Optimal. Leaf size=121 \[ -\frac {a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}-\frac {3 b \sqrt {1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac {b \sqrt {1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac {3 b \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{40 d e^6} \]
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Rubi [A] time = 0.09, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4805, 12, 4627, 266, 51, 63, 206} \[ -\frac {a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}-\frac {3 b \sqrt {1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac {b \sqrt {1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac {3 b \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{40 d e^6} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 206
Rule 266
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c+d x)}{(c e+d e x)^6} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{e^6 x^6} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{x^6} \, dx,x,c+d x\right )}{d e^6}\\ &=-\frac {a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {1-x^2}} \, dx,x,c+d x\right )}{5 d e^6}\\ &=-\frac {a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x^3} \, dx,x,(c+d x)^2\right )}{10 d e^6}\\ &=-\frac {b \sqrt {1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac {a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x^2} \, dx,x,(c+d x)^2\right )}{40 d e^6}\\ &=-\frac {b \sqrt {1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac {3 b \sqrt {1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac {a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,(c+d x)^2\right )}{80 d e^6}\\ &=-\frac {b \sqrt {1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac {3 b \sqrt {1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac {a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-(c+d x)^2}\right )}{40 d e^6}\\ &=-\frac {b \sqrt {1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac {3 b \sqrt {1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac {a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}-\frac {3 b \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{40 d e^6}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 65, normalized size = 0.54 \[ -\frac {\frac {a+b \sin ^{-1}(c+d x)}{(c+d x)^5}+b \sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-(c+d x)^2\right )}{5 d e^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 321, normalized size = 2.65 \[ -\frac {16 \, b \arcsin \left (d x + c\right ) + 3 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} + 1\right ) - 3 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} - 1\right ) + 2 \, {\left (3 \, b d^{3} x^{3} + 9 \, b c d^{2} x^{2} + 3 \, b c^{3} + {\left (9 \, b c^{2} + 2 \, b\right )} d x + 2 \, b c\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} + 16 \, a}{80 \, {\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.52, size = 580, normalized size = 4.79 \[ -\frac {{\left (d x + c\right )}^{5} b \arcsin \left (d x + c\right ) e^{\left (-6\right )}}{160 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{5}} - \frac {{\left (d x + c\right )}^{3} b \arcsin \left (d x + c\right ) e^{\left (-6\right )}}{32 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3}} - \frac {{\left (d x + c\right )} b \arcsin \left (d x + c\right ) e^{\left (-6\right )}}{16 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )} \arcsin \left (d x + c\right ) e^{\left (-6\right )}}{16 \, {\left (d x + c\right )} d} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3} \arcsin \left (d x + c\right ) e^{\left (-6\right )}}{32 \, {\left (d x + c\right )}^{3} d} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{5} \arcsin \left (d x + c\right ) e^{\left (-6\right )}}{160 \, {\left (d x + c\right )}^{5} d} - \frac {3 \, b e^{\left (-6\right )} \log \left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}{40 \, d} + \frac {3 \, b e^{\left (-6\right )} \log \left ({\left | d x + c \right |}\right )}{40 \, d} - \frac {{\left (d x + c\right )}^{5} a e^{\left (-6\right )}}{160 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{5}} + \frac {{\left (d x + c\right )}^{4} b e^{\left (-6\right )}}{320 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{4}} - \frac {{\left (d x + c\right )}^{3} a e^{\left (-6\right )}}{32 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3}} + \frac {{\left (d x + c\right )}^{2} b e^{\left (-6\right )}}{40 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} - \frac {{\left (d x + c\right )} a e^{\left (-6\right )}}{16 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )} e^{\left (-6\right )}}{16 \, {\left (d x + c\right )} d} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} e^{\left (-6\right )}}{40 \, {\left (d x + c\right )}^{2} d} - \frac {a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3} e^{\left (-6\right )}}{32 \, {\left (d x + c\right )}^{3} d} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{4} e^{\left (-6\right )}}{320 \, {\left (d x + c\right )}^{4} d} - \frac {a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{5} e^{\left (-6\right )}}{160 \, {\left (d x + c\right )}^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 100, normalized size = 0.83 \[ \frac {-\frac {a}{5 e^{6} \left (d x +c \right )^{5}}+\frac {b \left (-\frac {\arcsin \left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{20 \left (d x +c \right )^{4}}-\frac {3 \sqrt {1-\left (d x +c \right )^{2}}}{40 \left (d x +c \right )^{2}}-\frac {3 \arctanh \left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{40}\right )}{e^{6}}}{d} \]
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 \[ -\frac {{\left ({\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\right )} \int \frac {e^{\left (\frac {1}{2} \, \log \left (d x + c + 1\right ) + \frac {1}{2} \, \log \left (-d x - c + 1\right )\right )}}{d^{9} e^{6} x^{9} + 9 \, c d^{8} e^{6} x^{8} + {\left (36 \, c^{2} - 1\right )} d^{7} e^{6} x^{7} + 7 \, {\left (12 \, c^{3} - c\right )} d^{6} e^{6} x^{6} + 21 \, {\left (6 \, c^{4} - c^{2}\right )} d^{5} e^{6} x^{5} + 7 \, {\left (18 \, c^{5} - 5 \, c^{3}\right )} d^{4} e^{6} x^{4} + 7 \, {\left (12 \, c^{6} - 5 \, c^{4}\right )} d^{3} e^{6} x^{3} + 3 \, {\left (12 \, c^{7} - 7 \, c^{5}\right )} d^{2} e^{6} x^{2} + {\left (9 \, c^{8} - 7 \, c^{6}\right )} d e^{6} x + {\left (c^{9} - c^{7}\right )} e^{6} - {\left (d^{7} e^{6} x^{7} + 7 \, c d^{6} e^{6} x^{6} + {\left (21 \, c^{2} - 1\right )} d^{5} e^{6} x^{5} + 5 \, {\left (7 \, c^{3} - c\right )} d^{4} e^{6} x^{4} + 5 \, {\left (7 \, c^{4} - 2 \, c^{2}\right )} d^{3} e^{6} x^{3} + {\left (21 \, c^{5} - 10 \, c^{3}\right )} d^{2} e^{6} x^{2} + {\left (7 \, c^{6} - 5 \, c^{4}\right )} d e^{6} x + {\left (c^{7} - c^{5}\right )} e^{6}\right )} {\left (d x + c + 1\right )} {\left (d x + c - 1\right )}}\,{d x} + \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )\right )} b}{5 \, {\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\right )}} - \frac {a}{5 \, {\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^6} \,d x \]
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 \[ \frac {\int \frac {a}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx + \int \frac {b \operatorname {asin}{\left (c + d x \right )}}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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