Optimal. Leaf size=371 \[ \frac{\sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} \left (-5 a^2 d^2-52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{10 a^{3/2} c \sqrt [4]{a+b x^2} (b c-a d)^3}-\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} (11 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{7/2}}+\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} (11 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{7/2}}-\frac{d x}{2 c \left (a+b x^2\right )^{5/4} \left (c+d x^2\right ) (b c-a d)}+\frac{b x (5 a d+4 b c)}{10 a c \left (a+b x^2\right )^{5/4} (b c-a d)^2} \]
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Rubi [A] time = 0.551699, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{\sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} \left (-5 a^2 d^2-52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{10 a^{3/2} c \sqrt [4]{a+b x^2} (b c-a d)^3}-\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} (11 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{7/2}}+\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} (11 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{7/2}}-\frac{d x}{2 c \left (a+b x^2\right )^{5/4} \left (c+d x^2\right ) (b c-a d)}+\frac{b x (5 a d+4 b c)}{10 a c \left (a+b x^2\right )^{5/4} (b c-a d)^2} \]
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 )^2} \, dx &=-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac{\int \frac{2 b c-a d-\frac{7}{2} b d x^2}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=\frac{b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}-\frac{\int \frac{\frac{1}{2} \left (-6 b^2 c^2+20 a b c d-5 a^2 d^2\right )-\frac{3}{4} b d (4 b c+5 a d) x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx}{5 a c (b c-a d)^2}\\ &=\frac{b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}+\frac{b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x}{10 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac{2 \int \frac{\frac{1}{4} \left (-6 b^3 c^3+26 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right )-\frac{1}{8} b d \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x^2}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{5 a^2 c (b c-a d)^3}\\ &=\frac{b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}+\frac{b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x}{10 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac{\left (d^2 (11 b c-2 a d)\right ) \int \frac{1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)^3}-\frac{\left (b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right )\right ) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{20 a^2 c (b c-a d)^3}\\ &=\frac{b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}+\frac{b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x}{10 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac{\left (d^2 (11 b c-2 a d) \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 )}{2 c (b c-a d)^3 x}-\frac{\left (b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}\\ &=\frac{b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}-\frac{\left (d^{3/2} (11 b c-2 a d) \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 )}{4 c (b c-a d)^3 x}+\frac{\left (d^{3/2} (11 b c-2 a d) \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 )}{4 c (b c-a d)^3 x}+\frac{\left (b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}\\ &=\frac{b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac{\sqrt{b} \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \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 )}{10 a^{3/2} c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac{\sqrt [4]{a} d^{3/2} (11 b c-2 a d) \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 )}{4 c (-b c+a d)^{7/2} x}+\frac{\sqrt [4]{a} d^{3/2} (11 b c-2 a d) \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 )}{4 c (-b c+a d)^{7/2} x}\\ \end{align*}
Mathematica [C] time = 1.09193, size = 536, normalized size = 1.44 \[ \frac{b d x^3 \sqrt [4]{\frac{b x^2}{a}+1} \left (5 a^2 d^2+52 a b c d-12 b^2 c^2\right ) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+\frac{6 c \left (x^3 \left (a^2 b^2 d \left (56 c^2+56 c d x^2+5 d^2 x^4\right )+10 a^3 b d^3 x^2+5 a^4 d^3+4 a b^3 c \left (-4 c^2+9 c d x^2+13 d^2 x^4\right )-12 b^4 c^2 x^2 \left (c+d x^2\right )\right ) \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c x \left (a^2 b^2 d \left (30 c^2+26 c d x^2+5 d^2 x^4\right )+15 a^3 b d^2 \left (d x^2-2 c\right )+10 a^4 d^3+2 a b^3 c \left (-5 c^2+5 c d x^2+26 d^2 x^4\right )-6 b^4 c^2 x^2 \left (c+2 d x^2\right )\right ) F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}}{60 a^2 c^2 \sqrt [4]{a+b x^2} (b c-a d)^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \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 )}^{2}}\,{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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