Integrand size = 19, antiderivative size = 858 \[ \int \sqrt {a+b x} (c+d x)^{5/6} \, dx=\frac {15 (b c-a d) \sqrt {a+b x} (c+d x)^{5/6}}{56 b d}+\frac {3 (a+b x)^{3/2} (c+d x)^{5/6}}{7 b}+\frac {45 \left (1+\sqrt {3}\right ) (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}{112 b^{5/3} d \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {45 \sqrt [4]{3} (b c-a d)^{7/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{112 b^{5/3} d^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {15\ 3^{3/4} \left (1-\sqrt {3}\right ) (b c-a d)^{7/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{224 b^{5/3} d^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]
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Time = 0.61 (sec) , antiderivative size = 858, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {52, 65, 314, 231, 1895} \[ \int \sqrt {a+b x} (c+d x)^{5/6} \, dx=\frac {45 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (b c-a d)^{7/3}}{112 b^{5/3} d^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {15\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (b c-a d)^{7/3}}{224 b^{5/3} d^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {45 \left (1+\sqrt {3}\right ) \sqrt {a+b x} \sqrt [6]{c+d x} (b c-a d)^2}{112 b^{5/3} d \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {15 \sqrt {a+b x} (c+d x)^{5/6} (b c-a d)}{56 b d}+\frac {3 (a+b x)^{3/2} (c+d x)^{5/6}}{7 b} \]
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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)^{3/2} (c+d x)^{5/6}}{7 b}+\frac {(5 (b c-a d)) \int \frac {\sqrt {a+b x}}{\sqrt [6]{c+d x}} \, dx}{14 b} \\ & = \frac {15 (b c-a d) \sqrt {a+b x} (c+d x)^{5/6}}{56 b d}+\frac {3 (a+b x)^{3/2} (c+d x)^{5/6}}{7 b}-\frac {\left (15 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [6]{c+d x}} \, dx}{112 b d} \\ & = \frac {15 (b c-a d) \sqrt {a+b x} (c+d x)^{5/6}}{56 b d}+\frac {3 (a+b x)^{3/2} (c+d x)^{5/6}}{7 b}-\frac {\left (45 (b c-a 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 )}{56 b d^2} \\ & = \frac {15 (b c-a d) \sqrt {a+b x} (c+d x)^{5/6}}{56 b d}+\frac {3 (a+b x)^{3/2} (c+d x)^{5/6}}{7 b}+\frac {\left (45 (b c-a 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 )}{112 b^{5/3} d^2}+\frac {\left (45 \left (1-\sqrt {3}\right ) (b c-a d)^{8/3}\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 )}{112 b^{5/3} d^2} \\ & = \frac {15 (b c-a d) \sqrt {a+b x} (c+d x)^{5/6}}{56 b d}+\frac {3 (a+b x)^{3/2} (c+d x)^{5/6}}{7 b}+\frac {45 \left (1+\sqrt {3}\right ) (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}{112 b^{5/3} d \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {45 \sqrt [4]{3} (b c-a d)^{7/3} \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 )}{112 b^{5/3} d^2 \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 {15\ 3^{3/4} \left (1-\sqrt {3}\right ) (b c-a d)^{7/3} \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 )}{224 b^{5/3} d^2 \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 = 0.09 \[ \int \sqrt {a+b x} (c+d x)^{5/6} \, dx=\frac {2 (a+b x)^{3/2} (c+d x)^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {3}{2},\frac {5}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \]
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\[\int \sqrt {b x +a}\, \left (d x +c \right )^{\frac {5}{6}}d x\]
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\[ \int \sqrt {a+b x} (c+d x)^{5/6} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{6}} \,d x } \]
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\[ \int \sqrt {a+b x} (c+d x)^{5/6} \, dx=\int \sqrt {a + b x} \left (c + d x\right )^{\frac {5}{6}}\, dx \]
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\[ \int \sqrt {a+b x} (c+d x)^{5/6} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{6}} \,d x } \]
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\[ \int \sqrt {a+b x} (c+d x)^{5/6} \, dx=\int { \sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{6}} \,d x } \]
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Timed out. \[ \int \sqrt {a+b x} (c+d x)^{5/6} \, dx=\int \sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/6} \,d x \]
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