Optimal. Leaf size=494 \[ \frac {\left (17 i-36 a-24 i a^2\right ) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{24 b^3}+\frac {(3 i-8 a) (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{12 b^3}+\frac {x (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{3 b^2}+\frac {\left (17 i-36 a-24 i a^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}-\frac {\left (17 i-36 a-24 i a^2\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}-\frac {\left (17 i-36 a-24 i a^2\right ) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}+\frac {\left (17 i-36 a-24 i a^2\right ) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3} \]
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Rubi [A]
time = 0.29, antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5203, 92, 81,
52, 65, 338, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\left (-24 i a^2-36 a+17 i\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{8 \sqrt {2} b^3}-\frac {\left (-24 i a^2-36 a+17 i\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{8 \sqrt {2} b^3}+\frac {\left (-24 i a^2-36 a+17 i\right ) \sqrt [4]{i a+i b x+1} (-i a-i b x+1)^{3/4}}{24 b^3}-\frac {\left (-24 i a^2-36 a+17 i\right ) \log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{16 \sqrt {2} b^3}+\frac {\left (-24 i a^2-36 a+17 i\right ) \log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{16 \sqrt {2} b^3}+\frac {(-8 a+3 i) \sqrt [4]{i a+i b x+1} (-i a-i b x+1)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{i a+i b x+1} (-i a-i b x+1)^{7/4}}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5203
Rubi steps
\begin {align*} \int e^{-\frac {3}{2} i \tan ^{-1}(a+b x)} x^2 \, dx &=\int \frac {x^2 (1-i a-i b x)^{3/4}}{(1+i a+i b x)^{3/4}} \, dx\\ &=\frac {x (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{3 b^2}+\frac {\int \frac {(1-i a-i b x)^{3/4} \left (-1-a^2+\frac {1}{2} (3 i-8 a) b x\right )}{(1+i a+i b x)^{3/4}} \, dx}{3 b^2}\\ &=\frac {(3 i-8 a) (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{12 b^3}+\frac {x (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{3 b^2}-\frac {\left (17+36 i a-24 a^2\right ) \int \frac {(1-i a-i b x)^{3/4}}{(1+i a+i b x)^{3/4}} \, dx}{24 b^2}\\ &=-\frac {\left (36 a-i \left (17-24 a^2\right )\right ) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{24 b^3}+\frac {(3 i-8 a) (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{12 b^3}+\frac {x (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{3 b^2}-\frac {\left (17+36 i a-24 a^2\right ) \int \frac {1}{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}} \, dx}{16 b^2}\\ &=-\frac {\left (36 a-i \left (17-24 a^2\right )\right ) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{24 b^3}+\frac {(3 i-8 a) (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{12 b^3}+\frac {x (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{3 b^2}-\frac {\left (i \left (17+36 i a-24 a^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a-i b x}\right )}{4 b^3}\\ &=-\frac {\left (36 a-i \left (17-24 a^2\right )\right ) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{24 b^3}+\frac {(3 i-8 a) (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{12 b^3}+\frac {x (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{3 b^2}-\frac {\left (i \left (17+36 i a-24 a^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 b^3}\\ &=-\frac {\left (36 a-i \left (17-24 a^2\right )\right ) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{24 b^3}+\frac {(3 i-8 a) (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{12 b^3}+\frac {x (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{3 b^2}+\frac {\left (i \left (17+36 i a-24 a^2\right )\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 b^3}-\frac {\left (i \left (17+36 i a-24 a^2\right )\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 b^3}\\ &=-\frac {\left (36 a-i \left (17-24 a^2\right )\right ) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{24 b^3}+\frac {(3 i-8 a) (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{12 b^3}+\frac {x (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{3 b^2}-\frac {\left (i \left (17+36 i a-24 a^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 b^3}-\frac {\left (i \left (17+36 i a-24 a^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 b^3}-\frac {\left (i \left (17+36 i a-24 a^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}-\frac {\left (i \left (17+36 i a-24 a^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}\\ &=-\frac {\left (36 a-i \left (17-24 a^2\right )\right ) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{24 b^3}+\frac {(3 i-8 a) (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{12 b^3}+\frac {x (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{3 b^2}+\frac {\left (36 a-i \left (17-24 a^2\right )\right ) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}-\frac {\left (36 a-i \left (17-24 a^2\right )\right ) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}-\frac {\left (i \left (17+36 i a-24 a^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}+\frac {\left (i \left (17+36 i a-24 a^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}\\ &=-\frac {\left (36 a-i \left (17-24 a^2\right )\right ) (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{24 b^3}+\frac {(3 i-8 a) (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{12 b^3}+\frac {x (1-i a-i b x)^{7/4} \sqrt [4]{1+i a+i b x}}{3 b^2}-\frac {\left (36 a-i \left (17-24 a^2\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}+\frac {\left (36 a-i \left (17-24 a^2\right )\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}+\frac {\left (36 a-i \left (17-24 a^2\right )\right ) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}-\frac {\left (36 a-i \left (17-24 a^2\right )\right ) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.06, size = 98, normalized size = 0.20 \begin {gather*} \frac {(-i (i+a+b x))^{7/4} \left (7 \sqrt [4]{1+i a+i b x} (3 i-8 a+4 b x)+\sqrt [4]{2} \left (-17 i+36 a+24 i a^2\right ) \, _2F_1\left (\frac {3}{4},\frac {7}{4};\frac {11}{4};-\frac {1}{2} i (i+a+b x)\right )\right )}{84 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.35, size = 555, normalized size = 1.12 \begin {gather*} \frac {3 \, b^{3} \sqrt {\frac {576 i \, a^{4} + 1728 \, a^{3} - 2112 i \, a^{2} - 1224 \, a + 289 i}{b^{6}}} \log \left (\frac {i \, b^{3} \sqrt {\frac {576 i \, a^{4} + 1728 \, a^{3} - 2112 i \, a^{2} - 1224 \, a + 289 i}{b^{6}}} + {\left (24 \, a^{2} - 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} - 36 i \, a - 17}\right ) - 3 \, b^{3} \sqrt {\frac {576 i \, a^{4} + 1728 \, a^{3} - 2112 i \, a^{2} - 1224 \, a + 289 i}{b^{6}}} \log \left (\frac {-i \, b^{3} \sqrt {\frac {576 i \, a^{4} + 1728 \, a^{3} - 2112 i \, a^{2} - 1224 \, a + 289 i}{b^{6}}} + {\left (24 \, a^{2} - 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} - 36 i \, a - 17}\right ) + 3 \, b^{3} \sqrt {\frac {-576 i \, a^{4} - 1728 \, a^{3} + 2112 i \, a^{2} + 1224 \, a - 289 i}{b^{6}}} \log \left (\frac {i \, b^{3} \sqrt {\frac {-576 i \, a^{4} - 1728 \, a^{3} + 2112 i \, a^{2} + 1224 \, a - 289 i}{b^{6}}} + {\left (24 \, a^{2} - 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} - 36 i \, a - 17}\right ) - 3 \, b^{3} \sqrt {\frac {-576 i \, a^{4} - 1728 \, a^{3} + 2112 i \, a^{2} + 1224 \, a - 289 i}{b^{6}}} \log \left (\frac {-i \, b^{3} \sqrt {\frac {-576 i \, a^{4} - 1728 \, a^{3} + 2112 i \, a^{2} + 1224 \, a - 289 i}{b^{6}}} + {\left (24 \, a^{2} - 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} - 36 i \, a - 17}\right ) - 2 \, {\left (8 \, b^{3} x^{3} + 22 i \, b^{2} x^{2} + 8 \, a^{3} - {\left (40 i \, a + 37\right )} b x - 38 i \, a^{2} + 23 \, a - 23 i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{48 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x^2}{{\left (\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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