Integrand size = 26, antiderivative size = 281 \[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b}{6 c^7 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2}}{4 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 x^3 (a+b \text {arcsinh}(c x))}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 c^6 d^3}-\frac {5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^7 d^2 \sqrt {d+c^2 d x^2}}-\frac {7 b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{6 c^7 d^2 \sqrt {d+c^2 d x^2}} \]
[Out]
Time = 0.28 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5810, 5812, 5783, 30, 272, 45} \[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{4 b c^7 d^2 \sqrt {c^2 d x^2+d}}+\frac {5 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 c^6 d^3}-\frac {5 x^3 (a+b \text {arcsinh}(c x))}{3 c^4 d^2 \sqrt {c^2 d x^2+d}}-\frac {b}{6 c^7 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {7 b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{6 c^7 d^2 \sqrt {c^2 d x^2+d}}-\frac {b x^2 \sqrt {c^2 x^2+1}}{4 c^5 d^2 \sqrt {c^2 d x^2+d}} \]
[In]
[Out]
Rule 30
Rule 45
Rule 272
Rule 5783
Rule 5810
Rule 5812
Rubi steps \begin{align*} \text {integral}& = -\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {5 \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x^5}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 x^3 (a+b \text {arcsinh}(c x))}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx}{c^4 d^2}+\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 c d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 x^3 (a+b \text {arcsinh}(c x))}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 c^6 d^3}-\frac {5 \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+c^2 d x^2}} \, dx}{2 c^6 d^2}-\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \int x \, dx}{2 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{6 c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^4}+\frac {1}{c^4 \left (1+c^2 x\right )^2}-\frac {2}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b}{6 c^7 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {13 b x^2 \sqrt {1+c^2 x^2}}{12 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 x^3 (a+b \text {arcsinh}(c x))}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 c^6 d^3}-\frac {5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^7 d^2 \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{3 c^7 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c^3 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b}{6 c^7 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2}}{4 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^5 (a+b \text {arcsinh}(c x))}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 x^3 (a+b \text {arcsinh}(c x))}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 c^6 d^3}-\frac {5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^7 d^2 \sqrt {d+c^2 d x^2}}-\frac {7 b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{6 c^7 d^2 \sqrt {d+c^2 d x^2}} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.79 \[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {4 a c d x \left (15+20 c^2 x^2+3 c^4 x^4\right )+b d \left (4 c x \left (15+20 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)-30 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^2-\sqrt {1+c^2 x^2} \left (7+9 c^2 x^2+6 c^4 x^4+28 \left (1+c^2 x^2\right ) \log \left (1+c^2 x^2\right )\right )\right )-60 a \sqrt {d} \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{24 c^7 d^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}} \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.46
method | result | size |
default | \(\frac {a \,x^{5}}{2 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,x^{3}}{6 c^{4} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a x}{2 c^{6} d^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {5 a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{6} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (-12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+6 c^{6} x^{6}+30 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-56 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+56 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-80 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+15 c^{4} x^{4}+60 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-112 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+112 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-60 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+16 c^{2} x^{2}+30 \operatorname {arcsinh}\left (c x \right )^{2}-56 \,\operatorname {arcsinh}\left (c x \right )+56 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+7\right )}{24 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) c^{7} d^{3}}\) | \(410\) |
parts | \(\frac {a \,x^{5}}{2 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,x^{3}}{6 c^{4} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a x}{2 c^{6} d^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {5 a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{6} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (-12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+6 c^{6} x^{6}+30 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-56 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+56 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-80 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+15 c^{4} x^{4}+60 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-112 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+112 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-60 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+16 c^{2} x^{2}+30 \operatorname {arcsinh}\left (c x \right )^{2}-56 \,\operatorname {arcsinh}\left (c x \right )+56 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+7\right )}{24 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) c^{7} d^{3}}\) | \(410\) |
[In]
[Out]
\[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{6}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{6} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{6}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x^6 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^6\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]
[In]
[Out]