Optimal. Leaf size=97 \[ \frac {2}{11} x^{11/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {2}{3} a^2 c^2 x^{3/2}+\frac {4}{15} b d x^{15/2} (a d+b c)+\frac {4}{7} a c x^{7/2} (a d+b c)+\frac {2}{19} b^2 d^2 x^{19/2} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {448} \begin {gather*} \frac {2}{11} x^{11/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {2}{3} a^2 c^2 x^{3/2}+\frac {4}{15} b d x^{15/2} (a d+b c)+\frac {4}{7} a c x^{7/2} (a d+b c)+\frac {2}{19} b^2 d^2 x^{19/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 448
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\int \left (a^2 c^2 \sqrt {x}+2 a c (b c+a d) x^{5/2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{9/2}+2 b d (b c+a d) x^{13/2}+b^2 d^2 x^{17/2}\right ) \, dx\\ &=\frac {2}{3} a^2 c^2 x^{3/2}+\frac {4}{7} a c (b c+a d) x^{7/2}+\frac {2}{11} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{11/2}+\frac {4}{15} b d (b c+a d) x^{15/2}+\frac {2}{19} b^2 d^2 x^{19/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 83, normalized size = 0.86 \begin {gather*} \frac {2 x^{3/2} \left (1995 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+7315 a^2 c^2+2926 b d x^6 (a d+b c)+6270 a c x^2 (a d+b c)+1155 b^2 d^2 x^8\right )}{21945} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.05, size = 116, normalized size = 1.20 \begin {gather*} \frac {2 \left (7315 a^2 c^2 x^{3/2}+6270 a^2 c d x^{7/2}+1995 a^2 d^2 x^{11/2}+6270 a b c^2 x^{7/2}+7980 a b c d x^{11/2}+2926 a b d^2 x^{15/2}+1995 b^2 c^2 x^{11/2}+2926 b^2 c d x^{15/2}+1155 b^2 d^2 x^{19/2}\right )}{21945} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 88, normalized size = 0.91 \begin {gather*} \frac {2}{21945} \, {\left (1155 \, b^{2} d^{2} x^{9} + 2926 \, {\left (b^{2} c d + a b d^{2}\right )} x^{7} + 1995 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + 7315 \, a^{2} c^{2} x + 6270 \, {\left (a b c^{2} + a^{2} c d\right )} x^{3}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.42, size = 94, normalized size = 0.97 \begin {gather*} \frac {2}{19} \, b^{2} d^{2} x^{\frac {19}{2}} + \frac {4}{15} \, b^{2} c d x^{\frac {15}{2}} + \frac {4}{15} \, a b d^{2} x^{\frac {15}{2}} + \frac {2}{11} \, b^{2} c^{2} x^{\frac {11}{2}} + \frac {8}{11} \, a b c d x^{\frac {11}{2}} + \frac {2}{11} \, a^{2} d^{2} x^{\frac {11}{2}} + \frac {4}{7} \, a b c^{2} x^{\frac {7}{2}} + \frac {4}{7} \, a^{2} c d x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} c^{2} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 97, normalized size = 1.00 \begin {gather*} \frac {2 \left (1155 b^{2} d^{2} x^{8}+2926 a b \,d^{2} x^{6}+2926 b^{2} c d \,x^{6}+1995 a^{2} d^{2} x^{4}+7980 a b c d \,x^{4}+1995 b^{2} c^{2} x^{4}+6270 a^{2} c d \,x^{2}+6270 a b \,c^{2} x^{2}+7315 a^{2} c^{2}\right ) x^{\frac {3}{2}}}{21945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.06, size = 85, normalized size = 0.88 \begin {gather*} \frac {2}{19} \, b^{2} d^{2} x^{\frac {19}{2}} + \frac {4}{15} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, a^{2} c^{2} x^{\frac {3}{2}} + \frac {4}{7} \, {\left (a b c^{2} + a^{2} c d\right )} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 78, normalized size = 0.80 \begin {gather*} x^{11/2}\,\left (\frac {2\,a^2\,d^2}{11}+\frac {8\,a\,b\,c\,d}{11}+\frac {2\,b^2\,c^2}{11}\right )+\frac {2\,a^2\,c^2\,x^{3/2}}{3}+\frac {2\,b^2\,d^2\,x^{19/2}}{19}+\frac {4\,a\,c\,x^{7/2}\,\left (a\,d+b\,c\right )}{7}+\frac {4\,b\,d\,x^{15/2}\,\left (a\,d+b\,c\right )}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.38, size = 110, normalized size = 1.13 \begin {gather*} \frac {2 a^{2} c^{2} x^{\frac {3}{2}}}{3} + \frac {2 b^{2} d^{2} x^{\frac {19}{2}}}{19} + \frac {2 x^{\frac {15}{2}} \left (2 a b d^{2} + 2 b^{2} c d\right )}{15} + \frac {2 x^{\frac {11}{2}} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{11} + \frac {2 x^{\frac {7}{2}} \left (2 a^{2} c d + 2 a b c^{2}\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________