Optimal. Leaf size=161 \[ \frac {a^3 c x^{m+1} \sqrt {c \left (a+b x^2\right )^2}}{(m+1) \left (a+b x^2\right )}+\frac {3 a^2 b c x^{m+3} \sqrt {c \left (a+b x^2\right )^2}}{(m+3) \left (a+b x^2\right )}+\frac {b^3 c x^{m+7} \sqrt {c \left (a+b x^2\right )^2}}{(m+7) \left (a+b x^2\right )}+\frac {3 a b^2 c x^{m+5} \sqrt {c \left (a+b x^2\right )^2}}{(m+5) \left (a+b x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 205, normalized size of antiderivative = 1.27, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1989, 1112, 270} \[ \frac {a^3 c x^{m+1} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{(m+1) \left (a+b x^2\right )}+\frac {3 a^2 b c x^{m+3} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{(m+3) \left (a+b x^2\right )}+\frac {3 a b^2 c x^{m+5} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{(m+5) \left (a+b x^2\right )}+\frac {b^3 c x^{m+7} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{(m+7) \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 1112
Rule 1989
Rubi steps
\begin {align*} \int x^m \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx &=\int x^m \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2} \, dx\\ &=\frac {\sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \int x^m \left (a b c+b^2 c x^2\right )^3 \, dx}{b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac {\sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \int \left (a^3 b^3 c^3 x^m+3 a^2 b^4 c^3 x^{2+m}+3 a b^5 c^3 x^{4+m}+b^6 c^3 x^{6+m}\right ) \, dx}{b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac {a^3 c x^{1+m} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{(1+m) \left (a+b x^2\right )}+\frac {3 a^2 b c x^{3+m} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{(3+m) \left (a+b x^2\right )}+\frac {3 a b^2 c x^{5+m} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{(5+m) \left (a+b x^2\right )}+\frac {b^3 c x^{7+m} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{(7+m) \left (a+b x^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 132, normalized size = 0.82 \[ \frac {x^{m+1} \left (c \left (a+b x^2\right )^2\right )^{3/2} \left (a^3 \left (m^3+15 m^2+71 m+105\right )+3 a^2 b \left (m^3+13 m^2+47 m+35\right ) x^2+3 a b^2 \left (m^3+11 m^2+31 m+21\right ) x^4+b^3 \left (m^3+9 m^2+23 m+15\right ) x^6\right )}{(m+1) (m+3) (m+5) (m+7) \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.91, size = 233, normalized size = 1.45 \[ \frac {{\left ({\left (b^{3} c m^{3} + 9 \, b^{3} c m^{2} + 23 \, b^{3} c m + 15 \, b^{3} c\right )} x^{7} + 3 \, {\left (a b^{2} c m^{3} + 11 \, a b^{2} c m^{2} + 31 \, a b^{2} c m + 21 \, a b^{2} c\right )} x^{5} + 3 \, {\left (a^{2} b c m^{3} + 13 \, a^{2} b c m^{2} + 47 \, a^{2} b c m + 35 \, a^{2} b c\right )} x^{3} + {\left (a^{3} c m^{3} + 15 \, a^{3} c m^{2} + 71 \, a^{3} c m + 105 \, a^{3} c\right )} x\right )} \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c} x^{m}}{a m^{4} + 16 \, a m^{3} + 86 \, a m^{2} + {\left (b m^{4} + 16 \, b m^{3} + 86 \, b m^{2} + 176 \, b m + 105 \, b\right )} x^{2} + 176 \, a m + 105 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.38, size = 355, normalized size = 2.20 \[ \frac {{\left (b^{3} m^{3} x^{7} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 9 \, b^{3} m^{2} x^{7} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, a b^{2} m^{3} x^{5} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 23 \, b^{3} m x^{7} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 33 \, a b^{2} m^{2} x^{5} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 15 \, b^{3} x^{7} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, a^{2} b m^{3} x^{3} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 93 \, a b^{2} m x^{5} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 39 \, a^{2} b m^{2} x^{3} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 63 \, a b^{2} x^{5} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} m^{3} x x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 141 \, a^{2} b m x^{3} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 15 \, a^{3} m^{2} x x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, a^{2} b x^{3} x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 71 \, a^{3} m x x^{m} \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, a^{3} x x^{m} \mathrm {sgn}\left (b x^{2} + a\right )\right )} c^{\frac {3}{2}}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 200, normalized size = 1.24 \[ \frac {\left (b^{3} m^{3} x^{6}+9 b^{3} m^{2} x^{6}+3 a \,b^{2} m^{3} x^{4}+23 b^{3} m \,x^{6}+33 a \,b^{2} m^{2} x^{4}+15 b^{3} x^{6}+3 a^{2} b \,m^{3} x^{2}+93 a \,b^{2} m \,x^{4}+39 a^{2} b \,m^{2} x^{2}+63 a \,b^{2} x^{4}+a^{3} m^{3}+141 a^{2} b m \,x^{2}+15 a^{3} m^{2}+105 a^{2} b \,x^{2}+71 a^{3} m +105 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2} c \right )^{\frac {3}{2}} x^{m +1}}{\left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right ) \left (b \,x^{2}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.99, size = 119, normalized size = 0.74 \[ \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b^{3} c^{\frac {3}{2}} x^{7} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} a b^{2} c^{\frac {3}{2}} x^{5} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} a^{2} b c^{\frac {3}{2}} x^{3} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} a^{3} c^{\frac {3}{2}} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.97, size = 234, normalized size = 1.45 \[ \frac {x^m\,\left (\frac {3\,a^2\,c\,x^3\,\sqrt {c\,{\left (b\,x^2+a\right )}^2}\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {b^2\,c\,x^7\,\sqrt {c\,{\left (b\,x^2+a\right )}^2}\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {3\,a\,b\,c\,x^5\,\sqrt {c\,{\left (b\,x^2+a\right )}^2}\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {a^3\,c\,x\,\sqrt {c\,{\left (b\,x^2+a\right )}^2}\,\left (m^3+15\,m^2+71\,m+105\right )}{b\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}\right )}{\frac {a}{b}+x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________