Optimal. Leaf size=189 \[ \frac {\sqrt {a+b x^2} (8 b c-9 a d)}{63 a^2 x^7}-\frac {\sqrt {a+b x^2} \left (21 a^2 e-18 a b d+16 b^2 c\right )}{105 a^3 x^5}-\frac {2 b \sqrt {a+b x^2} \left (-105 a^3 f+84 a^2 b e-72 a b^2 d+64 b^3 c\right )}{315 a^5 x}+\frac {\sqrt {a+b x^2} \left (-105 a^3 f+84 a^2 b e-72 a b^2 d+64 b^3 c\right )}{315 a^4 x^3}-\frac {c \sqrt {a+b x^2}}{9 a x^9} \]
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Rubi [A] time = 0.25, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1803, 12, 271, 264} \begin {gather*} -\frac {2 b \sqrt {a+b x^2} \left (84 a^2 b e-105 a^3 f-72 a b^2 d+64 b^3 c\right )}{315 a^5 x}+\frac {\sqrt {a+b x^2} \left (84 a^2 b e-105 a^3 f-72 a b^2 d+64 b^3 c\right )}{315 a^4 x^3}-\frac {\sqrt {a+b x^2} \left (21 a^2 e-18 a b d+16 b^2 c\right )}{105 a^3 x^5}+\frac {\sqrt {a+b x^2} (8 b c-9 a d)}{63 a^2 x^7}-\frac {c \sqrt {a+b x^2}}{9 a x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 271
Rule 1803
Rubi steps
\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}-\frac {\int \frac {8 b c-9 a \left (d+e x^2+f x^4\right )}{x^8 \sqrt {a+b x^2}} \, dx}{9 a}\\ &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}+\frac {\int \frac {6 b (8 b c-9 a d)-7 a \left (-9 a e-9 a f x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx}{63 a^2}\\ &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}-\frac {\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^5}-\frac {\int \frac {4 b \left (48 b^2 c-54 a b d+63 a^2 e\right )-315 a^3 f}{x^4 \sqrt {a+b x^2}} \, dx}{315 a^3}\\ &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}-\frac {\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^5}-\frac {\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \int \frac {1}{x^4 \sqrt {a+b x^2}} \, dx}{105 a^3}\\ &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}-\frac {\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^5}+\frac {\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{315 a^4 x^3}+\frac {\left (2 b \left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right )\right ) \int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx}{315 a^4}\\ &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}-\frac {\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^5}+\frac {\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{315 a^4 x^3}-\frac {2 b \left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{315 a^5 x}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 134, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {a+b x^2} \left (a^4 \left (35 c+45 d x^2+63 e x^4+105 f x^6\right )-2 a^3 b x^2 \left (20 c+27 d x^2+42 e x^4+105 f x^6\right )+24 a^2 b^2 x^4 \left (2 c+3 d x^2+7 e x^4\right )-16 a b^3 x^6 \left (4 c+9 d x^2\right )+128 b^4 c x^8\right )}{315 a^5 x^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 160, normalized size = 0.85 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-35 a^4 c-45 a^4 d x^2-63 a^4 e x^4-105 a^4 f x^6+40 a^3 b c x^2+54 a^3 b d x^4+84 a^3 b e x^6+210 a^3 b f x^8-48 a^2 b^2 c x^4-72 a^2 b^2 d x^6-168 a^2 b^2 e x^8+64 a b^3 c x^6+144 a b^3 d x^8-128 b^4 c x^8\right )}{315 a^5 x^9} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.60, size = 141, normalized size = 0.75 \begin {gather*} -\frac {{\left (2 \, {\left (64 \, b^{4} c - 72 \, a b^{3} d + 84 \, a^{2} b^{2} e - 105 \, a^{3} b f\right )} x^{8} - {\left (64 \, a b^{3} c - 72 \, a^{2} b^{2} d + 84 \, a^{3} b e - 105 \, a^{4} f\right )} x^{6} + 35 \, a^{4} c + 3 \, {\left (16 \, a^{2} b^{2} c - 18 \, a^{3} b d + 21 \, a^{4} e\right )} x^{4} - 5 \, {\left (8 \, a^{3} b c - 9 \, a^{4} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, a^{5} x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 667, normalized size = 3.53 \begin {gather*} \frac {4 \, {\left (315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} b^{\frac {3}{2}} f - 1995 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a b^{\frac {3}{2}} f + 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} b^{\frac {5}{2}} e + 2520 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} b^{\frac {7}{2}} d + 5355 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{2} b^{\frac {3}{2}} f - 3780 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a b^{\frac {5}{2}} e + 8064 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {9}{2}} c - 6552 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {7}{2}} d - 7875 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {3}{2}} f + 6804 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {5}{2}} e - 5376 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {9}{2}} c + 6048 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {7}{2}} d + 6825 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {3}{2}} f - 6216 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} e + 2304 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {9}{2}} c - 2592 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {7}{2}} d - 3465 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {3}{2}} f + 3024 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {5}{2}} e - 576 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {9}{2}} c + 648 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {7}{2}} d + 945 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {3}{2}} f - 756 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} e + 64 \, a^{4} b^{\frac {9}{2}} c - 72 \, a^{5} b^{\frac {7}{2}} d - 105 \, a^{7} b^{\frac {3}{2}} f + 84 \, a^{6} b^{\frac {5}{2}} e\right )}}{315 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 157, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {b \,x^{2}+a}\, \left (-210 a^{3} b f \,x^{8}+168 a^{2} b^{2} e \,x^{8}-144 a \,b^{3} d \,x^{8}+128 b^{4} c \,x^{8}+105 a^{4} f \,x^{6}-84 a^{3} b e \,x^{6}+72 a^{2} b^{2} d \,x^{6}-64 a \,b^{3} c \,x^{6}+63 a^{4} e \,x^{4}-54 a^{3} b d \,x^{4}+48 a^{2} b^{2} c \,x^{4}+45 a^{4} d \,x^{2}-40 a^{3} b c \,x^{2}+35 c \,a^{4}\right )}{315 a^{5} x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 275, normalized size = 1.46 \begin {gather*} -\frac {128 \, \sqrt {b x^{2} + a} b^{4} c}{315 \, a^{5} x} + \frac {16 \, \sqrt {b x^{2} + a} b^{3} d}{35 \, a^{4} x} - \frac {8 \, \sqrt {b x^{2} + a} b^{2} e}{15 \, a^{3} x} + \frac {2 \, \sqrt {b x^{2} + a} b f}{3 \, a^{2} x} + \frac {64 \, \sqrt {b x^{2} + a} b^{3} c}{315 \, a^{4} x^{3}} - \frac {8 \, \sqrt {b x^{2} + a} b^{2} d}{35 \, a^{3} x^{3}} + \frac {4 \, \sqrt {b x^{2} + a} b e}{15 \, a^{2} x^{3}} - \frac {\sqrt {b x^{2} + a} f}{3 \, a x^{3}} - \frac {16 \, \sqrt {b x^{2} + a} b^{2} c}{105 \, a^{3} x^{5}} + \frac {6 \, \sqrt {b x^{2} + a} b d}{35 \, a^{2} x^{5}} - \frac {\sqrt {b x^{2} + a} e}{5 \, a x^{5}} + \frac {8 \, \sqrt {b x^{2} + a} b c}{63 \, a^{2} x^{7}} - \frac {\sqrt {b x^{2} + a} d}{7 \, a x^{7}} - \frac {\sqrt {b x^{2} + a} c}{9 \, a x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 171, normalized size = 0.90 \begin {gather*} \frac {\sqrt {b\,x^2+a}\,\left (-105\,f\,a^3+84\,e\,a^2\,b-72\,d\,a\,b^2+64\,c\,b^3\right )}{315\,a^4\,x^3}-\frac {\sqrt {b\,x^2+a}\,\left (9\,a\,d-8\,b\,c\right )}{63\,a^2\,x^7}-\frac {\sqrt {b\,x^2+a}\,\left (21\,e\,a^2-18\,d\,a\,b+16\,c\,b^2\right )}{105\,a^3\,x^5}-\frac {\sqrt {b\,x^2+a}\,\left (-210\,f\,a^3\,b+168\,e\,a^2\,b^2-144\,d\,a\,b^3+128\,c\,b^4\right )}{315\,a^5\,x}-\frac {c\,\sqrt {b\,x^2+a}}{9\,a\,x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.69, size = 1642, normalized size = 8.69
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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