Optimal. Leaf size=254 \[ \frac{2 \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{3 \sqrt{a} \left (a+b x^2\right )^{3/4} (b c-a d)}+\frac{2 b x}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)^2}-\frac{\sqrt [4]{a} d \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)^2} \]
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Rubi [A] time = 0.15701, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {403, 199, 233, 231, 401, 108, 409, 1218} \[ \frac{2 b x}{3 a \left (a+b x^2\right )^{3/4} (b c-a d)}+\frac{2 \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \left (a+b x^2\right )^{3/4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)^2}-\frac{\sqrt [4]{a} d \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 403
Rule 199
Rule 233
Rule 231
Rule 401
Rule 108
Rule 409
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx &=\frac{b \int \frac{1}{\left (a+b x^2\right )^{7/4}} \, dx}{b c-a d}-\frac{d \int \frac{1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{b c-a d}\\ &=\frac{2 b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac{b \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx}{3 a (b c-a d)}-\frac{\left (d \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-\frac{b x}{a}} (a+b x)^{3/4} (c+d x)} \, dx,x,x^2\right )}{2 (b c-a d) x}\\ &=\frac{2 b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac{\left (2 d \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\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) x}+\frac{\left (b \left (1+\frac{b x^2}{a}\right )^{3/4}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx}{3 a (b c-a d) \left (a+b x^2\right )^{3/4}}\\ &=\frac{2 b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac{2 \sqrt{b} \left (1+\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} (b c-a d) \left (a+b x^2\right )^{3/4}}-\frac{\left (d \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{-b c+a d}}\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 \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{-b c+a d}}\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{(b c-a d)^2 x}\\ &=\frac{2 b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac{2 \sqrt{b} \left (1+\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} (b c-a d) \left (a+b x^2\right )^{3/4}}-\frac{\sqrt [4]{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 )}{(b c-a d)^2 x}-\frac{\sqrt [4]{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 )}{(b c-a d)^2 x}\\ \end{align*}
Mathematica [C] time = 0.274444, size = 331, normalized size = 1.3 \[ \frac{x \left (\frac{6 \left (b x^2 \left (c+d x^2\right ) \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+3 a c \left (3 a d-3 b c-2 b d x^2\right ) F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{\left (c+d x^2\right ) \left (6 a c F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}-\frac{b d x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c}\right )}{9 a \left (a+b x^2\right )^{3/4} (a d-b c)} \]
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{7}{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{7}{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{7}{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{7}{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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