Optimal. Leaf size=229 \[ -\frac {\left (-6 i a^2-18 a+11 i\right ) \sqrt {i a+i b x+1} (-i a-i b x+1)^{3/2}}{6 b^3}-\frac {\left (-6 i a^2-18 a+11 i\right ) \sqrt {i a+i b x+1} \sqrt {-i a-i b x+1}}{2 b^3}+\frac {\left (-6 a^2+18 i a+11\right ) \sinh ^{-1}(a+b x)}{2 b^3}-\frac {i \sqrt {i a+i b x+1} (-i a-i b x+1)^{5/2}}{3 b^3}+\frac {i (-a+i)^2 (-i a-i b x+1)^{5/2}}{b^3 \sqrt {i a+i b x+1}} \]
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Rubi [A] time = 0.17, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5095, 89, 80, 50, 53, 619, 215} \[ -\frac {\left (-6 i a^2-18 a+11 i\right ) \sqrt {i a+i b x+1} (-i a-i b x+1)^{3/2}}{6 b^3}-\frac {\left (-6 i a^2-18 a+11 i\right ) \sqrt {i a+i b x+1} \sqrt {-i a-i b x+1}}{2 b^3}+\frac {\left (-6 a^2+18 i a+11\right ) \sinh ^{-1}(a+b x)}{2 b^3}-\frac {i \sqrt {i a+i b x+1} (-i a-i b x+1)^{5/2}}{3 b^3}+\frac {i (-a+i)^2 (-i a-i b x+1)^{5/2}}{b^3 \sqrt {i a+i b x+1}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 80
Rule 89
Rule 215
Rule 619
Rule 5095
Rubi steps
\begin {align*} \int e^{-3 i \tan ^{-1}(a+b x)} x^2 \, dx &=\int \frac {x^2 (1-i a-i b x)^{3/2}}{(1+i a+i b x)^{3/2}} \, dx\\ &=\frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {i \int \frac {(1-i a-i b x)^{3/2} \left (-(i-a) (3+2 i a) b-b^2 x\right )}{\sqrt {1+i a+i b x}} \, dx}{b^3}\\ &=\frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \int \frac {(1-i a-i b x)^{3/2}}{\sqrt {1+i a+i b x}} \, dx}{3 b^2}\\ &=\frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx}{2 b^2}\\ &=\frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 b^2}\\ &=\frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{2 b^2}\\ &=\frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{4 b^4}\\ &=\frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 198, normalized size = 0.86 \[ \frac {\sqrt [4]{-1} \left (-6 a^2+18 i a+11\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{b^{7/2}}+\frac {2 i a^4+a^3 (51+2 i b x)+a^2 (69 b x-50 i)+a \left (2 i b^3 x^3+9 b^2 x^2-106 i b x+51\right )+i \left (2 b^4 x^4+9 i b^3 x^3-26 b^2 x^2+33 i b x-52\right )}{6 b^3 \sqrt {a^2+2 a b x+b^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 174, normalized size = 0.76 \[ \frac {7 i \, a^{4} + 166 \, a^{3} + {\left (7 i \, a^{3} + 159 \, a^{2} - 249 i \, a - 96\right )} b x - 408 i \, a^{2} + {\left (72 \, a^{3} + 12 \, {\left (6 \, a^{2} - 18 i \, a - 11\right )} b x - 288 i \, a^{2} - 348 \, a + 132 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (8 i \, b^{3} x^{3} - 28 \, b^{2} x^{2} + 8 i \, a^{3} + {\left (64 \, a - 76 i\right )} b x + 212 \, a^{2} - 412 i \, a - 208\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 345 \, a + 96 i}{24 \, b^{4} x + {\left (24 \, a - 24 i\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 257, normalized size = 1.12 \[ \frac {1}{6} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (\frac {2 \, i x}{b} - \frac {2 \, a b^{6} i + 9 \, b^{6}}{b^{8}}\right )} x + \frac {2 \, a^{2} b^{5} i + 27 \, a b^{5} - 28 \, b^{5} i}{b^{8}}\right )} + \frac {{\left (6 \, a^{2} i + 18 \, a - 11 \, i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b i + a^{3} b i + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} i {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} i {\left | b \right |} + 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 \, a^{2} b - a b i + 4 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} i {\left | b \right |}\right )}{6 \, b^{2} i {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 1026, normalized size = 4.48 \[ \frac {2 i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}} a^{2}}{b^{3}}-\frac {11 i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3 b^{3}}-\frac {3 \ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right ) a^{2}}{b^{2} \sqrt {b^{2}}}+\frac {9 i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a^{2}}{b^{3}}+\frac {2 i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}} a}{b^{6} \left (x -\frac {i}{b}+\frac {a}{b}\right )^{3}}-\frac {2 i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}} a^{2}}{b^{5} \left (x -\frac {i}{b}+\frac {a}{b}\right )^{2}}-\frac {6 a \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b^{5} \left (x -\frac {i}{b}+\frac {a}{b}\right )^{2}}+\frac {9 i \ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right ) a}{b^{2} \sqrt {b^{2}}}+\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b^{6} \left (x -\frac {i}{b}+\frac {a}{b}\right )^{3}}-\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}} a^{2}}{b^{6} \left (x -\frac {i}{b}+\frac {a}{b}\right )^{3}}-\frac {3 \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, x \,a^{2}}{b^{2}}+\frac {9 i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, x a}{b^{2}}+\frac {4 i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b^{5} \left (x -\frac {i}{b}+\frac {a}{b}\right )^{2}}-\frac {3 \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a^{3}}{b^{3}}+\frac {11 \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, x}{2 b^{2}}+\frac {11 \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a}{2 b^{3}}+\frac {11 \ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 b^{2} \sqrt {b^{2}}}+\frac {6 a \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 624, normalized size = 2.72 \[ \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} - 2 i \, b^{4} x - 2 i \, a b^{3} - b^{3}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} - 2 i \, b^{4} x - 2 i \, a b^{3} - b^{3}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{2 i \, b^{4} x + 2 i \, a b^{3} + 2 \, b^{3}} + \frac {6 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{i \, b^{4} x + i \, a b^{3} + b^{3}} - \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} - 2 i \, b^{4} x - 2 i \, a b^{3} - b^{3}} - \frac {2 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{2 i \, b^{4} x + 2 i \, a b^{3} + 2 \, b^{3}} + \frac {12 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{i \, b^{4} x + i \, a b^{3} + b^{3}} - \frac {3 \, a^{2} \operatorname {arsinh}\left (b x + a\right )}{b^{3}} - \frac {6 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{i \, b^{4} x + i \, a b^{3} + b^{3}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} x}{2 \, b^{2}} + \frac {9 i \, a \operatorname {arsinh}\left (b x + a\right )}{b^{3}} + \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{3}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{b^{3}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} a}{2 \, b^{3}} + \frac {\arcsin \left (i \, b x + i \, a + 2\right )}{2 \, b^{3}} + \frac {6 \, \operatorname {arsinh}\left (b x + a\right )}{b^{3}} - \frac {3 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{3}} + \frac {i \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3}}{b^{3}} \]
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 \[ \int \frac {x^2\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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