Optimal. Leaf size=291 \[ \frac {b \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac {f \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^{3/2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}+\frac {f \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^{3/2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}-\frac {\sqrt {a+b x+c x^2}}{a d x} \]
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Rubi [A] time = 0.66, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6725, 730, 724, 206, 984} \begin {gather*} \frac {b \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac {f \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^{3/2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}+\frac {f \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^{3/2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}-\frac {\sqrt {a+b x+c x^2}}{a d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 730
Rule 984
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx &=\int \left (\frac {1}{d x^2 \sqrt {a+b x+c x^2}}+\frac {f}{d \sqrt {a+b x+c x^2} \left (d-f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {1}{x^2 \sqrt {a+b x+c x^2}} \, dx}{d}+\frac {f \int \frac {1}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{d}\\ &=-\frac {\sqrt {a+b x+c x^2}}{a d x}-\frac {b \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{2 a d}+\frac {f \int \frac {1}{\left (d-\sqrt {d} \sqrt {f} x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d}+\frac {f \int \frac {1}{\left (d+\sqrt {d} \sqrt {f} x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d}\\ &=-\frac {\sqrt {a+b x+c x^2}}{a d x}+\frac {b \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{a d}-\frac {f \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d^{3/2} \sqrt {f}+4 a d f-x^2} \, dx,x,\frac {-b d+2 a \sqrt {d} \sqrt {f}-\left (2 c d-b \sqrt {d} \sqrt {f}\right ) x}{\sqrt {a+b x+c x^2}}\right )}{d}-\frac {f \operatorname {Subst}\left (\int \frac {1}{4 c d^2+4 b d^{3/2} \sqrt {f}+4 a d f-x^2} \, dx,x,\frac {-b d-2 a \sqrt {d} \sqrt {f}-\left (2 c d+b \sqrt {d} \sqrt {f}\right ) x}{\sqrt {a+b x+c x^2}}\right )}{d}\\ &=-\frac {\sqrt {a+b x+c x^2}}{a d x}+\frac {b \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2} d}+\frac {f \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} \sqrt {c d-b \sqrt {d} \sqrt {f}+a f}}+\frac {f \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} \sqrt {c d+b \sqrt {d} \sqrt {f}+a f}}\\ \end {align*}
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Mathematica [A] time = 1.03, size = 325, normalized size = 1.12 \begin {gather*} \frac {\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{a^{3/2}}+\frac {f \tanh ^{-1}\left (\frac {2 a \sqrt {f}+b \sqrt {d}+b \sqrt {f} x+2 c \sqrt {d} x}{2 \sqrt {a+x (b+c x)} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{\sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}+\frac {f \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+b \left (\sqrt {d}-\sqrt {f} x\right )+2 c \sqrt {d} x}{2 \sqrt {a+x (b+c x)} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{\sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}-\frac {2 b \sqrt {d}}{a \sqrt {a+x (b+c x)}}-\frac {2 c \sqrt {d} x}{a \sqrt {a+x (b+c x)}}-\frac {2 \sqrt {d}}{x \sqrt {a+x (b+c x)}}}{2 d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.52, size = 227, normalized size = 0.78 \begin {gather*} -\frac {f \text {RootSum}\left [\text {$\#$1}^4 (-f)+2 \text {$\#$1}^2 a f+4 \text {$\#$1}^2 c d-4 \text {$\#$1} b \sqrt {c} d-a^2 f+b^2 d\&,\frac {b \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )-2 \text {$\#$1} \sqrt {c} \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )}{\text {$\#$1}^3 f-\text {$\#$1} a f-2 \text {$\#$1} c d+b \sqrt {c} d}\&\right ]}{2 d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\sqrt {a+b x+c x^2}}{a d x} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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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maple [A] time = 0.02, size = 427, normalized size = 1.47 \begin {gather*} -\frac {f \ln \left (\frac {\frac {2 a f +2 c d -2 \sqrt {d f}\, b}{f}+\frac {\left (b f -2 \sqrt {d f}\, c \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {a f +c d -\sqrt {d f}\, b}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (b f -2 \sqrt {d f}\, c \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {a f +c d -\sqrt {d f}\, b}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{2 \sqrt {d f}\, \sqrt {\frac {a f +c d -\sqrt {d f}\, b}{f}}\, d}+\frac {f \ln \left (\frac {\frac {2 a f +2 c d +2 \sqrt {d f}\, b}{f}+\frac {\left (b f +2 \sqrt {d f}\, c \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {a f +c d +\sqrt {d f}\, b}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (b f +2 \sqrt {d f}\, c \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {a f +c d +\sqrt {d f}\, b}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{2 \sqrt {d f}\, \sqrt {\frac {a f +c d +\sqrt {d f}\, b}{f}}\, d}+\frac {b \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {3}{2}} d}-\frac {\sqrt {c \,x^{2}+b x +a}}{a d x} \end {gather*}
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*} -\int \frac {1}{\sqrt {c x^{2} + b x + a} {\left (f x^{2} - d\right )} x^{2}}\,{d x} \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 {1}{x^2\,\left (d-f\,x^2\right )\,\sqrt {c\,x^2+b\,x+a}} \,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 {1}{- d x^{2} \sqrt {a + b x + c x^{2}} + f x^{4} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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