalized size = 1.3 \[{\frac{{a}^{2}x}{6} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{2}cx}{24} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{c}^{2}x}{16}\sqrt{d{x}^{2}+c}}+{\frac{5\,{a}^{2}{c}^{3}}{16}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}+{\frac{{b}^{2}{x}^{3}}{10\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{b}^{2}cx}{80\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{2}{c}^{2}x}{160\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{x{b}^{2}{c}^{3}}{128\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{2}{c}^{4}x}{256\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{b}^{2}{c}^{5}}{256}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{abx}{4\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{abcx}{24\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,ab{c}^{2}x}{96\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,ab{c}^{3}x}{64\,d}\sqrt{d{x}^{2}+c}}-{\frac{5\,ab{c}^{4}}{64}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^(5/2),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)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.531597, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (384 \, b^{2} d^{4} x^{9} + 48 \,{\left (21 \, b^{2} c d^{3} + 20 \, a b d^{4}\right )} x^{7} + 8 \,{\left (93 \, b^{2} c^{2} d^{2} + 340 \, a b c d^{3} + 80 \, a^{2} d^{4}\right )} x^{5} + 10 \,{\left (3 \, b^{2} c^{3} d + 236 \, a b c^{2} d^{2} + 208 \, a^{2} c d^{3}\right )} x^{3} - 15 \,{\left (3 \, b^{2} c^{4} - 20 \, a b c^{3} d - 176 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 15 \,{\left (3 \, b^{2} c^{5} - 20 \, a b c^{4} d + 80 \, a^{2} c^{3} d^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{7680 \, d^{\frac{5}{2}}}, \frac{{\left (384 \, b^{2} d^{4} x^{9} + 48 \,{\left (21 \, b^{2} c d^{3} + 20 \, a b d^{4}\right )} x^{7} + 8 \,{\left (93 \, b^{2} c^{2} d^{2} + 340 \, a b c d^{3} + 80 \, a^{2} d^{4}\right )} x^{5} + 10 \,{\left (3 \, b^{2} c^{3} d + 236 \, a b c^{2} d^{2} + 208 \, a^{2} c d^{3}\right )} x^{3} - 15 \,{\left (3 \, b^{2} c^{4} - 20 \, a b c^{3} d - 176 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} + 15 \,{\left (3 \, b^{2} c^{5} - 20 \, a b c^{4} d + 80 \, a^{2} c^{3} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{3840 \, \sqrt{-d} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 170.432, size = 537, normalized size = 2.24 \[ \frac{a^{2} c^{\frac{5}{2}} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} + \frac{3 a^{2} c^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{35 a^{2} c^{\frac{3}{2}} d x^{3}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 a^{2} \sqrt{c} d^{2} x^{5}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a^{2} c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{16 \sqrt{d}} + \frac{a^{2} d^{3} x^{7}}{6 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a b c^{\frac{7}{2}} x}{64 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{133 a b c^{\frac{5}{2}} x^{3}}{192 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{127 a b c^{\frac{3}{2}} d x^{5}}{96 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{23 a b \sqrt{c} d^{2} x^{7}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{5 a b c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{64 d^{\frac{3}{2}}} + \frac{a b d^{3} x^{9}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{3 b^{2} c^{\frac{9}{2}} x}{256 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{7}{2}} x^{3}}{256 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{129 b^{2} c^{\frac{5}{2}} x^{5}}{640 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{73 b^{2} c^{\frac{3}{2}} d x^{7}}{160 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{29 b^{2} \sqrt{c} d^{2} x^{9}}{80 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} c^{5} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{256 d^{\frac{5}{2}}} + \frac{b^{2} d^{3} x^{11}}{10 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.246865, size = 298, normalized size = 1.24 \[ \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, b^{2} d^{2} x^{2} + \frac{21 \, b^{2} c d^{9} + 20 \, a b d^{10}}{d^{8}}\right )} x^{2} + \frac{93 \, b^{2} c^{2} d^{8} + 340 \, a b c d^{9} + 80 \, a^{2} d^{10}}{d^{8}}\right )} x^{2} + \frac{5 \,{\left (3 \, b^{2} c^{3} d^{7} + 236 \, a b c^{2} d^{8} + 208 \, a^{2} c d^{9}\right )}}{d^{8}}\right )} x^{2} - \frac{15 \,{\left (3 \, b^{2} c^{4} d^{6} - 20 \, a b c^{3} d^{7} - 176 \, a^{2} c^{2} d^{8}\right )}}{d^{8}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (3 \, b^{2} c^{5} - 20 \, a b c^{4} d + 80 \, a^{2} c^{3} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{256 \, d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2),x, algorithm="giac")
[Out]