Optimal. Leaf size=164 \[ -\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{7/2}}+\frac{5 a^5 \sqrt{x} \sqrt{a+b x}}{512 b^3}-\frac{5 a^4 x^{3/2} \sqrt{a+b x}}{768 b^2}+\frac{a^3 x^{5/2} \sqrt{a+b x}}{192 b}+\frac{1}{32} a^2 x^{7/2} \sqrt{a+b x}+\frac{1}{12} a x^{7/2} (a+b x)^{3/2}+\frac{1}{6} x^{7/2} (a+b x)^{5/2} \]
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Rubi [A] time = 0.149244, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{7/2}}+\frac{5 a^5 \sqrt{x} \sqrt{a+b x}}{512 b^3}-\frac{5 a^4 x^{3/2} \sqrt{a+b x}}{768 b^2}+\frac{a^3 x^{5/2} \sqrt{a+b x}}{192 b}+\frac{1}{32} a^2 x^{7/2} \sqrt{a+b x}+\frac{1}{12} a x^{7/2} (a+b x)^{3/2}+\frac{1}{6} x^{7/2} (a+b x)^{5/2} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)*(a + b*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 24.118, size = 160, normalized size = 0.98 \[ - \frac{5 a^{6} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{512 b^{\frac{7}{2}}} - \frac{5 a^{5} \sqrt{x} \sqrt{a + b x}}{512 b^{3}} - \frac{5 a^{4} \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{768 b^{3}} - \frac{a^{3} \sqrt{x} \left (a + b x\right )^{\frac{5}{2}}}{192 b^{3}} + \frac{a^{2} \sqrt{x} \left (a + b x\right )^{\frac{7}{2}}}{32 b^{3}} - \frac{a x^{\frac{3}{2}} \left (a + b x\right )^{\frac{7}{2}}}{12 b^{2}} + \frac{x^{\frac{5}{2}} \left (a + b x\right )^{\frac{7}{2}}}{6 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)*(b*x+a)**(5/2),x)
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Mathematica [A] time = 0.0963994, size = 111, normalized size = 0.68 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (15 a^5-10 a^4 b x+8 a^3 b^2 x^2+432 a^2 b^3 x^3+640 a b^4 x^4+256 b^5 x^5\right )-15 a^6 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{1536 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)*(a + b*x)^(5/2),x]
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Maple [A] time = 0.007, size = 156, normalized size = 1. \[{\frac{1}{6\,b}{x}^{{\frac{5}{2}}} \left ( bx+a \right ) ^{{\frac{7}{2}}}}-{\frac{a}{12\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}}{32\,{b}^{3}}\sqrt{x} \left ( bx+a \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{3}}{192\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}-{\frac{5\,{a}^{4}}{768\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}-{\frac{5\,{a}^{5}}{512\,{b}^{3}}\sqrt{x}\sqrt{bx+a}}-{\frac{5\,{a}^{6}}{1024}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(b*x+a)^(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 + a)^(5/2)*x^(5/2),x, algorithm="maxima")
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Fricas [A] time = 0.2262, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{6} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (256 \, b^{5} x^{5} + 640 \, a b^{4} x^{4} + 432 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x + 15 \, a^{5}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{3072 \, b^{\frac{7}{2}}}, -\frac{15 \, a^{6} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (256 \, b^{5} x^{5} + 640 \, a b^{4} x^{4} + 432 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x + 15 \, a^{5}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{1536 \, \sqrt{-b} b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*x^(5/2),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)*(b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 24.8317, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*x^(5/2),x, algorithm="giac")
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