Optimal. Leaf size=424 \[ \frac {\sqrt {d} \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^2 c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {d x \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 a^2 c^3}+\frac {\left (c+d x^2\right ) \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 a^2 c^3 x}-\frac {b \sqrt {d} (b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^2 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\left (c+d x^2\right ) (b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 a c^2 x^3}-\frac {\left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 c x^5} \]
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Rubi [A] time = 0.63, antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6719, 475, 583, 531, 418, 492, 411} \[ -\frac {d x \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 a^2 c^3}+\frac {\left (c+d x^2\right ) \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 a^2 c^3 x}+\frac {\sqrt {d} \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^2 c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {d} (b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^2 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\left (c+d x^2\right ) (b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 a c^2 x^3}-\frac {\left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 c x^5} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 475
Rule 492
Rule 531
Rule 583
Rule 6719
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx &=\frac {\left (\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {\sqrt {a+b x^2}}{x^6 \sqrt {c+d x^2}} \, dx}{\sqrt {a+b x^2}}\\ &=-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c x^5}+\frac {\left (\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {b c-4 a d-3 b d x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 c \sqrt {a+b x^2}}\\ &=-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c x^5}-\frac {(b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a c^2 x^3}-\frac {\left (\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {2 b^2 c^2+3 a b c d-8 a^2 d^2+b d (b c-4 a d) x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 a c^2 \sqrt {a+b x^2}}\\ &=-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c x^5}-\frac {(b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a c^2 x^3}+\frac {\left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a^2 c^3 x}+\frac {\left (\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {-a b c d (b c-4 a d)-b d \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 a^2 c^3 \sqrt {a+b x^2}}\\ &=-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c x^5}-\frac {(b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a c^2 x^3}+\frac {\left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a^2 c^3 x}-\frac {\left (b d (b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 a c^2 \sqrt {a+b x^2}}-\frac {\left (b d \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 a^2 c^3 \sqrt {a+b x^2}}\\ &=-\frac {d \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 a^2 c^3}-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c x^5}-\frac {(b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a c^2 x^3}+\frac {\left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a^2 c^3 x}-\frac {b \sqrt {d} (b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^2 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\left (d \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 a^2 c^2 \sqrt {a+b x^2}}\\ &=-\frac {d \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 a^2 c^3}-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c x^5}-\frac {(b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a c^2 x^3}+\frac {\left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a^2 c^3 x}+\frac {\sqrt {d} \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^2 c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {d} (b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^2 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\\ \end {align*}
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Mathematica [C] time = 0.57, size = 302, normalized size = 0.71 \[ -\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-2 i b c x^5 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (2 a^2 d^2-a b c d-b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c x^5 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (8 a^2 d^2-3 a b c d-2 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a^2 \left (3 c^2-4 c d x^2+8 d^2 x^4\right )+a b c x^2 \left (c-3 d x^2\right )-2 b^2 c^2 x^4\right )\right )}{15 a^2 c^3 x^5 \sqrt {\frac {b}{a}} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{x^{6}}, x\right ) \]
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 \[ \int \frac {\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 708, normalized size = 1.67 \[ -\frac {\sqrt {\frac {\left (b \,x^{2}+a \right ) e}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (8 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{3} x^{8}-3 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{2} x^{8}-2 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d \,x^{8}+8 \sqrt {-\frac {b}{a}}\, a^{3} d^{3} x^{6}+\sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{2} x^{6}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} b c \,d^{2} x^{5} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} b c \,d^{2} x^{5} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-4 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d \,x^{6}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a \,b^{2} c^{2} d \,x^{5} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a \,b^{2} c^{2} d \,x^{5} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-2 \sqrt {-\frac {b}{a}}\, b^{3} c^{3} x^{6}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{3} c^{3} x^{5} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{3} c^{3} x^{5} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+4 \sqrt {-\frac {b}{a}}\, a^{3} c \,d^{2} x^{4}-3 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d \,x^{4}-\sqrt {-\frac {b}{a}}\, a \,b^{2} c^{3} x^{4}-\sqrt {-\frac {b}{a}}\, a^{3} c^{2} d \,x^{2}+4 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{3} x^{2}+3 \sqrt {-\frac {b}{a}}\, a^{3} c^{3}\right )}{15 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, a^{2} c^{3} x^{5}} \]
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 \[ \int \frac {\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{x^{6}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}}{x^6} \,d x \]
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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