Optimal. Leaf size=63 \[ \frac {2}{3} a^2 c x^{3/2}+\frac {2}{7} a (2 b c+a d) x^{7/2}+\frac {2}{11} b (b c+2 a d) x^{11/2}+\frac {2}{15} b^2 d x^{15/2} \]
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Rubi [A]
time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459}
\begin {gather*} \frac {2}{3} a^2 c x^{3/2}+\frac {2}{11} b x^{11/2} (2 a d+b c)+\frac {2}{7} a x^{7/2} (a d+2 b c)+\frac {2}{15} b^2 d x^{15/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\int \left (a^2 c \sqrt {x}+a (2 b c+a d) x^{5/2}+b (b c+2 a d) x^{9/2}+b^2 d x^{13/2}\right ) \, dx\\ &=\frac {2}{3} a^2 c x^{3/2}+\frac {2}{7} a (2 b c+a d) x^{7/2}+\frac {2}{11} b (b c+2 a d) x^{11/2}+\frac {2}{15} b^2 d x^{15/2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 59, normalized size = 0.94 \begin {gather*} \frac {2 x^{3/2} \left (385 a^2 c+330 a b c x^2+165 a^2 d x^2+105 b^2 c x^4+210 a b d x^4+77 b^2 d x^6\right )}{1155} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 52, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {2 b^{2} d \,x^{\frac {15}{2}}}{15}+\frac {2 \left (2 a b d +b^{2} c \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} d +2 a b c \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{2} c \,x^{\frac {3}{2}}}{3}\) | \(52\) |
default | \(\frac {2 b^{2} d \,x^{\frac {15}{2}}}{15}+\frac {2 \left (2 a b d +b^{2} c \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} d +2 a b c \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{2} c \,x^{\frac {3}{2}}}{3}\) | \(52\) |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (77 b^{2} d \,x^{6}+210 a b d \,x^{4}+105 b^{2} c \,x^{4}+165 a^{2} d \,x^{2}+330 a b c \,x^{2}+385 a^{2} c \right )}{1155}\) | \(56\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (77 b^{2} d \,x^{6}+210 a b d \,x^{4}+105 b^{2} c \,x^{4}+165 a^{2} d \,x^{2}+330 a b c \,x^{2}+385 a^{2} c \right )}{1155}\) | \(56\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (77 b^{2} d \,x^{6}+210 a b d \,x^{4}+105 b^{2} c \,x^{4}+165 a^{2} d \,x^{2}+330 a b c \,x^{2}+385 a^{2} c \right )}{1155}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{15} \, b^{2} d x^{\frac {15}{2}} + \frac {2}{11} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {11}{2}} + \frac {2}{3} \, a^{2} c x^{\frac {3}{2}} + \frac {2}{7} \, {\left (2 \, a b c + a^{2} d\right )} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.98, size = 54, normalized size = 0.86 \begin {gather*} \frac {2}{1155} \, {\left (77 \, b^{2} d x^{7} + 105 \, {\left (b^{2} c + 2 \, a b d\right )} x^{5} + 385 \, a^{2} c x + 165 \, {\left (2 \, a b c + a^{2} d\right )} x^{3}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.35, size = 66, normalized size = 1.05 \begin {gather*} \frac {2 a^{2} c x^{\frac {3}{2}}}{3} + \frac {2 b^{2} d x^{\frac {15}{2}}}{15} + \frac {2 x^{\frac {11}{2}} \cdot \left (2 a b d + b^{2} c\right )}{11} + \frac {2 x^{\frac {7}{2}} \left (a^{2} d + 2 a b c\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{15} \, b^{2} d x^{\frac {15}{2}} + \frac {2}{11} \, b^{2} c x^{\frac {11}{2}} + \frac {4}{11} \, a b d x^{\frac {11}{2}} + \frac {4}{7} \, a b c x^{\frac {7}{2}} + \frac {2}{7} \, a^{2} d x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} c x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 51, normalized size = 0.81 \begin {gather*} x^{7/2}\,\left (\frac {2\,d\,a^2}{7}+\frac {4\,b\,c\,a}{7}\right )+x^{11/2}\,\left (\frac {2\,c\,b^2}{11}+\frac {4\,a\,d\,b}{11}\right )+\frac {2\,a^2\,c\,x^{3/2}}{3}+\frac {2\,b^2\,d\,x^{15/2}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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