Optimal. Leaf size=629 \[ \frac {e \log \left (-\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac {e \log \left (\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac {e \tanh ^{-1}\left (\frac {-2 \sqrt {c} \sqrt {d+e x}+\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {c} \sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac {e \tanh ^{-1}\left (\frac {2 \sqrt {c} \sqrt {d+e x}+\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {c} \sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}} \]
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Rubi [A] time = 1.23, antiderivative size = 629, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {699, 1129, 634, 618, 206, 628} \[ \frac {e \log \left (-\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac {e \log \left (\sqrt {d+e x} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt {c d^2-e (-a e+i b d)}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac {e \tanh ^{-1}\left (\frac {-2 \sqrt {c} \sqrt {d+e x}+\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {c} \sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac {e \tanh ^{-1}\left (\frac {2 \sqrt {c} \sqrt {d+e x}+\sqrt {2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt {c} \sqrt {-2 \sqrt {c} \sqrt {c d^2-e (-a e+i b d)}-i b e+2 c d}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 699
Rule 1129
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{a+i b x+c x^2} \, dx &=(2 e) \operatorname {Subst}\left (\int \frac {x^2}{c d^2-i b d e+a e^2+(-2 c d+i b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )\\ &=\frac {e \operatorname {Subst}\left (\int \frac {x}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {e \operatorname {Subst}\left (\int \frac {x}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\\ &=\frac {e \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c}+\frac {e \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c}+\frac {e \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}}}{\sqrt {c}}+2 x}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {e \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}}}{\sqrt {c}}+2 x}{\frac {\sqrt {c d^2-i b d e+a e^2}}{\sqrt {c}}+\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}} x}{\sqrt {c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\\ &=\frac {e \log \left (\sqrt {c d^2-e (i b d-a e)}-\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {e \log \left (\sqrt {c d^2-e (i b d-a e)}+\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {e \operatorname {Subst}\left (\int \frac {1}{2 d-\frac {i b e}{c}-\frac {2 \sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-x^2} \, dx,x,-\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}}}{\sqrt {c}}+2 \sqrt {d+e x}\right )}{c}-\frac {e \operatorname {Subst}\left (\int \frac {1}{2 d-\frac {i b e}{c}-\frac {2 \sqrt {c d^2-e (i b d-a e)}}{\sqrt {c}}-x^2} \, dx,x,\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-i b d e+a e^2}}}{\sqrt {c}}+2 \sqrt {d+e x}\right )}{c}\\ &=\frac {e \tanh ^{-1}\left (\frac {\sqrt {c} \left (\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}{\sqrt {c}}-2 \sqrt {d+e x}\right )}{\sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )}{\sqrt {c} \sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {c} \left (\frac {\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}{\sqrt {c}}+2 \sqrt {d+e x}\right )}{\sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\right )}{\sqrt {c} \sqrt {2 c d-i b e-2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}+\frac {e \log \left (\sqrt {c d^2-e (i b d-a e)}-\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}-\frac {e \log \left (\sqrt {c d^2-e (i b d-a e)}+\sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {c} \sqrt {2 c d-i b e+2 \sqrt {c} \sqrt {c d^2-e (i b d-a e)}}}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 197, normalized size = 0.31 \[ \frac {\sqrt {2} \left (\sqrt {2 c d-e \left (\sqrt {-4 a c-b^2}+i b\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {-4 a c-b^2}+i b\right )}}\right )-\sqrt {e \sqrt {-4 a c-b^2}-i b e+2 c d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {-4 a c-b^2}-i b e+2 c d}}\right )\right )}{\sqrt {c} \sqrt {-4 a c-b^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 753, normalized size = 1.20 \[ -\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {2 \, c d - i \, b e + {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (\frac {\sqrt {2} {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt {-\frac {2 \, c d - i \, b e + {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e}{2 \, e}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {2 \, c d - i \, b e + {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (-\frac {\sqrt {2} {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt {-\frac {2 \, c d - i \, b e + {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} - 2 \, \sqrt {e x + d} e}{2 \, e}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {2 \, c d - i \, b e - {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (\frac {\sqrt {2} {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt {-\frac {2 \, c d - i \, b e - {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e}{2 \, e}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {2 \, c d - i \, b e - {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (-\frac {\sqrt {2} {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt {-\frac {2 \, c d - i \, b e - {\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt {-\frac {e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} - 2 \, \sqrt {e x + d} e}{2 \, e}\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.63, size = 609, normalized size = 0.97 \[ \frac {e \arctan \left (\frac {2 \sqrt {e x +d}\, \sqrt {c}-\sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}}{\sqrt {i b e -2 c d +4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}}\right )}{\sqrt {i b e -2 c d +4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}\, \sqrt {c}}+\frac {e \arctan \left (\frac {2 \sqrt {e x +d}\, \sqrt {c}+\sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}}{\sqrt {i b e -2 c d +4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}}\right )}{\sqrt {i b e -2 c d +4 \sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}\, \sqrt {c}}+\frac {e \ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}+\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}\, \sqrt {c}}-\frac {e \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}+\sqrt {-i b d e +a \,e^{2}+c \,d^{2}}\right )}{2 \sqrt {-i b e +2 c d +2 \sqrt {-\left (i b d e -a \,e^{2}-c \,d^{2}\right ) c}}\, \sqrt {c}} \]
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 \[ \int \frac {\sqrt {e x + d}}{c x^{2} + i \, b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.82, size = 711, normalized size = 1.13 \[ -2\,\mathrm {atanh}\left (\frac {\left (8\,c^2\,\sqrt {d+e\,x}\,\left (b^2\,e^4+b\,c\,d\,e^3\,2{}\mathrm {i}-2\,c^2\,d^2\,e^2+2\,a\,c\,e^4\right )-\frac {4\,c^2\,\sqrt {d+e\,x}\,\left (b^3\,c\,e^3\,1{}\mathrm {i}-2\,d\,b^2\,c^2\,e^2+4{}\mathrm {i}\,a\,b\,c^2\,e^3-8\,a\,d\,c^3\,e^2\right )\,\left (e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}-b^3\,e\,1{}\mathrm {i}+8\,a\,c^2\,d+2\,b^2\,c\,d-a\,b\,c\,e\,4{}\mathrm {i}\right )}{16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c}\right )\,\sqrt {-\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}-b^3\,e\,1{}\mathrm {i}+8\,a\,c^2\,d+2\,b^2\,c\,d-a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}}}{8\,c^2\,\left (c\,d^2\,e^3-1{}\mathrm {i}\,b\,d\,e^4+a\,e^5\right )}\right )\,\sqrt {-\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}-b^3\,e\,1{}\mathrm {i}+8\,a\,c^2\,d+2\,b^2\,c\,d-a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}}-2\,\mathrm {atanh}\left (\frac {\left (8\,c^2\,\sqrt {d+e\,x}\,\left (b^2\,e^4+b\,c\,d\,e^3\,2{}\mathrm {i}-2\,c^2\,d^2\,e^2+2\,a\,c\,e^4\right )+\frac {4\,c^2\,\sqrt {d+e\,x}\,\left (b^3\,c\,e^3\,1{}\mathrm {i}-2\,d\,b^2\,c^2\,e^2+4{}\mathrm {i}\,a\,b\,c^2\,e^3-8\,a\,d\,c^3\,e^2\right )\,\left (e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}+b^3\,e\,1{}\mathrm {i}-8\,a\,c^2\,d-2\,b^2\,c\,d+a\,b\,c\,e\,4{}\mathrm {i}\right )}{16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}+b^3\,e\,1{}\mathrm {i}-8\,a\,c^2\,d-2\,b^2\,c\,d+a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}}}{8\,c^2\,\left (c\,d^2\,e^3-1{}\mathrm {i}\,b\,d\,e^4+a\,e^5\right )}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (b^2+4\,a\,c\right )}^3}+b^3\,e\,1{}\mathrm {i}-8\,a\,c^2\,d-2\,b^2\,c\,d+a\,b\,c\,e\,4{}\mathrm {i}}{2\,\left (16\,a^2\,c^3+8\,a\,b^2\,c^2+b^4\,c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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