Optimal. Leaf size=207 \[ \frac {\left (1-4 a^2\right ) b^2 \left (a^2+a b x+1\right ) \sqrt {a^2+2 a b x+b^2 x^2+1}}{8 \left (a^2+1\right )^3 x^2}-\frac {\left (a^2+2 a b x+b^2 x^2+1\right )^{3/2}}{4 \left (a^2+1\right ) x^4}+\frac {5 a b \left (a^2+2 a b x+b^2 x^2+1\right )^{3/2}}{12 \left (a^2+1\right )^2 x^3}+\frac {\left (1-4 a^2\right ) b^4 \tanh ^{-1}\left (\frac {a^2+a b x+1}{\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}}\right )}{8 \left (a^2+1\right )^{7/2}}-\frac {a}{4 x^4}-\frac {b}{3 x^3} \]
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Rubi [A] time = 0.17, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5907, 14, 744, 806, 720, 724, 206} \[ \frac {\left (1-4 a^2\right ) b^2 \left (a^2+a b x+1\right ) \sqrt {a^2+2 a b x+b^2 x^2+1}}{8 \left (a^2+1\right )^3 x^2}+\frac {5 a b \left (a^2+2 a b x+b^2 x^2+1\right )^{3/2}}{12 \left (a^2+1\right )^2 x^3}-\frac {\left (a^2+2 a b x+b^2 x^2+1\right )^{3/2}}{4 \left (a^2+1\right ) x^4}+\frac {\left (1-4 a^2\right ) b^4 \tanh ^{-1}\left (\frac {a^2+a b x+1}{\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}}\right )}{8 \left (a^2+1\right )^{7/2}}-\frac {a}{4 x^4}-\frac {b}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 14
Rule 206
Rule 720
Rule 724
Rule 744
Rule 806
Rule 5907
Rubi steps
\begin {align*} \int \frac {e^{\sinh ^{-1}(a+b x)}}{x^5} \, dx &=\int \frac {a+b x+\sqrt {1+(a+b x)^2}}{x^5} \, dx\\ &=\int \left (\frac {a}{x^5}+\frac {b}{x^4}+\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x^5}\right ) \, dx\\ &=-\frac {a}{4 x^4}-\frac {b}{3 x^3}+\int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x^5} \, dx\\ &=-\frac {a}{4 x^4}-\frac {b}{3 x^3}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 \left (1+a^2\right ) x^4}-\frac {\int \frac {\left (5 a b+b^2 x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{x^4} \, dx}{4 \left (1+a^2\right )}\\ &=-\frac {a}{4 x^4}-\frac {b}{3 x^3}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 \left (1+a^2\right ) x^4}+\frac {5 a b \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{12 \left (1+a^2\right )^2 x^3}-\frac {\left (\left (1-4 a^2\right ) b^2\right ) \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x^3} \, dx}{4 \left (1+a^2\right )^2}\\ &=-\frac {a}{4 x^4}-\frac {b}{3 x^3}+\frac {\left (1-4 a^2\right ) b^2 \left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{8 \left (1+a^2\right )^3 x^2}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 \left (1+a^2\right ) x^4}+\frac {5 a b \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{12 \left (1+a^2\right )^2 x^3}-\frac {\left (\left (1-4 a^2\right ) b^4\right ) \int \frac {1}{x \sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx}{8 \left (1+a^2\right )^3}\\ &=-\frac {a}{4 x^4}-\frac {b}{3 x^3}+\frac {\left (1-4 a^2\right ) b^2 \left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{8 \left (1+a^2\right )^3 x^2}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 \left (1+a^2\right ) x^4}+\frac {5 a b \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{12 \left (1+a^2\right )^2 x^3}+\frac {\left (\left (1-4 a^2\right ) b^4\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (1+a^2\right )-x^2} \, dx,x,\frac {2 \left (1+a^2\right )+2 a b x}{\sqrt {1+a^2+2 a b x+b^2 x^2}}\right )}{4 \left (1+a^2\right )^3}\\ &=-\frac {a}{4 x^4}-\frac {b}{3 x^3}+\frac {\left (1-4 a^2\right ) b^2 \left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{8 \left (1+a^2\right )^3 x^2}-\frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 \left (1+a^2\right ) x^4}+\frac {5 a b \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{12 \left (1+a^2\right )^2 x^3}+\frac {\left (1-4 a^2\right ) b^4 \tanh ^{-1}\left (\frac {1+a^2+a b x}{\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}}\right )}{8 \left (1+a^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 192, normalized size = 0.93 \[ \frac {1}{24} \left (\frac {3 (2 a-1) (2 a+1) b^4 \log (x)}{\left (a^2+1\right )^{7/2}}-\frac {3 (2 a-1) (2 a+1) b^4 \log \left (\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right )}{\left (a^2+1\right )^{7/2}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2+1} \left (\frac {a \left (2 a^2-13\right ) b^3 x^3}{\left (a^2+1\right )^3}-\frac {\left (2 a^2-3\right ) b^2 x^2}{\left (a^2+1\right )^2}+\frac {2 a b x}{a^2+1}+6\right )}{x^4}-\frac {6 a}{x^4}-\frac {8 b}{x^3}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.69, size = 295, normalized size = 1.43 \[ \frac {3 \, {\left (4 \, a^{2} - 1\right )} \sqrt {a^{2} + 1} b^{4} x^{4} \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} - \sqrt {a^{2} + 1} a + 1\right )} - {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) - 6 \, a^{9} - {\left (2 \, a^{5} - 11 \, a^{3} - 13 \, a\right )} b^{4} x^{4} - 24 \, a^{7} - 36 \, a^{5} - 24 \, a^{3} - 8 \, {\left (a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} b x - {\left (6 \, a^{8} + {\left (2 \, a^{5} - 11 \, a^{3} - 13 \, a\right )} b^{3} x^{3} + 24 \, a^{6} - {\left (2 \, a^{6} + a^{4} - 4 \, a^{2} - 3\right )} b^{2} x^{2} + 36 \, a^{4} + 2 \, {\left (a^{7} + 3 \, a^{5} + 3 \, a^{3} + a\right )} b x + 24 \, a^{2} + 6\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 6 \, a}{24 \, {\left (a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 1173, normalized size = 5.67 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 841, normalized size = 4.06 \[ -\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{4 \left (a^{2}+1\right ) x^{4}}+\frac {5 a b \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{12 \left (a^{2}+1\right )^{2} x^{3}}-\frac {5 a^{2} b^{2} \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{8 \left (a^{2}+1\right )^{3} x^{2}}+\frac {5 a^{3} b^{3} \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{8 \left (a^{2}+1\right )^{4} x}-\frac {5 a^{4} b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 \left (a^{2}+1\right )^{4}}-\frac {5 a^{5} b^{5} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 \left (a^{2}+1\right )^{4} \sqrt {b^{2}}}+\frac {5 a^{4} b^{4} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{8 \left (a^{2}+1\right )^{\frac {7}{2}}}-\frac {5 a^{3} b^{5} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x}{8 \left (a^{2}+1\right )^{4}}-\frac {5 a^{3} b^{5} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 \left (a^{2}+1\right )^{4} \sqrt {b^{2}}}+\frac {7 a^{2} b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 \left (a^{2}+1\right )^{3}}+\frac {3 a^{3} b^{5} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{4 \left (a^{2}+1\right )^{3} \sqrt {b^{2}}}-\frac {3 a^{2} b^{4} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{4 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {b^{2} \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{8 \left (a^{2}+1\right )^{2} x^{2}}-\frac {b^{3} a \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{8 \left (a^{2}+1\right )^{3} x}+\frac {b^{5} a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x}{8 \left (a^{2}+1\right )^{3}}+\frac {b^{5} a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 \left (a^{2}+1\right )^{3} \sqrt {b^{2}}}-\frac {b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 \left (a^{2}+1\right )^{2}}-\frac {b^{5} a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 \left (a^{2}+1\right )^{2} \sqrt {b^{2}}}+\frac {b^{4} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{8 \left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {b}{3 x^{3}}-\frac {a}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 594, normalized size = 2.87 \[ \frac {5 \, a^{4} b^{4} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{8 \, {\left (a^{2} + 1\right )}^{\frac {7}{2}}} - \frac {3 \, a^{2} b^{4} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{4 \, {\left (a^{2} + 1\right )}^{\frac {5}{2}}} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} b^{4}}{8 \, {\left (a^{2} + 1\right )}^{3}} + \frac {b^{4} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{8 \, {\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{4}}{8 \, {\left (a^{2} + 1\right )}^{2}} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3} b^{3}}{8 \, {\left (a^{2} + 1\right )}^{3} x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b^{3}}{8 \, {\left (a^{2} + 1\right )}^{2} x} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2} b^{2}}{8 \, {\left (a^{2} + 1\right )}^{3} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} b^{2}}{8 \, {\left (a^{2} + 1\right )}^{2} x^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a b}{12 \, {\left (a^{2} + 1\right )}^{2} x^{3}} - \frac {b}{3 \, x^{3}} - \frac {a}{4 \, x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{4 \, {\left (a^{2} + 1\right )} x^{4}} \]
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 \[ \int \frac {a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x}{x^5} \,d x \]
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 \[ \int \frac {a + b x + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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