Optimal. Leaf size=202 \[ -\frac{45 a^8 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{32768 b^{7/2}}+\frac{45 a^7 x \sqrt{a+b x^2}}{32768 b^3}-\frac{15 a^6 x^3 \sqrt{a+b x^2}}{16384 b^2}+\frac{3 a^5 x^5 \sqrt{a+b x^2}}{4096 b}+\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2} \]
[Out]
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Rubi [A] time = 0.318463, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{45 a^8 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{32768 b^{7/2}}+\frac{45 a^7 x \sqrt{a+b x^2}}{32768 b^3}-\frac{15 a^6 x^3 \sqrt{a+b x^2}}{16384 b^2}+\frac{3 a^5 x^5 \sqrt{a+b x^2}}{4096 b}+\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2} \]
Antiderivative was successfully verified.
[In] Int[x^6*(a + b*x^2)^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 38.3372, size = 192, normalized size = 0.95 \[ - \frac{45 a^{8} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{32768 b^{\frac{7}{2}}} + \frac{45 a^{7} x \sqrt{a + b x^{2}}}{32768 b^{3}} - \frac{15 a^{6} x^{3} \sqrt{a + b x^{2}}}{16384 b^{2}} + \frac{3 a^{5} x^{5} \sqrt{a + b x^{2}}}{4096 b} + \frac{9 a^{4} x^{7} \sqrt{a + b x^{2}}}{2048} + \frac{3 a^{3} x^{7} \left (a + b x^{2}\right )^{\frac{3}{2}}}{256} + \frac{3 a^{2} x^{7} \left (a + b x^{2}\right )^{\frac{5}{2}}}{128} + \frac{9 a x^{7} \left (a + b x^{2}\right )^{\frac{7}{2}}}{224} + \frac{x^{7} \left (a + b x^{2}\right )^{\frac{9}{2}}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6*(b*x**2+a)**(9/2),x)
[Out]
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Mathematica [A] time = 0.125762, size = 131, normalized size = 0.65 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (315 a^7-210 a^6 b x^2+168 a^5 b^2 x^4+32624 a^4 b^3 x^6+98432 a^3 b^4 x^8+119040 a^2 b^5 x^{10}+66560 a b^6 x^{12}+14336 b^7 x^{14}\right )-315 a^8 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{229376 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^6*(a + b*x^2)^(9/2),x]
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Maple [A] time = 0.016, size = 169, normalized size = 0.8 \[{\frac{{x}^{5}}{16\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{5\,a{x}^{3}}{224\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{5\,{a}^{2}x}{896\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{a}^{3}x}{1792\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}-{\frac{9\,{a}^{4}x}{14336\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{a}^{5}x}{4096\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{15\,{a}^{6}x}{16384\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{45\,{a}^{7}x}{32768\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{45\,{a}^{8}}{32768}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6*(b*x^2+a)^(9/2),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)^(9/2)*x^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.666697, size = 1, normalized size = 0. \[ \left [\frac{315 \, a^{8} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (14336 \, b^{7} x^{15} + 66560 \, a b^{6} x^{13} + 119040 \, a^{2} b^{5} x^{11} + 98432 \, a^{3} b^{4} x^{9} + 32624 \, a^{4} b^{3} x^{7} + 168 \, a^{5} b^{2} x^{5} - 210 \, a^{6} b x^{3} + 315 \, a^{7} x\right )} \sqrt{b x^{2} + a} \sqrt{b}}{458752 \, b^{\frac{7}{2}}}, -\frac{315 \, a^{8} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (14336 \, b^{7} x^{15} + 66560 \, a b^{6} x^{13} + 119040 \, a^{2} b^{5} x^{11} + 98432 \, a^{3} b^{4} x^{9} + 32624 \, a^{4} b^{3} x^{7} + 168 \, a^{5} b^{2} x^{5} - 210 \, a^{6} b x^{3} + 315 \, a^{7} x\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{229376 \, \sqrt{-b} b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(9/2)*x^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 100.404, size = 258, normalized size = 1.28 \[ \frac{45 a^{\frac{15}{2}} x}{32768 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 a^{\frac{13}{2}} x^{3}}{32768 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{11}{2}} x^{5}}{16384 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{4099 a^{\frac{9}{2}} x^{7}}{28672 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{8191 a^{\frac{7}{2}} b x^{9}}{14336 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{1699 a^{\frac{5}{2}} b^{2} x^{11}}{1792 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{725 a^{\frac{3}{2}} b^{3} x^{13}}{896 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{79 \sqrt{a} b^{4} x^{15}}{224 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{45 a^{8} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{32768 b^{\frac{7}{2}}} + \frac{b^{5} x^{17}}{16 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6*(b*x**2+a)**(9/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214166, size = 180, normalized size = 0.89 \[ \frac{45 \, a^{8}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{32768 \, b^{\frac{7}{2}}} + \frac{1}{229376} \,{\left (\frac{315 \, a^{7}}{b^{3}} - 2 \,{\left (\frac{105 \, a^{6}}{b^{2}} - 4 \,{\left (\frac{21 \, a^{5}}{b} + 2 \,{\left (2039 \, a^{4} + 8 \,{\left (769 \, a^{3} b + 2 \,{\left (465 \, a^{2} b^{2} + 4 \,{\left (14 \, b^{4} x^{2} + 65 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(9/2)*x^6,x, algorithm="giac")
[Out]