Optimal. Leaf size=165 \[ \frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{128 a^{7/2}}-\frac{3 b^4 \sqrt{a x^2+b x^3}}{128 a^3 x^2}+\frac{b^3 \sqrt{a x^2+b x^3}}{64 a^2 x^3}-\frac{b^2 \sqrt{a x^2+b x^3}}{80 a x^4}-\frac{\left (a x^2+b x^3\right )^{3/2}}{5 x^8}-\frac{3 b \sqrt{a x^2+b x^3}}{40 x^5} \]
[Out]
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Rubi [A] time = 0.404075, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{128 a^{7/2}}-\frac{3 b^4 \sqrt{a x^2+b x^3}}{128 a^3 x^2}+\frac{b^3 \sqrt{a x^2+b x^3}}{64 a^2 x^3}-\frac{b^2 \sqrt{a x^2+b x^3}}{80 a x^4}-\frac{\left (a x^2+b x^3\right )^{3/2}}{5 x^8}-\frac{3 b \sqrt{a x^2+b x^3}}{40 x^5} \]
Antiderivative was successfully verified.
[In] Int[(a*x^2 + b*x^3)^(3/2)/x^9,x]
[Out]
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Rubi in Sympy [A] time = 41.6561, size = 150, normalized size = 0.91 \[ - \frac{3 b \sqrt{a x^{2} + b x^{3}}}{40 x^{5}} - \frac{\left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{5 x^{8}} - \frac{b^{2} \sqrt{a x^{2} + b x^{3}}}{80 a x^{4}} + \frac{b^{3} \sqrt{a x^{2} + b x^{3}}}{64 a^{2} x^{3}} - \frac{3 b^{4} \sqrt{a x^{2} + b x^{3}}}{128 a^{3} x^{2}} + \frac{3 b^{5} \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{128 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a*x**2)**(3/2)/x**9,x)
[Out]
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Mathematica [A] time = 0.119842, size = 116, normalized size = 0.7 \[ \frac{\sqrt{x^2 (a+b x)} \left (15 b^5 x^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\sqrt{a} \sqrt{a+b x} \left (128 a^4+176 a^3 b x+8 a^2 b^2 x^2-10 a b^3 x^3+15 b^4 x^4\right )\right )}{640 a^{7/2} x^6 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*x^2 + b*x^3)^(3/2)/x^9,x]
[Out]
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Maple [A] time = 0.02, size = 113, normalized size = 0.7 \[ -{\frac{1}{640\,{x}^{8}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 15\, \left ( bx+a \right ) ^{9/2}{a}^{7/2}-70\, \left ( bx+a \right ) ^{7/2}{a}^{9/2}+128\, \left ( bx+a \right ) ^{5/2}{a}^{11/2}-15\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ){a}^{3}{b}^{5}{x}^{5}+70\, \left ( bx+a \right ) ^{3/2}{a}^{13/2}-15\,\sqrt{bx+a}{a}^{15/2} \right ) \left ( bx+a \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{13}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a*x^2)^(3/2)/x^9,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^3 + a*x^2)^(3/2)/x^9,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232044, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{a} b^{5} x^{6} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) - 2 \,{\left (15 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} + 176 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt{b x^{3} + a x^{2}}}{1280 \, a^{4} x^{6}}, \frac{15 \, \sqrt{-a} b^{5} x^{6} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) -{\left (15 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} + 176 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt{b x^{3} + a x^{2}}}{640 \, a^{4} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x^2)^(3/2)/x^9,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}{x^{9}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a*x**2)**(3/2)/x**9,x)
[Out]
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GIAC/XCAS [A] time = 0.285097, size = 170, normalized size = 1.03 \[ -\frac{\frac{15 \, b^{6} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ){\rm sign}\left (x\right )}{\sqrt{-a} a^{3}} + \frac{15 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{6}{\rm sign}\left (x\right ) - 70 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{6}{\rm sign}\left (x\right ) + 128 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{6}{\rm sign}\left (x\right ) + 70 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{6}{\rm sign}\left (x\right ) - 15 \, \sqrt{b x + a} a^{4} b^{6}{\rm sign}\left (x\right )}{a^{3} b^{5} x^{5}}}{640 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x^2)^(3/2)/x^9,x, algorithm="giac")
[Out]