Optimal. Leaf size=115 \[ -\frac{\left (a+b x^3\right )^{3/2} (4 a B+A b)}{12 a x^3}+\frac{b \sqrt{a+b x^3} (4 a B+A b)}{4 a}-\frac{b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{A \left (a+b x^3\right )^{5/2}}{6 a x^6} \]
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Rubi [A] time = 0.0868472, antiderivative size = 115, 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, Rules used = {446, 78, 47, 50, 63, 208} \[ -\frac{\left (a+b x^3\right )^{3/2} (4 a B+A b)}{12 a x^3}+\frac{b \sqrt{a+b x^3} (4 a B+A b)}{4 a}-\frac{b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{A \left (a+b x^3\right )^{5/2}}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^7} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2} (A+B x)}{x^3} \, dx,x,x^3\right )\\ &=-\frac{A \left (a+b x^3\right )^{5/2}}{6 a x^6}+\frac{(A b+4 a B) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^2} \, dx,x,x^3\right )}{12 a}\\ &=-\frac{(A b+4 a B) \left (a+b x^3\right )^{3/2}}{12 a x^3}-\frac{A \left (a+b x^3\right )^{5/2}}{6 a x^6}+\frac{(b (A b+4 a B)) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^3\right )}{8 a}\\ &=\frac{b (A b+4 a B) \sqrt{a+b x^3}}{4 a}-\frac{(A b+4 a B) \left (a+b x^3\right )^{3/2}}{12 a x^3}-\frac{A \left (a+b x^3\right )^{5/2}}{6 a x^6}+\frac{1}{8} (b (A b+4 a B)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{b (A b+4 a B) \sqrt{a+b x^3}}{4 a}-\frac{(A b+4 a B) \left (a+b x^3\right )^{3/2}}{12 a x^3}-\frac{A \left (a+b x^3\right )^{5/2}}{6 a x^6}+\frac{1}{4} (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 )\\ &=\frac{b (A b+4 a B) \sqrt{a+b x^3}}{4 a}-\frac{(A b+4 a B) \left (a+b x^3\right )^{3/2}}{12 a x^3}-\frac{A \left (a+b x^3\right )^{5/2}}{6 a x^6}-\frac{b (A b+4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}\\ \end{align*}
Mathematica [C] time = 0.026296, size = 59, normalized size = 0.51 \[ \frac{\left (a+b x^3\right )^{5/2} \left (b x^6 (4 a B+A b) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x^3}{a}+1\right )-5 a^2 A\right )}{30 a^3 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 107, normalized size = 0.9 \begin{align*} A \left ( -{\frac{a}{6\,{x}^{6}}\sqrt{b{x}^{3}+a}}-{\frac{5\,b}{12\,{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{{b}^{2}}{4}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \right ) +B \left ( -{\frac{a}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{2\,b}{3}\sqrt{b{x}^{3}+a}}-\sqrt{a}b{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) \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 = 1.81928, size = 439, normalized size = 3.82 \begin{align*} \left [\frac{3 \,{\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 (8 \, B a b x^{6} -{\left (4 \, B a^{2} + 5 \, A a b\right )} x^{3} - 2 \, A a^{2}\right )} \sqrt{b x^{3} + a}}{24 \, a x^{6}}, \frac{3 \,{\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 (8 \, B a b x^{6} -{\left (4 \, B a^{2} + 5 \, A a b\right )} x^{3} - 2 \, A a^{2}\right )} \sqrt{b x^{3} + a}}{12 \, a x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 62.9708, size = 243, normalized size = 2.11 \begin{align*} - \frac{A a^{2}}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A a \sqrt{b}}{4 x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} - \frac{A b^{\frac{3}{2}}}{12 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 \sqrt{a}} - B \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )} - \frac{B a \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} + \frac{2 B a \sqrt{b}}{3 x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{2 B b^{\frac{3}{2}} x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x^{3}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15565, size = 177, normalized size = 1.54 \begin{align*} \frac{8 \, \sqrt{b x^{3} + a} B b^{2} + \frac{3 \,{\left (4 \, B a b^{2} + A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-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} + 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b^{3} - 3 \, \sqrt{b x^{3} + a} A a b^{3}}{b^{2} x^{6}}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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