Optimal. Leaf size=113 \[ \frac {-5 A b+2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac {A}{3 a x^3 \left (a+b x^3\right )^{3/2}}-\frac {5 A b-2 a B}{3 a^3 \sqrt {a+b x^3}}+\frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {457, 79, 53, 65,
214} \begin {gather*} \frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{7/2}}-\frac {5 A b-2 a B}{3 a^3 \sqrt {a+b x^3}}-\frac {5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac {A}{3 a x^3 \left (a+b x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 53
Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^4 \left (a+b x^3\right )^{5/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {A+B x}{x^2 (a+b x)^{5/2}} \, dx,x,x^3\right )\\ &=-\frac {A}{3 a x^3 \left (a+b x^3\right )^{3/2}}+\frac {\left (-\frac {5 A b}{2}+a B\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,x^3\right )}{3 a}\\ &=-\frac {5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac {A}{3 a x^3 \left (a+b x^3\right )^{3/2}}-\frac {(5 A b-2 a B) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^3\right )}{6 a^2}\\ &=-\frac {5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac {A}{3 a x^3 \left (a+b x^3\right )^{3/2}}-\frac {5 A b-2 a B}{3 a^3 \sqrt {a+b x^3}}-\frac {(5 A b-2 a B) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{6 a^3}\\ &=-\frac {5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac {A}{3 a x^3 \left (a+b x^3\right )^{3/2}}-\frac {5 A b-2 a B}{3 a^3 \sqrt {a+b x^3}}-\frac {(5 A b-2 a B) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 a^3 b}\\ &=-\frac {5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac {A}{3 a x^3 \left (a+b x^3\right )^{3/2}}-\frac {5 A b-2 a B}{3 a^3 \sqrt {a+b x^3}}+\frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 99, normalized size = 0.88 \begin {gather*} \frac {-3 a^2 A-20 a A b x^3+8 a^2 B x^3-15 A b^2 x^6+6 a b B x^6}{9 a^3 x^3 \left (a+b x^3\right )^{3/2}}+\frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.36, size = 157, normalized size = 1.39
method | result | size |
risch | \(-\frac {A \sqrt {b \,x^{3}+a}}{3 a^{3} x^{3}}-\frac {\frac {4 a \left (A b -B a \right )}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}}}-\frac {2 \left (5 A b -2 B a \right ) \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {a}}+\frac {\frac {8 A b}{3}-\frac {4 B a}{3}}{\sqrt {b \,x^{3}+a}}}{2 a^{3}}\) | \(94\) |
elliptic | \(-\frac {2 \left (A b -B a \right ) \sqrt {b \,x^{3}+a}}{9 a^{2} b^{2} \left (x^{3}+\frac {a}{b}\right )^{2}}-\frac {2 \left (2 A b -B a \right )}{3 a^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {A \sqrt {b \,x^{3}+a}}{3 a^{3} x^{3}}+\frac {\left (5 A b -2 B a \right ) \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {7}{2}}}\) | \(111\) |
default | \(A \left (-\frac {2 \sqrt {b \,x^{3}+a}}{9 a^{2} b \left (x^{3}+\frac {a}{b}\right )^{2}}-\frac {4 b}{3 a^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {\sqrt {b \,x^{3}+a}}{3 a^{3} x^{3}}+\frac {5 b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {7}{2}}}\right )+B \left (\frac {2 \sqrt {b \,x^{3}+a}}{9 a \,b^{2} \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {2}{3 a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {5}{2}}}\right )\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 170, normalized size = 1.50 \begin {gather*} -\frac {1}{18} \, A {\left (\frac {2 \, {\left (15 \, {\left (b x^{3} + a\right )}^{2} b - 10 \, {\left (b x^{3} + a\right )} a b - 2 \, a^{2} b\right )}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{3} - {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{4}} + \frac {15 \, b \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}{9} \, B {\left (\frac {3 \, \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x^{3} + 4 \, a\right )}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.91, size = 351, normalized size = 3.11 \begin {gather*} \left [-\frac {3 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\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 (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} - 3 \, A a^{3} + 4 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{18 \, {\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}}, \frac {3 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} - 3 \, A a^{3} + 4 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{9 \, {\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1608 vs.
\(2 (107) = 214\).
time = 87.16, size = 1608, normalized size = 14.23 \begin {gather*} A \left (- \frac {6 a^{17} \sqrt {1 + \frac {b x^{3}}{a}}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} - \frac {46 a^{16} b x^{3} \sqrt {1 + \frac {b x^{3}}{a}}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} - \frac {15 a^{16} b x^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} + \frac {30 a^{16} b x^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} - \frac {70 a^{15} b^{2} x^{6} \sqrt {1 + \frac {b x^{3}}{a}}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} - \frac {45 a^{15} b^{2} x^{6} \log {\left (\frac {b x^{3}}{a} \right )}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} + \frac {90 a^{15} b^{2} x^{6} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} - \frac {30 a^{14} b^{3} x^{9} \sqrt {1 + \frac {b x^{3}}{a}}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} - \frac {45 a^{14} b^{3} x^{9} \log {\left (\frac {b x^{3}}{a} \right )}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} + \frac {90 a^{14} b^{3} x^{9} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} - \frac {15 a^{13} b^{4} x^{12} \log {\left (\frac {b x^{3}}{a} \right )}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}} + \frac {30 a^{13} b^{4} x^{12} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{18 a^{\frac {39}{2}} x^{3} + 54 a^{\frac {37}{2}} b x^{6} + 54 a^{\frac {35}{2}} b^{2} x^{9} + 18 a^{\frac {33}{2}} b^{3} x^{12}}\right ) + B \left (\frac {8 a^{7} \sqrt {1 + \frac {b x^{3}}{a}}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}} + \frac {3 a^{7} \log {\left (\frac {b x^{3}}{a} \right )}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}} - \frac {6 a^{7} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}} + \frac {14 a^{6} b x^{3} \sqrt {1 + \frac {b x^{3}}{a}}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}} + \frac {9 a^{6} b x^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}} - \frac {18 a^{6} b x^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}} + \frac {6 a^{5} b^{2} x^{6} \sqrt {1 + \frac {b x^{3}}{a}}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}} + \frac {9 a^{5} b^{2} x^{6} \log {\left (\frac {b x^{3}}{a} \right )}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}} - \frac {18 a^{5} b^{2} x^{6} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}} + \frac {3 a^{4} b^{3} x^{9} \log {\left (\frac {b x^{3}}{a} \right )}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}} - \frac {6 a^{4} b^{3} x^{9} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{9 a^{\frac {19}{2}} + 27 a^{\frac {17}{2}} b x^{3} + 27 a^{\frac {15}{2}} b^{2} x^{6} + 9 a^{\frac {13}{2}} b^{3} x^{9}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.46, size = 101, normalized size = 0.89 \begin {gather*} \frac {{\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a} a^{3}} + \frac {2 \, {\left (3 \, {\left (b x^{3} + a\right )} B a + B a^{2} - 6 \, {\left (b x^{3} + a\right )} A b - A a b\right )}}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {\sqrt {b x^{3} + a} A}{3 \, a^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.97, size = 198, normalized size = 1.75 \begin {gather*} \frac {\ln \left (\frac {\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )\,{\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}^3}{x^6}\right )\,\left (5\,A\,b-2\,B\,a\right )}{6\,a^{7/2}}-\frac {\frac {2\,B\,a^2-5\,A\,a\,b}{2\,a^4}-\frac {a\,\left (\frac {A\,b^2}{3\,a^4}+\frac {5\,b\,\left (2\,B\,a^2-5\,A\,a\,b\right )}{6\,a^5}\right )}{b}}{\sqrt {b\,x^3+a}}-\frac {\frac {2\,B\,a^3-5\,A\,a^2\,b}{4\,a^4}-\frac {a\,\left (\frac {13\,b\,\left (2\,B\,a^3-5\,A\,a^2\,b\right )}{36\,a^5}+\frac {A\,b^2}{3\,a^3}\right )}{b}}{{\left (b\,x^3+a\right )}^{3/2}}-\frac {A\,\sqrt {b\,x^3+a}}{3\,a^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________