Optimal. Leaf size=321 \[ \frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (47 a^2 d^2-56 a b c d+21 b^2 c^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{21 d^3 \sqrt{a-b x^4}}-\frac{b x \sqrt{a-b x^4} (7 b c-13 a d)}{21 d^2}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^3 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^3 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt{a-b x^4}}+\frac{b x \left (a-b x^4\right )^{3/2}}{7 d} \]
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Rubi [A] time = 0.382883, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {416, 528, 523, 224, 221, 409, 1219, 1218} \[ \frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (47 a^2 d^2-56 a b c d+21 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 d^3 \sqrt{a-b x^4}}-\frac{b x \sqrt{a-b x^4} (7 b c-13 a d)}{21 d^2}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^3 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^3 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt{a-b x^4}}+\frac{b x \left (a-b x^4\right )^{3/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 416
Rule 528
Rule 523
Rule 224
Rule 221
Rule 409
Rule 1219
Rule 1218
Rubi steps
\begin{align*} \int \frac{\left (a-b x^4\right )^{5/2}}{c-d x^4} \, dx &=\frac{b x \left (a-b x^4\right )^{3/2}}{7 d}-\frac{\int \frac{\sqrt{a-b x^4} \left (a (b c-7 a d)-b (7 b c-13 a d) x^4\right )}{c-d x^4} \, dx}{7 d}\\ &=-\frac{b (7 b c-13 a d) x \sqrt{a-b x^4}}{21 d^2}+\frac{b x \left (a-b x^4\right )^{3/2}}{7 d}+\frac{\int \frac{a \left (7 b^2 c^2-16 a b c d+21 a^2 d^2\right )-b \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right ) x^4}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{21 d^2}\\ &=-\frac{b (7 b c-13 a d) x \sqrt{a-b x^4}}{21 d^2}+\frac{b x \left (a-b x^4\right )^{3/2}}{7 d}-\frac{(b c-a d)^3 \int \frac{1}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{d^3}+\frac{\left (b \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a-b x^4}} \, dx}{21 d^3}\\ &=-\frac{b (7 b c-13 a d) x \sqrt{a-b x^4}}{21 d^2}+\frac{b x \left (a-b x^4\right )^{3/2}}{7 d}-\frac{(b c-a d)^3 \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{2 c d^3}-\frac{(b c-a d)^3 \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{2 c d^3}+\frac{\left (b \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right ) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{21 d^3 \sqrt{a-b x^4}}\\ &=-\frac{b (7 b c-13 a d) x \sqrt{a-b x^4}}{21 d^2}+\frac{b x \left (a-b x^4\right )^{3/2}}{7 d}+\frac{\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right ) \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 d^3 \sqrt{a-b x^4}}-\frac{\left ((b c-a d)^3 \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{1-\frac{b x^4}{a}}} \, dx}{2 c d^3 \sqrt{a-b x^4}}-\frac{\left ((b c-a d)^3 \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{1-\frac{b x^4}{a}}} \, dx}{2 c d^3 \sqrt{a-b x^4}}\\ &=-\frac{b (7 b c-13 a d) x \sqrt{a-b x^4}}{21 d^2}+\frac{b x \left (a-b x^4\right )^{3/2}}{7 d}+\frac{\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right ) \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 d^3 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} (b c-a d)^3 \sqrt{1-\frac{b x^4}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} (b c-a d)^3 \sqrt{1-\frac{b x^4}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.715833, size = 290, normalized size = 0.9 \[ \frac{x \left (-\frac{b x^4 \sqrt{1-\frac{b x^4}{a}} \left (47 a^2 d^2-56 a b c d+21 b^2 c^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{c}+\frac{25 a^2 c \left (21 a^2 d^2-16 a b c d+7 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}+5 b \left (b x^4-a\right ) \left (-16 a d+7 b c+3 b d x^4\right )\right )}{105 d^2 \sqrt{a-b x^4}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.051, size = 408, normalized size = 1.3 \begin{align*} -{\frac{{b}^{2}{x}^{5}}{7\,d}\sqrt{-b{x}^{4}+a}}+{\frac{x}{3\,b} \left ({\frac{{b}^{2} \left ( 3\,ad-bc \right ) }{{d}^{2}}}-{\frac{5\,{b}^{2}a}{7\,d}} \right ) \sqrt{-b{x}^{4}+a}}-{ \left ( -{\frac{b \left ( 3\,{a}^{2}{d}^{2}-3\,cabd+{b}^{2}{c}^{2} \right ) }{{d}^{3}}}+{\frac{a}{3\,b} \left ({\frac{{b}^{2} \left ( 3\,ad-bc \right ) }{{d}^{2}}}-{\frac{5\,{b}^{2}a}{7\,d}} \right ) } \right ) \sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}+{\frac{1}{8\,{d}^{4}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{4}-c \right ) }{\frac{-{a}^{3}{d}^{3}+3\,cb{a}^{2}{d}^{2}-3\,a{b}^{2}{c}^{2}d+{b}^{3}{c}^{3}}{{{\it \_alpha}}^{3}} \left ( -{{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{{x}^{2}\sqrt{b}}{\sqrt{a}}}}\sqrt{1+{\frac{{x}^{2}\sqrt{b}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \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^{4} + a\right )}^{\frac{5}{2}}}{d x^{4} - 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{a^{2} \sqrt{a - b x^{4}}}{- c + d x^{4}}\, dx - \int \frac{b^{2} x^{8} \sqrt{a - b x^{4}}}{- c + d x^{4}}\, dx - \int - \frac{2 a b x^{4} \sqrt{a - b x^{4}}}{- c + d x^{4}}\, 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^{4} + a\right )}^{\frac{5}{2}}}{d x^{4} - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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