Optimal. Leaf size=711 \[ -\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {3 b^{4/3} (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {b (b d-4 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{12 a^{3/2}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (5 b c-14 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{560 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.75, antiderivative size = 711, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {14, 1839,
1849, 1846, 272, 65, 214, 1892, 224, 1891} \begin {gather*} -\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right )}{560 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 b c-14 a f) E\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {b (b d-4 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{12 a^{3/2}}-\frac {3 b^{4/3} \sqrt {a+b x^3} (5 b c-14 a f)}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {3 b \sqrt {a+b x^3} (5 b c-14 a f)}{112 a^2 x}-\frac {1}{420} \sqrt {a+b x^3} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 65
Rule 214
Rule 224
Rule 272
Rule 1839
Rule 1846
Rule 1849
Rule 1891
Rule 1892
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {1}{2} (3 b) \int \frac {-\frac {c}{7}-\frac {d x}{6}-\frac {e x^2}{5}-\frac {f x^3}{4}-\frac {g x^4}{3}}{x^5 \sqrt {a+b x^3}} \, dx\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}+\frac {(3 b) \int \frac {\frac {4 a d}{3}+\frac {8 a e x}{5}-\frac {1}{7} (5 b c-14 a f) x^2+\frac {8}{3} a g x^3}{x^4 \sqrt {a+b x^3}} \, dx}{16 a}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {b \int \frac {-\frac {48 a^2 e}{5}+\frac {6}{7} a (5 b c-14 a f) x+4 a (b d-4 a g) x^2}{x^3 \sqrt {a+b x^3}} \, dx}{32 a^2}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {b \int \frac {-\frac {24}{7} a^2 (5 b c-14 a f)-16 a^2 (b d-4 a g) x-\frac {48}{5} a^2 b e x^2}{x^2 \sqrt {a+b x^3}} \, dx}{128 a^3}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {b \int \frac {32 a^3 (b d-4 a g)+\frac {96}{5} a^3 b e x+\frac {24}{7} a^2 b (5 b c-14 a f) x^2}{x \sqrt {a+b x^3}} \, dx}{256 a^4}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {b \int \frac {\frac {96}{5} a^3 b e+\frac {24}{7} a^2 b (5 b c-14 a f) x}{\sqrt {a+b x^3}} \, dx}{256 a^4}-\frac {(b (b d-4 a g)) \int \frac {1}{x \sqrt {a+b x^3}} \, dx}{8 a}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {\left (3 b^{5/3} (5 b c-14 a f)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{224 a^2}-\frac {\left (3 b^{5/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{1120 a^{5/3}}-\frac {(b (b d-4 a g)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{24 a}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {3 b^{4/3} (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (5 b c-14 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{560 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {(b d-4 a g) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{12 a}\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \sqrt {a+b x^3}-\frac {3 b c \sqrt {a+b x^3}}{56 a x^4}-\frac {b d \sqrt {a+b x^3}}{12 a x^3}-\frac {3 b e \sqrt {a+b x^3}}{20 a x^2}+\frac {3 b (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 x}-\frac {3 b^{4/3} (5 b c-14 a f) \sqrt {a+b x^3}}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {b (b d-4 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{12 a^{3/2}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (5 b c-14 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (5 b c-14 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{560 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 11.52, size = 892, normalized size = 1.25 \begin {gather*} \frac {\sqrt {a+b x^3} \left (225 b^2 c x^6-2 a b x^3 (45 c+7 x (10 d+9 x (2 e+5 f x)))-4 a^2 (60 c+7 x (10 d+x (12 e+5 x (3 f+4 g x))))\right )}{1680 a^2 x^7}+\frac {b \left (140 \sqrt {a} b d \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-560 a^{3/2} g \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+252 a b^{2/3} e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )-225 \sqrt {2} \sqrt [3]{a} b^{4/3} c \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}} \left (-\left (\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )-F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )+630 \sqrt {2} a^{4/3} \sqrt [3]{b} f \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}} \left (-\left (\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )-F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )\right )}{1680 a^2 \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1375 vs. \(2 (549 ) = 1098\).
time = 0.42, size = 1376, normalized size = 1.94
method | result | size |
elliptic | \(-\frac {c \sqrt {b \,x^{3}+a}}{7 x^{7}}-\frac {d \sqrt {b \,x^{3}+a}}{6 x^{6}}-\frac {e \sqrt {b \,x^{3}+a}}{5 x^{5}}-\frac {\left (14 a f +3 b c \right ) \sqrt {b \,x^{3}+a}}{56 a \,x^{4}}-\frac {\left (4 a g +b d \right ) \sqrt {b \,x^{3}+a}}{12 a \,x^{3}}-\frac {3 b e \sqrt {b \,x^{3}+a}}{20 a \,x^{2}}-\frac {3 \left (14 a f -5 b c \right ) b \sqrt {b \,x^{3}+a}}{112 a^{2} x}+\frac {i b e \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{20 a \sqrt {b \,x^{3}+a}}-\frac {i \left (14 a f -5 b c \right ) b \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{112 a^{2} \sqrt {b \,x^{3}+a}}-\frac {\left (4 a g -b d \right ) b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{12 a^{\frac {3}{2}}}\) | \(901\) |
risch | \(\text {Expression too large to display}\) | \(1283\) |
default | \(\text {Expression too large to display}\) | \(1376\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.14, size = 439, normalized size = 0.62 \begin {gather*} \left [-\frac {252 \, a b^{\frac {3}{2}} e x^{7} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 35 \, {\left (b^{2} d - 4 \, a b g\right )} \sqrt {a} x^{7} \log \left (\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} \sqrt {b} x^{7} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (252 \, a b e x^{5} - 45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} x^{6} + 336 \, a^{2} e x^{2} + 140 \, {\left (a b d + 4 \, a^{2} g\right )} x^{4} + 280 \, a^{2} d x + 30 \, {\left (3 \, a b c + 14 \, a^{2} f\right )} x^{3} + 240 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{1680 \, a^{2} x^{7}}, -\frac {252 \, a b^{\frac {3}{2}} e x^{7} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 70 \, {\left (b^{2} d - 4 \, a b g\right )} \sqrt {-a} x^{7} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) - 45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} \sqrt {b} x^{7} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (252 \, a b e x^{5} - 45 \, {\left (5 \, b^{2} c - 14 \, a b f\right )} x^{6} + 336 \, a^{2} e x^{2} + 140 \, {\left (a b d + 4 \, a^{2} g\right )} x^{4} + 280 \, a^{2} d x + 30 \, {\left (3 \, a b c + 14 \, a^{2} f\right )} x^{3} + 240 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{1680 \, a^{2} x^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.35, size = 308, normalized size = 0.43 \begin {gather*} \frac {\sqrt {a} c \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {\sqrt {a} e \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {\sqrt {a} f \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} - \frac {a d}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b} d}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b} g \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} d}{12 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b g \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3 \sqrt {a}} + \frac {b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{12 a^{\frac {3}{2}}} \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 {\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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