Optimal. Leaf size=171 \[ -\frac {\left (-2 i a^2+2 a+i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 b^3}-\frac {\left (-2 a^2-2 i a+1\right ) \sinh ^{-1}(a+b x)}{2 b^3}-\frac {(4 a+i) \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{6 b^3}+\frac {x \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.13, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5095, 90, 80, 50, 53, 619, 215} \[ -\frac {\left (-2 i a^2+2 a+i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 b^3}-\frac {\left (-2 a^2-2 i a+1\right ) \sinh ^{-1}(a+b x)}{2 b^3}+\frac {x \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{3 b^2}-\frac {(4 a+i) \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 80
Rule 90
Rule 215
Rule 619
Rule 5095
Rubi steps
\begin {align*} \int e^{i \tan ^{-1}(a+b x)} x^2 \, dx &=\int \frac {x^2 \sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx\\ &=\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{3 b^2}+\frac {\int \frac {\sqrt {1+i a+i b x} \left (-1-a^2-(i+4 a) b x\right )}{\sqrt {1-i a-i b x}} \, dx}{3 b^2}\\ &=-\frac {(i+4 a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{3 b^2}-\frac {\left (1-2 i a-2 a^2\right ) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{2 b^2}\\ &=-\frac {\left (i+2 a-2 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}-\frac {(i+4 a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{3 b^2}-\frac {\left (1-2 i a-2 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 {\left (i+2 a-2 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}-\frac {(i+4 a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{3 b^2}-\frac {\left (1-2 i a-2 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 {\left (i+2 a-2 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}-\frac {(i+4 a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{3 b^2}-\frac {\left (1-2 i a-2 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 {\left (i+2 a-2 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}-\frac {(i+4 a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{3 b^2}-\frac {\left (1-2 i a-2 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 135, normalized size = 0.79 \[ \frac {\sqrt [4]{-1} \left (2 a^2+2 i a-1\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{b^{7/2}}+\frac {\sqrt {a^2+2 a b x+b^2 x^2+1} \left (2 i a^2+a (-9-2 i b x)+2 i b^2 x^2+3 b x-4 i\right )}{6 b^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 106, normalized size = 0.62 \[ \frac {7 i \, a^{3} - 21 \, a^{2} - 12 \, {\left (2 \, a^{2} + 2 i \, a - 1\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (8 i \, b^{2} x^{2} - 4 \, {\left (2 i \, a - 3\right )} b x + 8 i \, a^{2} - 36 \, a - 16 i\right )} - 9 i \, a}{24 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 117, normalized size = 0.68 \[ \frac {1}{6} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (\frac {2 \, i x}{b} - \frac {2 \, a b^{3} i - 3 \, b^{3}}{b^{5}}\right )} x + \frac {2 \, a^{2} b^{2} i - 9 \, a b^{2} - 4 \, b^{2} i}{b^{5}}\right )} - \frac {{\left (2 \, a^{2} + 2 \, a i - 1\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 302, normalized size = 1.77 \[ \frac {i x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 b}-\frac {i a x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 b^{2}}+\frac {i a^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 b^{3}}+\frac {i a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}-\frac {2 i \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 b^{3}}+\frac {x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {3 a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{3}}+\frac {a^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}-\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 351, normalized size = 2.05 \[ \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x^{2}}{3 \, b} - \frac {5 i \, a^{3} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{3}} + \frac {3 \, a^{2} {\left (i \, a + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{3}} - \frac {5 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{6 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-i \, a - 1\right )} x}{2 \, b^{2}} + \frac {3 i \, {\left (a^{2} + 1\right )} a \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{3}} - \frac {{\left (a^{2} + 1\right )} {\left (i \, a + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{3}} + \frac {5 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{3}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a {\left (i \, a + 1\right )}}{2 \, b^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 i \, a^{2} + 2 i\right )}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}{\sqrt {{\left (a+b\,x\right )}^2+1}} \,d x \]
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 \[ i \left (\int \left (- \frac {i x^{2}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a x^{2}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {b x^{3}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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