Optimal. Leaf size=195 \[ -\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}-\frac {\tanh ^{-1}\left (\frac {d x+e}{\sqrt {x^2-1} \sqrt {d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{2 \left (d^2-e^2\right )^{5/2}}+\frac {\sqrt {x^2-1} \left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right )}{2 e \left (d^2-e^2\right )^2 (d+e x)} \]
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Rubi [A] time = 0.21, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1651, 807, 725, 206} \begin {gather*} -\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac {\sqrt {x^2-1} \left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right )}{2 e \left (d^2-e^2\right )^2 (d+e x)}-\frac {\tanh ^{-1}\left (\frac {d x+e}{\sqrt {x^2-1} \sqrt {d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{2 \left (d^2-e^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 807
Rule 1651
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {-1+x^2}} \, dx &=-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right ) (d+e x)^2}-\frac {\int \frac {-2 (a d+c d-b e)-\left (b d+\frac {c d^2}{e}-a e-2 c e\right ) x}{(d+e x)^2 \sqrt {-1+x^2}} \, dx}{2 \left (d^2-e^2\right )}\\ &=-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac {\left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right )^2 (d+e x)}-\frac {\left (3 b d e-a \left (2 d^2+e^2\right )-c \left (d^2+2 e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {-1+x^2}} \, dx}{2 \left (d^2-e^2\right )^2}\\ &=-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac {\left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right )^2 (d+e x)}+\frac {\left (3 b d e-a \left (2 d^2+e^2\right )-c \left (d^2+2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{d^2-e^2-x^2} \, dx,x,\frac {-e-d x}{\sqrt {-1+x^2}}\right )}{2 \left (d^2-e^2\right )^2}\\ &=-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac {\left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right )^2 (d+e x)}-\frac {\left (3 b d e-a \left (2 d^2+e^2\right )-c \left (d^2+2 e^2\right )\right ) \tanh ^{-1}\left (\frac {e+d x}{\sqrt {d^2-e^2} \sqrt {-1+x^2}}\right )}{2 \left (d^2-e^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 240, normalized size = 1.23 \begin {gather*} \frac {1}{2} \left (-\frac {\log \left (-\sqrt {x^2-1} \sqrt {d^2-e^2}+d x+e\right ) \left (a \left (2 d^2+e^2\right )-3 b d e+c \left (d^2+2 e^2\right )\right )}{(d-e)^2 (d+e)^2 \sqrt {d^2-e^2}}+\frac {\log (d+e x) \left (a \left (2 d^2+e^2\right )-3 b d e+c \left (d^2+2 e^2\right )\right )}{(d-e)^2 (d+e)^2 \sqrt {d^2-e^2}}+\frac {\sqrt {x^2-1} \left (a e \left (-4 d^2-3 d e x+e^2\right )+b \left (2 d^3+d^2 e x+d e^2+2 e^3 x\right )+c d \left (d^2 x-3 d e-4 e^2 x\right )\right )}{\left (d^2-e^2\right )^2 (d+e x)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.43, size = 239, normalized size = 1.23 \begin {gather*} \frac {\tan ^{-1}\left (\frac {d-e \sqrt {x^2-1}+e x}{\sqrt {e^2-d^2}}\right ) \left (2 a d^2 \sqrt {e^2-d^2}+a e^2 \sqrt {e^2-d^2}-3 b d e \sqrt {e^2-d^2}+c d^2 \sqrt {e^2-d^2}+2 c e^2 \sqrt {e^2-d^2}\right )}{(d-e)^3 (d+e)^3}+\frac {\sqrt {x^2-1} \left (-4 a d^2 e-3 a d e^2 x+a e^3+2 b d^3+b d^2 e x+b d e^2+2 b e^3 x+c d^3 x-3 c d^2 e-4 c d e^2 x\right )}{2 (d-e)^2 (d+e)^2 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 1174, normalized size = 6.02
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 536, normalized size = 2.75 \begin {gather*} \frac {{\left (2 \, a d^{2} + c d^{2} - 3 \, b d e + a e^{2} + 2 \, c e^{2}\right )} \arctan \left (-\frac {{\left (x - \sqrt {x^{2} - 1}\right )} e + d}{\sqrt {-d^{2} + e^{2}}}\right )}{{\left (d^{4} - 2 \, d^{2} e^{2} + e^{4}\right )} \sqrt {-d^{2} + e^{2}}} + \frac {2 \, c d^{4} {\left (x - \sqrt {x^{2} - 1}\right )}^{3} e + 2 \, c d^{5} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 2 \, b d^{4} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e - 2 \, a d^{2} {\left (x - \sqrt {x^{2} - 1}\right )}^{3} e^{3} - 5 \, c d^{2} {\left (x - \sqrt {x^{2} - 1}\right )}^{3} e^{3} - 6 \, a d^{3} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{2} - 7 \, c d^{3} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{2} + 2 \, c d^{4} {\left (x - \sqrt {x^{2} - 1}\right )} e + 3 \, b d {\left (x - \sqrt {x^{2} - 1}\right )}^{3} e^{4} + 5 \, b d^{2} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{3} + 4 \, b d^{3} {\left (x - \sqrt {x^{2} - 1}\right )} e^{2} - a {\left (x - \sqrt {x^{2} - 1}\right )}^{3} e^{5} - 3 \, a d {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{4} - 4 \, c d {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{4} - 10 \, a d^{2} {\left (x - \sqrt {x^{2} - 1}\right )} e^{3} - 11 \, c d^{2} {\left (x - \sqrt {x^{2} - 1}\right )} e^{3} + c d^{3} e^{2} + 2 \, b {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{5} + 5 \, b d {\left (x - \sqrt {x^{2} - 1}\right )} e^{4} + b d^{2} e^{3} + a {\left (x - \sqrt {x^{2} - 1}\right )} e^{5} - 3 \, a d e^{4} - 4 \, c d e^{4} + 2 \, b e^{5}}{{\left (d^{4} e^{2} - 2 \, d^{2} e^{4} + e^{6}\right )} {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{2} e + 2 \, d {\left (x - \sqrt {x^{2} - 1}\right )} + e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1407, normalized size = 7.22
result too large to display
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 \begin {gather*} \text {Exception raised: ValueError} \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 = -0.01 \begin {gather*} \int \frac {c\,x^2+b\,x+a}{\sqrt {x^2-1}\,{\left (d+e\,x\right )}^3} \,d x \end {gather*}
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 \begin {gather*} \int \frac {a + b x + c x^{2}}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \left (d + e x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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