Optimal. Leaf size=90 \[ \frac{\sqrt{a+b x^3} (3 A b-4 a B)}{12 a^2 x^3}-\frac{b (3 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{5/2}}-\frac{A \sqrt{a+b x^3}}{6 a x^6} \]
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Rubi [A] time = 0.0695414, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \[ \frac{\sqrt{a+b x^3} (3 A b-4 a B)}{12 a^2 x^3}-\frac{b (3 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{5/2}}-\frac{A \sqrt{a+b x^3}}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^7 \sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x^3 \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=-\frac{A \sqrt{a+b x^3}}{6 a x^6}+\frac{\left (-\frac{3 A b}{2}+2 a B\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^3\right )}{6 a}\\ &=-\frac{A \sqrt{a+b x^3}}{6 a x^6}+\frac{(3 A b-4 a B) \sqrt{a+b x^3}}{12 a^2 x^3}+\frac{(b (3 A b-4 a B)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{24 a^2}\\ &=-\frac{A \sqrt{a+b x^3}}{6 a x^6}+\frac{(3 A b-4 a B) \sqrt{a+b x^3}}{12 a^2 x^3}+\frac{(3 A b-4 a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{12 a^2}\\ &=-\frac{A \sqrt{a+b x^3}}{6 a x^6}+\frac{(3 A b-4 a B) \sqrt{a+b x^3}}{12 a^2 x^3}-\frac{b (3 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.205633, size = 81, normalized size = 0.9 \[ \frac{\sqrt{a+b x^3} \left (b \left (2 a B-\frac{3 A b}{2}\right ) \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\frac{a}{b x^3}\right )-\frac{a^2 A}{x^6}\right )}{6 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 102, normalized size = 1.1 \begin{align*} A \left ( -{\frac{1}{6\,a{x}^{6}}\sqrt{b{x}^{3}+a}}+{\frac{b}{4\,{a}^{2}{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{{b}^{2}}{4}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}} \right ) +B \left ( -{\frac{1}{3\,a{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{b}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \right ) \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 = 1.82213, size = 409, normalized size = 4.54 \begin{align*} \left [-\frac{{\left (4 \, B a b - 3 \, A b^{2}\right )} \sqrt{a} x^{6} \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + 2 \,{\left ({\left (4 \, B a^{2} - 3 \, A a b\right )} x^{3} + 2 \, A a^{2}\right )} \sqrt{b x^{3} + a}}{24 \, a^{3} x^{6}}, -\frac{{\left (4 \, B a b - 3 \, A b^{2}\right )} \sqrt{-a} x^{6} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) +{\left ({\left (4 \, B a^{2} - 3 \, A a b\right )} x^{3} + 2 \, A a^{2}\right )} \sqrt{b x^{3} + a}}{12 \, a^{3} x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 31.3192, size = 163, normalized size = 1.81 \begin{align*} - \frac{A}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{A \sqrt{b}}{12 a x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{A b^{\frac{3}{2}}}{4 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{4 a^{\frac{5}{2}}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 a x^{\frac{3}{2}}} + \frac{B b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1216, size = 163, normalized size = 1.81 \begin{align*} -\frac{\frac{{\left (4 \, B a b^{2} - 3 \, A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x^{3} + a} B a^{2} b^{2} - 3 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b^{3} + 5 \, \sqrt{b x^{3} + a} A a b^{3}}{a^{2} b^{2} x^{6}}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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