Optimal. Leaf size=302 \[ \frac{2 a^{3/2} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{3 d \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} (b c-a d) \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^2 x}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^2 x}+\frac{2 b x \sqrt [4]{a+b x^2}}{3 d} \]
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Rubi [A] time = 0.206693, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {402, 195, 233, 231, 401, 108, 409, 1218} \[ \frac{2 a^{3/2} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 d \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} (b c-a d) F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^2 x}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^2 x}+\frac{2 b x \sqrt [4]{a+b x^2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 402
Rule 195
Rule 233
Rule 231
Rule 401
Rule 108
Rule 409
Rule 1218
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx &=\frac{b \int \sqrt [4]{a+b x^2} \, dx}{d}-\frac{(b c-a d) \int \frac{\sqrt [4]{a+b x^2}}{c+d x^2} \, dx}{d}\\ &=\frac{2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac{(a b) \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx}{3 d}-\frac{(b (b c-a d)) \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx}{d^2}+\frac{(b c-a d)^2 \int \frac{1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{d^2}\\ &=\frac{2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac{\left ((b c-a d)^2 \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 d^2 x}+\frac{\left (a 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 d \left (a+b x^2\right )^{3/4}}-\frac{\left (b (b c-a d) \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}{d^2 \left (a+b x^2\right )^{3/4}}\\ &=\frac{2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac{2 a^{3/2} \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 d \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt{a} \sqrt{b} (b c-a d) \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 )}{d^2 \left (a+b x^2\right )^{3/4}}-\frac{\left (2 (b c-a d)^2 \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 )}{d^2 x}\\ &=\frac{2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac{2 a^{3/2} \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 d \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt{a} \sqrt{b} (b c-a d) \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 )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac{\left ((b c-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 )}{d^2 x}+\frac{\left ((b c-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 )}{d^2 x}\\ &=\frac{2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac{2 a^{3/2} \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 d \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt{a} \sqrt{b} (b c-a d) \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 )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac{\sqrt [4]{a} (b c-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 )}{d^2 x}+\frac{\sqrt [4]{a} (b c-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 )}{d^2 x}\\ \end{align*}
Mathematica [C] time = 0.480034, size = 348, normalized size = 1.15 \[ \frac{x \left (\frac{6 \left (b x^2 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-3 a c \left (3 a^2 d+2 a b d x^2+2 b^2 x^2 \left (c+d x^2\right )\right ) F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{\left (c+d x^2\right ) \left (x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}+\frac{b x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} (4 a d-3 b c) F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c}\right )}{9 d \left (a+b x^2\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{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{5}{4}}}{d x^{2} + c}\,{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{5}{4}}}{c + d x^{2}}\, 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{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}{d x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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