Optimal. Leaf size=73 \[ \frac{2 \left (a+b x^3\right )^{3/2} (A b-2 a B)}{9 b^3}-\frac{2 a \sqrt{a+b x^3} (A b-a B)}{3 b^3}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0546994, 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} \[ \frac{2 \left (a+b x^3\right )^{3/2} (A b-2 a B)}{9 b^3}-\frac{2 a \sqrt{a+b x^3} (A b-a B)}{3 b^3}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^5 \left (A+B x^3\right )}{\sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x (A+B x)}{\sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a (-A b+a B)}{b^2 \sqrt{a+b x}}+\frac{(A b-2 a B) \sqrt{a+b x}}{b^2}+\frac{B (a+b x)^{3/2}}{b^2}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 a (A b-a B) \sqrt{a+b x^3}}{3 b^3}+\frac{2 (A b-2 a B) \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b^3}\\ \end{align*}
Mathematica [A] time = 0.039873, size = 56, normalized size = 0.77 \[ \frac{2 \sqrt{a+b x^3} \left (8 a^2 B-2 a b \left (5 A+2 B x^3\right )+b^2 x^3 \left (5 A+3 B x^3\right )\right )}{45 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 53, normalized size = 0.7 \begin{align*} -{\frac{-6\,{b}^{2}B{x}^{6}-10\,A{x}^{3}{b}^{2}+8\,B{x}^{3}ab+20\,abA-16\,{a}^{2}B}{45\,{b}^{3}}\sqrt{b{x}^{3}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.929006, size = 112, normalized size = 1.53 \begin{align*} \frac{2}{45} \, B{\left (\frac{3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{b^{3}} - \frac{10 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{b^{3}} + \frac{15 \, \sqrt{b x^{3} + a} a^{2}}{b^{3}}\right )} + \frac{2}{9} \, A{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{b^{2}} - \frac{3 \, \sqrt{b x^{3} + a} a}{b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6849, size = 117, normalized size = 1.6 \begin{align*} \frac{2 \,{\left (3 \, B b^{2} x^{6} -{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 10 \, A a b\right )} \sqrt{b x^{3} + a}}{45 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.59172, size = 124, normalized size = 1.7 \begin{align*} \begin{cases} - \frac{4 A a \sqrt{a + b x^{3}}}{9 b^{2}} + \frac{2 A x^{3} \sqrt{a + b x^{3}}}{9 b} + \frac{16 B a^{2} \sqrt{a + b x^{3}}}{45 b^{3}} - \frac{8 B a x^{3} \sqrt{a + b x^{3}}}{45 b^{2}} + \frac{2 B x^{6} \sqrt{a + b x^{3}}}{15 b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{9}}{9}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11859, size = 99, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} B - 10 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B a + 15 \, \sqrt{b x^{3} + a} B a^{2} + 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b - 15 \, \sqrt{b x^{3} + a} A a b\right )}}{45 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]