Optimal. Leaf size=115 \[ \frac {b (A b+4 a B) \sqrt {a+b x^3}}{4 a}-\frac {(A b+4 a B) \left (a+b x^3\right )^{3/2}}{12 a x^3}-\frac {A \left (a+b x^3\right )^{5/2}}{6 a x^6}-\frac {b (A b+4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]
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Rubi [A]
time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, 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 = {457, 79, 43, 52,
65, 214} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 79
Rule 214
Rule 457
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} \text {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) \text {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)) \text {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)) \text {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) \text {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*}
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Mathematica [A]
time = 0.14, size = 81, normalized size = 0.70 \begin {gather*} \frac {\sqrt {a+b x^3} \left (-2 a A-5 A b x^3-4 a B x^3+8 b B x^6\right )}{12 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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 107, normalized size = 0.93
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (5 A b \,x^{3}+4 B a \,x^{3}+2 A a \right )}{12 x^{6}}+\frac {b \left (\frac {16 B \sqrt {b \,x^{3}+a}}{3}-\frac {2 \left (3 A b +12 B a \right ) \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {a}}\right )}{8}\) | \(79\) |
elliptic | \(-\frac {A a \sqrt {b \,x^{3}+a}}{6 x^{6}}-\frac {\left (\frac {5 A b}{4}+B a \right ) \sqrt {b \,x^{3}+a}}{3 x^{3}}+\frac {2 B b \sqrt {b \,x^{3}+a}}{3}-\frac {2 \left (\frac {3}{8} b^{2} A +\frac {3}{2} a b B \right ) \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {a}}\) | \(84\) |
default | \(A \left (-\frac {a \sqrt {b \,x^{3}+a}}{6 x^{6}}-\frac {5 b \sqrt {b \,x^{3}+a}}{12 x^{3}}-\frac {b^{2} \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 \sqrt {a}}\right )+B \left (-\frac {a \sqrt {b \,x^{3}+a}}{3 x^{3}}+\frac {2 b \sqrt {b \,x^{3}+a}}{3}-b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right ) \sqrt {a}\right )\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 171, normalized size = 1.49 \begin {gather*} \frac {1}{24} \, {\left (\frac {3 \, b^{2} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{\sqrt {a}} - \frac {2 \, {\left (5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {b x^{3} + a} a b^{2}\right )}}{{\left (b x^{3} + a\right )}^{2} - 2 \, {\left (b x^{3} + a\right )} a + a^{2}}\right )} A + \frac {1}{6} \, {\left (3 \, \sqrt {a} b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right ) + 4 \, \sqrt {b x^{3} + a} b - \frac {2 \, \sqrt {b x^{3} + a} a}{x^{3}}\right )} B \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.99, size = 191, normalized size = 1.66 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs.
\(2 (102) = 204\).
time = 49.27, size = 243, normalized size = 2.11 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.33, size = 131, normalized size = 1.14 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.47, size = 110, normalized size = 0.96 \begin {gather*} \frac {2\,B\,b\,\sqrt {b\,x^3+a}}{3}-\frac {\sqrt {b\,x^3+a}\,\left (4\,B\,a^3+5\,A\,b\,a^2\right )}{12\,a^2\,x^3}-\frac {A\,a\,\sqrt {b\,x^3+a}}{6\,x^6}+\frac {b\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )\,\left (A\,b+4\,B\,a\right )}{8\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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