Optimal. Leaf size=113 \[ \frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{7/2}}-\frac {5 A b-2 a B}{3 a^3 \sqrt {a+b x^3}}+\frac {2 a B-5 A b}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac {A}{3 a x^3 \left (a+b x^3\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, 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 = {446, 78, 51, 63, 208} \begin {gather*} -\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 {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{7/2}}-\frac {A}{3 a x^3 \left (a+b x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^4 \left (a+b x^3\right )^{5/2}} \, dx &=\frac {1}{3} \operatorname {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 ) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 [C] time = 0.03, size = 57, normalized size = 0.50 \begin {gather*} \frac {x^3 (2 a B-5 A b) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b x^3}{a}+1\right )-3 a A}{9 a^2 x^3 \left (a+b x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.14, size = 99, normalized size = 0.88 \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 {-3 a^2 A+8 a^2 B x^3-20 a A b x^3+6 a b B x^6-15 A b^2 x^6}{9 a^3 x^3 \left (a+b x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.75, 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]
________________________________________________________________________________________
giac [A] time = 0.19, 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]
________________________________________________________________________________________
maple [A] time = 0.09, size = 157, normalized size = 1.39 \begin {gather*} \left (\frac {5 b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {7}{2}}}-\frac {4 b}{3 \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}\, a^{3}}-\frac {2 \sqrt {b \,x^{3}+a}}{9 \left (x^{3}+\frac {a}{b}\right )^{2} a^{2} b}-\frac {\sqrt {b \,x^{3}+a}}{3 a^{3} x^{3}}\right ) A +\left (-\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {5}{2}}}+\frac {2}{3 \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}\, a^{2}}+\frac {2 \sqrt {b \,x^{3}+a}}{9 \left (x^{3}+\frac {a}{b}\right )^{2} a \,b^{2}}\right ) B \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.25, 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]
________________________________________________________________________________________
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]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________