Optimal. Leaf size=203 \[ \frac{b \left (4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right ) \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}} \]
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Rubi [A] time = 0.265115, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {546, 377, 205, 527, 12} \[ \frac{b \left (4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right ) \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac{b^2 x \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}} \]
Antiderivative was successfully verified.
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Rule 546
Rule 377
Rule 205
Rule 527
Rule 12
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^2 \left (c+d x^2\right ) \sqrt{e+f x^2}} \, dx &=-\frac{b \int \frac{-b c+2 a d+b d x^2}{\left (a+b x^2\right )^2 \sqrt{e+f x^2}} \, dx}{(b c-a d)^2}+\frac{d^2 \int \frac{1}{\left (c+d x^2\right ) \sqrt{e+f x^2}} \, dx}{(b c-a d)^2}\\ &=\frac{b^2 x \sqrt{e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac{d^2 \operatorname{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)^2}+\frac{b \int \frac{b^2 c e-3 a b d e-2 a b c f+4 a^2 d f}{\left (a+b x^2\right ) \sqrt{e+f x^2}} \, dx}{2 a (b c-a d)^2 (b e-a f)}\\ &=\frac{b^2 x \sqrt{e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac{d^2 \tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}}+\frac{\left (b \left (b^2 c e-3 a b d e-2 a b c f+4 a^2 d f\right )\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{e+f x^2}} \, dx}{2 a (b c-a d)^2 (b e-a f)}\\ &=\frac{b^2 x \sqrt{e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac{d^2 \tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}}+\frac{\left (b \left (b^2 c e-3 a b d e-2 a b c f+4 a^2 d f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-(-b e+a f) x^2} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{2 a (b c-a d)^2 (b e-a f)}\\ &=\frac{b^2 x \sqrt{e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac{b \left (b^2 c e-3 a b d e-2 a b c f+4 a^2 d f\right ) \tan ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{a} \sqrt{e+f x^2}}\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac{d^2 \tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d)^2 \sqrt{d e-c f}}\\ \end{align*}
Mathematica [C] time = 2.5579, size = 531, normalized size = 2.62 \[ \frac{\frac{b x \sqrt{e+f x^2} (b c-a d) \left (-30 f x^2 \sqrt{\frac{a x^2 \left (e+f x^2\right ) (b e-a f)}{e^2 \left (a+b x^2\right )^2}}-45 e \sqrt{\frac{a x^2 \left (e+f x^2\right ) (b e-a f)}{e^2 \left (a+b x^2\right )^2}}+16 f x^2 \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \left (\frac{x^2 (b e-a f)}{e \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,3;\frac{7}{2};\frac{(b e-a f) x^2}{e \left (b x^2+a\right )}\right )+16 e \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \left (\frac{x^2 (b e-a f)}{e \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,3;\frac{7}{2};\frac{(b e-a f) x^2}{e \left (b x^2+a\right )}\right )+30 f x^2 \sin ^{-1}\left (\sqrt{\frac{x^2 (b e-a f)}{e \left (a+b x^2\right )}}\right )+45 e \sin ^{-1}\left (\sqrt{\frac{x^2 (b e-a f)}{e \left (a+b x^2\right )}}\right )\right )}{e^2 \left (a+b x^2\right )^2 \left (\frac{x^2 (b e-a f)}{e \left (a+b x^2\right )}\right )^{3/2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}-\frac{30 b d \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} \sqrt{b e-a f}}+\frac{30 d^2 \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}}}{30 (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.056, size = 1865, normalized size = 9.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )} \sqrt{f x^{2} + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 16.9323, size = 647, normalized size = 3.19 \begin{align*} -\frac{1}{2} \,{\left (\frac{2 \, d^{2} \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 )}{{\left (b^{2} c^{2} f^{2} - 2 \, a b c d f^{2} + a^{2} d^{2} f^{2}\right )} \sqrt{-c^{2} f^{2} + c d f e}} + \frac{{\left (2 \, a b^{2} c f - 4 \, a^{2} b d f - b^{3} c e + 3 \, a b^{2} d e\right )} \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 )}{{\left (a^{2} b^{2} c^{2} f^{3} - 2 \, a^{3} b c d f^{3} + a^{4} d^{2} f^{3} - a b^{3} c^{2} f^{2} e + 2 \, a^{2} b^{2} c d f^{2} e - a^{3} b d^{2} f^{2} e\right )} \sqrt{-a^{2} f^{2} + a b f e}} + \frac{2 \,{\left (2 \,{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} a b f -{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} b^{2} e + b^{2} e^{2}\right )}}{{\left (a^{2} b c f^{3} - a^{3} d f^{3} - a b^{2} c f^{2} e + a^{2} b d f^{2} e\right )}{\left ({\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{4} b + 4 \,{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} a f - 2 \,{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} b e + b e^{2}\right )}}\right )} f^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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