Integrand size = 19, antiderivative size = 457 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{7/2}} \, dx=-\frac {2 \sqrt [3]{c+d x}}{5 b (a+b x)^{5/2}}-\frac {4 d \sqrt [3]{c+d x}}{45 b (b c-a d) (a+b x)^{3/2}}+\frac {28 d^2 \sqrt [3]{c+d x}}{135 b (b c-a d)^2 \sqrt {a+b x}}-\frac {28 \sqrt {2-\sqrt {3}} d^2 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{135 \sqrt [4]{3} b^{4/3} (b c-a d)^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]
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Time = 0.25 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {49, 53, 65, 225} \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{7/2}} \, dx=-\frac {28 \sqrt {2-\sqrt {3}} d^2 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{135 \sqrt [4]{3} b^{4/3} \sqrt {a+b x} (b c-a d)^2 \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {28 d^2 \sqrt [3]{c+d x}}{135 b \sqrt {a+b x} (b c-a d)^2}-\frac {4 d \sqrt [3]{c+d x}}{45 b (a+b x)^{3/2} (b c-a d)}-\frac {2 \sqrt [3]{c+d x}}{5 b (a+b x)^{5/2}} \]
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Rule 49
Rule 53
Rule 65
Rule 225
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt [3]{c+d x}}{5 b (a+b x)^{5/2}}+\frac {(2 d) \int \frac {1}{(a+b x)^{5/2} (c+d x)^{2/3}} \, dx}{15 b} \\ & = -\frac {2 \sqrt [3]{c+d x}}{5 b (a+b x)^{5/2}}-\frac {4 d \sqrt [3]{c+d x}}{45 b (b c-a d) (a+b x)^{3/2}}-\frac {\left (14 d^2\right ) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{2/3}} \, dx}{135 b (b c-a d)} \\ & = -\frac {2 \sqrt [3]{c+d x}}{5 b (a+b x)^{5/2}}-\frac {4 d \sqrt [3]{c+d x}}{45 b (b c-a d) (a+b x)^{3/2}}+\frac {28 d^2 \sqrt [3]{c+d x}}{135 b (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (14 d^3\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{2/3}} \, dx}{405 b (b c-a d)^2} \\ & = -\frac {2 \sqrt [3]{c+d x}}{5 b (a+b x)^{5/2}}-\frac {4 d \sqrt [3]{c+d x}}{45 b (b c-a d) (a+b x)^{3/2}}+\frac {28 d^2 \sqrt [3]{c+d x}}{135 b (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (14 d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^3}{d}}} \, dx,x,\sqrt [3]{c+d x}\right )}{135 b (b c-a d)^2} \\ & = -\frac {2 \sqrt [3]{c+d x}}{5 b (a+b x)^{5/2}}-\frac {4 d \sqrt [3]{c+d x}}{45 b (b c-a d) (a+b x)^{3/2}}+\frac {28 d^2 \sqrt [3]{c+d x}}{135 b (b c-a d)^2 \sqrt {a+b x}}-\frac {28 \sqrt {2-\sqrt {3}} d^2 \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 (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt {3}\right )}{135 \sqrt [4]{3} b^{4/3} (b c-a d)^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\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 = 0.16 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{7/2}} \, dx=-\frac {2 \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {1}{3},-\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )}{5 b (a+b x)^{5/2} \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \]
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\[\int \frac {\left (d x +c \right )^{\frac {1}{3}}}{\left (b x +a \right )^{\frac {7}{2}}}d x\]
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\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{7/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{7/2}} \, dx=\int \frac {\sqrt [3]{c + d x}}{\left (a + b x\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{7/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{7/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{7/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{1/3}}{{\left (a+b\,x\right )}^{7/2}} \,d x \]
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