Optimal. Leaf size=286 \[ \frac{2 x^{7/2} (a+b x) (A b-a B)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 x^{3/2} (a+b x) (A b-a B)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{5/2} (a+b x) (A b-a B)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{7/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^3 \sqrt{x} (a+b x) (A b-a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
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Rubi [A] time = 0.375064, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{2 x^{7/2} (a+b x) (A b-a B)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 x^{3/2} (a+b x) (A b-a B)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{5/2} (a+b x) (A b-a B)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{7/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^3 \sqrt{x} (a+b x) (A b-a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^(7/2)*(A + B*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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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**(7/2)*(B*x+A)/((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.17593, size = 139, normalized size = 0.49 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{x} \left (315 a^4 B-105 a^3 b (3 A+B x)+21 a^2 b^2 x (5 A+3 B x)-9 a b^3 x^2 (7 A+5 B x)+5 b^4 x^3 (9 A+7 B x)\right )-315 a^{7/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{315 b^{11/2} \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(7/2)*(A + B*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.013, size = 197, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{315\,{b}^{5}} \left ( 35\,B\sqrt{ab}{x}^{9/2}{b}^{4}+45\,A\sqrt{ab}{x}^{7/2}{b}^{4}-45\,B\sqrt{ab}{x}^{7/2}a{b}^{3}-63\,A\sqrt{ab}{x}^{5/2}a{b}^{3}+63\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}+105\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}-105\,B\sqrt{ab}{x}^{3/2}{a}^{3}b-315\,A\sqrt{ab}\sqrt{x}{a}^{3}b+315\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{4}b+315\,B\sqrt{ab}\sqrt{x}{a}^{4}-315\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(7/2)*(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^(7/2)/sqrt((b*x + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.291187, size = 1, normalized size = 0. \[ \left [-\frac{315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{315 \, b^{5}}, -\frac{2 \,{\left (315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}\right )}}{315 \, b^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(7/2)/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**(7/2)*(B*x+A)/((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.276741, size = 277, normalized size = 0.97 \[ -\frac{2 \,{\left (B a^{5}{\rm sign}\left (b x + a\right ) - A a^{4} b{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} + \frac{2 \,{\left (35 \, B b^{8} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) - 45 \, B a b^{7} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + 45 \, A b^{8} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + 63 \, B a^{2} b^{6} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) - 63 \, A a b^{7} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) - 105 \, B a^{3} b^{5} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 105 \, A a^{2} b^{6} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 315 \, B a^{4} b^{4} \sqrt{x}{\rm sign}\left (b x + a\right ) - 315 \, A a^{3} b^{5} \sqrt{x}{\rm sign}\left (b x + a\right )\right )}}{315 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(7/2)/sqrt((b*x + a)^2),x, algorithm="giac")
[Out]