Optimal. Leaf size=109 \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{3/2}}-\frac{b^2 \sqrt{a x^2+b x^3}}{8 a x^2}-\frac{b \sqrt{a x^2+b x^3}}{4 x^3}-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6} \]
[Out]
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Rubi [A] time = 0.23508, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{3/2}}-\frac{b^2 \sqrt{a x^2+b x^3}}{8 a x^2}-\frac{b \sqrt{a x^2+b x^3}}{4 x^3}-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6} \]
Antiderivative was successfully verified.
[In] Int[(a*x^2 + b*x^3)^(3/2)/x^7,x]
[Out]
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Rubi in Sympy [A] time = 24.603, size = 94, normalized size = 0.86 \[ - \frac{b \sqrt{a x^{2} + b x^{3}}}{4 x^{3}} - \frac{\left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{3 x^{6}} - \frac{b^{2} \sqrt{a x^{2} + b x^{3}}}{8 a x^{2}} + \frac{b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{8 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a*x**2)**(3/2)/x**7,x)
[Out]
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Mathematica [A] time = 0.0874827, size = 94, normalized size = 0.86 \[ \frac{\sqrt{x^2 (a+b x)} \left (3 b^3 x^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\sqrt{a} \sqrt{a+b x} \left (8 a^2+14 a b x+3 b^2 x^2\right )\right )}{24 a^{3/2} x^4 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*x^2 + b*x^3)^(3/2)/x^7,x]
[Out]
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Maple [A] time = 0.019, size = 87, normalized size = 0.8 \[ -{\frac{1}{24\,{x}^{6}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3\, \left ( bx+a \right ) ^{5/2}{a}^{3/2}-3\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ){x}^{3}a{b}^{3}+8\, \left ( bx+a \right ) ^{3/2}{a}^{5/2}-3\,\sqrt{bx+a}{a}^{7/2} \right ) \left ( bx+a \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a*x^2)^(3/2)/x^7,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^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241384, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} b^{3} x^{4} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) - 2 \,{\left (3 \, a b^{2} x^{2} + 14 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{48 \, a^{2} x^{4}}, \frac{3 \, \sqrt{-a} b^{3} x^{4} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) -{\left (3 \, a b^{2} x^{2} + 14 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{24 \, a^{2} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x^2)^(3/2)/x^7,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a*x**2)**(3/2)/x**7,x)
[Out]
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GIAC/XCAS [A] time = 0.266586, size = 124, normalized size = 1.14 \[ -\frac{\frac{3 \, b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ){\rm sign}\left (x\right )}{\sqrt{-a} a} + \frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{4}{\rm sign}\left (x\right ) + 8 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{4}{\rm sign}\left (x\right ) - 3 \, \sqrt{b x + a} a^{2} b^{4}{\rm sign}\left (x\right )}{a b^{3} x^{3}}}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x^2)^(3/2)/x^7,x, algorithm="giac")
[Out]