Optimal. Leaf size=362 \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} (a d+2 b c) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{4 a^{3/4} c \sqrt{a-b x^4} (b c-a d)^2}-\frac{3 \sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} (3 b c-a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^2}-\frac{3 \sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} (3 b c-a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^2}+\frac{b x (a d+2 b c)}{4 a c \sqrt{a-b x^4} (b c-a d)^2}-\frac{d x}{4 c \sqrt{a-b x^4} \left (c-d x^4\right ) (b c-a d)} \]
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Rubi [A] time = 0.40423, antiderivative size = 362, 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 = {414, 527, 523, 224, 221, 409, 1219, 1218} \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} (a d+2 b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{3/4} c \sqrt{a-b x^4} (b c-a d)^2}-\frac{3 \sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} (3 b c-a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^2}-\frac{3 \sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} (3 b c-a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^2}+\frac{b x (a d+2 b c)}{4 a c \sqrt{a-b x^4} (b c-a d)^2}-\frac{d x}{4 c \sqrt{a-b x^4} \left (c-d x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 414
Rule 527
Rule 523
Rule 224
Rule 221
Rule 409
Rule 1219
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )^2} \, dx &=-\frac{d x}{4 c (b c-a d) \sqrt{a-b x^4} \left (c-d x^4\right )}-\frac{\int \frac{-4 b c+3 a d-5 b d x^4}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=\frac{b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt{a-b x^4}}-\frac{d x}{4 c (b c-a d) \sqrt{a-b x^4} \left (c-d x^4\right )}-\frac{\int \frac{-2 \left (2 b^2 c^2-8 a b c d+3 a^2 d^2\right )+2 b d (2 b c+a d) x^4}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{8 a c (b c-a d)^2}\\ &=\frac{b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt{a-b x^4}}-\frac{d x}{4 c (b c-a d) \sqrt{a-b x^4} \left (c-d x^4\right )}-\frac{(3 d (3 b c-a d)) \int \frac{1}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)^2}+\frac{(b (2 b c+a d)) \int \frac{1}{\sqrt{a-b x^4}} \, dx}{4 a c (b c-a d)^2}\\ &=\frac{b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt{a-b x^4}}-\frac{d x}{4 c (b c-a d) \sqrt{a-b x^4} \left (c-d x^4\right )}-\frac{(3 d (3 b c-a d)) \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{8 c^2 (b c-a d)^2}-\frac{(3 d (3 b c-a d)) \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{8 c^2 (b c-a d)^2}+\frac{\left (b (2 b c+a d) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{4 a c (b c-a d)^2 \sqrt{a-b x^4}}\\ &=\frac{b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt{a-b x^4}}-\frac{d x}{4 c (b c-a d) \sqrt{a-b x^4} \left (c-d x^4\right )}+\frac{b^{3/4} (2 b c+a d) \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 )}{4 a^{3/4} c (b c-a d)^2 \sqrt{a-b x^4}}-\frac{\left (3 d (3 b c-a d) \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}{8 c^2 (b c-a d)^2 \sqrt{a-b x^4}}-\frac{\left (3 d (3 b c-a d) \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}{8 c^2 (b c-a d)^2 \sqrt{a-b x^4}}\\ &=\frac{b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt{a-b x^4}}-\frac{d x}{4 c (b c-a d) \sqrt{a-b x^4} \left (c-d x^4\right )}+\frac{b^{3/4} (2 b c+a d) \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 )}{4 a^{3/4} c (b c-a d)^2 \sqrt{a-b x^4}}-\frac{3 \sqrt [4]{a} d (3 b c-a d) \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 )}{8 \sqrt [4]{b} c^2 (b c-a d)^2 \sqrt{a-b x^4}}-\frac{3 \sqrt [4]{a} d (3 b c-a d) \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 )}{8 \sqrt [4]{b} c^2 (b c-a d)^2 \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.523988, size = 374, normalized size = 1.03 \[ \frac{x \left (\frac{c \left (25 a c \left (4 a^2 d^2-a b d \left (8 c+d x^4\right )+2 b^2 c \left (2 c-d x^4\right )\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )-10 x^4 \left (-a^2 d^2+a b d^2 x^4-2 b^2 c \left (c-d x^4\right )\right ) \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}-b d x^4 \sqrt{1-\frac{b x^4}{a}} (a d+2 b c) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}{20 a c^2 \sqrt{a-b x^4} (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.038, size = 375, normalized size = 1. \begin{align*} -{\frac{{d}^{2}x}{4\, \left ( ad-bc \right ) ^{2}c \left ( d{x}^{4}-c \right ) }\sqrt{-b{x}^{4}+a}}+{\frac{{b}^{2}x}{2\,a \left ( ad-bc \right ) ^{2}}{\frac{1}{\sqrt{- \left ({x}^{4}-{\frac{a}{b}} \right ) b}}}}+{ \left ({\frac{bd}{4\, \left ( ad-bc \right ) ^{2}c}}+{\frac{{b}^{2}}{2\,a \left ( ad-bc \right ) ^{2}}} \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{3}{32\,c}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{ad-3\,bc}{ \left ( ad-bc \right ) ^{2}{{\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{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}{\left (d x^{4} - 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^{4} + a\right )}^{\frac{3}{2}}{\left (d x^{4} - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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