Optimal. Leaf size=118 \[ \frac {b (5 A b-4 a B)}{4 a^3 \sqrt {a+b x^3}}-\frac {A}{6 a x^6 \sqrt {a+b x^3}}+\frac {5 A b-4 a B}{12 a^2 x^3 \sqrt {a+b x^3}}-\frac {b (5 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 a^{7/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 118, 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, 44, 53,
65, 214} \begin {gather*} -\frac {b (5 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 a^{7/2}}+\frac {b (5 A b-4 a B)}{4 a^3 \sqrt {a+b x^3}}+\frac {5 A b-4 a B}{12 a^2 x^3 \sqrt {a+b x^3}}-\frac {A}{6 a x^6 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {A+B x}{x^3 (a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^3}}+\frac {\left (-\frac {5 A b}{2}+2 a B\right ) \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,x^3\right )}{6 a}\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^3}}-\frac {5 A b-4 a B}{6 a^2 x^3 \sqrt {a+b x^3}}-\frac {(5 A b-4 a B) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^3\right )}{4 a^2}\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^3}}-\frac {5 A b-4 a B}{6 a^2 x^3 \sqrt {a+b x^3}}+\frac {(5 A b-4 a B) \sqrt {a+b x^3}}{4 a^3 x^3}+\frac {(b (5 A b-4 a B)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{8 a^3}\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^3}}-\frac {5 A b-4 a B}{6 a^2 x^3 \sqrt {a+b x^3}}+\frac {(5 A b-4 a B) \sqrt {a+b x^3}}{4 a^3 x^3}+\frac {(5 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 )}{4 a^3}\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^3}}-\frac {5 A b-4 a B}{6 a^2 x^3 \sqrt {a+b x^3}}+\frac {(5 A b-4 a B) \sqrt {a+b x^3}}{4 a^3 x^3}-\frac {b (5 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 a^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 100, normalized size = 0.85 \begin {gather*} \frac {-2 a^2 A+5 a A b x^3-4 a^2 B x^3+15 A b^2 x^6-12 a b B x^6}{12 a^3 x^6 \sqrt {a+b x^3}}+\frac {b (-5 A b+4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 141, normalized size = 1.19
method | result | size |
elliptic | \(-\frac {A \sqrt {b \,x^{3}+a}}{6 a^{2} x^{6}}+\frac {\left (7 A b -4 B a \right ) \sqrt {b \,x^{3}+a}}{12 a^{3} x^{3}}+\frac {2 b \left (A b -B a \right )}{3 a^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {b \left (5 A b -4 B a \right ) \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 a^{\frac {7}{2}}}\) | \(102\) |
risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (-7 A b \,x^{3}+4 B a \,x^{3}+2 A a \right )}{12 a^{3} x^{6}}+\frac {b \left (-\frac {2 \left (7 A b -4 B a \right )}{3 \sqrt {b \,x^{3}+a}}+3 a \left (5 A b -4 B a \right ) \left (\frac {2}{3 a \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {3}{2}}}\right )\right )}{8 a^{3}}\) | \(115\) |
default | \(A \left (-\frac {\sqrt {b \,x^{3}+a}}{6 a^{2} x^{6}}+\frac {7 b \sqrt {b \,x^{3}+a}}{12 a^{3} x^{3}}+\frac {2 b^{2}}{3 a^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {5 b^{2} \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 a^{\frac {7}{2}}}\right )+B \left (-\frac {2 b}{3 a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {\sqrt {b \,x^{3}+a}}{3 a^{2} x^{3}}+\frac {b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 215 vs.
\(2 (98) = 196\).
time = 0.49, size = 215, normalized size = 1.82 \begin {gather*} \frac {1}{24} \, A {\left (\frac {2 \, {\left (15 \, {\left (b x^{3} + a\right )}^{2} b^{2} - 25 \, {\left (b x^{3} + a\right )} a b^{2} + 8 \, a^{2} b^{2}\right )}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{3} - 2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{4} + \sqrt {b x^{3} + a} a^{5}} + \frac {15 \, b^{2} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )} - \frac {1}{6} \, B {\left (\frac {2 \, {\left (3 \, {\left (b x^{3} + a\right )} b - 2 \, a b\right )}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2} - \sqrt {b x^{3} + a} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.22, size = 289, normalized size = 2.45 \begin {gather*} \left [-\frac {3 \, {\left ({\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6}\right )} \sqrt {a} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (3 \, {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} + 2 \, A a^{3} + {\left (4 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{24 \, {\left (a^{4} b x^{9} + a^{5} x^{6}\right )}}, -\frac {3 \, {\left ({\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} + 2 \, A a^{3} + {\left (4 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{12 \, {\left (a^{4} b x^{9} + a^{5} x^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 50.66, size = 192, normalized size = 1.63 \begin {gather*} A \left (- \frac {1}{6 a \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {5 \sqrt {b}}{12 a^{2} x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {5 b^{\frac {3}{2}}}{4 a^{3} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {5 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 a^{\frac {7}{2}}}\right ) + B \left (- \frac {1}{3 a \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b}}{a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{a^{\frac {5}{2}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.69, size = 137, normalized size = 1.16 \begin {gather*} -\frac {{\left (4 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{3}} - \frac {2 \, {\left (B a b - A b^{2}\right )}}{3 \, \sqrt {b x^{3} + a} a^{3}} - \frac {4 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} B a b - 4 \, \sqrt {b x^{3} + a} B a^{2} b - 7 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} A b^{2} + 9 \, \sqrt {b x^{3} + a} A a b^{2}}{12 \, a^{3} b^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.18, size = 167, normalized size = 1.42 \begin {gather*} \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 (5\,A\,b-4\,B\,a\right )}{8\,a^{7/2}}-\frac {\left (4\,B\,a^2-7\,A\,a\,b\right )\,\sqrt {b\,x^3+a}}{12\,a^4\,x^3}-\frac {A\,\sqrt {b\,x^3+a}}{6\,a^2\,x^6}-\frac {\frac {a\,\left (\frac {7\,A\,b^3-4\,B\,a\,b^2}{12\,a^4}-\frac {5\,b^2\,\left (5\,A\,b-4\,B\,a\right )}{8\,a^4}\right )}{b}+\frac {3\,b\,\left (5\,A\,b-4\,B\,a\right )}{8\,a^3}}{\sqrt {b\,x^3+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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