Optimal. Leaf size=292 \[ -\frac{\sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{2 c \left (a+b x^2\right )^{3/4} (b c-a d)}-\frac{d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2} \]
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Rubi [A] time = 0.23078, antiderivative size = 292, 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 = {414, 530, 233, 231, 401, 108, 409, 1218} \[ -\frac{d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c \left (a+b x^2\right )^{3/4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 414
Rule 530
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 )^{3/4} \left (c+d x^2\right )^2} \, dx &=-\frac{d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{\int \frac{2 b c-a d-\frac{1}{2} b d x^2}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=-\frac{d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac{b \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx}{4 c (b c-a d)}+\frac{(5 b c-2 a d) \int \frac{1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)}\\ &=-\frac{d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{\left ((5 b c-2 a 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 )}{8 c (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}{4 c (b c-a d) \left (a+b x^2\right )^{3/4}}\\ &=-\frac{d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac{\sqrt{a} \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 )}{2 c (b c-a d) \left (a+b x^2\right )^{3/4}}-\frac{\left ((5 b c-2 a 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 )}{2 c (b c-a d) x}\\ &=-\frac{d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac{\sqrt{a} \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 )}{2 c (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac{\left ((5 b c-2 a 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 )}{4 c (b c-a d)^2 x}+\frac{\left ((5 b c-2 a 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 )}{4 c (b c-a d)^2 x}\\ &=-\frac{d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac{\sqrt{a} \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 )}{2 c (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac{\sqrt [4]{a} (5 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)^2 x}+\frac{\sqrt [4]{a} (5 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)^2 x}\\ \end{align*}
Mathematica [C] time = 0.345522, size = 336, normalized size = 1.15 \[ \frac{x \left (\frac{b d x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{a d-b c}+\frac{c \left (36 a c \left (2 a d-2 b c+b d x^2\right ) F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-6 d x^2 \left (a+b x^2\right ) \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) (b c-a d) \left (x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}\right )}{12 c^2 \left (a+b x^2\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{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{3}{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{3}{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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