Integrand size = 25, antiderivative size = 594 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {2 x \left (a d+a e x-b c x^2\right )}{3 b^2 \sqrt {a+b x^3}}+\frac {4 c \sqrt {a+b x^3}}{3 b^2}+\frac {2 d x \sqrt {a+b x^3}}{5 b^2}+\frac {2 e x^2 \sqrt {a+b x^3}}{7 b^2}-\frac {80 a e \sqrt {a+b x^3}}{21 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {40 \sqrt {2-\sqrt {3}} a^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{7\ 3^{3/4} b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {16 \sqrt {2+\sqrt {3}} a \left (14 \sqrt [3]{b} d-25 \left (1-\sqrt {3}\right ) \sqrt [3]{a} e\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{105 \sqrt [4]{3} b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Time = 0.46 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1842, 1902, 1900, 267, 1892, 224, 1891} \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=-\frac {16 \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (14 \sqrt [3]{b} d-25 \left (1-\sqrt {3}\right ) \sqrt [3]{a} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{105 \sqrt [4]{3} b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {40 \sqrt {2-\sqrt {3}} a^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{7\ 3^{3/4} b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {80 a e \sqrt {a+b x^3}}{21 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x \left (a d+a e x-b c x^2\right )}{3 b^2 \sqrt {a+b x^3}}+\frac {4 c \sqrt {a+b x^3}}{3 b^2}+\frac {2 d x \sqrt {a+b x^3}}{5 b^2}+\frac {2 e x^2 \sqrt {a+b x^3}}{7 b^2} \]
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Rule 224
Rule 267
Rule 1842
Rule 1891
Rule 1892
Rule 1900
Rule 1902
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 b^2 \sqrt {a+b x^3}}-\frac {2 \int \frac {a^2 b d+2 a^2 b e x-3 a b^2 c x^2-\frac {3}{2} a b^2 d x^3-\frac {3}{2} a b^2 e x^4}{\sqrt {a+b x^3}} \, dx}{3 a b^3} \\ & = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 b^2 \sqrt {a+b x^3}}+\frac {2 e x^2 \sqrt {a+b x^3}}{7 b^2}-\frac {4 \int \frac {\frac {7}{2} a^2 b^2 d+10 a^2 b^2 e x-\frac {21}{2} a b^3 c x^2-\frac {21}{4} a b^3 d x^3}{\sqrt {a+b x^3}} \, dx}{21 a b^4} \\ & = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 b^2 \sqrt {a+b x^3}}+\frac {2 d x \sqrt {a+b x^3}}{5 b^2}+\frac {2 e x^2 \sqrt {a+b x^3}}{7 b^2}-\frac {8 \int \frac {14 a^2 b^3 d+25 a^2 b^3 e x-\frac {105}{4} a b^4 c x^2}{\sqrt {a+b x^3}} \, dx}{105 a b^5} \\ & = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 b^2 \sqrt {a+b x^3}}+\frac {2 d x \sqrt {a+b x^3}}{5 b^2}+\frac {2 e x^2 \sqrt {a+b x^3}}{7 b^2}-\frac {8 \int \frac {14 a^2 b^3 d+25 a^2 b^3 e x}{\sqrt {a+b x^3}} \, dx}{105 a b^5}+\frac {(2 c) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{b} \\ & = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 b^2 \sqrt {a+b x^3}}+\frac {4 c \sqrt {a+b x^3}}{3 b^2}+\frac {2 d x \sqrt {a+b x^3}}{5 b^2}+\frac {2 e x^2 \sqrt {a+b x^3}}{7 b^2}-\frac {(40 a e) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{21 b^{7/3}}-\frac {\left (8 a \left (14 \sqrt [3]{b} d-25 \left (1-\sqrt {3}\right ) \sqrt [3]{a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{105 b^{7/3}} \\ & = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 b^2 \sqrt {a+b x^3}}+\frac {4 c \sqrt {a+b x^3}}{3 b^2}+\frac {2 d x \sqrt {a+b x^3}}{5 b^2}+\frac {2 e x^2 \sqrt {a+b x^3}}{7 b^2}-\frac {80 a e \sqrt {a+b x^3}}{21 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {40 \sqrt {2-\sqrt {3}} a^{4/3} e \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 )}{7\ 3^{3/4} b^{8/3} \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 {16 \sqrt {2+\sqrt {3}} a \left (14 \sqrt [3]{b} d-25 \left (1-\sqrt {3}\right ) \sqrt [3]{a} e\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 )}{105 \sqrt [4]{3} b^{8/3} \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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.23 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {2 \left (70 a c+56 a d x-150 a e x^2+35 b c x^3+21 b d x^4+15 b e x^5-56 a d x \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )+150 a e x^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{2},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{105 b^2 \sqrt {a+b x^3}} \]
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Time = 2.05 (sec) , antiderivative size = 813, normalized size of antiderivative = 1.37
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(813\) |
default | \(\text {Expression too large to display}\) | \(836\) |
risch | \(\text {Expression too large to display}\) | \(1587\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.23 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 \, {\left (112 \, {\left (a b d x^{3} + a^{2} d\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 200 \, {\left (a b e x^{3} + a^{2} e\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (15 \, b^{2} e x^{5} + 21 \, b^{2} d x^{4} + 35 \, b^{2} c x^{3} + 50 \, a b e x^{2} + 56 \, a b d x + 70 \, a b c\right )} \sqrt {b x^{3} + a}\right )}}{105 \, {\left (b^{4} x^{3} + a b^{3}\right )}} \]
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Time = 6.88 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.22 \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=c \left (\begin {cases} \frac {4 a}{3 b^{2} \sqrt {a + b x^{3}}} + \frac {2 x^{3}}{3 b \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {d x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {10}{3}\right )} + \frac {e x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {11}{3}\right )} \]
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\[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d x + c\right )} x^{5}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d x + c\right )} x^{5}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^5 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^{3/2}} \, dx=\int \frac {x^5\,\left (e\,x^2+d\,x+c\right )}{{\left (b\,x^3+a\right )}^{3/2}} \,d x \]
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