Optimal. Leaf size=362 \[ \frac {d^{3/2} (a c+2 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{3/2} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {d^2 x (a c+2 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 c^2 (a c+b)}+\frac {d (a c+2 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 c^2 x (a c+b)}-\frac {a d^{3/2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {\left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 c x^3} \]
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Rubi [A] time = 0.63, antiderivative size = 472, normalized size of antiderivative = 1.30, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6722, 1975, 475, 583, 531, 418, 492, 411} \[ -\frac {d^2 x (a c+2 b) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (a c+b) \sqrt {a \left (c+d x^2\right )+b}}+\frac {d^{3/2} (a c+2 b) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{3/2} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a \left (c+d x^2\right )+b}}+\frac {d (a c+2 b) \left (c+d x^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 x (a c+b) \sqrt {a \left (c+d x^2\right )+b}}-\frac {a d^{3/2} \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt {a \left (c+d x^2\right )+b}}-\frac {\left (c+d x^2\right ) \sqrt {a c+a d x^2+b} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \sqrt {a \left (c+d x^2\right )+b}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 475
Rule 492
Rule 531
Rule 583
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^4} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a \left (c+d x^2\right )}}{x^4 \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 {\sqrt {b+a c+a d x^2}}{x^4 \sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \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 {-(2 b+a c) d-a d^2 x^2}{x^2 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 c \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(2 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) x \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 {a c (b+a c) d^2+a (2 b+a c) d^3 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(2 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (a d^2 \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 c \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (a (2 b+a c) d^3 \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 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {(2 b+a c) d^2 x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(2 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}-\frac {a d^{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 \sqrt {c} (b+a c) \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 ((2 b+a c) d^2 \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 c (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {(2 b+a c) d^2 x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c x^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(2 b+a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{3 c^2 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}+\frac {(2 b+a c) d^{3/2} \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 c^{3/2} (b+a c) \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 {a d^{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 \sqrt {c} (b+a c) \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 = 0.99, size = 314, normalized size = 0.87 \[ -\frac {\sqrt {\frac {a d}{a c+b}} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\left (c+d x^2\right ) \sqrt {\frac {a d}{a c+b}} \left (a^2 c \left (c^2-d^2 x^4\right )+2 a b \left (c^2-c d x^2-d^2 x^4\right )+b^2 \left (c-2 d x^2\right )\right )+i a b c d^2 x^3 \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )-i a c d^2 x^3 (a c+2 b) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|\frac {b}{a c}+1\right )\right )}{3 a c^2 d x^3 \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\sqrt {a + \frac {b}{d x^{2} + c}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 571, normalized size = 1.58 \[ \frac {\left (\sqrt {-\frac {a d}{a c +b}}\, a^{2} c \,d^{3} x^{6}+2 \sqrt {-\frac {a d}{a c +b}}\, a b \,d^{3} x^{6}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} d^{2} x^{4}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} c^{2} d^{2} x^{3} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )+4 \sqrt {-\frac {a d}{a c +b}}\, a b c \,d^{2} x^{4}-2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b c \,d^{2} x^{3} \EllipticE \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )+\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b c \,d^{2} x^{3} \EllipticF \left (\sqrt {-\frac {a d}{a c +b}}\, x , \sqrt {\frac {a c +b}{a c}}\right )-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{3} d \,x^{2}+2 \sqrt {-\frac {a d}{a c +b}}\, b^{2} d^{2} x^{4}-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{4}+\sqrt {-\frac {a d}{a c +b}}\, b^{2} c d \,x^{2}-2 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{3}-\sqrt {-\frac {a d}{a c +b}}\, b^{2} c^{2}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \left (a c +b \right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, c^{2} x^{3}} \]
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 \[ \int \frac {\sqrt {a + \frac {b}{d x^{2} + c}}}{x^{4}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^4} \,d x \]
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 \[ \int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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