Optimal. Leaf size=149 \[ \frac{5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 A b-a B)}{128 b}+\frac{x \left (a+b x^2\right )^{5/2} (8 A b-a B)}{48 b}+\frac{5 a x \left (a+b x^2\right )^{3/2} (8 A b-a B)}{192 b}+\frac{B x \left (a+b x^2\right )^{7/2}}{8 b} \]
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Rubi [A] time = 0.137666, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 A b-a B)}{128 b}+\frac{x \left (a+b x^2\right )^{5/2} (8 A b-a B)}{48 b}+\frac{5 a x \left (a+b x^2\right )^{3/2} (8 A b-a B)}{192 b}+\frac{B x \left (a+b x^2\right )^{7/2}}{8 b} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(5/2)*(A + B*x^2),x]
[Out]
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Rubi in Sympy [A] time = 14.5118, size = 134, normalized size = 0.9 \[ \frac{B x \left (a + b x^{2}\right )^{\frac{7}{2}}}{8 b} + \frac{5 a^{3} \left (8 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{3}{2}}} + \frac{5 a^{2} x \sqrt{a + b x^{2}} \left (8 A b - B a\right )}{128 b} + \frac{5 a x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (8 A b - B a\right )}{192 b} + \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (8 A b - B a\right )}{48 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(5/2)*(B*x**2+A),x)
[Out]
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Mathematica [A] time = 0.121125, size = 121, normalized size = 0.81 \[ \sqrt{a+b x^2} \left (\frac{a^2 x (5 a B+88 A b)}{128 b}+\frac{1}{48} b x^5 (17 a B+8 A b)+\frac{1}{192} a x^3 (59 a B+104 A b)+\frac{1}{8} b^2 B x^7\right )-\frac{5 a^3 (a B-8 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{128 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(5/2)*(A + B*x^2),x]
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Maple [A] time = 0.008, size = 166, normalized size = 1.1 \[{\frac{Ax}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,aAx}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}Ax}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,A{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{Bx}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Bxa}{48\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,Bx{a}^{2}}{192\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{a}^{3}x}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,B{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(5/2)*(B*x^2+A),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^2 + A)*(b*x^2 + a)^(5/2),x, algorithm="maxima")
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Fricas [A] time = 0.308742, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B b^{3} x^{7} + 8 \,{\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{5} + 2 \,{\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{3} + 88 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{768 \, b^{\frac{3}{2}}}, \frac{{\left (48 \, B b^{3} x^{7} + 8 \,{\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{5} + 2 \,{\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{3} + 88 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{384 \, \sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(5/2),x, algorithm="fricas")
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Sympy [A] time = 87.4629, size = 316, normalized size = 2.12 \[ \frac{A a^{\frac{5}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 A a^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 A a^{\frac{3}{2}} b x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 A \sqrt{a} b^{2} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{A b^{3} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{\frac{7}{2}} x}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 B a^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 B a^{\frac{3}{2}} b x^{5}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 B \sqrt{a} b^{2} x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{3}{2}}} + \frac{B b^{3} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(5/2)*(B*x**2+A),x)
[Out]
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GIAC/XCAS [A] time = 0.249302, size = 181, normalized size = 1.21 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, B b^{2} x^{2} + \frac{17 \, B a b^{7} + 8 \, A b^{8}}{b^{6}}\right )} x^{2} + \frac{59 \, B a^{2} b^{6} + 104 \, A a b^{7}}{b^{6}}\right )} x^{2} + \frac{3 \,{\left (5 \, B a^{3} b^{5} + 88 \, A a^{2} b^{6}\right )}}{b^{6}}\right )} \sqrt{b x^{2} + a} x + \frac{5 \,{\left (B a^{4} - 8 \, A a^{3} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(5/2),x, algorithm="giac")
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