Integrand size = 16, antiderivative size = 89 \[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}+\frac {4 b E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {c} \sqrt {d}}-\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{\sqrt {c} \sqrt {d}} \]
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Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4724, 335, 313, 227, 1213, 435} \[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}-\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{\sqrt {c} \sqrt {d}}+\frac {4 b E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {c} \sqrt {d}} \]
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Rule 227
Rule 313
Rule 335
Rule 435
Rule 1213
Rule 4724
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}+\frac {(2 b c) \int \frac {\sqrt {d x}}{\sqrt {1-c^2 x^2}} \, dx}{d} \\ & = \frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}+\frac {(4 b c) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{d^2} \\ & = \frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}-\frac {(4 b) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{d}+\frac {(4 b) \text {Subst}\left (\int \frac {1+\frac {c x^2}{d}}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{d} \\ & = \frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}-\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{\sqrt {c} \sqrt {d}}+\frac {(4 b) \text {Subst}\left (\int \frac {\sqrt {1+\frac {c x^2}{d}}}{\sqrt {1-\frac {c x^2}{d}}} \, dx,x,\sqrt {d x}\right )}{d} \\ & = \frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}+\frac {4 b E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {c} \sqrt {d}}-\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{\sqrt {c} \sqrt {d}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51 \[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\frac {2 x \left (3 (a+b \arccos (c x))+2 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right )\right )}{3 \sqrt {d x}} \]
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Time = 1.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10
method | result | size |
derivativedivides | \(\frac {2 \sqrt {d x}\, a +2 b \left (\sqrt {d x}\, \arccos \left (c x \right )-\frac {2 \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{\sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\) | \(98\) |
default | \(\frac {2 \sqrt {d x}\, a +2 b \left (\sqrt {d x}\, \arccos \left (c x \right )-\frac {2 \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{\sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\) | \(98\) |
parts | \(\frac {2 a \sqrt {d x}}{d}+\frac {2 b \left (\sqrt {d x}\, \arccos \left (c x \right )-\frac {2 \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{\sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\) | \(101\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.58 \[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\frac {2 \, {\left (2 \, \sqrt {-c^{2} d} b {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) + {\left (b c \arccos \left (c x\right ) + a c\right )} \sqrt {d x}\right )}}{c d} \]
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Exception generated. \[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\sqrt {d x}} \,d x } \]
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\[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\sqrt {d x}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{\sqrt {d\,x}} \,d x \]
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