Integrand size = 19, antiderivative size = 890 \[ \int \frac {(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx=\frac {81 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 d^3}-\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}+\frac {243 \left (1+\sqrt {3}\right ) (b c-a d)^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {243 \sqrt [4]{3} (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{448 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {81\ 3^{3/4} \left (1-\sqrt {3}\right ) (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{896 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]
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Time = 0.71 (sec) , antiderivative size = 890, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {52, 65, 314, 231, 1895} \[ \int \frac {(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx=\frac {243 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (b c-a d)^{10/3}}{448 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {81\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (b c-a d)^{10/3}}{896 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {243 \left (1+\sqrt {3}\right ) \sqrt {a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {81 \sqrt {a+b x} (c+d x)^{5/6} (b c-a d)^2}{224 d^3}-\frac {9 (a+b x)^{3/2} (c+d x)^{5/6} (b c-a d)}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d} \]
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Rule 52
Rule 65
Rule 231
Rule 314
Rule 1895
Rubi steps \begin{align*} \text {integral}& = \frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}-\frac {(3 (b c-a d)) \int \frac {(a+b x)^{3/2}}{\sqrt [6]{c+d x}} \, dx}{4 d} \\ & = -\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}+\frac {\left (27 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt [6]{c+d x}} \, dx}{56 d^2} \\ & = \frac {81 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 d^3}-\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}-\frac {\left (81 (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [6]{c+d x}} \, dx}{448 d^3} \\ & = \frac {81 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 d^3}-\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}-\frac {\left (243 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{224 d^4} \\ & = \frac {81 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 d^3}-\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}+\frac {\left (243 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\left (-1+\sqrt {3}\right ) (b c-a d)^{2/3}-2 b^{2/3} x^4}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{448 b^{2/3} d^4}+\frac {\left (243 \left (1-\sqrt {3}\right ) (b c-a d)^{11/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{448 b^{2/3} d^4} \\ & = \frac {81 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 d^3}-\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}+\frac {243 \left (1+\sqrt {3}\right ) (b c-a d)^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {243 \sqrt [4]{3} (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{448 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {81\ 3^{3/4} \left (1-\sqrt {3}\right ) (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{896 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.08 \[ \int \frac {(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx=\frac {2 (a+b x)^{7/2} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {7}{2},\frac {9}{2},\frac {d (a+b x)}{-b c+a d}\right )}{7 b \sqrt [6]{c+d x}} \]
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\[\int \frac {\left (b x +a \right )^{\frac {5}{2}}}{\left (d x +c \right )^{\frac {1}{6}}}d x\]
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\[ \int \frac {(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{2}}}{{\left (d x + c\right )}^{\frac {1}{6}}} \,d x } \]
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\[ \int \frac {(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{\sqrt [6]{c + d x}}\, dx \]
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\[ \int \frac {(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{2}}}{{\left (d x + c\right )}^{\frac {1}{6}}} \,d x } \]
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\[ \int \frac {(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{2}}}{{\left (d x + c\right )}^{\frac {1}{6}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{1/6}} \,d x \]
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