Integrand size = 28, antiderivative size = 405 \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {7 b^2 d x \sqrt {d+c^2 d x^2}}{1152 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}+\frac {7 b^2 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{1152 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{48 b c^3 \sqrt {1+c^2 x^2}} \]
[Out]
Time = 0.48 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5808, 5806, 5812, 5783, 5776, 327, 221, 14, 5803, 12, 470} \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {b d x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{16 c \sqrt {c^2 x^2+1}}+\frac {d x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{16 c^2}-\frac {7 b c d x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{48 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{8} d x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{48 b c^3 \sqrt {c^2 x^2+1}}-\frac {b c^3 d x^6 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{18 \sqrt {c^2 x^2+1}}+\frac {7 b^2 d \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{1152 c^3 \sqrt {c^2 x^2+1}}-\frac {7 b^2 d x \sqrt {c^2 d x^2+d}}{1152 c^2}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {c^2 d x^2+d}+\frac {43 b^2 d x^3 \sqrt {c^2 d x^2+d}}{1728} \]
[In]
[Out]
Rule 12
Rule 14
Rule 221
Rule 327
Rule 470
Rule 5776
Rule 5783
Rule 5803
Rule 5806
Rule 5808
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x)) \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c d x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{12 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{18 \sqrt {1+c^2 x^2}}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int x^3 (a+b \text {arcsinh}(c x)) \, dx}{4 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4 \left (3+2 c^2 x^2\right )}{12 \sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{16 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx}{8 c \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4 \left (3+2 c^2 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx}{36 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}} \\ & = \frac {1}{64} b^2 d x^3 \sqrt {d+c^2 d x^2}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}-\frac {b d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{48 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{64 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{27 \sqrt {1+c^2 x^2}} \\ & = \frac {b^2 d x \sqrt {d+c^2 d x^2}}{128 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}-\frac {b d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{48 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{36 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{128 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{32 c^2 \sqrt {1+c^2 x^2}} \\ & = -\frac {7 b^2 d x \sqrt {d+c^2 d x^2}}{1152 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}-\frac {b^2 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{128 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{48 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{72 c^2 \sqrt {1+c^2 x^2}} \\ & = -\frac {7 b^2 d x \sqrt {d+c^2 d x^2}}{1152 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}+\frac {7 b^2 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{1152 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{48 b c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 2.11 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.25 \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {864 a^2 c d x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+4032 a^2 c^3 d x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+2304 a^2 c^5 d x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-288 b^2 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^3+216 a b d \sqrt {d+c^2 d x^2} \cosh (2 \text {arcsinh}(c x))-108 a b d \sqrt {d+c^2 d x^2} \cosh (4 \text {arcsinh}(c x))-24 a b d \sqrt {d+c^2 d x^2} \cosh (6 \text {arcsinh}(c x))-864 a^2 d^{3/2} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-108 b^2 d \sqrt {d+c^2 d x^2} \sinh (2 \text {arcsinh}(c x))+27 b^2 d \sqrt {d+c^2 d x^2} \sinh (4 \text {arcsinh}(c x))+4 b^2 d \sqrt {d+c^2 d x^2} \sinh (6 \text {arcsinh}(c x))+12 b d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) (18 b \cosh (2 \text {arcsinh}(c x))-9 b \cosh (4 \text {arcsinh}(c x))-2 b \cosh (6 \text {arcsinh}(c x))-36 a \sinh (2 \text {arcsinh}(c x))+36 a \sinh (4 \text {arcsinh}(c x))+12 a \sinh (6 \text {arcsinh}(c x)))+72 b d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2 (-12 a-3 b \sinh (2 \text {arcsinh}(c x))+3 b \sinh (4 \text {arcsinh}(c x))+b \sinh (6 \text {arcsinh}(c x)))}{13824 c^3 \sqrt {1+c^2 x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1551\) vs. \(2(351)=702\).
Time = 0.36 (sec) , antiderivative size = 1552, normalized size of antiderivative = 3.83
method | result | size |
default | \(\text {Expression too large to display}\) | \(1552\) |
parts | \(\text {Expression too large to display}\) | \(1552\) |
[In]
[Out]
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
[In]
[Out]
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^{2} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
[In]
[Out]
Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
[In]
[Out]