Optimal. Leaf size=705 \[ -\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{4/3} (b d+14 a g) \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac {b^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (b d+14 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (7 \sqrt [3]{b} (b c-16 a f)+20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d+14 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{2240 a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.66, antiderivative size = 705, normalized size of antiderivative = 1.00, number of steps
used = 11, 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 {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 (7 \sqrt [3]{b} (b c-16 a f)+20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (14 a g+b d)\right )}{2240 a \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 g+b d) 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 g+b d)}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}-\frac {b^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {1}{560} b \sqrt {a+b x^3} \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right )-\frac {1}{840} \left (a+b x^3\right )^{3/2} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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 {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^9} \, dx &=-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac {1}{2} (9 b) \int \frac {\sqrt {a+b x^3} \left (-\frac {c}{8}-\frac {d x}{7}-\frac {e x^2}{6}-\frac {f x^3}{5}-\frac {g x^4}{4}\right )}{x^6} \, dx\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac {1}{4} \left (27 b^2\right ) \int \frac {\frac {c}{40}+\frac {d x}{28}+\frac {e x^2}{18}+\frac {f x^3}{10}+\frac {g x^4}{4}}{x^3 \sqrt {a+b x^3}} \, dx\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac {\left (27 b^2\right ) \int \frac {-\frac {a d}{7}-\frac {2 a e x}{9}+\frac {1}{40} (b c-16 a f) x^2-a g x^3}{x^2 \sqrt {a+b x^3}} \, dx}{16 a}\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac {\left (27 b^2\right ) \int \frac {\frac {4 a^2 e}{9}-\frac {1}{20} a (b c-16 a f) x+\frac {1}{7} a (b d+14 a g) x^2}{x \sqrt {a+b x^3}} \, dx}{32 a^2}\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac {\left (27 b^2\right ) \int \frac {-\frac {1}{20} a (b c-16 a f)+\frac {1}{7} a (b d+14 a g) x}{\sqrt {a+b x^3}} \, dx}{32 a^2}+\frac {1}{8} \left (3 b^2 e\right ) \int \frac {1}{x \sqrt {a+b x^3}} \, dx\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac {1}{8} \left (b^2 e\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )+\frac {\left (27 b^{5/3} (b d+14 a g)\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^2 \left (7 (b c-16 a f)+\frac {20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d+14 a g)}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{4480 a}\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{4/3} (b d+14 a g) \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (b d+14 a g) \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^{5/3} \left (7 (b c-16 a f)+\frac {20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d+14 a g)}{\sqrt [3]{b}}\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 )}{2240 a \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 e) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{4/3} (b d+14 a g) \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac {b^2 e \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 d+14 a g) \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^{5/3} \left (7 (b c-16 a f)+\frac {20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d+14 a g)}{\sqrt [3]{b}}\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 )}{2240 a \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 = 11.74, size = 978, normalized size = 1.39 \begin {gather*} -\frac {\sqrt {a+b x^3} \left (81 b^2 x^6 (7 c+20 d x)+4 a b x^3 \left (399 c+2 x \left (255 d+7 x \left (50 e+78 f x+165 g x^2\right )\right )\right )+8 a^2 (105 c+2 x (60 d+7 x (10 e+3 x (4 f+5 g x))))\right )}{6720 a x^8}-\frac {b^{4/3} \left (560 \sqrt {a} b^{2/3} e \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 )-189 b^{4/3} c \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 )+3024 a \sqrt [3]{b} f \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 )-540 \sqrt {2} \sqrt [3]{a} b d \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 )-7560 \sqrt {2} a^{4/3} g \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 )}{2240 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. 1662 vs. \(2 (549 ) = 1098\).
time = 0.43, size = 1663, normalized size = 2.36
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(949\) |
risch | \(\text {Expression too large to display}\) | \(1579\) |
default | \(\text {Expression too large to display}\) | \(1663\) |
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 = 0.25, size = 470, normalized size = 0.67 \begin {gather*} \left [\frac {420 \, \sqrt {a} b^{2} e x^{8} \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 ) - 567 \, {\left (b^{2} c - 16 \, a b f\right )} \sqrt {b} x^{8} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 1620 \, {\left (b^{2} d + 14 \, a b g\right )} \sqrt {b} x^{8} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (2800 \, a b e x^{5} + 60 \, {\left (27 \, b^{2} d + 154 \, a b g\right )} x^{7} + 21 \, {\left (27 \, b^{2} c + 208 \, a b f\right )} x^{6} + 1120 \, a^{2} e x^{2} + 120 \, {\left (17 \, a b d + 14 \, a^{2} g\right )} x^{4} + 960 \, a^{2} d x + 84 \, {\left (19 \, a b c + 16 \, a^{2} f\right )} x^{3} + 840 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{6720 \, a x^{8}}, \frac {840 \, \sqrt {-a} b^{2} e x^{8} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) - 567 \, {\left (b^{2} c - 16 \, a b f\right )} \sqrt {b} x^{8} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 1620 \, {\left (b^{2} d + 14 \, a b g\right )} \sqrt {b} x^{8} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (2800 \, a b e x^{5} + 60 \, {\left (27 \, b^{2} d + 154 \, a b g\right )} x^{7} + 21 \, {\left (27 \, b^{2} c + 208 \, a b f\right )} x^{6} + 1120 \, a^{2} e x^{2} + 120 \, {\left (17 \, a b d + 14 \, a^{2} g\right )} x^{4} + 960 \, a^{2} d x + 84 \, {\left (19 \, a b c + 16 \, a^{2} f\right )} x^{3} + 840 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{6720 \, a x^{8}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 8.30, size = 527, normalized size = 0.75 \begin {gather*} \frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, - \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{8} \Gamma \left (- \frac {5}{3}\right )} + \frac {a^{\frac {3}{2}} d \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}} f \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}} g \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 {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} b d \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 f \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 g \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 )} - \frac {a^{2} e}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} e}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} e}{12 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} e \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^9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________