Integrand size = 25, antiderivative size = 395 \[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {245 b^2 d^2 x \sqrt {d+c^2 d x^2}}{1152}+\frac {65 b^2 d x \left (d+c^2 d x^2\right )^{3/2}}{1728}+\frac {1}{108} b^2 x \left (d+c^2 d x^2\right )^{5/2}-\frac {115 b^2 d^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{1152 c \sqrt {1+c^2 x^2}}-\frac {5 b c d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{48 c}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{18 c}+\frac {5}{16} d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{24} d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{6} x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2+\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{48 b c \sqrt {1+c^2 x^2}} \] Output:
245/1152*b^2*d^2*x*(c^2*d*x^2+d)^(1/2)+65/1728*b^2*d*x*(c^2*d*x^2+d)^(3/2) +1/108*b^2*x*(c^2*d*x^2+d)^(5/2)-115/1152*b^2*d^2*(c^2*d*x^2+d)^(1/2)*arcs inh(c*x)/c/(c^2*x^2+1)^(1/2)-5/16*b*c*d^2*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arc sinh(c*x))/(c^2*x^2+1)^(1/2)-5/48*b*d^2*(c^2*x^2+1)^(3/2)*(c^2*d*x^2+d)^(1 /2)*(a+b*arcsinh(c*x))/c-1/18*b*d^2*(c^2*x^2+1)^(5/2)*(c^2*d*x^2+d)^(1/2)* (a+b*arcsinh(c*x))/c+5/16*d^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2+5 /24*d*x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2+1/6*x*(c^2*d*x^2+d)^(5/2) *(a+b*arcsinh(c*x))^2+5/48*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^3/b/ c/(c^2*x^2+1)^(1/2)
Time = 2.23 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.26 \[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (9504 a^2 c x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+7488 a^2 c^3 x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+2304 a^2 c^5 x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+1440 b^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^3-3240 a b \sqrt {d+c^2 d x^2} \cosh (2 \text {arcsinh}(c x))-324 a b \sqrt {d+c^2 d x^2} \cosh (4 \text {arcsinh}(c x))-24 a b \sqrt {d+c^2 d x^2} \cosh (6 \text {arcsinh}(c x))+4320 a^2 \sqrt {d} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+1620 b^2 \sqrt {d+c^2 d x^2} \sinh (2 \text {arcsinh}(c x))+81 b^2 \sqrt {d+c^2 d x^2} \sinh (4 \text {arcsinh}(c x))+4 b^2 \sqrt {d+c^2 d x^2} \sinh (6 \text {arcsinh}(c x))-12 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) (270 b \cosh (2 \text {arcsinh}(c x))+27 b \cosh (4 \text {arcsinh}(c x))+2 b \cosh (6 \text {arcsinh}(c x))-540 a \sinh (2 \text {arcsinh}(c x))-108 a \sinh (4 \text {arcsinh}(c x))-12 a \sinh (6 \text {arcsinh}(c x)))+72 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2 (60 a+45 b \sinh (2 \text {arcsinh}(c x))+9 b \sinh (4 \text {arcsinh}(c x))+b \sinh (6 \text {arcsinh}(c x)))\right )}{13824 c \sqrt {1+c^2 x^2}} \] Input:
Integrate[(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^2*(9504*a^2*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 7488*a^2*c^3*x^ 3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 2304*a^2*c^5*x^5*Sqrt[1 + c^2*x^ 2]*Sqrt[d + c^2*d*x^2] + 1440*b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 - 324 0*a*b*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 324*a*b*Sqrt[d + c^2*d*x^ 2]*Cosh[4*ArcSinh[c*x]] - 24*a*b*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] + 4320*a^2*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^ 2]] + 1620*b^2*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + 81*b^2*Sqrt[d + c^2*d*x^2]*Sinh[4*ArcSinh[c*x]] + 4*b^2*Sqrt[d + c^2*d*x^2]*Sinh[6*ArcSinh [c*x]] - 12*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(270*b*Cosh[2*ArcSinh[c*x]] + 27*b*Cosh[4*ArcSinh[c*x]] + 2*b*Cosh[6*ArcSinh[c*x]] - 540*a*Sinh[2*Arc Sinh[c*x]] - 108*a*Sinh[4*ArcSinh[c*x]] - 12*a*Sinh[6*ArcSinh[c*x]]) + 72* b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(60*a + 45*b*Sinh[2*ArcSinh[c*x]] + 9 *b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]])))/(13824*c*Sqrt[1 + c^2* x^2])
Time = 2.18 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.27, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {6201, 6201, 6200, 6191, 262, 222, 6198, 6213, 211, 211, 211, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx\) |
\(\Big \downarrow \) 6201 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \int \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 6201 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \int \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 6200 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {c^2 d x^2+d} \int x (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx\right )}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )\right )}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {b \int \left (c^2 x^2+1\right )^{5/2}dx}{6 c}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \int \left (c^2 x^2+1\right )^{3/2}dx}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {b \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2}dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )}{6 c}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )}{6 c}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )}{6 c}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )}{6 c}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2+\frac {5}{6} d \left (\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\right )\) |
Input:
Int[(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]
Output:
(x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/6 - (b*c*d^2*Sqrt[d + c^2 *d*x^2]*(((1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]))/(6*c^2) - (b*((x*(1 + c^2* x^2)^(5/2))/6 + (5*((x*(1 + c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 + c^2*x^2])/ 2 + ArcSinh[c*x]/(2*c)))/4))/6))/(6*c)))/(3*Sqrt[1 + c^2*x^2]) + (5*d*((x* (d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + (3*d*((x*Sqrt[d + c^2*d* x^2]*(a + b*ArcSinh[c*x])^2)/2 + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) ^3)/(6*b*c*Sqrt[1 + c^2*x^2]) - (b*c*Sqrt[d + c^2*d*x^2]*((x^2*(a + b*ArcS inh[c*x]))/2 - (b*c*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)) )/2))/Sqrt[1 + c^2*x^2]))/4 - (b*c*d*Sqrt[d + c^2*d*x^2]*(((1 + c^2*x^2)^2 *(a + b*ArcSinh[c*x]))/(4*c^2) - (b*((x*(1 + c^2*x^2)^(3/2))/4 + (3*((x*Sq rt[1 + c^2*x^2])/2 + ArcSinh[c*x]/(2*c)))/4))/(4*c)))/(2*Sqrt[1 + c^2*x^2] )))/6
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1567\) vs. \(2(341)=682\).
Time = 0.97 (sec) , antiderivative size = 1568, normalized size of antiderivative = 3.97
method | result | size |
default | \(\text {Expression too large to display}\) | \(1568\) |
parts | \(\text {Expression too large to display}\) | \(1568\) |
Input:
int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/6*x*(c^2*d*x^2+d)^(5/2)*a^2+5/24*a^2*d*x*(c^2*d*x^2+d)^(3/2)+5/16*a^2*d^ 2*x*(c^2*d*x^2+d)^(1/2)+5/16*a^2*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d )^(1/2))/(c^2*d)^(1/2)+b^2*(5/48*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c *arcsinh(x*c)^3*d^2+1/6912*(d*(c^2*x^2+1))^(1/2)*(32*x^7*c^7+32*x^6*c^6*(c ^2*x^2+1)^(1/2)+64*x^5*c^5+48*x^4*c^4*(c^2*x^2+1)^(1/2)+38*x^3*c^3+18*x^2* c^2*(c^2*x^2+1)^(1/2)+6*x*c+(c^2*x^2+1)^(1/2))*(18*arcsinh(x*c)^2-6*arcsin h(x*c)+1)*d^2/(c^2*x^2+1)/c+3/1024*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5+8*x^4* c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^ 2+1)^(1/2))*(8*arcsinh(x*c)^2-4*arcsinh(x*c)+1)*d^2/(c^2*x^2+1)/c+15/256*( d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2 +1)^(1/2))*(2*arcsinh(x*c)^2-2*arcsinh(x*c)+1)*d^2/(c^2*x^2+1)/c+15/256*(d *(c^2*x^2+1))^(1/2)*(2*x^3*c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+ 1)^(1/2))*(2*arcsinh(x*c)^2+2*arcsinh(x*c)+1)*d^2/(c^2*x^2+1)/c+3/1024*(d* (c^2*x^2+1))^(1/2)*(8*x^5*c^5-8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3-8*x^2 *c^2*(c^2*x^2+1)^(1/2)+4*x*c-(c^2*x^2+1)^(1/2))*(8*arcsinh(x*c)^2+4*arcsin h(x*c)+1)*d^2/(c^2*x^2+1)/c+1/6912*(d*(c^2*x^2+1))^(1/2)*(32*x^7*c^7-32*x^ 6*c^6*(c^2*x^2+1)^(1/2)+64*x^5*c^5-48*x^4*c^4*(c^2*x^2+1)^(1/2)+38*x^3*c^3 -18*x^2*c^2*(c^2*x^2+1)^(1/2)+6*x*c-(c^2*x^2+1)^(1/2))*(18*arcsinh(x*c)^2+ 6*arcsinh(x*c)+1)*d^2/(c^2*x^2+1)/c)+2*a*b*(5/32*(d*(c^2*x^2+1))^(1/2)/(c^ 2*x^2+1)^(1/2)/c*arcsinh(x*c)^2*d^2+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*x^...
\[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
integral((a^2*c^4*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 + 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsinh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 + 2*a* b*c^2*d^2*x^2 + a*b*d^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)
\[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \] Input:
integrate((c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2,x)
Output:
Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))**2, x)
Exception generated. \[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \] Input:
int((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2),x)
Output:
int((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2), x)
\[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (8 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{5} x^{5}+26 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}+33 \sqrt {c^{2} x^{2}+1}\, a^{2} c x +96 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{4}d x \right ) a b \,c^{5}+192 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+96 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) a b c +48 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+96 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+48 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}d x \right ) b^{2} c +15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2}\right )}{48 c} \] Input:
int((c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x))^2,x)
Output:
(sqrt(d)*d**2*(8*sqrt(c**2*x**2 + 1)*a**2*c**5*x**5 + 26*sqrt(c**2*x**2 + 1)*a**2*c**3*x**3 + 33*sqrt(c**2*x**2 + 1)*a**2*c*x + 96*int(sqrt(c**2*x** 2 + 1)*asinh(c*x)*x**4,x)*a*b*c**5 + 192*int(sqrt(c**2*x**2 + 1)*asinh(c*x )*x**2,x)*a*b*c**3 + 96*int(sqrt(c**2*x**2 + 1)*asinh(c*x),x)*a*b*c + 48*i nt(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x**4,x)*b**2*c**5 + 96*int(sqrt(c**2* x**2 + 1)*asinh(c*x)**2*x**2,x)*b**2*c**3 + 48*int(sqrt(c**2*x**2 + 1)*asi nh(c*x)**2,x)*b**2*c + 15*log(sqrt(c**2*x**2 + 1) + c*x)*a**2))/(48*c)