Optimal. Leaf size=122 \[ \frac {b \tan ^{-1}\left (\frac {\sqrt {b e-a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )}{\sqrt {a} (b c-a d) \sqrt {b e-a f}}-\frac {d \tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} (b c-a d) \sqrt {d e-c f}} \]
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Rubi [A]
time = 0.08, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {546, 385, 211}
\begin {gather*} \frac {b \text {ArcTan}\left (\frac {x \sqrt {b e-a f}}{\sqrt {a} \sqrt {e+f x^2}}\right )}{\sqrt {a} (b c-a d) \sqrt {b e-a f}}-\frac {d \text {ArcTan}\left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} (b c-a d) \sqrt {d e-c f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 385
Rule 546
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx &=\frac {b \int \frac {1}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx}{b c-a d}-\frac {d \int \frac {1}{\left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx}{b c-a d}\\ &=\frac {b \text {Subst}\left (\int \frac {1}{a-(-b e+a f) x^2} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{b c-a d}-\frac {d \text {Subst}\left (\int \frac {1}{c-(-d e+c f) x^2} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{b c-a d}\\ &=\frac {b \tan ^{-1}\left (\frac {\sqrt {b e-a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )}{\sqrt {a} (b c-a d) \sqrt {b e-a f}}-\frac {d \tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} (b c-a d) \sqrt {d e-c f}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 153, normalized size = 1.25 \begin {gather*} \frac {-\frac {b \tan ^{-1}\left (\frac {a \sqrt {f}+b x \left (\sqrt {f} x-\sqrt {e+f x^2}\right )}{\sqrt {a} \sqrt {b e-a f}}\right )}{\sqrt {a} \sqrt {b e-a f}}+\frac {d \tan ^{-1}\left (\frac {c \sqrt {f}+d x \left (\sqrt {f} x-\sqrt {e+f x^2}\right )}{\sqrt {c} \sqrt {d e-c f}}\right )}{\sqrt {c} \sqrt {d e-c f}}}{b c-a 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. \(781\) vs.
\(2(102)=204\).
time = 0.12, size = 782, normalized size = 6.41
method | result | size |
default | \(\frac {b \,d^{2} \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}+\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}-\frac {b^{2} d \ln \left (\frac {-\frac {2 \left (a f -b e \right )}{b}+\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} f +\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-\frac {a f -b e}{b}}}+\frac {b^{2} d \ln \left (\frac {-\frac {2 \left (a f -b e \right )}{b}-\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} f -\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-\frac {a f -b e}{b}}}-\frac {b \,d^{2} \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}-\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}\) | \(782\) |
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. 307 vs.
\(2 (108) = 216\).
time = 58.27, size = 1377, normalized size = 11.29 \begin {gather*} \left [-\frac {{\left (b c^{2} f - b c d e\right )} \sqrt {a^{2} f - a b e} \log \left (\frac {8 \, a^{2} f^{2} x^{4} - 4 \, {\left (2 \, a f x^{3} - {\left (b x^{3} - a x\right )} e\right )} \sqrt {a^{2} f - a b e} \sqrt {f x^{2} + e} + {\left (b^{2} x^{4} - 6 \, a b x^{2} + a^{2}\right )} e^{2} - 8 \, {\left (a b f x^{4} - a^{2} f x^{2}\right )} e}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + {\left (a^{2} d f - a b d e\right )} \sqrt {c^{2} f - c d e} \log \left (\frac {8 \, c^{2} f^{2} x^{4} + 4 \, {\left (2 \, c f x^{3} - {\left (d x^{3} - c x\right )} e\right )} \sqrt {c^{2} f - c d e} \sqrt {f x^{2} + e} + {\left (d^{2} x^{4} - 6 \, c d x^{2} + c^{2}\right )} e^{2} - 8 \, {\left (c d f x^{4} - c^{2} f x^{2}\right )} e}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}, \frac {2 \, {\left (a^{2} d f - a b d e\right )} \sqrt {-c^{2} f + c d e} \arctan \left (\frac {{\left (2 \, c f x^{2} - {\left (d x^{2} - c\right )} e\right )} \sqrt {-c^{2} f + c d e} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} f^{2} x^{3} - c d x e^{2} - {\left (c d f x^{3} - c^{2} f x\right )} e\right )}}\right ) - {\left (b c^{2} f - b c d e\right )} \sqrt {a^{2} f - a b e} \log \left (\frac {8 \, a^{2} f^{2} x^{4} - 4 \, {\left (2 \, a f x^{3} - {\left (b x^{3} - a x\right )} e\right )} \sqrt {a^{2} f - a b e} \sqrt {f x^{2} + e} + {\left (b^{2} x^{4} - 6 \, a b x^{2} + a^{2}\right )} e^{2} - 8 \, {\left (a b f x^{4} - a^{2} f x^{2}\right )} e}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}, -\frac {2 \, {\left (b c^{2} f - b c d e\right )} \sqrt {-a^{2} f + a b e} \arctan \left (\frac {{\left (2 \, a f x^{2} - {\left (b x^{2} - a\right )} e\right )} \sqrt {-a^{2} f + a b e} \sqrt {f x^{2} + e}}{2 \, {\left (a^{2} f^{2} x^{3} - a b x e^{2} - {\left (a b f x^{3} - a^{2} f x\right )} e\right )}}\right ) + {\left (a^{2} d f - a b d e\right )} \sqrt {c^{2} f - c d e} \log \left (\frac {8 \, c^{2} f^{2} x^{4} + 4 \, {\left (2 \, c f x^{3} - {\left (d x^{3} - c x\right )} e\right )} \sqrt {c^{2} f - c d e} \sqrt {f x^{2} + e} + {\left (d^{2} x^{4} - 6 \, c d x^{2} + c^{2}\right )} e^{2} - 8 \, {\left (c d f x^{4} - c^{2} f x^{2}\right )} e}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}, -\frac {{\left (b c^{2} f - b c d e\right )} \sqrt {-a^{2} f + a b e} \arctan \left (\frac {{\left (2 \, a f x^{2} - {\left (b x^{2} - a\right )} e\right )} \sqrt {-a^{2} f + a b e} \sqrt {f x^{2} + e}}{2 \, {\left (a^{2} f^{2} x^{3} - a b x e^{2} - {\left (a b f x^{3} - a^{2} f x\right )} e\right )}}\right ) - {\left (a^{2} d f - a b d e\right )} \sqrt {-c^{2} f + c d e} \arctan \left (\frac {{\left (2 \, c f x^{2} - {\left (d x^{2} - c\right )} e\right )} \sqrt {-c^{2} f + c d e} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} f^{2} x^{3} - c d x e^{2} - {\left (c d f x^{3} - c^{2} f x\right )} e\right )}}\right )}{2 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}\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 {1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right ) \sqrt {e + f x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.27, size = 173, normalized size = 1.42 \begin {gather*} -f^{\frac {3}{2}} {\left (\frac {b \arctan \left (\frac {{\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} b + 2 \, a f - b e}{2 \, \sqrt {-a^{2} f^{2} + a b f e}}\right )}{\sqrt {-a^{2} f^{2} + a b f e} {\left (b c f - a d f\right )}} - \frac {d \arctan \left (\frac {{\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt {-c^{2} f^{2} + c d f e}}\right )}{\sqrt {-c^{2} f^{2} + c d f e} {\left (b c f - a d f\right )}}\right )} \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 {1}{\left (b\,x^2+a\right )\,\left (d\,x^2+c\right )\,\sqrt {f\,x^2+e}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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