Optimal. Leaf size=113 \[ \frac {x \sqrt {c+d x^2} (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}-\frac {(a c f-2 a d e+b c e) \tanh ^{-1}\left (\frac {x \sqrt {d e-c f}}{\sqrt {e} \sqrt {c+d x^2}}\right )}{2 e^{3/2} (d e-c f)^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {527, 12, 377, 208} \begin {gather*} \frac {x \sqrt {c+d x^2} (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}-\frac {(a c f-2 a d e+b c e) \tanh ^{-1}\left (\frac {x \sqrt {d e-c f}}{\sqrt {e} \sqrt {c+d x^2}}\right )}{2 e^{3/2} (d e-c f)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 208
Rule 377
Rule 527
Rubi steps
\begin {align*} \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx &=\frac {(b e-a f) x \sqrt {c+d x^2}}{2 e (d e-c f) \left (e+f x^2\right )}+\frac {\int \frac {-b c e+2 a d e-a c f}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx}{2 e (d e-c f)}\\ &=\frac {(b e-a f) x \sqrt {c+d x^2}}{2 e (d e-c f) \left (e+f x^2\right )}-\frac {(b c e-2 a d e+a c f) \int \frac {1}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx}{2 e (d e-c f)}\\ &=\frac {(b e-a f) x \sqrt {c+d x^2}}{2 e (d e-c f) \left (e+f x^2\right )}-\frac {(b c e-2 a d e+a c f) \operatorname {Subst}\left (\int \frac {1}{e-(d e-c f) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 e (d e-c f)}\\ &=\frac {(b e-a f) x \sqrt {c+d x^2}}{2 e (d e-c f) \left (e+f x^2\right )}-\frac {(b c e-2 a d e+a c f) \tanh ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )}{2 e^{3/2} (d e-c f)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 191, normalized size = 1.69 \begin {gather*} \frac {x (b e-a f) \left (f \left (c+d x^2\right )-\frac {\left (e+f x^2\right ) (2 d e-c f) \tanh ^{-1}\left (\sqrt {\frac {x^2 (d e-c f)}{e \left (c+d x^2\right )}}\right )}{e \sqrt {\frac {x^2 (d e-c f)}{e \left (c+d x^2\right )}}}\right )}{2 e f \sqrt {c+d x^2} \left (e+f x^2\right ) (d e-c f)}+\frac {b \tanh ^{-1}\left (\frac {x \sqrt {d e-c f}}{\sqrt {e} \sqrt {c+d x^2}}\right )}{\sqrt {e} f \sqrt {d e-c f}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 146, normalized size = 1.29 \begin {gather*} \frac {\sqrt {c f-d e} (-a c f+2 a d e-b c e) \tan ^{-1}\left (\frac {-f x \sqrt {c+d x^2}+\sqrt {d} e+\sqrt {d} f x^2}{\sqrt {e} \sqrt {c f-d e}}\right )}{2 e^{3/2} (d e-c f)^2}+\frac {x \sqrt {c+d x^2} (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 5.77, size = 513, normalized size = 4.54 \begin {gather*} \left [\frac {4 \, {\left (b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f\right )} \sqrt {d x^{2} + c} x - {\left (a c e f + {\left (b c - 2 \, a d\right )} e^{2} + {\left (a c f^{2} + {\left (b c - 2 \, a d\right )} e f\right )} x^{2}\right )} \sqrt {d e^{2} - c e f} \log \left (\frac {{\left (8 \, d^{2} e^{2} - 8 \, c d e f + c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} + 2 \, {\left (4 \, c d e^{2} - 3 \, c^{2} e f\right )} x^{2} + 4 \, {\left ({\left (2 \, d e - c f\right )} x^{3} + c e x\right )} \sqrt {d e^{2} - c e f} \sqrt {d x^{2} + c}}{f^{2} x^{4} + 2 \, e f x^{2} + e^{2}}\right )}{8 \, {\left (d^{2} e^{5} - 2 \, c d e^{4} f + c^{2} e^{3} f^{2} + {\left (d^{2} e^{4} f - 2 \, c d e^{3} f^{2} + c^{2} e^{2} f^{3}\right )} x^{2}\right )}}, \frac {2 \, {\left (b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f\right )} \sqrt {d x^{2} + c} x + {\left (a c e f + {\left (b c - 2 \, a d\right )} e^{2} + {\left (a c f^{2} + {\left (b c - 2 \, a d\right )} e f\right )} x^{2}\right )} \sqrt {-d e^{2} + c e f} \arctan \left (\frac {\sqrt {-d e^{2} + c e f} {\left ({\left (2 \, d e - c f\right )} x^{2} + c e\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (d^{2} e^{2} - c d e f\right )} x^{3} + {\left (c d e^{2} - c^{2} e f\right )} x\right )}}\right )}{4 \, {\left (d^{2} e^{5} - 2 \, c d e^{4} f + c^{2} e^{3} f^{2} + {\left (d^{2} e^{4} f - 2 \, c d e^{3} f^{2} + c^{2} e^{2} f^{3}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.40, size = 336, normalized size = 2.97 \begin {gather*} -\frac {{\left (a c \sqrt {d} f + b c \sqrt {d} e - 2 \, a d^{\frac {3}{2}} e\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} f - c f + 2 \, d e}{2 \, \sqrt {c d f e - d^{2} e^{2}}}\right )}{2 \, \sqrt {c d f e - d^{2} e^{2}} {\left (c f e - d e^{2}\right )}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c \sqrt {d} f^{2} - {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c \sqrt {d} f e - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d^{\frac {3}{2}} f e - a c^{2} \sqrt {d} f^{2} + 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b d^{\frac {3}{2}} e^{2} + b c^{2} \sqrt {d} f e}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} f - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} c f + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} d e + c^{2} f\right )} {\left (c f^{2} e - d f e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1622, normalized size = 14.35
result too large to display
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 {b x^{2} + a}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{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 = -0.01 \begin {gather*} \int \frac {b\,x^2+a}{\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2} \,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^{2}}{\sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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