Optimal. Leaf size=71 \[ \frac{A b^2 x^{m+5}}{m+5}+\frac{b x^{m+7} (2 A c+b B)}{m+7}+\frac{c x^{m+9} (A c+2 b B)}{m+9}+\frac{B c^2 x^{m+11}}{m+11} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0519871, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1584, 448} \[ \frac{A b^2 x^{m+5}}{m+5}+\frac{b x^{m+7} (2 A c+b B)}{m+7}+\frac{c x^{m+9} (A c+2 b B)}{m+9}+\frac{B c^2 x^{m+11}}{m+11} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1584
Rule 448
Rubi steps
\begin{align*} \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx &=\int x^{4+m} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{4+m}+b (b B+2 A c) x^{6+m}+c (2 b B+A c) x^{8+m}+B c^2 x^{10+m}\right ) \, dx\\ &=\frac{A b^2 x^{5+m}}{5+m}+\frac{b (b B+2 A c) x^{7+m}}{7+m}+\frac{c (2 b B+A c) x^{9+m}}{9+m}+\frac{B c^2 x^{11+m}}{11+m}\\ \end{align*}
Mathematica [A] time = 0.0571561, size = 66, normalized size = 0.93 \[ x^{m+5} \left (\frac{A b^2}{m+5}+\frac{c x^4 (A c+2 b B)}{m+9}+\frac{b x^2 (2 A c+b B)}{m+7}+\frac{B c^2 x^6}{m+11}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.006, size = 262, normalized size = 3.7 \begin{align*}{\frac{{x}^{5+m} \left ( B{c}^{2}{m}^{3}{x}^{6}+21\,B{c}^{2}{m}^{2}{x}^{6}+A{c}^{2}{m}^{3}{x}^{4}+2\,Bbc{m}^{3}{x}^{4}+143\,B{c}^{2}m{x}^{6}+23\,A{c}^{2}{m}^{2}{x}^{4}+46\,Bbc{m}^{2}{x}^{4}+315\,B{c}^{2}{x}^{6}+2\,Abc{m}^{3}{x}^{2}+167\,A{c}^{2}m{x}^{4}+B{b}^{2}{m}^{3}{x}^{2}+334\,Bbcm{x}^{4}+50\,Abc{m}^{2}{x}^{2}+385\,A{c}^{2}{x}^{4}+25\,B{b}^{2}{m}^{2}{x}^{2}+770\,B{x}^{4}bc+A{b}^{2}{m}^{3}+398\,Abcm{x}^{2}+199\,B{b}^{2}m{x}^{2}+27\,A{b}^{2}{m}^{2}+990\,Abc{x}^{2}+495\,B{x}^{2}{b}^{2}+239\,A{b}^{2}m+693\,A{b}^{2} \right ) }{ \left ( 11+m \right ) \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.30448, size = 512, normalized size = 7.21 \begin{align*} \frac{{\left ({\left (B c^{2} m^{3} + 21 \, B c^{2} m^{2} + 143 \, B c^{2} m + 315 \, B c^{2}\right )} x^{11} +{\left ({\left (2 \, B b c + A c^{2}\right )} m^{3} + 770 \, B b c + 385 \, A c^{2} + 23 \,{\left (2 \, B b c + A c^{2}\right )} m^{2} + 167 \,{\left (2 \, B b c + A c^{2}\right )} m\right )} x^{9} +{\left ({\left (B b^{2} + 2 \, A b c\right )} m^{3} + 495 \, B b^{2} + 990 \, A b c + 25 \,{\left (B b^{2} + 2 \, A b c\right )} m^{2} + 199 \,{\left (B b^{2} + 2 \, A b c\right )} m\right )} x^{7} +{\left (A b^{2} m^{3} + 27 \, A b^{2} m^{2} + 239 \, A b^{2} m + 693 \, A b^{2}\right )} x^{5}\right )} x^{m}}{m^{4} + 32 \, m^{3} + 374 \, m^{2} + 1888 \, m + 3465} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 5.04163, size = 1051, normalized size = 14.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.34679, size = 459, normalized size = 6.46 \begin{align*} \frac{B c^{2} m^{3} x^{11} x^{m} + 21 \, B c^{2} m^{2} x^{11} x^{m} + 2 \, B b c m^{3} x^{9} x^{m} + A c^{2} m^{3} x^{9} x^{m} + 143 \, B c^{2} m x^{11} x^{m} + 46 \, B b c m^{2} x^{9} x^{m} + 23 \, A c^{2} m^{2} x^{9} x^{m} + 315 \, B c^{2} x^{11} x^{m} + B b^{2} m^{3} x^{7} x^{m} + 2 \, A b c m^{3} x^{7} x^{m} + 334 \, B b c m x^{9} x^{m} + 167 \, A c^{2} m x^{9} x^{m} + 25 \, B b^{2} m^{2} x^{7} x^{m} + 50 \, A b c m^{2} x^{7} x^{m} + 770 \, B b c x^{9} x^{m} + 385 \, A c^{2} x^{9} x^{m} + A b^{2} m^{3} x^{5} x^{m} + 199 \, B b^{2} m x^{7} x^{m} + 398 \, A b c m x^{7} x^{m} + 27 \, A b^{2} m^{2} x^{5} x^{m} + 495 \, B b^{2} x^{7} x^{m} + 990 \, A b c x^{7} x^{m} + 239 \, A b^{2} m x^{5} x^{m} + 693 \, A b^{2} x^{5} x^{m}}{m^{4} + 32 \, m^{3} + 374 \, m^{2} + 1888 \, m + 3465} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]