Optimal. Leaf size=227 \[ \frac{\left (-6 i a^2+18 a+11 i\right ) \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{6 b^3}+\frac{\left (-6 i a^2+18 a+11 i\right ) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{2 b^3}+\frac{\left (-6 a^2-18 i a+11\right ) \sinh ^{-1}(a+b x)}{2 b^3}+\frac{i \sqrt{-i a-i b x+1} (i a+i b x+1)^{5/2}}{3 b^3}-\frac{i (a+i)^2 (i a+i b x+1)^{5/2}}{b^3 \sqrt{-i a-i b x+1}} \]
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Rubi [A] time = 0.169239, antiderivative size = 227, normalized size of antiderivative = 1., 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 5095
Rule 89
Rule 80
Rule 50
Rule 53
Rule 619
Rule 215
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 ((3-2 i a) (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 \sqrt{1-i a-i b x} (1+i a+i b x)^{5/2}}{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{\left (11 i+18 a-6 i a^2\right ) \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{6 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 \sqrt{1-i a-i b x} (1+i a+i b x)^{5/2}}{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{\left (11 i+18 a-6 i a^2\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^3}+\frac{\left (11 i+18 a-6 i a^2\right ) \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{6 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 \sqrt{1-i a-i b x} (1+i a+i b x)^{5/2}}{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{\left (11 i+18 a-6 i a^2\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^3}+\frac{\left (11 i+18 a-6 i a^2\right ) \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{6 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 \sqrt{1-i a-i b x} (1+i a+i b x)^{5/2}}{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{\left (11 i+18 a-6 i a^2\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^3}+\frac{\left (11 i+18 a-6 i a^2\right ) \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{6 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 \sqrt{1-i a-i b x} (1+i a+i b x)^{5/2}}{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{\left (11 i+18 a-6 i a^2\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^3}+\frac{\left (11 i+18 a-6 i a^2\right ) \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{6 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 \sqrt{1-i a-i b x} (1+i a+i b x)^{5/2}}{3 b^3}+\frac{\left (11-18 i a-6 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.236446, size = 160, normalized size = 0.7 \[ \frac{\sqrt{i a+i b x+1} \left (-2 a^3-53 i a^2+a (103-16 i b x)-2 b^3 x^3+7 i b^2 x^2+19 b x+52 i\right )}{6 b^3 \sqrt{-i (a+b x+i)}}+\frac{(-1)^{3/4} \left (6 a^2+18 i a-11\right ) \sinh ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right )}{\sqrt{-i b} b^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.12, size = 519, normalized size = 2.3 \begin{align*}{\frac{17\,a}{2\,{b}^{3}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}+{\frac{17\,{a}^{3}}{2\,{b}^{3}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}-{{\frac{i}{3}}a{x}^{3}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}-{\frac{9\,ia}{{b}^{2}}\ln \left ({({b}^{2}x+ab){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{{\frac{26\,i}{3}}}{{b}^{3}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}+{\frac{11}{2\,{b}^{2}}\ln \left ({({b}^{2}x+ab){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{11\,x}{2\,{b}^{2}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}-{\frac{{\frac{i}{3}}{a}^{3}x}{{b}^{2}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}-{{\frac{i}{3}}b{x}^{4}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}+{\frac{{\frac{53\,i}{3}}ax}{{b}^{2}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}+{\frac{{\frac{25\,i}{3}}{a}^{2}}{{b}^{3}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}-{\frac{{\frac{i}{3}}{a}^{4}}{{b}^{3}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}+{\frac{{\frac{13\,i}{3}}{x}^{2}}{b}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}-3\,{\frac{{a}^{2}}{{b}^{2}\sqrt{{b}^{2}}}\ln \left ({\frac{{b}^{2}x+ab}{\sqrt{{b}^{2}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ) }+{\frac{3\,a{x}^{2}}{2\,b}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}-{\frac{3\,{x}^{3}}{2}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}+{\frac{23\,{a}^{2}x}{2\,{b}^{2}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01836, size = 489, normalized size = 2.15 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (i a + i b x + 1\right )^{3}}{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16247, size = 375, normalized size = 1.65 \begin{align*} -\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} + 18 \, a i - 11\right )} \log \left (3 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + 2 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}^{2} b i + 2 \, a^{2} b i +{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}^{3}{\left | b \right |} + 3 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )} a^{2}{\left | b \right |} + 4 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )} a i{\left | b \right |} - a b -{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}{\left | b \right |}\right )}{6 \, b^{2}{\left | b \right |}} + \frac{{\left (6 \, a^{2}{\left | b \right |} + 18 \, a i{\left | b \right |} - 11 \,{\left | b \right |}\right )} \log \left (8 \, b^{3}\right )}{3 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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