Optimal. Leaf size=399 \[ -\frac {b \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right ) \sqrt {a+b x^4}}{1680}-\frac {b^2 c \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 d \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} d x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-\frac {3 b^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {4 b^{9/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {2 b^{7/4} \left (7 \sqrt {b} d+15 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 a^{3/4} \sqrt {a+b x^4}} \]
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Rubi [A]
time = 0.28, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {14, 1839,
1847, 1266, 821, 272, 65, 214, 1296, 1212, 226, 1210} \begin {gather*} \frac {2 b^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (15 \sqrt {a} f+7 \sqrt {b} d\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 a^{3/4} \sqrt {a+b x^4}}-\frac {4 b^{9/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {4 b^{5/2} d x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {b^2 c \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 d \sqrt {a+b x^4}}{15 a x}-\frac {3 b^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {b \sqrt {a+b x^4} \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right )}{1680}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right )}{2520} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 65
Rule 214
Rule 226
Rule 272
Rule 821
Rule 1210
Rule 1212
Rule 1266
Rule 1296
Rule 1839
Rule 1847
Rubi steps
\begin {align*} \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{11}} \, dx &=-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-(6 b) \int \frac {\left (-\frac {c}{10}-\frac {d x}{9}-\frac {e x^2}{8}-\frac {f x^3}{7}\right ) \sqrt {a+b x^4}}{x^7} \, dx\\ &=-\frac {b \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right ) \sqrt {a+b x^4}}{1680}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}+\left (12 b^2\right ) \int \frac {\frac {c}{60}+\frac {d x}{45}+\frac {e x^2}{32}+\frac {f x^3}{21}}{x^3 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right ) \sqrt {a+b x^4}}{1680}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}+\left (12 b^2\right ) \int \left (\frac {\frac {c}{60}+\frac {e x^2}{32}}{x^3 \sqrt {a+b x^4}}+\frac {\frac {d}{45}+\frac {f x^2}{21}}{x^2 \sqrt {a+b x^4}}\right ) \, dx\\ &=-\frac {b \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right ) \sqrt {a+b x^4}}{1680}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}+\left (12 b^2\right ) \int \frac {\frac {c}{60}+\frac {e x^2}{32}}{x^3 \sqrt {a+b x^4}} \, dx+\left (12 b^2\right ) \int \frac {\frac {d}{45}+\frac {f x^2}{21}}{x^2 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right ) \sqrt {a+b x^4}}{1680}-\frac {4 b^2 d \sqrt {a+b x^4}}{15 a x}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}+\left (6 b^2\right ) \text {Subst}\left (\int \frac {\frac {c}{60}+\frac {e x}{32}}{x^2 \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {\left (12 b^2\right ) \int \frac {-\frac {a f}{21}-\frac {1}{45} b d x^2}{\sqrt {a+b x^4}} \, dx}{a}\\ &=-\frac {b \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right ) \sqrt {a+b x^4}}{1680}-\frac {b^2 c \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 d \sqrt {a+b x^4}}{15 a x}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-\frac {\left (4 b^{5/2} d\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 \sqrt {a}}+\frac {1}{16} \left (3 b^2 e\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )+\frac {1}{105} \left (4 b^2 \left (\frac {7 \sqrt {b} d}{\sqrt {a}}+15 f\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right ) \sqrt {a+b x^4}}{1680}-\frac {b^2 c \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 d \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} d x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-\frac {4 b^{9/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {2 b^{7/4} \left (7 \sqrt {b} d+15 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 a^{3/4} \sqrt {a+b x^4}}+\frac {1}{32} \left (3 b^2 e\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )\\ &=-\frac {b \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right ) \sqrt {a+b x^4}}{1680}-\frac {b^2 c \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 d \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} d x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-\frac {4 b^{9/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {2 b^{7/4} \left (7 \sqrt {b} d+15 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 a^{3/4} \sqrt {a+b x^4}}+\frac {1}{16} (3 b e) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )\\ &=-\frac {b \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right ) \sqrt {a+b x^4}}{1680}-\frac {b^2 c \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 d \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} d x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-\frac {3 b^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {4 b^{9/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {2 b^{7/4} \left (7 \sqrt {b} d+15 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 a^{3/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.43, size = 314, normalized size = 0.79 \begin {gather*} \frac {-\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \left (\left (a+b x^4\right ) \left (168 b^2 x^8 (3 c+8 d x)+a^2 (504 c+10 x (56 d+9 x (7 e+8 f x)))+a b x^4 (1008 c+x (1232 d+45 x (35 e+48 f x)))\right )+945 \sqrt {a} b^2 e x^{10} \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )+1344 \sqrt {a} b^{5/2} d x^{10} \sqrt {1+\frac {b x^4}{a}} E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )-192 i \sqrt {a} b^2 \left (-7 i \sqrt {b} d+15 \sqrt {a} f\right ) x^{10} \sqrt {1+\frac {b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )}{5040 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x^{10} \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.42, size = 357, normalized size = 0.89
method | result | size |
default | \(d \left (-\frac {a \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {11 b \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 i b^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+e \left (-\frac {a \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {5 b \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\right )+f \left (-\frac {a \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {3 b \sqrt {b \,x^{4}+a}}{7 x^{3}}+\frac {4 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {c \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \sqrt {b \,x^{4}+a}}{10 a \,x^{10}}\) | \(357\) |
risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (1344 b^{2} d \,x^{9}+504 b^{2} c \,x^{8}+2160 a b f \,x^{7}+1575 a b e \,x^{6}+1232 a b d \,x^{5}+1008 a b c \,x^{4}+720 a^{2} f \,x^{3}+630 a^{2} e \,x^{2}+560 a^{2} d x +504 a^{2} c \right )}{5040 x^{10} a}+\frac {4 i b^{\frac {5}{2}} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {4 i b^{\frac {5}{2}} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4 b^{2} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\) | \(363\) |
elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{10 x^{10}}-\frac {a d \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {a e \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {a f \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {b c \sqrt {b \,x^{4}+a}}{5 x^{6}}-\frac {11 b d \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {5 b e \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {3 b f \sqrt {b \,x^{4}+a}}{7 x^{3}}-\frac {b^{2} c \sqrt {b \,x^{4}+a}}{10 a \,x^{2}}-\frac {4 b^{2} d \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 b^{2} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4 i b^{\frac {5}{2}} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} e \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{16 \sqrt {a}}\) | \(366\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.12, size = 217, normalized size = 0.54 \begin {gather*} -\frac {2688 \, \sqrt {a} b^{2} d x^{10} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 945 \, \sqrt {a} b^{2} e x^{10} \log \left (-\frac {b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 384 \, {\left (7 \, b^{2} d - 15 \, a b f\right )} \sqrt {a} x^{10} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (1344 \, b^{2} d x^{9} + 504 \, b^{2} c x^{8} + 2160 \, a b f x^{7} + 1575 \, a b e x^{6} + 1232 \, a b d x^{5} + 1008 \, a b c x^{4} + 720 \, a^{2} f x^{3} + 630 \, a^{2} e x^{2} + 560 \, a^{2} d x + 504 \, a^{2} c\right )} \sqrt {b x^{4} + a}}{10080 \, a x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 7.31, size = 398, normalized size = 1.00 \begin {gather*} \frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {\sqrt {a} b d \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {\sqrt {a} b f \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {a^{2} e}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} c \sqrt {\frac {a}{b x^{4}} + 1}}{10 x^{8}} - \frac {3 a \sqrt {b} e}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{4}} + 1}}{5 x^{4}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} - \frac {b^{\frac {3}{2}} e}{16 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {5}{2}} c \sqrt {\frac {a}{b x^{4}} + 1}}{10 a} - \frac {3 b^{2} e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{11}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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