Optimal. Leaf size=265 \[ \frac {b d^3 \left (8 a^2 c^2+12 a b c+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{16 c^{7/2} (a c+b)^{5/2}}-\frac {d^2 \left (8 a^2 c^2+20 a b c+11 b^2\right ) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{16 c^3 x^2 (a c+b)^2}+\frac {d (4 a c+3 b) \left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{8 c^3 x^4 (a c+b)}-\frac {\left (c+d x^2\right )^3 \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}}{6 c^2 x^6 (a c+b)} \]
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Rubi [A] time = 0.61, antiderivative size = 271, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6722, 1975, 446, 99, 151, 12, 93, 208} \[ \frac {b d^3 \left (8 a^2 c^2+12 a b c+5 b^2\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {a c+b} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a \left (c+d x^2\right )+b}}\right )}{16 c^{7/2} (a c+b)^{5/2} \sqrt {a \left (c+d x^2\right )+b}}-\frac {d^2 (2 a c+5 b) (4 a c+3 b) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 x^2 (a c+b)^2}+\frac {d (4 a c+5 b) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4 (a c+b)}-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 208
Rule 446
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^7} \, 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^7 \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^7 \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 ) \operatorname {Subst}\left (\int \frac {\sqrt {b+a c+a d x}}{x^4 \sqrt {c+d x}} \, dx,x,x^2\right )}{2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {-\frac {1}{2} (5 b+4 a c) d-2 a d^2 x}{x^3 \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{6 c \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {(5 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 (b+a c) x^4}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {-\frac {1}{4} (5 b+2 a c) (3 b+4 a c) d^2-\frac {1}{2} a (5 b+4 a c) d^3 x}{x^2 \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{12 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {(5 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 (b+a c) x^4}-\frac {(5 b+2 a c) (3 b+4 a c) d^2 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c)^2 x^2}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int -\frac {3 b \left (5 b^2+12 a b c+8 a^2 c^2\right ) d^3}{8 x \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{12 c^3 (b+a c)^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {(5 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 (b+a c) x^4}-\frac {(5 b+2 a c) (3 b+4 a c) d^2 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c)^2 x^2}-\frac {\left (b \left (5 b^2+12 a b c+8 a^2 c^2\right ) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{32 c^3 (b+a c)^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {(5 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 (b+a c) x^4}-\frac {(5 b+2 a c) (3 b+4 a c) d^2 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c)^2 x^2}-\frac {\left (b \left (5 b^2+12 a b c+8 a^2 c^2\right ) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{-c-(-b-a c) x^2} \, dx,x,\frac {\sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{16 c^3 (b+a c)^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {(5 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 (b+a c) x^4}-\frac {(5 b+2 a c) (3 b+4 a c) d^2 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c)^2 x^2}+\frac {b \left (5 b^2+12 a b c+8 a^2 c^2\right ) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {b+a c} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {b+a \left (c+d x^2\right )}}\right )}{16 c^{7/2} (b+a c)^{5/2} \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 245, normalized size = 0.92 \[ -\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {3 b d^3 \left (8 a^2 c^2+12 a b c+5 b^2\right ) \left (c+d x^2\right ) \left (2 \log (x)-\log \left (2 \sqrt {c (a c+b)} \sqrt {\left (c+d x^2\right ) \left (a c+a d x^2+b\right )}+2 a c \left (c+d x^2\right )+b \left (2 c+d x^2\right )\right )\right )}{\sqrt {c (a c+b)} \sqrt {\left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )}}+2 d^3 \left (8 a^2 c^2+26 a b c+15 b^2\right )+\frac {16 c^3 (a c+b)^2}{x^6}-\frac {4 b c^2 d (a c+b)}{x^4}+\frac {2 b c d^2 (8 a c+5 b)}{x^2}\right )}{96 c^3 (a c+b)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.41, size = 755, normalized size = 2.85 \[ \left [\frac {3 \, {\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} \sqrt {a c^{2} + b c} d^{3} x^{6} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} + 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (8 \, a^{3} c^{7} + {\left (8 \, a^{3} c^{4} + 34 \, a^{2} b c^{3} + 41 \, a b^{2} c^{2} + 15 \, b^{3} c\right )} d^{3} x^{6} + 24 \, a^{2} b c^{6} + 24 \, a b^{2} c^{5} + 8 \, b^{3} c^{4} + {\left (8 \, a^{2} b c^{4} + 13 \, a b^{2} c^{3} + 5 \, b^{3} c^{2}\right )} d^{2} x^{4} - 2 \, {\left (a^{2} b c^{5} + 2 \, a b^{2} c^{4} + b^{3} c^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, {\left (a^{3} c^{7} + 3 \, a^{2} b c^{6} + 3 \, a b^{2} c^{5} + b^{3} c^{4}\right )} x^{6}}, -\frac {3 \, {\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} \sqrt {-a c^{2} - b c} d^{3} x^{6} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{7} + {\left (8 \, a^{3} c^{4} + 34 \, a^{2} b c^{3} + 41 \, a b^{2} c^{2} + 15 \, b^{3} c\right )} d^{3} x^{6} + 24 \, a^{2} b c^{6} + 24 \, a b^{2} c^{5} + 8 \, b^{3} c^{4} + {\left (8 \, a^{2} b c^{4} + 13 \, a b^{2} c^{3} + 5 \, b^{3} c^{2}\right )} d^{2} x^{4} - 2 \, {\left (a^{2} b c^{5} + 2 \, a b^{2} c^{4} + b^{3} c^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, {\left (a^{3} c^{7} + 3 \, a^{2} b c^{6} + 3 \, a b^{2} c^{5} + b^{3} c^{4}\right )} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.77, size = 1414, normalized size = 5.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1518, normalized size = 5.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.91, size = 557, normalized size = 2.10 \[ -\frac {{\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} d^{3} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{32 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {3 \, {\left (8 \, a^{2} b c^{4} + 20 \, a b^{2} c^{3} + 11 \, b^{3} c^{2}\right )} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (6 \, a^{3} b c^{4} + 18 \, a^{2} b^{2} c^{3} + 17 \, a b^{3} c^{2} + 5 \, b^{4} c\right )} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + 3 \, {\left (8 \, a^{4} b c^{4} + 28 \, a^{3} b^{2} c^{3} + 37 \, a^{2} b^{3} c^{2} + 22 \, a b^{4} c + 5 \, b^{5}\right )} d^{3} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{48 \, {\left (a^{5} c^{8} + 5 \, a^{4} b c^{7} + 10 \, a^{3} b^{2} c^{6} + 10 \, a^{2} b^{3} c^{5} + 5 \, a b^{4} c^{4} + b^{5} c^{3} - \frac {{\left (a^{2} c^{8} + 2 \, a b c^{7} + b^{2} c^{6}\right )} {\left (a d x^{2} + a c + b\right )}^{3}}{{\left (d x^{2} + c\right )}^{3}} + \frac {3 \, {\left (a^{3} c^{8} + 3 \, a^{2} b c^{7} + 3 \, a b^{2} c^{6} + b^{3} c^{5}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {3 \, {\left (a^{4} c^{8} + 4 \, a^{3} b c^{7} + 6 \, a^{2} b^{2} c^{6} + 4 \, a b^{3} c^{5} + b^{4} c^{4}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \]
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 \[ \int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^7} \,d x \]
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 \[ \int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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