Optimal. Leaf size=120 \[ -\frac{b^2 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}+\frac{b \sqrt{a+b x^2} (A b-2 a B)}{16 a^2 x^2}+\frac{\sqrt{a+b x^2} (A b-2 a B)}{8 a x^4}-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6} \]
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Rubi [A] time = 0.253423, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{b^2 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}+\frac{b \sqrt{a+b x^2} (A b-2 a B)}{16 a^2 x^2}+\frac{\sqrt{a+b x^2} (A b-2 a B)}{8 a x^4}-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^7,x]
[Out]
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Rubi in Sympy [A] time = 21.0993, size = 107, normalized size = 0.89 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{6 a x^{6}} + \frac{\sqrt{a + b x^{2}} \left (\frac{A b}{2} - B a\right )}{4 a x^{4}} + \frac{b \sqrt{a + b x^{2}} \left (A b - 2 B a\right )}{16 a^{2} x^{2}} - \frac{b^{2} \left (\frac{A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**7,x)
[Out]
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Mathematica [A] time = 0.165432, size = 123, normalized size = 1.02 \[ -\frac{b^2 (A b-2 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{16 a^{5/2}}+\frac{b^2 \log (x) (A b-2 a B)}{16 a^{5/2}}+\sqrt{a+b x^2} \left (-\frac{b (2 a B-A b)}{16 a^2 x^2}+\frac{-6 a B-A b}{24 a x^4}-\frac{A}{6 x^6}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^7,x]
[Out]
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Maple [A] time = 0.013, size = 197, normalized size = 1.6 \[ -{\frac{A}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ab}{8\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}A}{16\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{A{b}^{3}}{16\,{a}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{B}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B{b}^{2}}{8\,{a}^{2}}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)*(b*x^2+a)^(1/2)/x^7,x)
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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^2 + A)*sqrt(b*x^2 + a)/x^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247209, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{6} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (2 \, B a b - A b^{2}\right )} x^{4} + 8 \, A a^{2} + 2 \,{\left (6 \, B a^{2} + A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{96 \, a^{\frac{5}{2}} x^{6}}, \frac{3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \,{\left (2 \, B a b - A b^{2}\right )} x^{4} + 8 \, A a^{2} + 2 \,{\left (6 \, B a^{2} + A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{48 \, \sqrt{-a} a^{2} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 101.49, size = 226, normalized size = 1.88 \[ - \frac{A a}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{3}{2}}}{48 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{5}{2}}}{16 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{5}{2}}} - \frac{B a}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 B \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{B b^{\frac{3}{2}}}{8 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**7,x)
[Out]
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GIAC/XCAS [A] time = 0.229216, size = 189, normalized size = 1.58 \[ -\frac{\frac{3 \,{\left (2 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{6 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a b^{3} - 6 \, \sqrt{b x^{2} + a} B a^{3} b^{3} - 3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{4} + 8 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{4} + 3 \, \sqrt{b x^{2} + a} A a^{2} b^{4}}{a^{2} b^{3} x^{6}}}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^7,x, algorithm="giac")
[Out]