Optimal. Leaf size=122 \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{7/2}}+\frac{x^3 \sqrt{a+b x^2} (6 A b-5 a B)}{24 b^2}-\frac{a x \sqrt{a+b x^2} (6 A b-5 a B)}{16 b^3}+\frac{B x^5 \sqrt{a+b x^2}}{6 b} \]
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Rubi [A] time = 0.0529656, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {459, 321, 217, 206} \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{7/2}}+\frac{x^3 \sqrt{a+b x^2} (6 A b-5 a B)}{24 b^2}-\frac{a x \sqrt{a+b x^2} (6 A b-5 a B)}{16 b^3}+\frac{B x^5 \sqrt{a+b x^2}}{6 b} \]
Antiderivative was successfully verified.
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Rule 459
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4 \left (A+B x^2\right )}{\sqrt{a+b x^2}} \, dx &=\frac{B x^5 \sqrt{a+b x^2}}{6 b}-\frac{(-6 A b+5 a B) \int \frac{x^4}{\sqrt{a+b x^2}} \, dx}{6 b}\\ &=\frac{(6 A b-5 a B) x^3 \sqrt{a+b x^2}}{24 b^2}+\frac{B x^5 \sqrt{a+b x^2}}{6 b}-\frac{(a (6 A b-5 a B)) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{8 b^2}\\ &=-\frac{a (6 A b-5 a B) x \sqrt{a+b x^2}}{16 b^3}+\frac{(6 A b-5 a B) x^3 \sqrt{a+b x^2}}{24 b^2}+\frac{B x^5 \sqrt{a+b x^2}}{6 b}+\frac{\left (a^2 (6 A b-5 a B)\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{16 b^3}\\ &=-\frac{a (6 A b-5 a B) x \sqrt{a+b x^2}}{16 b^3}+\frac{(6 A b-5 a B) x^3 \sqrt{a+b x^2}}{24 b^2}+\frac{B x^5 \sqrt{a+b x^2}}{6 b}+\frac{\left (a^2 (6 A b-5 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{16 b^3}\\ &=-\frac{a (6 A b-5 a B) x \sqrt{a+b x^2}}{16 b^3}+\frac{(6 A b-5 a B) x^3 \sqrt{a+b x^2}}{24 b^2}+\frac{B x^5 \sqrt{a+b x^2}}{6 b}+\frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0952979, size = 100, normalized size = 0.82 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (15 a^2 B-2 a b \left (9 A+5 B x^2\right )+4 b^2 x^2 \left (3 A+2 B x^2\right )\right )-3 a^2 (5 a B-6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{48 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 143, normalized size = 1.2 \begin{align*}{\frac{{x}^{5}B}{6\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,Ba{x}^{3}}{24\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,{a}^{2}Bx}{16\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{A{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,aAx}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71548, size = 490, normalized size = 4.02 \begin{align*} \left [-\frac{3 \,{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (8 \, B b^{3} x^{5} - 2 \,{\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{96 \, b^{4}}, \frac{3 \,{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (8 \, B b^{3} x^{5} - 2 \,{\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{48 \, b^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 9.834, size = 235, normalized size = 1.93 \begin{align*} - \frac{3 A a^{\frac{3}{2}} x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A \sqrt{a} x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} + \frac{A x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{\frac{5}{2}} x}{16 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{\frac{3}{2}} x^{3}}{48 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B \sqrt{a} x^{5}}{24 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{7}{2}}} + \frac{B x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13658, size = 144, normalized size = 1.18 \begin{align*} \frac{1}{48} \,{\left (2 \,{\left (\frac{4 \, B x^{2}}{b} - \frac{5 \, B a b^{3} - 6 \, A b^{4}}{b^{5}}\right )} x^{2} + \frac{3 \,{\left (5 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )}}{b^{5}}\right )} \sqrt{b x^{2} + a} x + \frac{{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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