Optimal. Leaf size=309 \[ \frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+b c) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} x \sqrt{a d-b c}}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+b c) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} x \sqrt{a d-b c}}-\frac{b x}{2 c d \sqrt [4]{a+b x^2}}+\frac{x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac{\sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c d \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.222382, antiderivative size = 309, 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 = {412, 530, 229, 227, 196, 399, 490, 1218} \[ \frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+b c) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} x \sqrt{a d-b c}}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (2 a d+b c) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} x \sqrt{a d-b c}}-\frac{b x}{2 c d \sqrt [4]{a+b x^2}}+\frac{x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac{\sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c d \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 412
Rule 530
Rule 229
Rule 227
Rule 196
Rule 399
Rule 490
Rule 1218
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx &=\frac{x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}-\frac{\int \frac{-a+\frac{b x^2}{2}}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{2 c}\\ &=\frac{x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}-\frac{b \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{4 c d}+\frac{(b c+2 a d) \int \frac{1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c d}\\ &=\frac{x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac{\left ((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 d x}-\frac{\left (b \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{4 c d \sqrt [4]{a+b x^2}}\\ &=-\frac{b x}{2 c d \sqrt [4]{a+b x^2}}+\frac{x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}-\frac{\left ((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 d^{3/2} x}+\frac{\left ((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 d^{3/2} x}+\frac{\left (b \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{4 c d \sqrt [4]{a+b x^2}}\\ &=-\frac{b x}{2 c d \sqrt [4]{a+b x^2}}+\frac{x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac{\sqrt{a} \sqrt{b} \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 )}{2 c d \sqrt [4]{a+b x^2}}+\frac{\sqrt [4]{a} (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 d^{3/2} \sqrt{-b c+a d} x}-\frac{\sqrt [4]{a} (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 d^{3/2} \sqrt{-b c+a d} x}\\ \end{align*}
Mathematica [C] time = 0.202731, size = 232, normalized size = 0.75 \[ \frac{x \left (\frac{6 \left (\frac{a+b x^2}{c}-\frac{6 a^2 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}\right )}{c+d x^2}-\frac{b x^2 \sqrt [4]{\frac{b x^2}{a}+1} F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c^2}\right )}{12 \sqrt [4]{a+b x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.046, 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{{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right )^{\frac{3}{4}}}{\left (c + d x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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