Integrand size = 30, antiderivative size = 235 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=-\frac {51 (a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}+\frac {45 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{64 b}+\frac {3 (a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)}{32 b}-\frac {27 \text {arcsinh}(a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \text {arcsinh}(a+b x)^2}{16 b}-\frac {3 \left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^3}{4 b}+\frac {3 \text {arcsinh}(a+b x)^4}{32 b} \]
-51/128*(b*x+a)^2/b-3/128*(b*x+a)^4/b+3/32*(b*x+a)*(1+(b*x+a)^2)^(3/2)*arc sinh(b*x+a)/b-27/128*arcsinh(b*x+a)^2/b-9/16*(b*x+a)^2*arcsinh(b*x+a)^2/b- 3/16*(1+(b*x+a)^2)^2*arcsinh(b*x+a)^2/b+1/4*(b*x+a)*(1+(b*x+a)^2)^(3/2)*ar csinh(b*x+a)^3/b+3/32*arcsinh(b*x+a)^4/b+45/64*(b*x+a)*arcsinh(b*x+a)*(1+( b*x+a)^2)^(1/2)/b+3/8*(b*x+a)*arcsinh(b*x+a)^3*(1+(b*x+a)^2)^(1/2)/b
Time = 0.16 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.13 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=-\frac {6 a \left (17+2 a^2\right ) b x+3 \left (17+6 a^2\right ) b^2 x^2+12 a b^3 x^3+3 b^4 x^4-6 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (17 a+2 a^3+17 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \text {arcsinh}(a+b x)+3 \left (17+8 a^4+32 a^3 b x+40 b^2 x^2+8 b^4 x^4+16 a b x \left (5+2 b^2 x^2\right )+8 a^2 \left (5+6 b^2 x^2\right )\right ) \text {arcsinh}(a+b x)^2-16 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (5 a+2 a^3+5 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \text {arcsinh}(a+b x)^3-12 \text {arcsinh}(a+b x)^4}{128 b} \]
-1/128*(6*a*(17 + 2*a^2)*b*x + 3*(17 + 6*a^2)*b^2*x^2 + 12*a*b^3*x^3 + 3*b ^4*x^4 - 6*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(17*a + 2*a^3 + 17*b*x + 6*a^ 2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSinh[a + b*x] + 3*(17 + 8*a^4 + 32*a^3 *b*x + 40*b^2*x^2 + 8*b^4*x^4 + 16*a*b*x*(5 + 2*b^2*x^2) + 8*a^2*(5 + 6*b^ 2*x^2))*ArcSinh[a + b*x]^2 - 16*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(5*a + 2 *a^3 + 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSinh[a + b*x]^3 - 1 2*ArcSinh[a + b*x]^4)/b
Time = 1.62 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.28, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6275, 6201, 6200, 6191, 6198, 6213, 6201, 244, 2009, 6200, 15, 6198, 6227, 15, 6198}
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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx\) |
| \(\Big \downarrow \) 6275 |
| \(\displaystyle \frac {\int \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3d(a+b x)}{b}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {-\frac {3}{4} \int (a+b x) \left ((a+b x)^2+1\right ) \text {arcsinh}(a+b x)^2d(a+b x)+\frac {3}{4} \int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3d(a+b x)+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3}{b}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle \frac {-\frac {3}{4} \int (a+b x) \left ((a+b x)^2+1\right ) \text {arcsinh}(a+b x)^2d(a+b x)+\frac {3}{4} \left (-\frac {3}{2} \int (a+b x) \text {arcsinh}(a+b x)^2d(a+b x)+\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)^3}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3}{b}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {-\frac {3}{4} \int (a+b x) \left ((a+b x)^2+1\right ) \text {arcsinh}(a+b x)^2d(a+b x)+\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)^3}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3}{b}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )-\frac {3}{4} \int (a+b x) \left ((a+b x)^2+1\right ) \text {arcsinh}(a+b x)^2d(a+b x)+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3}{b}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )-\frac {3}{4} \left (\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2-\frac {1}{2} \int \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)d(a+b x)\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3}{b}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )-\frac {3}{4} \left (\frac {1}{2} \left (-\frac {3}{4} \int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)d(a+b x)+\frac {1}{4} \int (a+b x) \left ((a+b x)^2+1\right )d(a+b x)-\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)\right )+\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3}{b}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )-\frac {3}{4} \left (\frac {1}{2} \left (-\frac {3}{4} \int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)d(a+b x)+\frac {1}{4} \int \left ((a+b x)^3+a+b x\right )d(a+b x)-\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)\right )+\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )-\frac {3}{4} \left (\frac {1}{2} \left (-\frac {3}{4} \int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)d(a+b x)-\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4+\frac {1}{2} (a+b x)^2\right )\right )+\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3}{b}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle \frac {-\frac {3}{4} \left (\frac {1}{2} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)-\frac {1}{2} \int (a+b x)d(a+b x)+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)\right )-\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4+\frac {1}{2} (a+b x)^2\right )\right )+\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2\right )+\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3}{b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {-\frac {3}{4} \left (\frac {1}{2} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)-\frac {1}{4} (a+b x)^2\right )-\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4+\frac {1}{2} (a+b x)^2\right )\right )+\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2\right )+\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3}{b}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\int \frac {(a+b x)^2 \text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3-\frac {3}{4} \left (\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2+\frac {1}{2} \left (-\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)-\frac {3}{4} \left (\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} \text {arcsinh}(a+b x)^2-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4+\frac {1}{2} (a+b x)^2\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} \int (a+b x)d(a+b x)+\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3-\frac {3}{4} \left (\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2+\frac {1}{2} \left (-\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)-\frac {3}{4} \left (\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} \text {arcsinh}(a+b x)^2-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4+\frac {1}{2} (a+b x)^2\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} (a+b x)^2\right )+\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3-\frac {3}{4} \left (\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2+\frac {1}{2} \left (-\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)-\frac {3}{4} \left (\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} \text {arcsinh}(a+b x)^2-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4+\frac {1}{2} (a+b x)^2\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^3+\frac {3}{4} \left (\frac {1}{8} \text {arcsinh}(a+b x)^4+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^3-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)^2-\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} \text {arcsinh}(a+b x)^2+\frac {1}{4} (a+b x)^2\right )\right )-\frac {3}{4} \left (\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2+\frac {1}{2} \left (-\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)-\frac {3}{4} \left (\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} \text {arcsinh}(a+b x)^2-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4+\frac {1}{2} (a+b x)^2\right )\right )\right )}{b}\) |
(((a + b*x)*(1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x]^3)/4 + (3*(((a + b*x) *Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^3)/2 + ArcSinh[a + b*x]^4/8 - (3*( (a + b*x)^2/4 - ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/2 + Arc Sinh[a + b*x]^2/4 + ((a + b*x)^2*ArcSinh[a + b*x]^2)/2))/2))/4 - (3*(((1 + (a + b*x)^2)^2*ArcSinh[a + b*x]^2)/4 + (((a + b*x)^2/2 + (a + b*x)^4/4)/4 - ((a + b*x)*(1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x])/4 - (3*(-1/4*(a + b*x)^2 + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/2 + ArcSinh[a + b*x]^2/4))/4)/2))/4)/b
3.3.66.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/d Subst[Int[(C/d^2 + (C/d^2)*x^2 )^p*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C , n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(591\) vs. \(2(207)=414\).
Time = 0.73 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.52
| method | result | size |
| default | \(\frac {-48-102 a b x -51 a^{2}-51 b^{2} x^{2}+12 \operatorname {arcsinh}\left (b x +a \right )^{4}-51 \operatorname {arcsinh}\left (b x +a \right )^{2}-12 a \,b^{3} x^{3}-3 b^{4} x^{4}+96 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a \,b^{2} x^{2}+96 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b x +36 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a \,b^{2} x^{2}+36 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b x -3 a^{4}+12 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{3} x^{3}+32 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{3} x^{3}-96 \operatorname {arcsinh}\left (b x +a \right )^{2} a \,b^{3} x^{3}-144 \operatorname {arcsinh}\left (b x +a \right )^{2} a^{2} b^{2} x^{2}-96 \operatorname {arcsinh}\left (b x +a \right )^{2} a^{3} b x +80 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x -240 \operatorname {arcsinh}\left (b x +a \right )^{2} a b x +102 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x -24 a^{4} \operatorname {arcsinh}\left (b x +a \right )^{2}-120 a^{2} \operatorname {arcsinh}\left (b x +a \right )^{2}-120 \operatorname {arcsinh}\left (b x +a \right )^{2} b^{2} x^{2}+80 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a +102 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a +12 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}+32 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}-24 \operatorname {arcsinh}\left (b x +a \right )^{2} b^{4} x^{4}-18 a^{2} b^{2} x^{2}-12 a^{3} b x}{128 b}\) | \(592\) |
1/128*(-48-102*a*b*x-51*a^2-51*b^2*x^2+12*arcsinh(b*x+a)^4-51*arcsinh(b*x+ a)^2-12*a*b^3*x^3-3*b^4*x^4+96*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1 /2)*a*b^2*x^2+96*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^2*b*x+36 *arcsinh(b*x+a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a*b^2*x^2+36*arcsinh(b*x+a)* (b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^2*b*x-3*a^4+12*arcsinh(b*x+a)*(b^2*x^2+2*a *b*x+a^2+1)^(1/2)*b^3*x^3+32*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2 )*b^3*x^3-96*arcsinh(b*x+a)^2*a*b^3*x^3-144*arcsinh(b*x+a)^2*a^2*b^2*x^2-9 6*arcsinh(b*x+a)^2*a^3*b*x+80*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/ 2)*b*x-240*arcsinh(b*x+a)^2*a*b*x+102*arcsinh(b*x+a)*(b^2*x^2+2*a*b*x+a^2+ 1)^(1/2)*b*x-24*a^4*arcsinh(b*x+a)^2-120*a^2*arcsinh(b*x+a)^2-120*arcsinh( b*x+a)^2*b^2*x^2+80*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a+102*a rcsinh(b*x+a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a+12*arcsinh(b*x+a)*(b^2*x^2+2 *a*b*x+a^2+1)^(1/2)*a^3+32*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)* a^3-24*arcsinh(b*x+a)^2*b^4*x^4-18*a^2*b^2*x^2-12*a^3*b*x)/b
Time = 0.26 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.41 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=-\frac {3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} + 3 \, {\left (6 \, a^{2} + 17\right )} b^{2} x^{2} - 16 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 5\right )} b x + 5 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 12 \, \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{4} + 6 \, {\left (2 \, a^{3} + 17 \, a\right )} b x + 3 \, {\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \, {\left (6 \, a^{2} + 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \, {\left (2 \, a^{3} + 5 \, a\right )} b x + 40 \, a^{2} + 17\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} - 6 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 17\right )} b x + 17 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{128 \, b} \]
-1/128*(3*b^4*x^4 + 12*a*b^3*x^3 + 3*(6*a^2 + 17)*b^2*x^2 - 16*(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 + 5)*b*x + 5*a)*sqrt(b^2*x^2 + 2*a*b*x + a^ 2 + 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - 12*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^4 + 6*(2*a^3 + 17*a)*b*x + 3*(8*b^4* x^4 + 32*a*b^3*x^3 + 8*(6*a^2 + 5)*b^2*x^2 + 8*a^4 + 16*(2*a^3 + 5*a)*b*x + 40*a^2 + 17)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 - 6*(2*b ^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 + 17)*b*x + 17*a)*sqrt(b^2*x^2 + 2*a *b*x + a^2 + 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)))/b
Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (223) = 446\).
Time = 1.15 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.95 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=\begin {cases} - \frac {3 a^{4} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16 b} - \frac {3 a^{3} x \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} - \frac {3 a^{3} x}{32} + \frac {a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4 b} + \frac {3 a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32 b} - \frac {9 a^{2} b x^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{8} - \frac {9 a^{2} b x^{2}}{64} + \frac {3 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4} + \frac {9 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32} - \frac {15 a^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16 b} - \frac {3 a b^{2} x^{3} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} - \frac {3 a b^{2} x^{3}}{32} + \frac {3 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4} + \frac {9 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32} - \frac {15 a x \operatorname {asinh}^{2}{\left (a + b x \right )}}{8} - \frac {51 a x}{64} + \frac {5 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{8 b} + \frac {51 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{64 b} - \frac {3 b^{3} x^{4} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16} - \frac {3 b^{3} x^{4}}{128} + \frac {b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4} + \frac {3 b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32} - \frac {15 b x^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16} - \frac {51 b x^{2}}{128} + \frac {5 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{8} + \frac {51 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{64} + \frac {3 \operatorname {asinh}^{4}{\left (a + b x \right )}}{32 b} - \frac {51 \operatorname {asinh}^{2}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \left (a^{2} + 1\right )^{\frac {3}{2}} \operatorname {asinh}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((-3*a**4*asinh(a + b*x)**2/(16*b) - 3*a**3*x*asinh(a + b*x)**2/4 - 3*a**3*x/32 + a**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)* *3/(4*b) + 3*a**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(32* b) - 9*a**2*b*x**2*asinh(a + b*x)**2/8 - 9*a**2*b*x**2/64 + 3*a**2*x*sqrt( a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**3/4 + 9*a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/32 - 15*a**2*asinh(a + b*x)**2/(1 6*b) - 3*a*b**2*x**3*asinh(a + b*x)**2/4 - 3*a*b**2*x**3/32 + 3*a*b*x**2*s qrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**3/4 + 9*a*b*x**2*sqrt( a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/32 - 15*a*x*asinh(a + b*x)* *2/8 - 51*a*x/64 + 5*a*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x) **3/(8*b) + 51*a*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(64*b ) - 3*b**3*x**4*asinh(a + b*x)**2/16 - 3*b**3*x**4/128 + b**2*x**3*sqrt(a* *2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**3/4 + 3*b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/32 - 15*b*x**2*asinh(a + b*x)**2 /16 - 51*b*x**2/128 + 5*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b *x)**3/8 + 51*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/64 + 3 *asinh(a + b*x)**4/(32*b) - 51*asinh(a + b*x)**2/(128*b), Ne(b, 0)), (x*(a **2 + 1)**(3/2)*asinh(a)**3, True))
\[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=\int { {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]
\[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=\int { {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=\int {\mathrm {asinh}\left (a+b\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2} \,d x \]