Optimal. Leaf size=238 \[ \frac{2 x^{5/2} (a+b x) (A b-a B)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 \sqrt{x} (a+b x) (A b-a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{3/2} (a+b x) (A b-a B)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.310522, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{2 x^{5/2} (a+b x) (A b-a B)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 \sqrt{x} (a+b x) (A b-a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{3/2} (a+b x) (A b-a B)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^(5/2)*(A + B*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
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Rubi in Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)*(B*x+A)/((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.139106, size = 120, normalized size = 0.5 \[ \frac{2 (a+b x) \left (105 a^{5/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )+\sqrt{b} \sqrt{x} \left (-105 a^3 B+35 a^2 b (3 A+B x)-7 a b^2 x (5 A+3 B x)+3 b^3 x^2 (7 A+5 B x)\right )\right )}{105 b^{9/2} \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(5/2)*(A + B*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
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Maple [A] time = 0.01, size = 163, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{105\,{b}^{4}} \left ( 15\,B\sqrt{ab}{x}^{7/2}{b}^{3}+21\,A\sqrt{ab}{x}^{5/2}{b}^{3}-21\,B\sqrt{ab}{x}^{5/2}a{b}^{2}-35\,A\sqrt{ab}{x}^{3/2}a{b}^{2}+35\,B\sqrt{ab}{x}^{3/2}{a}^{2}b+105\,A\sqrt{ab}\sqrt{x}{a}^{2}b-105\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{3}b-105\,B\sqrt{ab}\sqrt{x}{a}^{3}+105\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{4} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(B*x+A)/((b*x+a)^2)^(1/2),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)*x^(5/2)/sqrt((b*x + a)^2),x, algorithm="maxima")
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Fricas [A] time = 0.282254, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{x}}{105 \, b^{4}}, \frac{2 \,{\left (105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{x}\right )}}{105 \, b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(5/2)/sqrt((b*x + a)^2),x, algorithm="fricas")
[Out]
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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)/((b*x+a)**2)**(1/2),x)
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GIAC/XCAS [A] time = 0.277153, size = 228, normalized size = 0.96 \[ \frac{2 \,{\left (B a^{4}{\rm sign}\left (b x + a\right ) - A a^{3} b{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{2 \,{\left (15 \, B b^{6} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) - 21 \, B a b^{5} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + 21 \, A b^{6} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + 35 \, B a^{2} b^{4} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) - 35 \, A a b^{5} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) - 105 \, B a^{3} b^{3} \sqrt{x}{\rm sign}\left (b x + a\right ) + 105 \, A a^{2} b^{4} \sqrt{x}{\rm sign}\left (b x + a\right )\right )}}{105 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(5/2)/sqrt((b*x + a)^2),x, algorithm="giac")
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