Optimal. Leaf size=746 \[ \frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{4/3} (b c+14 a f) \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {b (b d+4 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (b c+14 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (b c+14 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{560 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.84, antiderivative size = 746, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {14, 1839,
1840, 1849, 1846, 272, 65, 214, 1892, 224, 1891} \begin {gather*} \frac {9\ 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 ) (14 a f+b c)\right )}{560 a^{2/3} \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 {27 \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}} (14 a f+b c) 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^{2/3} \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 {27 b^{4/3} \sqrt {a+b x^3} (14 a f+b c)}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {1}{420} \left (a+b x^3\right )^{3/2} \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 {27 b \sqrt {a+b x^3} (14 a f+b c)}{112 a x}+\frac {27 b c \sqrt {a+b x^3}}{280 x^4}-\frac {b (4 a g+b d) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 65
Rule 214
Rule 224
Rule 272
Rule 1839
Rule 1840
Rule 1846
Rule 1849
Rule 1891
Rule 1892
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{3/2} \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 ) \left (a+b x^3\right )^{3/2}-\frac {1}{2} (9 b) \int \frac {\sqrt {a+b x^3} \left (-\frac {c}{7}-\frac {d x}{6}-\frac {e x^2}{5}-\frac {f x^3}{4}-\frac {g x^4}{3}\right )}{x^5} \, 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 ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {1}{4} (27 a b) \int \frac {\frac {2 c}{35}+\frac {d x}{9}+\frac {2 e x^2}{5}-\frac {f x^3}{2}-\frac {2 g x^4}{9}}{x^5 \sqrt {a+b x^3}} \, dx\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^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 ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}+\frac {1}{32} (27 b) \int \frac {-\frac {8 a d}{9}-\frac {16 a e x}{5}+\frac {2}{7} (b c+14 a f) x^2+\frac {16}{9} a g x^3}{x^4 \sqrt {a+b x^3}} \, dx\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^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 ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {(9 b) \int \frac {\frac {96 a^2 e}{5}-\frac {12}{7} a (b c+14 a f) x-\frac {8}{3} a (b d+4 a g) x^2}{x^3 \sqrt {a+b x^3}} \, dx}{64 a}\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^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 ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}+\frac {(9 b) \int \frac {\frac {48}{7} a^2 (b c+14 a f)+\frac {32}{3} a^2 (b d+4 a g) x+\frac {96}{5} a^2 b e x^2}{x^2 \sqrt {a+b x^3}} \, dx}{256 a^2}\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}-\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 ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {(9 b) \int \frac {-\frac {64}{3} a^3 (b d+4 a g)-\frac {192}{5} a^3 b e x-\frac {48}{7} a^2 b (b c+14 a f) x^2}{x \sqrt {a+b x^3}} \, dx}{512 a^3}\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}-\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 ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {(9 b) \int \frac {-\frac {192}{5} a^3 b e-\frac {48}{7} a^2 b (b c+14 a f) x}{\sqrt {a+b x^3}} \, dx}{512 a^3}+\frac {1}{8} (3 b (b d+4 a g)) \int \frac {1}{x \sqrt {a+b x^3}} \, dx\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}-\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 ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}+\frac {\left (27 b^{5/3} (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}+\frac {\left (27 b^{5/3} \left (28 \sqrt [3]{b} e-\frac {5 \left (1-\sqrt {3}\right ) (b c+14 a f)}{a^{2/3}}\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{1120}+\frac {1}{8} (b (b d+4 a g)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{4/3} (b c+14 a f) \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\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 ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (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^{2/3} \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 {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (28 \sqrt [3]{b} e-\frac {5 \left (1-\sqrt {3}\right ) (b c+14 a f)}{a^{2/3}}\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 \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 {1}{4} (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 )\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{4/3} (b c+14 a f) \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\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 ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {b (b d+4 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (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^{2/3} \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 {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (28 \sqrt [3]{b} e-\frac {5 \left (1-\sqrt {3}\right ) (b c+14 a f)}{a^{2/3}}\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 \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*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 11.51, size = 897, normalized size = 1.20 \begin {gather*} -\frac {\sqrt {a+b x^3} \left (405 b^2 c x^6+2 a b x^3 \left (255 c+7 x \left (50 d+x \left (78 e+165 f x-80 g x^2\right )\right )\right )+4 a^2 (60 c+7 x (10 d+x (12 e+5 x (3 f+4 g x))))\right )}{1680 a 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 )+756 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 )-135 \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 )-1890 \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 )}{560 a \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.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1374 vs. \(2 (580 ) = 1160\).
time = 0.45, size = 1375, normalized size = 1.84
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(916\) |
risch | \(\text {Expression too large to display}\) | \(1295\) |
default | \(\text {Expression too large to display}\) | \(1375\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.39, size = 446, normalized size = 0.60 \begin {gather*} \left [\frac {2268 \, a b^{\frac {3}{2}} e x^{7} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 105 \, {\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 ) - 405 \, {\left (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 (1120 \, a b g x^{7} - 1092 \, a b e x^{5} - 15 \, {\left (27 \, b^{2} c + 154 \, a b f\right )} x^{6} - 336 \, a^{2} e x^{2} - 140 \, {\left (5 \, a b d + 4 \, a^{2} g\right )} x^{4} - 280 \, a^{2} d x - 30 \, {\left (17 \, a b c + 14 \, a^{2} f\right )} x^{3} - 240 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{1680 \, a x^{7}}, \frac {2268 \, a b^{\frac {3}{2}} e x^{7} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 210 \, {\left (b^{2} d + 4 \, a b g\right )} \sqrt {-a} x^{7} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) - 405 \, {\left (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 (1120 \, a b g x^{7} - 1092 \, a b e x^{5} - 15 \, {\left (27 \, b^{2} c + 154 \, a b f\right )} x^{6} - 336 \, a^{2} e x^{2} - 140 \, {\left (5 \, a b d + 4 \, a^{2} g\right )} x^{4} - 280 \, a^{2} d x - 30 \, {\left (17 \, a b c + 14 \, a^{2} f\right )} x^{3} - 240 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{1680 \, a x^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 8.98, size = 536, normalized size = 0.72 \begin {gather*} \frac {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 {\sqrt {a} b c \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 {\sqrt {a} b e \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {\sqrt {a} b f \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \sqrt {a} b g \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )} - \frac {a^{2} d}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} d}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} g \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} + \frac {2 a \sqrt {b} g}{3 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} d}{12 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 b^{\frac {3}{2}} g x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\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.
[In]
[Out]
________________________________________________________________________________________