Optimal. Leaf size=243 \[ \frac {c^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{5/2} \left (a+b x^2\right )}-\frac {c x \sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2} (b c-2 a d)}{16 d^2 \left (a+b x^2\right )}+\frac {b x^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{6 d \left (a+b x^2\right )}-\frac {x^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2} (b c-2 a d)}{8 d \left (a+b x^2\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1250, 459, 279, 321, 217, 206} \[ \frac {c^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{5/2} \left (a+b x^2\right )}-\frac {c x \sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2} (b c-2 a d)}{16 d^2 \left (a+b x^2\right )}+\frac {b x^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{6 d \left (a+b x^2\right )}-\frac {x^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2} (b c-2 a d)}{8 d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 279
Rule 321
Rule 459
Rule 1250
Rubi steps
\begin {align*} \int x^2 \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int x^2 \left (a b+b^2 x^2\right ) \sqrt {c+d x^2} \, dx}{a b+b^2 x^2}\\ &=\frac {b x^3 \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 d \left (a+b x^2\right )}-\frac {\left (b (b c-2 a d) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \int x^2 \sqrt {c+d x^2} \, dx}{2 d \left (a b+b^2 x^2\right )}\\ &=-\frac {(b c-2 a d) x^3 \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 d \left (a+b x^2\right )}+\frac {b x^3 \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 d \left (a+b x^2\right )}-\frac {\left (b c (b c-2 a d) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx}{8 d \left (a b+b^2 x^2\right )}\\ &=-\frac {c (b c-2 a d) x \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 d^2 \left (a+b x^2\right )}-\frac {(b c-2 a d) x^3 \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 d \left (a+b x^2\right )}+\frac {b x^3 \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 d \left (a+b x^2\right )}+\frac {\left (b c^2 (b c-2 a d) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{16 d^2 \left (a b+b^2 x^2\right )}\\ &=-\frac {c (b c-2 a d) x \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 d^2 \left (a+b x^2\right )}-\frac {(b c-2 a d) x^3 \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 d \left (a+b x^2\right )}+\frac {b x^3 \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 d \left (a+b x^2\right )}+\frac {\left (b c^2 (b c-2 a d) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{16 d^2 \left (a b+b^2 x^2\right )}\\ &=-\frac {c (b c-2 a d) x \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 d^2 \left (a+b x^2\right )}-\frac {(b c-2 a d) x^3 \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 d \left (a+b x^2\right )}+\frac {b x^3 \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 d \left (a+b x^2\right )}+\frac {c^2 (b c-2 a d) \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{5/2} \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 142, normalized size = 0.58 \[ \frac {\sqrt {\left (a+b x^2\right )^2} \sqrt {c+d x^2} \left (3 c^{3/2} (b c-2 a d) \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )+\sqrt {d} x \sqrt {\frac {d x^2}{c}+1} \left (6 a d \left (c+2 d x^2\right )+b \left (-3 c^2+2 c d x^2+8 d^2 x^4\right )\right )\right )}{48 d^{5/2} \left (a+b x^2\right ) \sqrt {\frac {d x^2}{c}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 206, normalized size = 0.85 \[ \left [-\frac {3 \, {\left (b c^{3} - 2 \, a c^{2} d\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (8 \, b d^{3} x^{5} + 2 \, {\left (b c d^{2} + 6 \, a d^{3}\right )} x^{3} - 3 \, {\left (b c^{2} d - 2 \, a c d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{96 \, d^{3}}, -\frac {3 \, {\left (b c^{3} - 2 \, a c^{2} d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (8 \, b d^{3} x^{5} + 2 \, {\left (b c d^{2} + 6 \, a d^{3}\right )} x^{3} - 3 \, {\left (b c^{2} d - 2 \, a c d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{48 \, d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 156, normalized size = 0.64 \[ \frac {1}{48} \, {\left (2 \, {\left (4 \, b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {b c d^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a d^{4} \mathrm {sgn}\left (b x^{2} + a\right )}{d^{4}}\right )} x^{2} - \frac {3 \, {\left (b c^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right ) - 2 \, a c d^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )}}{d^{4}}\right )} \sqrt {d x^{2} + c} x - \frac {{\left (b c^{3} \mathrm {sgn}\left (b x^{2} + a\right ) - 2 \, a c^{2} d \mathrm {sgn}\left (b x^{2} + a\right )\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{16 \, d^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 159, normalized size = 0.65 \[ \frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (8 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b \,d^{\frac {3}{2}} x^{3}-6 a \,c^{2} d \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+3 b \,c^{3} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )-6 \sqrt {d \,x^{2}+c}\, a c \,d^{\frac {3}{2}} x +3 \sqrt {d \,x^{2}+c}\, b \,c^{2} \sqrt {d}\, x +12 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a \,d^{\frac {3}{2}} x -6 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b c \sqrt {d}\, x \right )}{48 \left (b \,x^{2}+a \right ) d^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 124, normalized size = 0.51 \[ \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b x^{3}}{6 \, d} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b c x}{8 \, d^{2}} + \frac {\sqrt {d x^{2} + c} b c^{2} x}{16 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a x}{4 \, d} - \frac {\sqrt {d x^{2} + c} a c x}{8 \, d} + \frac {b c^{3} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{16 \, d^{\frac {5}{2}}} - \frac {a c^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, d^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\sqrt {d\,x^2+c}\,\sqrt {{\left (b\,x^2+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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