Optimal. Leaf size=741 \[ \frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (8 a f+7 b c)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{280 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (8 a f+7 b c) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{112 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}-\frac {27 \sqrt {a+b x^3} (8 a f+7 b c)}{56 x}+\frac {27 \sqrt [3]{b} \sqrt {a+b x^3} (8 a f+7 b c)}{56 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {27 a c \sqrt {a+b x^3}}{20 x^4}-\frac {1}{3} \sqrt {a} (2 a g+3 b d) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {a d \sqrt {a+b x^3}}{x^3}+\frac {27 a e \sqrt {a+b x^3}}{10 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.24, antiderivative size = 741, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {1826, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ \frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (8 a f+7 b c)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{280 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (8 a f+7 b c) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{112 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}-\frac {27 \sqrt {a+b x^3} (8 a f+7 b c)}{56 x}+\frac {27 \sqrt [3]{b} \sqrt {a+b x^3} (8 a f+7 b c)}{56 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {27 a c \sqrt {a+b x^3}}{20 x^4}-\frac {1}{3} \sqrt {a} (2 a g+3 b d) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {a d \sqrt {a+b x^3}}{x^3}+\frac {27 a e \sqrt {a+b x^3}}{10 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 218
Rule 266
Rule 1826
Rule 1832
Rule 1835
Rule 1877
Rule 1878
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^5} \, dx &=\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}+\frac {1}{2} (9 a) \int \frac {\sqrt {a+b x^3} \left (2 c+\frac {2 d x}{3}+\frac {2 e x^2}{5}+\frac {2 f x^3}{7}+\frac {2 g x^4}{9}\right )}{x^5} \, dx\\ &=-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}+\frac {1}{4} \left (27 a^2\right ) \int \frac {-\frac {4 c}{5}-\frac {4 d x}{9}-\frac {4 e x^2}{5}+\frac {4 f x^3}{7}+\frac {4 g x^4}{27}}{x^5 \sqrt {a+b x^3}} \, dx\\ &=\frac {27 a c \sqrt {a+b x^3}}{20 x^4}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}-\frac {1}{32} (27 a) \int \frac {\frac {32 a d}{9}+\frac {32 a e x}{5}-\frac {4}{7} (7 b c+8 a f) x^2-\frac {32}{27} a g x^3}{x^4 \sqrt {a+b x^3}} \, dx\\ &=\frac {27 a c \sqrt {a+b x^3}}{20 x^4}+\frac {a d \sqrt {a+b x^3}}{x^3}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}+\frac {9}{64} \int \frac {-\frac {192 a^2 e}{5}+\frac {24}{7} a (7 b c+8 a f) x+\frac {32}{9} a (3 b d+2 a g) x^2}{x^3 \sqrt {a+b x^3}} \, dx\\ &=\frac {27 a c \sqrt {a+b x^3}}{20 x^4}+\frac {a d \sqrt {a+b x^3}}{x^3}+\frac {27 a e \sqrt {a+b x^3}}{10 x^2}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}-\frac {9 \int \frac {-\frac {96}{7} a^2 (7 b c+8 a f)-\frac {128}{9} a^2 (3 b d+2 a g) x-\frac {192}{5} a^2 b e x^2}{x^2 \sqrt {a+b x^3}} \, dx}{256 a}\\ &=\frac {27 a c \sqrt {a+b x^3}}{20 x^4}+\frac {a d \sqrt {a+b x^3}}{x^3}+\frac {27 a e \sqrt {a+b x^3}}{10 x^2}-\frac {27 (7 b c+8 a f) \sqrt {a+b x^3}}{56 x}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}+\frac {9 \int \frac {\frac {256}{9} a^3 (3 b d+2 a g)+\frac {384}{5} a^3 b e x+\frac {96}{7} a^2 b (7 b c+8 a f) x^2}{x \sqrt {a+b x^3}} \, dx}{512 a^2}\\ &=\frac {27 a c \sqrt {a+b x^3}}{20 x^4}+\frac {a d \sqrt {a+b x^3}}{x^3}+\frac {27 a e \sqrt {a+b x^3}}{10 x^2}-\frac {27 (7 b c+8 a f) \sqrt {a+b x^3}}{56 x}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}+\frac {9 \int \frac {\frac {384}{5} a^3 b e+\frac {96}{7} a^2 b (7 b c+8 a f) x}{\sqrt {a+b x^3}} \, dx}{512 a^2}+\frac {1}{2} (a (3 b d+2 a g)) \int \frac {1}{x \sqrt {a+b x^3}} \, dx\\ &=\frac {27 a c \sqrt {a+b x^3}}{20 x^4}+\frac {a d \sqrt {a+b x^3}}{x^3}+\frac {27 a e \sqrt {a+b x^3}}{10 x^2}-\frac {27 (7 b c+8 a f) \sqrt {a+b x^3}}{56 x}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}+\frac {1}{112} \left (27 b^{2/3} (7 b c+8 a f)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx+\frac {1}{560} \left (27 \sqrt [3]{a} b^{2/3} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (7 b c+8 a f)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx+\frac {1}{6} (a (3 b d+2 a g)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {27 a c \sqrt {a+b x^3}}{20 x^4}+\frac {a d \sqrt {a+b x^3}}{x^3}+\frac {27 a e \sqrt {a+b x^3}}{10 x^2}-\frac {27 (7 b c+8 a f) \sqrt {a+b x^3}}{56 x}+\frac {27 \sqrt [3]{b} (7 b c+8 a f) \sqrt {a+b x^3}}{56 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} (7 b c+8 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 )}{112 \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}} \sqrt [3]{a} \sqrt [3]{b} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (7 b c+8 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 )}{280 \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 {(a (3 b d+2 a g)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 b}\\ &=\frac {27 a c \sqrt {a+b x^3}}{20 x^4}+\frac {a d \sqrt {a+b x^3}}{x^3}+\frac {27 a e \sqrt {a+b x^3}}{10 x^2}-\frac {27 (7 b c+8 a f) \sqrt {a+b x^3}}{56 x}+\frac {27 \sqrt [3]{b} (7 b c+8 a f) \sqrt {a+b x^3}}{56 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 a \sqrt {a+b x^3} \left (189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right )}{105 x^5}+\frac {2 \left (a+b x^3\right )^{3/2} \left (315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right )}{315 x^5}-\frac {1}{3} \sqrt {a} (3 b d+2 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} (7 b c+8 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 )}{112 \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}} \sqrt [3]{a} \sqrt [3]{b} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (7 b c+8 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 )}{280 \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] time = 0.76, size = 246, normalized size = 0.33 \[ \frac {-45 a^3 c \sqrt {a+b x^3} \, _2F_1\left (-\frac {3}{2},-\frac {4}{3};-\frac {1}{3};-\frac {b x^3}{a}\right )-90 a^3 e x^2 \sqrt {a+b x^3} \, _2F_1\left (-\frac {3}{2},-\frac {2}{3};\frac {1}{3};-\frac {b x^3}{a}\right )+4 x^3 \left (2 x \sqrt {\frac {b x^3}{a}+1} \left (5 a^2 g \left (\sqrt {a+b x^3} \left (4 a+b x^3\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )+3 b d \left (a+b x^3\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {b x^3}{a}+1\right )\right )-45 a^3 f \sqrt {a+b x^3} \, _2F_1\left (-\frac {3}{2},-\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )\right )}{180 a^2 x^4 \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 1342, normalized size = 1.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 14.64, size = 495, normalized size = 0.67 \[ \frac {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 )} - \frac {2 a^{\frac {3}{2}} g \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3} + \frac {\sqrt {a} b c \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 d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )} + \frac {\sqrt {a} b e x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {\sqrt {a} b f x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {2 a^{2} g}{3 \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} d \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} + \frac {2 a \sqrt {b} d}{3 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 a \sqrt {b} g x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 b^{\frac {3}{2}} d x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} + b g \left (\begin {cases} \frac {\sqrt {a} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{9 b} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________