Optimal. Leaf size=73 \[ \frac {2 \left (a+b x^3\right )^{7/2} (A b-2 a B)}{21 b^3}-\frac {2 a \left (a+b x^3\right )^{5/2} (A b-a B)}{15 b^3}+\frac {2 B \left (a+b x^3\right )^{9/2}}{27 b^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 77} \begin {gather*} \frac {2 \left (a+b x^3\right )^{7/2} (A b-2 a B)}{21 b^3}-\frac {2 a \left (a+b x^3\right )^{5/2} (A b-a B)}{15 b^3}+\frac {2 B \left (a+b x^3\right )^{9/2}}{27 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rule 446
Rubi steps
\begin {align*} \int x^5 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x (a+b x)^{3/2} (A+B x) \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a (-A b+a B) (a+b x)^{3/2}}{b^2}+\frac {(A b-2 a B) (a+b x)^{5/2}}{b^2}+\frac {B (a+b x)^{7/2}}{b^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 a (A b-a B) \left (a+b x^3\right )^{5/2}}{15 b^3}+\frac {2 (A b-2 a B) \left (a+b x^3\right )^{7/2}}{21 b^3}+\frac {2 B \left (a+b x^3\right )^{9/2}}{27 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 57, normalized size = 0.78 \begin {gather*} \frac {2 \left (a+b x^3\right )^{5/2} \left (8 a^2 B-2 a b \left (9 A+10 B x^3\right )+5 b^2 x^3 \left (9 A+7 B x^3\right )\right )}{945 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.04, size = 56, normalized size = 0.77 \begin {gather*} \frac {2 \left (a+b x^3\right )^{5/2} \left (8 a^2 B-18 a A b-20 a b B x^3+45 A b^2 x^3+35 b^2 B x^6\right )}{945 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.82, size = 99, normalized size = 1.36 \begin {gather*} \frac {2 \, {\left (35 \, B b^{4} x^{12} + 5 \, {\left (10 \, B a b^{3} + 9 \, A b^{4}\right )} x^{9} + 3 \, {\left (B a^{2} b^{2} + 24 \, A a b^{3}\right )} x^{6} + 8 \, B a^{4} - 18 \, A a^{3} b - {\left (4 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{945 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 73, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (35 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}} B - 90 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} B a + 63 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} B a^{2} + 45 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} A b - 63 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} A a b\right )}}{945 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 53, normalized size = 0.73 \begin {gather*} -\frac {2 \left (b \,x^{3}+a \right )^{\frac {5}{2}} \left (-35 B \,b^{2} x^{6}-45 A \,b^{2} x^{3}+20 B a b \,x^{3}+18 A a b -8 B \,a^{2}\right )}{945 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.61, size = 84, normalized size = 1.15 \begin {gather*} \frac {2}{105} \, {\left (\frac {5 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}}}{b^{2}} - \frac {7 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a}{b^{2}}\right )} A + \frac {2}{945} \, {\left (\frac {35 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}}}{b^{3}} - \frac {90 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a}{b^{3}} + \frac {63 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{2}}{b^{3}}\right )} B \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.72, size = 211, normalized size = 2.89 \begin {gather*} \frac {x^6\,\sqrt {b\,x^3+a}\,\left (2\,B\,a^2+4\,A\,a\,b-\frac {6\,a\,\left (2\,A\,b^2+\frac {20\,B\,a\,b}{9}\right )}{7\,b}\right )}{15\,b}-\frac {2\,a\,\left (2\,A\,a^2-\frac {4\,a\,\left (2\,B\,a^2+4\,A\,a\,b-\frac {6\,a\,\left (2\,A\,b^2+\frac {20\,B\,a\,b}{9}\right )}{7\,b}\right )}{5\,b}\right )\,\sqrt {b\,x^3+a}}{9\,b^2}+\frac {2\,B\,b\,x^{12}\,\sqrt {b\,x^3+a}}{27}+\frac {x^3\,\left (2\,A\,a^2-\frac {4\,a\,\left (2\,B\,a^2+4\,A\,a\,b-\frac {6\,a\,\left (2\,A\,b^2+\frac {20\,B\,a\,b}{9}\right )}{7\,b}\right )}{5\,b}\right )\,\sqrt {b\,x^3+a}}{9\,b}+\frac {x^9\,\left (2\,A\,b^2+\frac {20\,B\,a\,b}{9}\right )\,\sqrt {b\,x^3+a}}{21\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.75, size = 216, normalized size = 2.96 \begin {gather*} \begin {cases} - \frac {4 A a^{3} \sqrt {a + b x^{3}}}{105 b^{2}} + \frac {2 A a^{2} x^{3} \sqrt {a + b x^{3}}}{105 b} + \frac {16 A a x^{6} \sqrt {a + b x^{3}}}{105} + \frac {2 A b x^{9} \sqrt {a + b x^{3}}}{21} + \frac {16 B a^{4} \sqrt {a + b x^{3}}}{945 b^{3}} - \frac {8 B a^{3} x^{3} \sqrt {a + b x^{3}}}{945 b^{2}} + \frac {2 B a^{2} x^{6} \sqrt {a + b x^{3}}}{315 b} + \frac {20 B a x^{9} \sqrt {a + b x^{3}}}{189} + \frac {2 B b x^{12} \sqrt {a + b x^{3}}}{27} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{6}}{6} + \frac {B x^{9}}{9}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________