Optimal. Leaf size=115 \[ -\frac {a+b \sinh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {3 b \sqrt {(c+d x)^2+1}}{40 d e^6 (c+d x)^2}-\frac {b \sqrt {(c+d x)^2+1}}{20 d e^6 (c+d x)^4}-\frac {3 b \tanh ^{-1}\left (\sqrt {(c+d x)^2+1}\right )}{40 d e^6} \]
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Rubi [A] time = 0.09, antiderivative size = 115, 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 = {5865, 12, 5661, 266, 51, 63, 207} \[ -\frac {a+b \sinh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {3 b \sqrt {(c+d x)^2+1}}{40 d e^6 (c+d x)^2}-\frac {b \sqrt {(c+d x)^2+1}}{20 d e^6 (c+d x)^4}-\frac {3 b \tanh ^{-1}\left (\sqrt {(c+d x)^2+1}\right )}{40 d e^6} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 207
Rule 266
Rule 5661
Rule 5865
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^6} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{e^6 x^6} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x^6} \, dx,x,c+d x\right )}{d e^6}\\ &=-\frac {a+b \sinh ^{-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 \sinh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x}} \, 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 \sinh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, 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 \sinh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+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 \sinh ^{-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 \sinh ^{-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 = 0.08, size = 61, normalized size = 0.53 \[ -\frac {\frac {a+b \sinh ^{-1}(c+d x)}{(c+d x)^5}+b \sqrt {(c+d x)^2+1} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};(c+d x)^2+1\right )}{5 d e^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 509, normalized size = 4.43 \[ -\frac {8 \, a c^{5} - 8 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + 3 \, {\left (b c^{5} d^{5} x^{5} + 5 \, b c^{6} d^{4} x^{4} + 10 \, b c^{7} d^{3} x^{3} + 10 \, b c^{8} d^{2} x^{2} + 5 \, b c^{9} d x + b c^{10}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right ) - 8 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 3 \, {\left (b c^{5} d^{5} x^{5} + 5 \, b c^{6} d^{4} x^{4} + 10 \, b c^{7} d^{3} x^{3} + 10 \, b c^{8} d^{2} x^{2} + 5 \, b c^{9} d x + b c^{10}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} - 1\right ) - {\left (3 \, b c^{5} d^{3} x^{3} + 9 \, b c^{6} d^{2} x^{2} + 3 \, b c^{8} - 2 \, b c^{6} + {\left (9 \, b c^{7} - 2 \, b c^{5}\right )} d x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{40 \, {\left (c^{5} d^{6} e^{6} x^{5} + 5 \, c^{6} d^{5} e^{6} x^{4} + 10 \, c^{7} d^{4} e^{6} x^{3} + 10 \, c^{8} d^{3} e^{6} x^{2} + 5 \, c^{9} d^{2} e^{6} x + c^{10} d e^{6}\right )}} \]
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 \[ \int \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 0.82 \[ \frac {-\frac {a}{5 e^{6} \left (d x +c \right )^{5}}+\frac {b \left (-\frac {\arcsinh \left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{20 \left (d x +c \right )^{4}}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}}{40 \left (d x +c \right )^{2}}-\frac {3 \arctanh \left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{40}\right )}{e^{6}}}{d} \]
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 \[ \frac {1}{30} \, b {\left (\frac {2 \, {\left (3 \, d^{4} x^{4} + 12 \, c d^{3} x^{3} + 3 \, c^{4} + {\left (18 \, c^{2} d^{2} - d^{2}\right )} x^{2} - c^{2} + 2 \, {\left (6 \, c^{3} d - c d\right )} x - 3 \, \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )\right )}}{d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}} - \frac {3 i \, {\left (\log \left (\frac {i \, {\left (d^{2} x + c d\right )}}{d} + 1\right ) - \log \left (-\frac {i \, {\left (d^{2} x + c d\right )}}{d} + 1\right )\right )}}{d e^{6}} + 30 \, \int \frac {1}{5 \, {\left (d^{8} e^{6} x^{8} + 8 \, c d^{7} e^{6} x^{7} + c^{8} e^{6} + c^{6} e^{6} + {\left (28 \, c^{2} d^{6} e^{6} + d^{6} e^{6}\right )} x^{6} + 2 \, {\left (28 \, c^{3} d^{5} e^{6} + 3 \, c d^{5} e^{6}\right )} x^{5} + 5 \, {\left (14 \, c^{4} d^{4} e^{6} + 3 \, c^{2} d^{4} e^{6}\right )} x^{4} + 4 \, {\left (14 \, c^{5} d^{3} e^{6} + 5 \, c^{3} d^{3} e^{6}\right )} x^{3} + {\left (28 \, c^{6} d^{2} e^{6} + 15 \, c^{4} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (4 \, c^{7} d e^{6} + 3 \, c^{5} d e^{6}\right )} x + {\left (d^{7} e^{6} x^{7} + 7 \, c d^{6} e^{6} x^{6} + c^{7} e^{6} + c^{5} e^{6} + {\left (21 \, c^{2} d^{5} e^{6} + d^{5} e^{6}\right )} x^{5} + 5 \, {\left (7 \, c^{3} d^{4} e^{6} + c d^{4} e^{6}\right )} x^{4} + 5 \, {\left (7 \, c^{4} d^{3} e^{6} + 2 \, c^{2} d^{3} e^{6}\right )} x^{3} + {\left (21 \, c^{5} d^{2} e^{6} + 10 \, c^{3} d^{2} e^{6}\right )} x^{2} + {\left (7 \, c^{6} d e^{6} + 5 \, c^{4} d e^{6}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )}}\,{d x}\right )} - \frac {a}{5 \, {\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\right )}} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^6} \,d x \]
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 \[ \frac {\int \frac {a}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx + \int \frac {b \operatorname {asinh}{\left (c + d x \right )}}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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