Optimal. Leaf size=142 \[ -\frac {\sqrt {a^2 x^2+1}}{2 a^3 c (a x+i) \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \log (-a x+i)}{4 a^3 c \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \log (a x+i)}{4 a^3 c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.22, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {5085, 5082, 88} \[ -\frac {\sqrt {a^2 x^2+1}}{2 a^3 c (a x+i) \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \log (-a x+i)}{4 a^3 c \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \log (a x+i)}{4 a^3 c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 5082
Rule 5085
Rubi steps
\begin {align*} \int \frac {e^{i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{i \tan ^{-1}(a x)} x^2}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \frac {x^2}{(1-i a x)^2 (1+i a x)} \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \left (\frac {i}{4 a^2 (-i+a x)}+\frac {1}{2 a^2 (i+a x)^2}+\frac {3 i}{4 a^2 (i+a x)}\right ) \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {1+a^2 x^2}}{2 a^3 c (i+a x) \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \log (i-a x)}{4 a^3 c \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \log (i+a x)}{4 a^3 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 74, normalized size = 0.52 \[ \frac {\sqrt {a^2 x^2+1} \left (-\frac {2}{a x+i}+i \log (-a x+i)+3 i \log (a x+i)\right )}{4 a^3 c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ \frac {{\left (-3 i \, a^{5} c^{2} x^{3} + 3 \, a^{4} c^{2} x^{2} - 3 i \, a^{3} c^{2} x + 3 \, a^{2} c^{2}\right )} \sqrt {\frac {1}{a^{6} c^{3}}} \log \left (\frac {i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a^{3} c x \sqrt {\frac {1}{a^{6} c^{3}}} + i \, a^{2} x^{3} + i \, x}{a^{3} x^{3} + i \, a^{2} x^{2} + a x + i}\right ) + {\left (3 i \, a^{5} c^{2} x^{3} - 3 \, a^{4} c^{2} x^{2} + 3 i \, a^{3} c^{2} x - 3 \, a^{2} c^{2}\right )} \sqrt {\frac {1}{a^{6} c^{3}}} \log \left (\frac {-i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a^{3} c x \sqrt {\frac {1}{a^{6} c^{3}}} + i \, a^{2} x^{3} + i \, x}{a^{3} x^{3} + i \, a^{2} x^{2} + a x + i}\right ) + {\left (i \, a^{5} c^{2} x^{3} - a^{4} c^{2} x^{2} + i \, a^{3} c^{2} x - a^{2} c^{2}\right )} \sqrt {\frac {1}{a^{6} c^{3}}} \log \left (\frac {i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a^{3} c x \sqrt {\frac {1}{a^{6} c^{3}}} - i \, a^{2} x^{3} - i \, x}{a^{3} x^{3} - i \, a^{2} x^{2} + a x - i}\right ) + {\left (-i \, a^{5} c^{2} x^{3} + a^{4} c^{2} x^{2} - i \, a^{3} c^{2} x + a^{2} c^{2}\right )} \sqrt {\frac {1}{a^{6} c^{3}}} \log \left (\frac {-i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a^{3} c x \sqrt {\frac {1}{a^{6} c^{3}}} - i \, a^{2} x^{3} - i \, x}{a^{3} x^{3} - i \, a^{2} x^{2} + a x - i}\right ) + {\left (4 i \, a^{5} c^{2} x^{3} - 4 \, a^{4} c^{2} x^{2} + 4 i \, a^{3} c^{2} x - 4 \, a^{2} c^{2}\right )} \sqrt {\frac {1}{a^{6} c^{3}}} \log \left (\frac {\sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a^{3} c x \sqrt {\frac {1}{a^{6} c^{3}}} + a^{2} x^{3} + x}{a^{2} x^{2} + 1}\right ) + {\left (-4 i \, a^{5} c^{2} x^{3} + 4 \, a^{4} c^{2} x^{2} - 4 i \, a^{3} c^{2} x + 4 \, a^{2} c^{2}\right )} \sqrt {\frac {1}{a^{6} c^{3}}} \log \left (-\frac {\sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a^{3} c x \sqrt {\frac {1}{a^{6} c^{3}}} - a^{2} x^{3} - x}{a^{2} x^{2} + 1}\right ) - 4 i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} x + 2 \, {\left (4 \, a^{5} c^{2} x^{3} + 4 i \, a^{4} c^{2} x^{2} + 4 \, a^{3} c^{2} x + 4 i \, a^{2} c^{2}\right )} {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} {\left (2 i \, a x + 1\right )}}{2 \, {\left (a^{6} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}}, x\right )}{2 \, {\left (4 \, a^{5} c^{2} x^{3} + 4 i \, a^{4} c^{2} x^{2} + 4 \, a^{3} c^{2} x + 4 i \, a^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a x + 1\right )} x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 87, normalized size = 0.61 \[ \frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (3 i \ln \left (a x +i\right ) x a +i \ln \left (-a x +i\right ) x a -3 \ln \left (a x +i\right )-\ln \left (-a x +i\right )-2\right )}{4 \sqrt {a^{2} x^{2}+1}\, c^{2} a^{3} \left (a x +i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {x^2\,\left (1+a\,x\,1{}\mathrm {i}\right )}{{\left (c\,a^2\,x^2+c\right )}^{3/2}\,\sqrt {a^2\,x^2+1}} \,d x \]
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 \[ i \left (\int \left (- \frac {i x^{2}}{a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a x^{3}}{a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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