Integrand size = 28, antiderivative size = 512 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {11 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {11 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}} \]
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Time = 0.65 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5810, 5798, 5772, 267, 5812, 5789, 4265, 2317, 2438, 272, 45} \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {22 b \sqrt {c^2 x^2+1} \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {8 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^6 d^3}+\frac {11 b x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {4 x^2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {c^2 d x^2+d}}-\frac {b x^3 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {16 a b x \sqrt {c^2 x^2+1}}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}+\frac {11 i b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {11 i b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {16 b^2 x \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 b^2 \left (c^2 x^2+1\right )}{c^6 d^2 \sqrt {c^2 d x^2+d}}+\frac {b^2}{3 c^6 d^2 \sqrt {c^2 d x^2+d}} \]
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Rule 45
Rule 267
Rule 272
Rule 2317
Rule 2438
Rule 4265
Rule 5772
Rule 5789
Rule 5798
Rule 5810
Rule 5812
Rubi steps \begin{align*} \text {integral}& = -\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b x^3 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{3 c^4 d^2}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b x^3 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^6 d^3}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (16 b \sqrt {1+c^2 x^2}\right ) \int (a+b \text {arcsinh}(c x)) \, dx}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{\left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 c^2 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {11 b^2 \left (1+c^2 x^2\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^6 d^3}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {1+c^2 x^2}\right ) \int \text {arcsinh}(c x) \, dx}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2 \left (1+c^2 x\right )^{3/2}}+\frac {1}{c^2 \sqrt {1+c^2 x}}\right ) \, dx,x,x^2\right )}{6 c^2 d^2 \sqrt {d+c^2 d x^2}} \\ & = \frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {10 b^2 \left (1+c^2 x^2\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt {d+c^2 d x^2}} \\ & = \frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}} \\ & = \frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {11 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {11 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}} \\ \end{align*}
Time = 1.59 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.65 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+c^2 d x^2} \left (a^2 \left (8+12 c^2 x^2+3 c^4 x^4\right )+a b \left (2 \left (8+12 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)-\sqrt {1+c^2 x^2} \left (c x \left (5+6 c^2 x^2\right )+22 \left (1+c^2 x^2\right ) \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )\right )+b^2 \left (c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-6 c x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)-\text {arcsinh}(c x)^2+3 \left (1+c^2 x^2\right )^2 \left (2+\text {arcsinh}(c x)^2\right )+\left (1+c^2 x^2\right ) \left (1+6 \text {arcsinh}(c x)^2\right )+11 i \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \left (\log \left (1-i e^{-\text {arcsinh}(c x)}\right )-\log \left (1+i e^{-\text {arcsinh}(c x)}\right )\right )+11 i \left (1+c^2 x^2\right )^{3/2} \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )\right )\right )\right )}{3 c^6 d^3 \left (1+c^2 x^2\right )^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1041 vs. \(2 (475 ) = 950\).
Time = 0.40 (sec) , antiderivative size = 1042, normalized size of antiderivative = 2.04
method | result | size |
default | \(\text {Expression too large to display}\) | \(1042\) |
parts | \(\text {Expression too large to display}\) | \(1042\) |
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\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]
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