Optimal. Leaf size=88 \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}+\frac{\sqrt{a+b x^3} (A b-4 a B)}{12 a x^3}-\frac{A \left (a+b x^3\right )^{3/2}}{6 a x^6} \]
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Rubi [A] time = 0.0696097, antiderivative size = 88, 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, 47, 63, 208} \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}+\frac{\sqrt{a+b x^3} (A b-4 a B)}{12 a x^3}-\frac{A \left (a+b x^3\right )^{3/2}}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^3} \left (A+B x^3\right )}{x^7} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x} (A+B x)}{x^3} \, dx,x,x^3\right )\\ &=-\frac{A \left (a+b x^3\right )^{3/2}}{6 a x^6}+\frac{\left (-\frac{A b}{2}+2 a B\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,x^3\right )}{6 a}\\ &=\frac{(A b-4 a B) \sqrt{a+b x^3}}{12 a x^3}-\frac{A \left (a+b x^3\right )^{3/2}}{6 a x^6}-\frac{(b (A b-4 a B)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{24 a}\\ &=\frac{(A b-4 a B) \sqrt{a+b x^3}}{12 a x^3}-\frac{A \left (a+b x^3\right )^{3/2}}{6 a x^6}-\frac{(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}\\ &=\frac{(A b-4 a B) \sqrt{a+b x^3}}{12 a x^3}-\frac{A \left (a+b x^3\right )^{3/2}}{6 a x^6}+\frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0683401, size = 93, normalized size = 1.06 \[ \frac{-\left (a+b x^3\right ) \left (2 a \left (A+2 B x^3\right )+A b x^3\right )-b x^6 \sqrt{\frac{b x^3}{a}+1} (4 a B-A b) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{12 a x^6 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 96, normalized size = 1.1 \begin{align*} A \left ( -{\frac{1}{6\,{x}^{6}}\sqrt{b{x}^{3}+a}}-{\frac{b}{12\,a{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{{b}^{2}}{12}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \right ) +B \left ( -{\frac{1}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{b}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \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 = 1.62361, size = 397, normalized size = 4.51 \begin{align*} \left [-\frac{{\left (4 \, B a b - 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} + A a b\right )} x^{3} + 2 \, A a^{2}\right )} \sqrt{b x^{3} + a}}{24 \, a^{2} x^{6}}, \frac{{\left (4 \, B a b - 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} + A a b\right )} x^{3} + 2 \, A a^{2}\right )} \sqrt{b x^{3} + a}}{12 \, a^{2} x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 50.8912, size = 160, normalized size = 1.82 \begin{align*} - \frac{A a}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A \sqrt{b}}{4 x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A b^{\frac{3}{2}}}{12 a 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 )}}{12 a^{\frac{3}{2}}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} - \frac{B b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11189, size = 162, normalized size = 1.84 \begin{align*} \frac{\frac{{\left (4 \, B a b^{2} - A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \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} +{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b^{3} + \sqrt{b x^{3} + a} A a b^{3}}{a b^{2} x^{6}}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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