Optimal. Leaf size=73 \[ \frac{2 \left (a+b x^3\right )^{5/2} (A b-2 a B)}{15 b^3}-\frac{2 a \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^3}+\frac{2 B \left (a+b x^3\right )^{7/2}}{21 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0598291, antiderivative size = 73, normalized size of antiderivative = 1., 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} \[ \frac{2 \left (a+b x^3\right )^{5/2} (A b-2 a B)}{15 b^3}-\frac{2 a \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^3}+\frac{2 B \left (a+b x^3\right )^{7/2}}{21 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 77
Rubi steps
\begin{align*} \int x^5 \sqrt{a+b x^3} \left (A+B x^3\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x \sqrt{a+b x} (A+B x) \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a (-A b+a B) \sqrt{a+b x}}{b^2}+\frac{(A b-2 a B) (a+b x)^{3/2}}{b^2}+\frac{B (a+b x)^{5/2}}{b^2}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 a (A b-a B) \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac{2 (A b-2 a B) \left (a+b x^3\right )^{5/2}}{15 b^3}+\frac{2 B \left (a+b x^3\right )^{7/2}}{21 b^3}\\ \end{align*}
Mathematica [A] time = 0.0387175, size = 57, normalized size = 0.78 \[ \frac{2 \left (a+b x^3\right )^{3/2} \left (8 a^2 B-2 a b \left (7 A+6 B x^3\right )+3 b^2 x^3 \left (7 A+5 B x^3\right )\right )}{315 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 53, normalized size = 0.7 \begin{align*} -{\frac{-30\,{b}^{2}B{x}^{6}-42\,A{x}^{3}{b}^{2}+24\,B{x}^{3}ab+28\,abA-16\,{a}^{2}B}{315\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.937005, size = 113, normalized size = 1.55 \begin{align*} \frac{2}{315} \, B{\left (\frac{15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{b^{3}} - \frac{42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{b^{3}} + \frac{35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}}{b^{3}}\right )} + \frac{2}{45} \, A{\left (\frac{3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69199, size = 166, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (15 \, B b^{3} x^{9} + 3 \,{\left (B a b^{2} + 7 \, A b^{3}\right )} x^{6} + 8 \, B a^{3} - 14 \, A a^{2} b -{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt{b x^{3} + a}}{315 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.61222, size = 168, normalized size = 2.3 \begin{align*} \begin{cases} - \frac{4 A a^{2} \sqrt{a + b x^{3}}}{45 b^{2}} + \frac{2 A a x^{3} \sqrt{a + b x^{3}}}{45 b} + \frac{2 A x^{6} \sqrt{a + b x^{3}}}{15} + \frac{16 B a^{3} \sqrt{a + b x^{3}}}{315 b^{3}} - \frac{8 B a^{2} x^{3} \sqrt{a + b x^{3}}}{315 b^{2}} + \frac{2 B a x^{6} \sqrt{a + b x^{3}}}{105 b} + \frac{2 B x^{9} \sqrt{a + b x^{3}}}{21} & \text{for}\: b \neq 0 \\\sqrt{a} \left (\frac{A x^{6}}{6} + \frac{B x^{9}}{9}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10447, size = 107, normalized size = 1.47 \begin{align*} \frac{2 \,{\left (\frac{7 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a\right )} A}{b} + \frac{{\left (15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}\right )} B}{b^{2}}\right )}}{315 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]