Optimal. Leaf size=96 \[ -\frac{\left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{63 c^2 x^3}+\frac{2 b \left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{315 c^3 x^5}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{9 c x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.069621, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1145, 2002, 2014} \[ -\frac{\left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{63 c^2 x^3}+\frac{2 b \left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{315 c^3 x^5}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{9 c x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1145
Rule 2002
Rule 2014
Rubi steps
\begin{align*} \int \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac{B \left (b x^2+c x^4\right )^{5/2}}{9 c x}-\frac{(4 b B-9 A c) \int \left (b x^2+c x^4\right )^{3/2} \, dx}{9 c}\\ &=-\frac{(4 b B-9 A c) \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{9 c x}+\frac{(2 b (4 b B-9 A c)) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx}{63 c^2}\\ &=\frac{2 b (4 b B-9 A c) \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac{(4 b B-9 A c) \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{9 c x}\\ \end{align*}
Mathematica [A] time = 0.047036, size = 71, normalized size = 0.74 \[ \frac{x \left (b+c x^2\right )^3 \left (-2 b c \left (9 A+10 B x^2\right )+5 c^2 x^2 \left (9 A+7 B x^2\right )+8 b^2 B\right )}{315 c^3 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 67, normalized size = 0.7 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -35\,B{c}^{2}{x}^{4}-45\,A{x}^{2}{c}^{2}+20\,B{x}^{2}bc+18\,Abc-8\,B{b}^{2} \right ) }{315\,{c}^{3}{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.27695, size = 142, normalized size = 1.48 \begin{align*} \frac{{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{2} + b} A}{35 \, c^{2}} + \frac{{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt{c x^{2} + b} B}{315 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.08831, size = 228, normalized size = 2.38 \begin{align*} \frac{{\left (35 \, B c^{4} x^{8} + 5 \,{\left (10 \, B b c^{3} + 9 \, A c^{4}\right )} x^{6} + 8 \, B b^{4} - 18 \, A b^{3} c + 3 \,{\left (B b^{2} c^{2} + 24 \, A b c^{3}\right )} x^{4} -{\left (4 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, c^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.13837, size = 288, normalized size = 3. \begin{align*} \frac{\frac{21 \,{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} A b \mathrm{sgn}\left (x\right )}{c} + \frac{3 \,{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )} B b \mathrm{sgn}\left (x\right )}{c^{2}} + \frac{3 \,{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )} A \mathrm{sgn}\left (x\right )}{c} + \frac{{\left (35 \,{\left (c x^{2} + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{3}\right )} B \mathrm{sgn}\left (x\right )}{c^{2}}}{315 \, c} - \frac{2 \,{\left (4 \, B b^{\frac{9}{2}} - 9 \, A b^{\frac{7}{2}} c\right )} \mathrm{sgn}\left (x\right )}{315 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]