Optimal. Leaf size=507 \[ -\frac {\sqrt {a+c x^2}}{2 d x^2}+\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {f \left (c d^2 \left (e+\sqrt {e^2-4 d f}\right )+a \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}-\frac {f \left (c d^2 \left (e-\sqrt {e^2-4 d f}\right )+a \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d}-\frac {\sqrt {a} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3} \]
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Rubi [A]
time = 1.18, antiderivative size = 507, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {6860, 272,
43, 65, 214, 283, 223, 212, 52, 1034, 1094, 1048, 739} \begin {gather*} -\frac {\sqrt {a} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3}+\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {f \left (a \left (e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )+c d^2 \left (\sqrt {e^2-4 d f}+e\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {f \left (a \left (-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )+c d^2 \left (e-\sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\sqrt {a+c x^2}}{2 d x^2}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 212
Rule 214
Rule 223
Rule 272
Rule 283
Rule 739
Rule 1034
Rule 1048
Rule 1094
Rule 6860
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^2}}{x^3 \left (d+e x+f x^2\right )} \, dx &=\int \left (\frac {\sqrt {a+c x^2}}{d x^3}-\frac {e \sqrt {a+c x^2}}{d^2 x^2}+\frac {\left (e^2-d f\right ) \sqrt {a+c x^2}}{d^3 x}+\frac {\left (-e \left (e^2-2 d f\right )-f \left (e^2-d f\right ) x\right ) \sqrt {a+c x^2}}{d^3 \left (d+e x+f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (-e \left (e^2-2 d f\right )-f \left (e^2-d f\right ) x\right ) \sqrt {a+c x^2}}{d+e x+f x^2} \, dx}{d^3}+\frac {\int \frac {\sqrt {a+c x^2}}{x^3} \, dx}{d}-\frac {e \int \frac {\sqrt {a+c x^2}}{x^2} \, dx}{d^2}+\frac {\left (e^2-d f\right ) \int \frac {\sqrt {a+c x^2}}{x} \, dx}{d^3}\\ &=-\frac {\left (e^2-d f\right ) \sqrt {a+c x^2}}{d^3}+\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )}{2 d}-\frac {(c e) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d^2}+\frac {\int \frac {-a e f \left (e^2-2 d f\right )+f (c d-a f) \left (e^2-d f\right ) x+c d e f^2 x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{d^3 f}+\frac {\left (e^2-d f\right ) \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{2 d^3}\\ &=-\frac {\sqrt {a+c x^2}}{2 d x^2}+\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {c \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 d}+\frac {(c e) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d^2}-\frac {(c e) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d^2}+\frac {\int \frac {-c d^2 e f^2-a e f^2 \left (e^2-2 d f\right )+\left (-c d e^2 f^2+f^2 (c d-a f) \left (e^2-d f\right )\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{d^3 f^2}+\frac {\left (a \left (e^2-d f\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^3}\\ &=-\frac {\sqrt {a+c x^2}}{2 d x^2}+\frac {e \sqrt {a+c x^2}}{d^2 x}-\frac {\sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{d^2}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 d}+\frac {(c e) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d^2}+\frac {\left (a \left (e^2-d f\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^3}-\frac {\left (f \left (c d^2 \left (e+\sqrt {e^2-4 d f}\right )+a \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d^3 \sqrt {e^2-4 d f}}+\frac {\left (f \left (c d^2 \left (e-\sqrt {e^2-4 d f}\right )+a \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}\right )\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d^3 \sqrt {e^2-4 d f}}\\ &=-\frac {\sqrt {a+c x^2}}{2 d x^2}+\frac {e \sqrt {a+c x^2}}{d^2 x}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d}-\frac {\sqrt {a} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3}+\frac {\left (f \left (c d^2 \left (e+\sqrt {e^2-4 d f}\right )+a \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{d^3 \sqrt {e^2-4 d f}}-\frac {\left (f \left (c d^2 \left (e-\sqrt {e^2-4 d f}\right )+a \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{d^3 \sqrt {e^2-4 d f}}\\ &=-\frac {\sqrt {a+c x^2}}{2 d x^2}+\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {f \left (c d^2 \left (e+\sqrt {e^2-4 d f}\right )+a \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}-\frac {f \left (c d^2 \left (e-\sqrt {e^2-4 d f}\right )+a \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d}-\frac {\sqrt {a} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.81, size = 533, normalized size = 1.05 \begin {gather*} \frac {\frac {d (-d+2 e x) \sqrt {a+c x^2}}{x^2}+\frac {2 c d^2 \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-4 \sqrt {a} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac {-\sqrt {c} x+\sqrt {a+c x^2}}{\sqrt {a}}\right )-2 \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {-a c d^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-a^2 e^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+a^2 d f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-2 c^{3/2} d^2 e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a \sqrt {c} e^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a \sqrt {c} d e f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a e^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a d f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{2 d^3} \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. \(1520\) vs.
\(2(442)=884\).
time = 0.16, size = 1521, normalized size = 3.00
method | result | size |
default | \(\text {Expression too large to display}\) | \(1521\) |
risch | \(\text {Expression too large to display}\) | \(2711\) |
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 [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + c x^{2}}}{x^{3} \left (d + e x + f x^{2}\right )}\, dx \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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,x^2+a}}{x^3\,\left (f\,x^2+e\,x+d\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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