Optimal. Leaf size=201 \[ -\frac {\sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2} \left (-18 a^2+2 (6 a+i) b x-10 i a+7\right )}{24 b^4}-\frac {\left (8 i a^3-12 a^2-12 i a+3\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^4}+\frac {\left (-8 a^3-12 i a^2+12 a+3 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}+\frac {x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b^2} \]
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Rubi [A] time = 0.19, antiderivative size = 201, 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, 100, 147, 50, 53, 619, 215} \[ -\frac {\sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2} \left (-18 a^2+2 (6 a+i) b x-10 i a+7\right )}{24 b^4}-\frac {\left (8 i a^3-12 a^2-12 i a+3\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^4}+\frac {\left (-8 a^3-12 i a^2+12 a+3 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}+\frac {x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 100
Rule 147
Rule 215
Rule 619
Rule 5095
Rubi steps
\begin {align*} \int e^{i \tan ^{-1}(a+b x)} x^3 \, dx &=\int \frac {x^3 \sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx\\ &=\frac {x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}+\frac {\int \frac {x \sqrt {1+i a+i b x} \left (-2 \left (1+a^2\right )-(i+6 a) b x\right )}{\sqrt {1-i a-i b x}} \, dx}{4 b^2}\\ &=\frac {x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (7-10 i a-18 a^2+2 (i+6 a) b x\right )}{24 b^4}+\frac {\left (3 i+12 a-12 i a^2-8 a^3\right ) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{8 b^3}\\ &=-\frac {\left (3-12 i a-12 a^2+8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (7-10 i a-18 a^2+2 (i+6 a) b x\right )}{24 b^4}+\frac {\left (3 i+12 a-12 i a^2-8 a^3\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^3}\\ &=-\frac {\left (3-12 i a-12 a^2+8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (7-10 i a-18 a^2+2 (i+6 a) b x\right )}{24 b^4}+\frac {\left (3 i+12 a-12 i a^2-8 a^3\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^3}\\ &=-\frac {\left (3-12 i a-12 a^2+8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (7-10 i a-18 a^2+2 (i+6 a) b x\right )}{24 b^4}+\frac {\left (3 i+12 a-12 i a^2-8 a^3\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 )}{16 b^5}\\ &=-\frac {\left (3-12 i a-12 a^2+8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (7-10 i a-18 a^2+2 (i+6 a) b x\right )}{24 b^4}+\frac {\left (3 i+12 a-12 i a^2-8 a^3\right ) \sinh ^{-1}(a+b x)}{8 b^4}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 176, normalized size = 0.88 \[ \frac {\sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2+1} \left (-6 i a^3+a^2 (44+6 i b x)+a \left (-6 i b^2 x^2-20 b x+39 i\right )+6 i b^3 x^3+8 b^2 x^2-9 i b x-16\right )-6 \sqrt [4]{-1} \left (8 a^3+12 i a^2-12 a-3 i\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{24 b^{9/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.63, size = 137, normalized size = 0.68 \[ \frac {-45 i \, a^{4} + 224 \, a^{3} + 192 i \, a^{2} + {\left (192 \, a^{3} + 288 i \, a^{2} - 288 \, a - 72 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (48 i \, b^{3} x^{3} - 16 \, {\left (3 i \, a - 4\right )} b^{2} x^{2} - 48 i \, a^{3} + {\left (48 i \, a^{2} - 160 \, a - 72 i\right )} b x + 352 \, a^{2} + 312 i \, a - 128\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 72 \, a}{192 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 163, normalized size = 0.81 \[ \frac {1}{24} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, {\left (\frac {3 \, i x}{b} - \frac {3 \, a b^{5} i - 4 \, b^{5}}{b^{7}}\right )} x + \frac {6 \, a^{2} b^{4} i - 20 \, a b^{4} - 9 \, b^{4} i}{b^{7}}\right )} x - \frac {6 \, a^{3} b^{3} i - 44 \, a^{2} b^{3} - 39 \, a b^{3} i + 16 \, b^{3}}{b^{7}}\right )} + \frac {{\left (8 \, a^{3} + 12 \, a^{2} i - 12 \, a - 3 \, i\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{3} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 465, normalized size = 2.31 \[ \frac {3 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{3} \sqrt {b^{2}}}-\frac {3 i x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 b^{3}}+\frac {13 i a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 b^{4}}-\frac {i a^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{4}}+\frac {i x^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b}-\frac {i a \,x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {i a^{2} x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{3}}-\frac {3 i a^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{3} \sqrt {b^{2}}}+\frac {x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 b^{2}}-\frac {5 a x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{6 b^{3}}+\frac {11 a^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{6 b^{4}}-\frac {a^{3} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{3} \sqrt {b^{2}}}+\frac {3 a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{3} \sqrt {b^{2}}}-\frac {2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 529, normalized size = 2.63 \[ \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x^{3}}{4 \, b} - \frac {7 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x^{2}}{12 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-i \, a - 1\right )} x^{2}}{3 \, b^{2}} + \frac {35 i \, a^{4} \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 )}{8 \, b^{4}} - \frac {5 \, a^{3} {\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^{4}} + \frac {35 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} x}{24 \, b^{3}} - \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a {\left (i \, a + 1\right )} x}{6 \, b^{3}} - \frac {15 i \, {\left (a^{2} + 1\right )} a^{2} \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 )}{4 \, b^{4}} + \frac {3 \, {\left (a^{2} + 1\right )} a {\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^{4}} - \frac {35 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3}}{8 \, b^{4}} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} {\left (i \, a + 1\right )}}{2 \, b^{4}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (3 i \, a^{2} + 3 i\right )} x}{8 \, b^{3}} + \frac {3 i \, {\left (a^{2} + 1\right )}^{2} \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 )}{8 \, b^{4}} + \frac {55 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a}{24 \, b^{4}} - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} {\left (i \, a + 1\right )}}{3 \, b^{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 \[ \int \frac {x^3\,\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}{\sqrt {{\left (a+b\,x\right )}^2+1}} \,d x \]
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 \[ i \left (\int \left (- \frac {i x^{3}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a x^{3}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {b x^{4}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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