Optimal. Leaf size=274 \[ \frac{2 \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (3 b c-8 a d) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a+b x^2} (b c-a d)^2}+\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{5/2}}-\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{5/2}}+\frac{2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)} \]
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Rubi [A] time = 0.381708, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {414, 527, 530, 229, 227, 196, 399, 490, 1218} \[ \frac{2 \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (3 b c-8 a d) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a+b x^2} (b c-a d)^2}+\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{5/2}}-\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{5/2}}+\frac{2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 414
Rule 527
Rule 530
Rule 229
Rule 227
Rule 196
Rule 399
Rule 490
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx &=\frac{2 b x}{5 a (b c-a d) \left (a+b x^2\right )^{5/4}}-\frac{2 \int \frac{\frac{1}{2} (-3 b c+5 a d)-\frac{3}{2} b d x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx}{5 a (b c-a d)}\\ &=\frac{2 b x}{5 a (b c-a d) \left (a+b x^2\right )^{5/4}}+\frac{2 b (3 b c-8 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^2}}+\frac{4 \int \frac{\frac{1}{4} \left (-3 b^2 c^2+8 a b c d+5 a^2 d^2\right )-\frac{1}{4} b d (3 b c-8 a d) x^2}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{5 a^2 (b c-a d)^2}\\ &=\frac{2 b x}{5 a (b c-a d) \left (a+b x^2\right )^{5/4}}+\frac{2 b (3 b c-8 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^2}}+\frac{d^2 \int \frac{1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{(b c-a d)^2}-\frac{(b (3 b c-8 a d)) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{5 a^2 (b c-a d)^2}\\ &=\frac{2 b x}{5 a (b c-a d) \left (a+b x^2\right )^{5/4}}+\frac{2 b (3 b c-8 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^2}}+\frac{\left (2 d^2 \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^4}{a}} \left (b c-a d+d x^4\right )} \, dx,x,\sqrt [4]{a+b x^2}\right )}{(b c-a d)^2 x}-\frac{\left (b (3 b c-8 a d) \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^2}}\\ &=\frac{2 b x}{5 a (b c-a d) \left (a+b x^2\right )^{5/4}}-\frac{\left (d^{3/2} \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-b c+a d}-\sqrt{d} x^2\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{(b c-a d)^2 x}+\frac{\left (d^{3/2} \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-b c+a d}+\sqrt{d} x^2\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{(b c-a d)^2 x}+\frac{\left (b (3 b c-8 a d) \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^2}}\\ &=\frac{2 b x}{5 a (b c-a d) \left (a+b x^2\right )^{5/4}}+\frac{2 \sqrt{b} (3 b c-8 a d) \sqrt [4]{1+\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} (b c-a d)^2 \sqrt [4]{a+b x^2}}+\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{-b c+a d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{(-b c+a d)^{5/2} x}-\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{-b c+a d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{(-b c+a d)^{5/2} x}\\ \end{align*}
Mathematica [C] time = 0.703583, size = 419, normalized size = 1.53 \[ \frac{x \left (\frac{b d x^2 \sqrt [4]{\frac{b x^2}{a}+1} (8 a d-3 b c) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c}-\frac{6 \left (3 a c \left (-a^2 b d \left (10 c+13 d x^2\right )+5 a^3 d^2+a b^2 \left (5 c^2-16 d^2 x^4\right )+3 b^3 c x^2 \left (c+2 d x^2\right )\right ) F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b x^2 \left (c+d x^2\right ) \left (9 a^2 d-4 a b \left (c-2 d x^2\right )-3 b^2 c x^2\right ) \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}\right )}{15 a^2 \sqrt [4]{a+b x^2} (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \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 )}^{\frac{9}{4}}{\left (d x^{2} + c\right )}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{2}\right )^{\frac{9}{4}} \left (c + d x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}{\left (d x^{2} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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