Optimal. Leaf size=169 \[ -\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{9/2}}+\frac{5 \sqrt{x} \sqrt{a+b x} (4 A b-7 a B)}{4 b^4}-\frac{5 x^{3/2} \sqrt{a+b x} (4 A b-7 a B)}{6 a b^3}+\frac{2 x^{5/2} (4 A b-7 a B)}{3 a b^2 \sqrt{a+b x}}+\frac{2 x^{7/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.192091, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{9/2}}+\frac{5 \sqrt{x} \sqrt{a+b x} (4 A b-7 a B)}{4 b^4}-\frac{5 x^{3/2} \sqrt{a+b x} (4 A b-7 a B)}{6 a b^3}+\frac{2 x^{5/2} (4 A b-7 a B)}{3 a b^2 \sqrt{a+b x}}+\frac{2 x^{7/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(x^(5/2)*(A + B*x))/(a + b*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 19.0439, size = 162, normalized size = 0.96 \[ - \frac{5 a \left (4 A b - 7 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{4 b^{\frac{9}{2}}} + \frac{5 \sqrt{x} \sqrt{a + b x} \left (4 A b - 7 B a\right )}{4 b^{4}} + \frac{2 x^{\frac{7}{2}} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{\frac{3}{2}}} + \frac{2 x^{\frac{5}{2}} \left (4 A b - 7 B a\right )}{3 a b^{2} \sqrt{a + b x}} - \frac{5 x^{\frac{3}{2}} \sqrt{a + b x} \left (4 A b - 7 B a\right )}{6 a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)*(B*x+A)/(b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.187098, size = 115, normalized size = 0.68 \[ \frac{\sqrt{x} \left (-105 a^3 B+20 a^2 b (3 A-7 B x)+a b^2 x (80 A-21 B x)+6 b^3 x^2 (2 A+B x)\right )}{12 b^4 (a+b x)^{3/2}}+\frac{5 a (7 a B-4 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{4 b^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(5/2)*(A + B*x))/(a + b*x)^(5/2),x]
[Out]
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Maple [B] time = 0.024, size = 362, normalized size = 2.1 \[ -{\frac{1}{24} \left ( -12\,B{x}^{3}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+60\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}a{b}^{3}-24\,A{x}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-105\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}{a}^{2}{b}^{2}+42\,B{x}^{2}a{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+120\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}{b}^{2}-160\,Aax\sqrt{x \left ( bx+a \right ) }{b}^{5/2}-210\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{3}b+280\,B{a}^{2}x\sqrt{x \left ( bx+a \right ) }{b}^{3/2}+60\,A{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-120\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-105\,B{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +210\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ) \sqrt{x}{b}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(B*x+A)/(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)*x^(5/2)/(b*x + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24889, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B a^{3} - 4 \, A a^{2} b +{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{x} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (6 \, B b^{3} x^{4} - 3 \,{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 20 \,{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} - 15 \,{\left (7 \, B a^{3} - 4 \, A a^{2} b\right )} x\right )} \sqrt{b}}{24 \,{\left (b^{5} x + a b^{4}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}, \frac{15 \,{\left (7 \, B a^{3} - 4 \, A a^{2} b +{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (6 \, B b^{3} x^{4} - 3 \,{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 20 \,{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} - 15 \,{\left (7 \, B a^{3} - 4 \, A a^{2} b\right )} x\right )} \sqrt{-b}}{12 \,{\left (b^{5} x + a b^{4}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(5/2)/(b*x + a)^(5/2),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)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.28652, size = 467, normalized size = 2.76 \[ \frac{1}{4} \, \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} B{\left | b \right |}}{b^{6}} - \frac{13 \, B a b^{11}{\left | b \right |} - 4 \, A b^{12}{\left | b \right |}}{b^{17}}\right )} - \frac{5 \,{\left (7 \, B a^{2} \sqrt{b}{\left | b \right |} - 4 \, A a b^{\frac{3}{2}}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{8 \, b^{6}} - \frac{4 \,{\left (12 \, B a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b}{\left | b \right |} + 18 \, B a^{4}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}}{\left | b \right |} - 9 \, A a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{3}{2}}{\left | b \right |} + 10 \, B a^{5} b^{\frac{5}{2}}{\left | b \right |} - 12 \, A a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{5}{2}}{\left | b \right |} - 7 \, A a^{4} b^{\frac{7}{2}}{\left | b \right |}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(5/2)/(b*x + a)^(5/2),x, algorithm="giac")
[Out]