Integrand size = 19, antiderivative size = 896 \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx=-\frac {2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac {2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}+\frac {8 d^2 (c+d x)^{5/6}}{27 b (b c-a d)^2 \sqrt {a+b x}}+\frac {8 \left (1+\sqrt {3}\right ) d^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{27 b^{5/3} (b c-a d)^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {8 d^2 \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 )}{9\ 3^{3/4} b^{5/3} (b c-a d)^{5/3} \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 {4 \left (1-\sqrt {3}\right ) d^2 \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 )}{27 \sqrt [4]{3} b^{5/3} (b c-a d)^{5/3} \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 = 0.67 (sec) , antiderivative size = 896, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {49, 53, 65, 314, 231, 1895} \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx=\frac {8 \left (1+\sqrt {3}\right ) \sqrt {a+b x} \sqrt [6]{c+d x} d^3}{27 b^{5/3} (b c-a d)^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {8 \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 ) d^2}{9\ 3^{3/4} b^{5/3} (b c-a d)^{5/3} \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 {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 ) d^2}{27 \sqrt [4]{3} b^{5/3} (b c-a d)^{5/3} \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 {8 (c+d x)^{5/6} d^2}{27 b (b c-a d)^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/6} d}{9 b (b c-a d) (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}} \]
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Rule 49
Rule 53
Rule 65
Rule 231
Rule 314
Rule 1895
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}+\frac {d \int \frac {1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx}{3 b} \\ & = -\frac {2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac {2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}-\frac {\left (4 d^2\right ) \int \frac {1}{(a+b x)^{3/2} \sqrt [6]{c+d x}} \, dx}{27 b (b c-a d)} \\ & = -\frac {2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac {2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}+\frac {8 d^2 (c+d x)^{5/6}}{27 b (b c-a d)^2 \sqrt {a+b x}}-\frac {\left (8 d^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [6]{c+d x}} \, dx}{81 b (b c-a d)^2} \\ & = -\frac {2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac {2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}+\frac {8 d^2 (c+d x)^{5/6}}{27 b (b c-a d)^2 \sqrt {a+b x}}-\frac {\left (16 d^2\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 )}{27 b (b c-a d)^2} \\ & = -\frac {2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac {2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}+\frac {8 d^2 (c+d x)^{5/6}}{27 b (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (8 d^2\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 )}{27 b^{5/3} (b c-a d)^2}+\frac {\left (8 \left (1-\sqrt {3}\right ) d^2\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 )}{27 b^{5/3} (b c-a d)^{4/3}} \\ & = -\frac {2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac {2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}+\frac {8 d^2 (c+d x)^{5/6}}{27 b (b c-a d)^2 \sqrt {a+b x}}+\frac {8 \left (1+\sqrt {3}\right ) d^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{27 b^{5/3} (b c-a d)^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {8 d^2 \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 )}{9\ 3^{3/4} b^{5/3} (b c-a d)^{5/3} \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 {4 \left (1-\sqrt {3}\right ) d^2 \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 )}{27 \sqrt [4]{3} b^{5/3} (b c-a d)^{5/3} \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 \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx=-\frac {2 (c+d x)^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{6},-\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )}{5 b (a+b x)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \]
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\[\int \frac {\left (d x +c \right )^{\frac {5}{6}}}{\left (b x +a \right )^{\frac {7}{2}}}d x\]
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\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{6}}}{\left (a + b x\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/6}}{{\left (a+b\,x\right )}^{7/2}} \,d x \]
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