Optimal. Leaf size=267 \[ -\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}-\frac {\sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d \sqrt {f}}+\frac {\sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d \sqrt {f}} \]
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Rubi [A]
time = 0.50, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {6857, 748,
857, 635, 212, 738, 1035, 1092, 1047} \begin {gather*} -\frac {\sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d} \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 \sqrt {f}}+\frac {\sqrt {a f+b \sqrt {d} \sqrt {f}+c d} \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 \sqrt {f}}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 748
Rule 857
Rule 1035
Rule 1047
Rule 1092
Rule 6857
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x+c x^2}}{x \left (d-f x^2\right )} \, dx &=\int \left (\frac {\sqrt {a+b x+c x^2}}{d x}-\frac {f x \sqrt {a+b x+c x^2}}{d \left (-d+f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\sqrt {a+b x+c x^2}}{x} \, dx}{d}-\frac {f \int \frac {x \sqrt {a+b x+c x^2}}{-d+f x^2} \, dx}{d}\\ &=-\frac {\int \frac {-2 a-b x}{x \sqrt {a+b x+c x^2}} \, dx}{2 d}+\frac {\int \frac {-\frac {b d}{2}-(c d+a f) x-\frac {1}{2} b f x^2}{\sqrt {a+b x+c x^2} \left (-d+f x^2\right )} \, dx}{d}\\ &=\frac {a \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{d}+\frac {\int \frac {-b d f+f (-c d-a f) x}{\sqrt {a+b x+c x^2} \left (-d+f x^2\right )} \, dx}{d f}\\ &=-\frac {(2 a) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{d}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right ) \int \frac {1}{\left (\sqrt {d} \sqrt {f}+f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d}-\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right ) \int \frac {1}{\left (-\sqrt {d} \sqrt {f}+f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d}\\ &=-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}+\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right ) \text {Subst}\left (\int \frac {1}{4 c d f-4 b \sqrt {d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac {-b \sqrt {d} \sqrt {f}+2 a f-\left (2 c \sqrt {d} \sqrt {f}-b f\right ) x}{\sqrt {a+b x+c x^2}}\right )}{d}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right ) \text {Subst}\left (\int \frac {1}{4 c d f+4 b \sqrt {d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac {b \sqrt {d} \sqrt {f}+2 a f-\left (-2 c \sqrt {d} \sqrt {f}-b f\right ) x}{\sqrt {a+b x+c x^2}}\right )}{d}\\ &=-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}-\frac {\sqrt {c d-b \sqrt {d} \sqrt {f}+a 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 \sqrt {f}}+\frac {\sqrt {c d+b \sqrt {d} \sqrt {f}+a 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 \sqrt {f}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.25, size = 334, normalized size = 1.25 \begin {gather*} -\frac {-4 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+\text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b^2 d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 b \sqrt {c} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(849\) vs.
\(2(203)=406\).
time = 0.14, size = 850, normalized size = 3.18
method | result | size |
default | \(-\frac {\sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \ln \left (\frac {\frac {-2 c \sqrt {d f}+b f}{2 f}+c \left (x +\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}\right )}{2 f \sqrt {c}}-\frac {\left (-b \sqrt {d f}+f a +c d \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{2 d}+\frac {\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d}-\frac {\sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}+\frac {\left (2 c \sqrt {d f}+b f \right ) \ln \left (\frac {\frac {2 c \sqrt {d f}+b f}{2 f}+c \left (x -\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}\right )}{2 f \sqrt {c}}-\frac {\left (b \sqrt {d f}+f a +c d \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{2 d}\) | \(850\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 622 vs.
\(2 (203) = 406\).
time = 10.12, size = 1253, normalized size = 4.69 \begin {gather*} \left [\frac {d \sqrt {\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} + c d + a f}{d^{2} f}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} \sqrt {\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} + c d + a f}{d^{2} f}} + 2 \, b c x + b^{2} + {\left (b d f x + 2 \, a d f\right )} \sqrt {\frac {b^{2}}{d^{3} f}}}{x}\right ) - d \sqrt {\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} + c d + a f}{d^{2} f}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} \sqrt {\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} + c d + a f}{d^{2} f}} - 2 \, b c x - b^{2} - {\left (b d f x + 2 \, a d f\right )} \sqrt {\frac {b^{2}}{d^{3} f}}}{x}\right ) - d \sqrt {-\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} - c d - a f}{d^{2} f}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} \sqrt {-\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} - c d - a f}{d^{2} f}} + 2 \, b c x + b^{2} - {\left (b d f x + 2 \, a d f\right )} \sqrt {\frac {b^{2}}{d^{3} f}}}{x}\right ) + d \sqrt {-\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} - c d - a f}{d^{2} f}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} \sqrt {-\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} - c d - a f}{d^{2} f}} - 2 \, b c x - b^{2} + {\left (b d f x + 2 \, a d f\right )} \sqrt {\frac {b^{2}}{d^{3} f}}}{x}\right ) + 2 \, \sqrt {a} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right )}{4 \, d}, \frac {d \sqrt {\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} + c d + a f}{d^{2} f}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} \sqrt {\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} + c d + a f}{d^{2} f}} + 2 \, b c x + b^{2} + {\left (b d f x + 2 \, a d f\right )} \sqrt {\frac {b^{2}}{d^{3} f}}}{x}\right ) - d \sqrt {\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} + c d + a f}{d^{2} f}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} \sqrt {\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} + c d + a f}{d^{2} f}} - 2 \, b c x - b^{2} - {\left (b d f x + 2 \, a d f\right )} \sqrt {\frac {b^{2}}{d^{3} f}}}{x}\right ) - d \sqrt {-\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} - c d - a f}{d^{2} f}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} \sqrt {-\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} - c d - a f}{d^{2} f}} + 2 \, b c x + b^{2} - {\left (b d f x + 2 \, a d f\right )} \sqrt {\frac {b^{2}}{d^{3} f}}}{x}\right ) + d \sqrt {-\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} - c d - a f}{d^{2} f}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} \sqrt {-\frac {d^{2} f \sqrt {\frac {b^{2}}{d^{3} f}} - c d - a f}{d^{2} f}} - 2 \, b c x - b^{2} + {\left (b d f x + 2 \, a d f\right )} \sqrt {\frac {b^{2}}{d^{3} f}}}{x}\right ) + 4 \, \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right )}{4 \, d}\right ] \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 {\sqrt {a + b x + c x^{2}}}{- d x + f x^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {\sqrt {c\,x^2+b\,x+a}}{x\,\left (d-f\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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