Integrand size = 23, antiderivative size = 298 \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x)) \, dx=\frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1+(c+d x)^2}}{405 d}-\frac {4 b (e (c+d x))^{7/2} \sqrt {1+(c+d x)^2}}{81 d}-\frac {28 b e^3 \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{135 d (1+c+d x)}+\frac {2 (e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))}{9 d e}+\frac {28 b e^{7/2} (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} E\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )|\frac {1}{2}\right )}{135 d \sqrt {1+(c+d x)^2}}-\frac {14 b e^{7/2} (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{135 d \sqrt {1+(c+d x)^2}} \]
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Time = 0.21 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5859, 5776, 327, 335, 311, 226, 1210} \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x)) \, dx=\frac {2 (e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))}{9 d e}-\frac {14 b e^{7/2} (c+d x+1) \sqrt {\frac {(c+d x)^2+1}{(c+d x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{135 d \sqrt {(c+d x)^2+1}}+\frac {28 b e^{7/2} (c+d x+1) \sqrt {\frac {(c+d x)^2+1}{(c+d x+1)^2}} E\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )|\frac {1}{2}\right )}{135 d \sqrt {(c+d x)^2+1}}-\frac {28 b e^3 \sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}}{135 d (c+d x+1)}+\frac {28 b e^2 \sqrt {(c+d x)^2+1} (e (c+d x))^{3/2}}{405 d}-\frac {4 b \sqrt {(c+d x)^2+1} (e (c+d x))^{7/2}}{81 d} \]
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Rule 226
Rule 311
Rule 327
Rule 335
Rule 1210
Rule 5776
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (e x)^{7/2} (a+b \text {arcsinh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {2 (e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))}{9 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{9/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d e} \\ & = -\frac {4 b (e (c+d x))^{7/2} \sqrt {1+(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))}{9 d e}+\frac {(14 b e) \text {Subst}\left (\int \frac {(e x)^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{81 d} \\ & = \frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1+(c+d x)^2}}{405 d}-\frac {4 b (e (c+d x))^{7/2} \sqrt {1+(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))}{9 d e}-\frac {\left (14 b e^3\right ) \text {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{135 d} \\ & = \frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1+(c+d x)^2}}{405 d}-\frac {4 b (e (c+d x))^{7/2} \sqrt {1+(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))}{9 d e}-\frac {\left (28 b e^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{135 d} \\ & = \frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1+(c+d x)^2}}{405 d}-\frac {4 b (e (c+d x))^{7/2} \sqrt {1+(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))}{9 d e}-\frac {\left (28 b e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{135 d}+\frac {\left (28 b e^3\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{e}}{\sqrt {1+\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{135 d} \\ & = \frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1+(c+d x)^2}}{405 d}-\frac {4 b (e (c+d x))^{7/2} \sqrt {1+(c+d x)^2}}{81 d}-\frac {28 b e^3 \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{135 d (1+c+d x)}+\frac {2 (e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))}{9 d e}+\frac {28 b e^{7/2} (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} E\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )|\frac {1}{2}\right )}{135 d \sqrt {1+(c+d x)^2}}-\frac {14 b e^{7/2} (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{135 d \sqrt {1+(c+d x)^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.16 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.38 \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x)) \, dx=\frac {2 (e (c+d x))^{7/2} \left (45 a (c+d x)^3+14 b \sqrt {1+(c+d x)^2}-10 b (c+d x)^2 \sqrt {1+(c+d x)^2}+45 b (c+d x)^3 \text {arcsinh}(c+d x)-14 b \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-(c+d x)^2\right )\right )}{405 d (c+d x)^2} \]
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Result contains complex when optimal does not.
Time = 2.98 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {\frac {2 a \left (d e x +c e \right )^{\frac {9}{2}}}{9}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {9}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{9}-\frac {2 \left (\frac {e^{2} \left (d e x +c e \right )^{\frac {7}{2}} \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{9}-\frac {7 e^{4} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{45}+\frac {7 i e^{5} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )\right )}{15 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{9 e}\right )}{d e}\) | \(238\) |
default | \(\frac {\frac {2 a \left (d e x +c e \right )^{\frac {9}{2}}}{9}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {9}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{9}-\frac {2 \left (\frac {e^{2} \left (d e x +c e \right )^{\frac {7}{2}} \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{9}-\frac {7 e^{4} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{45}+\frac {7 i e^{5} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )\right )}{15 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{9 e}\right )}{d e}\) | \(238\) |
parts | \(\frac {2 a \left (d e x +c e \right )^{\frac {9}{2}}}{9 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {9}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{9}-\frac {2 \left (\frac {e^{2} \left (d e x +c e \right )^{\frac {7}{2}} \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{9}-\frac {7 e^{4} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{45}+\frac {7 i e^{5} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )\right )}{15 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{9 e}\right )}{d e}\) | \(243\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.03 \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x)) \, dx=\frac {2 \, {\left (42 \, \sqrt {d^{3} e} b e^{3} {\rm weierstrassZeta}\left (-\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + 45 \, {\left (b d^{5} e^{3} x^{4} + 4 \, b c d^{4} e^{3} x^{3} + 6 \, b c^{2} d^{3} e^{3} x^{2} + 4 \, b c^{3} d^{2} e^{3} x + b c^{4} d e^{3}\right )} \sqrt {d e x + c e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \, {\left (5 \, b d^{4} e^{3} x^{3} + 15 \, b c d^{3} e^{3} x^{2} + {\left (15 \, b c^{2} - 7 \, b\right )} d^{2} e^{3} x + {\left (5 \, b c^{3} - 7 \, b c\right )} d e^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d e x + c e} + 45 \, {\left (a d^{5} e^{3} x^{4} + 4 \, a c d^{4} e^{3} x^{3} + 6 \, a c^{2} d^{3} e^{3} x^{2} + 4 \, a c^{3} d^{2} e^{3} x + a c^{4} d e^{3}\right )} \sqrt {d e x + c e}\right )}}{405 \, d^{2}} \]
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Timed out. \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x)) \, dx=\text {Timed out} \]
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Exception generated. \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x)) \, dx=\text {Exception raised: ValueError} \]
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\[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x)) \, dx=\int { {\left (d e x + c e\right )}^{\frac {7}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^{7/2}\,\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right ) \,d x \]
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