Optimal. Leaf size=108 \[ -\frac{32 b^3 \left (b x+c x^2\right )^{5/2}}{1155 c^4 x^{5/2}}+\frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{231 c^3 x^{3/2}}-\frac{4 b \left (b x+c x^2\right )^{5/2}}{33 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0429493, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ -\frac{32 b^3 \left (b x+c x^2\right )^{5/2}}{1155 c^4 x^{5/2}}+\frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{231 c^3 x^{3/2}}-\frac{4 b \left (b x+c x^2\right )^{5/2}}{33 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 656
Rule 648
Rubi steps
\begin{align*} \int x^{3/2} \left (b x+c x^2\right )^{3/2} \, dx &=\frac{2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac{(6 b) \int \sqrt{x} \left (b x+c x^2\right )^{3/2} \, dx}{11 c}\\ &=-\frac{4 b \left (b x+c x^2\right )^{5/2}}{33 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c}+\frac{\left (8 b^2\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx}{33 c^2}\\ &=\frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{231 c^3 x^{3/2}}-\frac{4 b \left (b x+c x^2\right )^{5/2}}{33 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac{\left (16 b^3\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{231 c^3}\\ &=-\frac{32 b^3 \left (b x+c x^2\right )^{5/2}}{1155 c^4 x^{5/2}}+\frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{231 c^3 x^{3/2}}-\frac{4 b \left (b x+c x^2\right )^{5/2}}{33 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c}\\ \end{align*}
Mathematica [A] time = 0.0324024, size = 53, normalized size = 0.49 \[ \frac{2 (x (b+c x))^{5/2} \left (40 b^2 c x-16 b^3-70 b c^2 x^2+105 c^3 x^3\right )}{1155 c^4 x^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 55, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -105\,{x}^{3}{c}^{3}+70\,b{x}^{2}{c}^{2}-40\,{b}^{2}xc+16\,{b}^{3} \right ) }{1155\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.15566, size = 167, normalized size = 1.55 \begin{align*} \frac{2 \,{\left ({\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} x^{4} + 11 \,{\left (35 \, b c^{4} x^{5} + 5 \, b^{2} c^{3} x^{4} - 6 \, b^{3} c^{2} x^{3} + 8 \, b^{4} c x^{2} - 16 \, b^{5} x\right )} x^{3}\right )} \sqrt{c x + b}}{3465 \, c^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.97309, size = 166, normalized size = 1.54 \begin{align*} \frac{2 \,{\left (105 \, c^{5} x^{5} + 140 \, b c^{4} x^{4} + 5 \, b^{2} c^{3} x^{3} - 6 \, b^{3} c^{2} x^{2} + 8 \, b^{4} c x - 16 \, b^{5}\right )} \sqrt{c x^{2} + b x}}{1155 \, c^{4} \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19372, size = 181, normalized size = 1.68 \begin{align*} -\frac{2}{3465} \, c{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} + \frac{2}{315} \, b{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]