Integrand size = 32, antiderivative size = 585 \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {810 a^3 d \sqrt {a+b x^3}}{1729 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {54 a^2 \left (1729 c x+935 d x^2\right ) \sqrt {a+b x^3}}{323323}+\frac {30 a \left (247 c x+187 d x^2\right ) \left (a+b x^3\right )^{3/2}}{46189}+\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}-\frac {405 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1729 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (1729 \sqrt [3]{b} c-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{323323 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Time = 0.31 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1866, 1867, 1892, 224, 1891} \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=-\frac {405 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {810 a^3 d \sqrt {a+b x^3}}{1729 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {54 a^2 \sqrt {a+b x^3} \left (1729 c x+935 d x^2\right )}{323323}+\frac {54\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (1729 \sqrt [3]{b} c-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{323323 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {30 a \left (a+b x^3\right )^{3/2} \left (247 c x+187 d x^2\right )}{46189}+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right ) \]
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Rule 224
Rule 1866
Rule 1867
Rule 1891
Rule 1892
Rubi steps \begin{align*} \text {integral}& = \int (c+d x) \left (a+b x^3\right )^{5/2} \, dx \\ & = \frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}+\frac {1}{2} (15 a) \int \left (\frac {2 c}{17}+\frac {2 d x}{19}\right ) \left (a+b x^3\right )^{3/2} \, dx \\ & = \frac {30 a \left (247 c x+187 d x^2\right ) \left (a+b x^3\right )^{3/2}}{46189}+\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}+\frac {1}{4} \left (135 a^2\right ) \int \left (\frac {4 c}{187}+\frac {4 d x}{247}\right ) \sqrt {a+b x^3} \, dx \\ & = \frac {54 a^2 \left (1729 c x+935 d x^2\right ) \sqrt {a+b x^3}}{323323}+\frac {30 a \left (247 c x+187 d x^2\right ) \left (a+b x^3\right )^{3/2}}{46189}+\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}+\frac {1}{8} \left (405 a^3\right ) \int \frac {\frac {8 c}{935}+\frac {8 d x}{1729}}{\sqrt {a+b x^3}} \, dx \\ & = \frac {54 a^2 \left (1729 c x+935 d x^2\right ) \sqrt {a+b x^3}}{323323}+\frac {30 a \left (247 c x+187 d x^2\right ) \left (a+b x^3\right )^{3/2}}{46189}+\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}+\frac {\left (405 a^3 d\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{1729 \sqrt [3]{b}}+\frac {\left (81 a^3 \left (1729 c-\frac {935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{323323} \\ & = \frac {810 a^3 d \sqrt {a+b x^3}}{1729 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {54 a^2 \left (1729 c x+935 d x^2\right ) \sqrt {a+b x^3}}{323323}+\frac {30 a \left (247 c x+187 d x^2\right ) \left (a+b x^3\right )^{3/2}}{46189}+\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}-\frac {405 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} d \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 )}{1729 b^{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 {54\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (1729 \sqrt [3]{b} c-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\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 )}{323323 b^{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}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.59 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.13 \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {a^2 x \sqrt {a+b x^3} \left (2 c \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+d x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{2 \sqrt {1+\frac {b x^3}{a}}} \]
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Time = 1.72 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.34
method | result | size |
risch | \(\text {Expression too large to display}\) | \(786\) |
elliptic | \(\text {Expression too large to display}\) | \(830\) |
default | \(\text {Expression too large to display}\) | \(1618\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.20 \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {2 \, {\left (140049 \, a^{3} \sqrt {b} c {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 75735 \, a^{3} \sqrt {b} d {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (17017 \, b^{3} d x^{8} + 19019 \, b^{3} c x^{7} + 53669 \, a b^{2} d x^{5} + 63973 \, a b^{2} c x^{4} + 61897 \, a^{2} b d x^{2} + 91637 \, a^{2} b c x\right )} \sqrt {b x^{3} + a}\right )}}{323323 \, b} \]
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Time = 2.64 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.45 \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {a^{\frac {5}{2}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {5}{2}} d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {2 a^{\frac {3}{2}} b c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {2 a^{\frac {3}{2}} b d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} b^{2} c x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt {a} b^{2} d x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \]
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\[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\int { {\left (b d x^{4} + b c x^{3} + a d x + a c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\int { {\left (b d x^{4} + b c x^{3} + a d x + a c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\int {\left (b\,x^3+a\right )}^{3/2}\,\left (b\,d\,x^4+b\,c\,x^3+a\,d\,x+a\,c\right ) \,d x \]
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