Optimal. Leaf size=282 \[ -\frac {c^{3/2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {\sqrt {c} (b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a d^{3/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x (b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 a d}+\frac {x \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 d} \]
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Rubi [A] time = 0.52, antiderivative size = 370, normalized size of antiderivative = 1.31, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6722, 1975, 478, 531, 418, 492, 411} \[ -\frac {c^{3/2} \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a \left (c+d x^2\right )+b}}-\frac {\sqrt {c} (b-a c) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a d^{3/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a \left (c+d x^2\right )+b}}+\frac {x (b-a c) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{3 a d \sqrt {a \left (c+d x^2\right )+b}}+\frac {x \left (c+d x^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{3 d \sqrt {a \left (c+d x^2\right )+b}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 478
Rule 492
Rule 531
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^2 \sqrt {b+a \left (c+d x^2\right )}}{\sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^2 \sqrt {b+a c+a d x^2}}{\sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {c (b+a c)-(b-a c) d x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 d \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left ((b-a c) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c (b+a c) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(b-a c) x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 a d \sqrt {b+a \left (c+d x^2\right )}}+\frac {x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {c^{3/2} \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c (b-a c) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(b-a c) x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 a d \sqrt {b+a \left (c+d x^2\right )}}+\frac {x \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 d \sqrt {b+a \left (c+d x^2\right )}}-\frac {\sqrt {c} (b-a c) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}-\frac {c^{3/2} \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}
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Mathematica [C] time = 0.58, size = 250, normalized size = 0.89 \[ \frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (x \left (c+d x^2\right ) \sqrt {\frac {a d}{a c+b}} \left (a c+a d x^2+b\right )+2 i b c \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )+i c (a c-b) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )\right )}{3 d \sqrt {\frac {a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + \frac {b}{d x^{2} + c}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 406, normalized size = 1.44 \[ \frac {\left (\sqrt {-\frac {a d}{a c +b}}\, a \,d^{2} x^{5}+2 \sqrt {-\frac {a d}{a c +b}}\, a c d \,x^{3}+\sqrt {-\frac {a d}{a c +b}}\, b d \,x^{3}+\sqrt {-\frac {a d}{a c +b}}\, a \,c^{2} x -\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a \,c^{2} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )+\sqrt {-\frac {a d}{a c +b}}\, b c x +\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b c \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )-2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b c \EllipticF \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + \frac {b}{d x^{2} + c}} x^{2}\,{d x} \]
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 \[ \int x^2\,\sqrt {a+\frac {b}{d\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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