Integrand size = 19, antiderivative size = 144 \[ \int \frac {(c+d x)^{5/4}}{\sqrt {a+b x}} \, dx=\frac {20 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {20 (b c-a d)^{9/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{21 b^{9/4} d \sqrt {a+b x}} \]
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Time = 0.07 (sec) , antiderivative size = 144, 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 = {52, 65, 230, 227} \[ \int \frac {(c+d x)^{5/4}}{\sqrt {a+b x}} \, dx=\frac {20 (b c-a d)^{9/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{21 b^{9/4} d \sqrt {a+b x}}+\frac {20 \sqrt {a+b x} \sqrt [4]{c+d x} (b c-a d)}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b} \]
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Rule 52
Rule 65
Rule 227
Rule 230
Rubi steps \begin{align*} \text {integral}& = \frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {(5 (b c-a d)) \int \frac {\sqrt [4]{c+d x}}{\sqrt {a+b x}} \, dx}{7 b} \\ & = \frac {20 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {\left (5 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{21 b^2} \\ & = \frac {20 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {\left (20 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{21 b^2 d} \\ & = \frac {20 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {\left (20 (b c-a d)^2 \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{21 b^2 d \sqrt {a+b x}} \\ & = \frac {20 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {20 (b c-a d)^{9/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{21 b^{9/4} d \sqrt {a+b x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.49 \[ \int \frac {(c+d x)^{5/4}}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} (c+d x)^{5/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4}} \]
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\[\int \frac {\left (d x +c \right )^{\frac {5}{4}}}{\sqrt {b x +a}}d x\]
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\[ \int \frac {(c+d x)^{5/4}}{\sqrt {a+b x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{\sqrt {b x + a}} \,d x } \]
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\[ \int \frac {(c+d x)^{5/4}}{\sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{4}}}{\sqrt {a + b x}}\, dx \]
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\[ \int \frac {(c+d x)^{5/4}}{\sqrt {a+b x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{\sqrt {b x + a}} \,d x } \]
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\[ \int \frac {(c+d x)^{5/4}}{\sqrt {a+b x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{\sqrt {b x + a}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^{5/4}}{\sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/4}}{\sqrt {a+b\,x}} \,d x \]
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