Integrand size = 22, antiderivative size = 406 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d-128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^2 x}-\frac {\left (3 b^3 c^3+109 a b^2 c^2 d-19 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x^2}-\frac {\left (3 b^2 c^2+16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 c^2 x^3}-\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}-\frac {(b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{5/2}}+2 b^{5/2} d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
-1/8*(a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(5/2)/c/x^4-1/5*(b*x+a)^(5/2)*(d*x+c) ^(5/2)/x^5-1/128*(a*d+b*c)*(3*a^4*d^4-28*a^3*b*c*d^3+178*a^2*b^2*c^2*d^2-2 8*a*b^3*c^3*d+3*b^4*c^4)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/ 2))/a^(5/2)/c^(5/2)+2*b^(5/2)*d^(5/2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2 )/(d*x+c)^(1/2))-1/192*(3*a^3*d^3-19*a^2*b*c*d^2+109*a*b^2*c^2*d+3*b^3*c^3 )*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a/c^2/x^2-1/48*(-3*a^2*d^2+16*a*b*c*d+3*b^2* c^2)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/c^2/x^3+1/128*(-3*a^4*d^4+22*a^3*b*c*d^3- 128*a^2*b^2*c^2*d^2-22*a*b^3*c^3*d+3*b^4*c^4)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/ a^2/c^2/x
Time = 1.22 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 b^4 c^4 x^4+30 a b^3 c^3 x^3 (c+12 d x)+2 a^2 b^2 c^2 x^2 \left (372 c^2+1289 c d x+1877 d^2 x^2\right )+2 a^3 b c x \left (504 c^3+1448 c^2 d x+1289 c d^2 x^2+180 d^3 x^3\right )+3 a^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )\right )}{1920 a^2 c^2 x^5}-\frac {\left (3 b^5 c^5-25 a b^4 c^4 d+150 a^2 b^3 c^3 d^2+150 a^3 b^2 c^2 d^3-25 a^4 b c d^4+3 a^5 d^5\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{5/2} c^{5/2}}+2 b^{5/2} d^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \]
-1/1920*(Sqrt[a + b*x]*Sqrt[c + d*x]*(-45*b^4*c^4*x^4 + 30*a*b^3*c^3*x^3*( c + 12*d*x) + 2*a^2*b^2*c^2*x^2*(372*c^2 + 1289*c*d*x + 1877*d^2*x^2) + 2* a^3*b*c*x*(504*c^3 + 1448*c^2*d*x + 1289*c*d^2*x^2 + 180*d^3*x^3) + 3*a^4* (128*c^4 + 336*c^3*d*x + 248*c^2*d^2*x^2 + 10*c*d^3*x^3 - 15*d^4*x^4)))/(a ^2*c^2*x^5) - ((3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3 *b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x]) /(Sqrt[c]*Sqrt[a + b*x])])/(128*a^(5/2)*c^(5/2)) + 2*b^(5/2)*d^(5/2)*ArcTa nh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])]
Time = 0.59 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {108, 27, 166, 27, 166, 27, 166, 27, 166, 27, 175, 66, 104, 221}
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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{5} \int \frac {5 (a+b x)^{3/2} (c+d x)^{3/2} (b c+a d+2 b d x)}{2 x^5}dx-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2} (b c+a d+2 b d x)}{x^5}dx-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 c^2 b^2+16 c d x b^2+16 a c d b-3 a^2 d^2\right )}{2 x^4}dx}{4 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 c^2 b^2+16 c d x b^2+16 a c d b-3 a^2 d^2\right )}{x^4}dx}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {(c+d x)^{3/2} \left (3 c^3 b^3+96 c^2 d x b^3+109 a c^2 d b^2-19 a^2 c d^2 b+3 a^3 d^3\right )}{2 x^3 \sqrt {a+b x}}dx}{3 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{3 c x^3}}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {(c+d x)^{3/2} \left (3 c^3 b^3+96 c^2 d x b^3+109 a c^2 d b^2-19 a^2 c d^2 b+3 a^3 d^3\right )}{x^3 \sqrt {a+b x}}dx}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{3 c x^3}}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {\int -\frac {3 \sqrt {c+d x} \left (3 b^4 c^4-22 a b^3 d c^3-128 a^2 b^2 d^2 c^2-128 a b^3 d^2 x c^2+22 a^3 b d^3 c-3 a^4 d^4\right )}{2 x^2 \sqrt {a+b x}}dx}{2 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{3 c x^3}}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {-\frac {3 \int \frac {\sqrt {c+d x} \left (3 b^4 c^4-22 a b^3 d c^3-128 a^2 b^2 d^2 c^2-128 a b^3 d^2 x c^2+22 a^3 b d^3 c-3 a^4 d^4\right )}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{3 c x^3}}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {-\frac {3 \left (\frac {\int -\frac {256 a^2 b^3 c^2 x d^3+(b c+a d) \left (3 b^4 c^4-28 a b^3 d c^3+178 a^2 b^2 d^2 c^2-28 a^3 b d^3 c+3 a^4 d^4\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3-128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{3 c x^3}}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {-\frac {3 \left (-\frac {\int \frac {256 a^2 b^3 c^2 x d^3+(b c+a d) \left (3 b^4 c^4-28 a b^3 d c^3+178 a^2 b^2 d^2 c^2-28 a^3 b d^3 c+3 a^4 d^4\right )}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3-128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{3 c x^3}}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {-\frac {3 \left (-\frac {256 a^2 b^3 c^2 d^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx+(a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3-128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{3 c x^3}}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {-\frac {3 \left (-\frac {512 a^2 b^3 c^2 d^3 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+(a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3-128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{3 c x^3}}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {-\frac {3 \left (-\frac {512 a^2 b^3 c^2 d^3 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+2 (a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3-128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{3 c x^3}}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{2 a x^2}-\frac {3 \left (-\frac {512 a^2 b^{5/2} c^2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {2 (a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} \sqrt {c}}}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3-128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{a x}\right )}{4 a}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{3 c x^3}}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x^4}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}\) |
-1/5*((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^5 + (-1/4*((b*c + a*d)*(a + b*x)^ (3/2)*(c + d*x)^(5/2))/(c*x^4) + (-1/3*((3*b^2*c^2 + 16*a*b*c*d - 3*a^2*d^ 2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(c*x^3) + (-1/2*((3*b^3*c^3 + 109*a*b^2* c^2*d - 19*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(a*x^2) - (3*(-(((3*b^4*c^4 - 22*a*b^3*c^3*d - 128*a^2*b^2*c^2*d^2 + 22*a^3*b*c*d ^3 - 3*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) - ((-2*(b*c + a*d)*(3* b^4*c^4 - 28*a*b^3*c^3*d + 178*a^2*b^2*c^2*d^2 - 28*a^3*b*c*d^3 + 3*a^4*d^ 4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*Sqrt [c]) + 512*a^2*b^(5/2)*c^2*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b ]*Sqrt[c + d*x])])/(2*a)))/(4*a))/(6*c))/(8*c))/2
3.7.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Leaf count of result is larger than twice the leaf count of optimal. \(984\) vs. \(2(350)=700\).
Time = 0.75 (sec) , antiderivative size = 985, normalized size of antiderivative = 2.43
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{5} d^{5} x^{5} \sqrt {b d}-375 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{4} b c \,d^{4} x^{5} \sqrt {b d}+2250 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} b^{2} c^{2} d^{3} x^{5} \sqrt {b d}+2250 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b^{3} c^{3} d^{2} x^{5} \sqrt {b d}-375 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{4} c^{4} d \,x^{5} \sqrt {b d}+45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{5} c^{5} x^{5} \sqrt {b d}-3840 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{3} c^{2} d^{3} x^{5} \sqrt {a c}-90 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} d^{4} x^{4}+720 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} b c \,d^{3} x^{4}+7508 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}+720 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{3} c^{3} d \,x^{4}-90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, b^{4} c^{4} x^{4}+60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{4} c \,d^{3} x^{3}+5156 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} b \,c^{2} d^{2} x^{3}+5156 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b^{2} c^{3} d \,x^{3}+60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{3} c^{4} x^{3}+1488 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{4} c^{2} d^{2} x^{2}+5792 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} b \,c^{3} d \,x^{2}+1488 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b^{2} c^{4} x^{2}+2016 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{4} c^{3} d x +2016 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} b \,c^{4} x +768 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{4} c^{4}\right )}{3840 a^{2} c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{5} \sqrt {b d}\, \sqrt {a c}}\) | \(985\) |
-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^2*(45*ln((a*d*x+b*c*x+2*(a*c)^(1 /2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^5*d^5*x^5*(b*d)^(1/2)-375*ln((a*d* x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^4*b*c*d^4*x^5*(b *d)^(1/2)+2250*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c )/x)*a^3*b^2*c^2*d^3*x^5*(b*d)^(1/2)+2250*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(( b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b^3*c^3*d^2*x^5*(b*d)^(1/2)-375*ln((a* d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b^4*c^4*d*x^5* (b*d)^(1/2)+45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c )/x)*b^5*c^5*x^5*(b*d)^(1/2)-3840*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2 )*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^3*c^2*d^3*x^5*(a*c)^(1/2)-90*(b* d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*d^4*x^4+720*((b*x+a)*(d*x +c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^3*b*c*d^3*x^4+7508*((b*x+a)*(d*x+c))^ (1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^2*b^2*c^2*d^2*x^4+720*((b*x+a)*(d*x+c))^(1 /2)*(b*d)^(1/2)*(a*c)^(1/2)*a*b^3*c^3*d*x^4-90*((b*x+a)*(d*x+c))^(1/2)*(b* d)^(1/2)*(a*c)^(1/2)*b^4*c^4*x^4+60*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a *c)^(1/2)*a^4*c*d^3*x^3+5156*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/ 2)*a^3*b*c^2*d^2*x^3+5156*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)* a^2*b^2*c^3*d*x^3+60*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a*b^3 *c^4*x^3+1488*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^4*c^2*d^2* x^2+5792*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^3*b*c^3*d*x^...
Time = 10.20 (sec) , antiderivative size = 1849, normalized size of antiderivative = 4.55 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx=\text {Too large to display} \]
[1/7680*(3840*sqrt(b*d)*a^3*b^2*c^3*d^2*x^5*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt (d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 15*(3*b^5*c^5 - 25*a*b^4*c^4*d + 15 0*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt (a*c)*x^5*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2* c*d)*x)/x^2) - 4*(384*a^5*c^5 - (45*a*b^4*c^5 - 360*a^2*b^3*c^4*d - 3754*a ^3*b^2*c^3*d^2 - 360*a^4*b*c^2*d^3 + 45*a^5*c*d^4)*x^4 + 2*(15*a^2*b^3*c^5 + 1289*a^3*b^2*c^4*d + 1289*a^4*b*c^3*d^2 + 15*a^5*c^2*d^3)*x^3 + 8*(93*a ^3*b^2*c^5 + 362*a^4*b*c^4*d + 93*a^5*c^3*d^2)*x^2 + 1008*(a^4*b*c^5 + a^5 *c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^3*x^5), -1/7680*(7680*sqrt( -b*d)*a^3*b^2*c^3*d^2*x^5*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt (b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 15*(3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^ 3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(a*c)*x^5*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(384*a^5*c^5 - (45*a* b^4*c^5 - 360*a^2*b^3*c^4*d - 3754*a^3*b^2*c^3*d^2 - 360*a^4*b*c^2*d^3 + 4 5*a^5*c*d^4)*x^4 + 2*(15*a^2*b^3*c^5 + 1289*a^3*b^2*c^4*d + 1289*a^4*b*c^3 *d^2 + 15*a^5*c^2*d^3)*x^3 + 8*(93*a^3*b^2*c^5 + 362*a^4*b*c^4*d + 93*a...
\[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{6}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 5983 vs. \(2 (350) = 700\).
Time = 2.16 (sec) , antiderivative size = 5983, normalized size of antiderivative = 14.74 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx=\text {Too large to display} \]
-1/1920*(1920*sqrt(b*d)*b^2*d^2*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt (b^2*c + (b*x + a)*b*d - a*b*d))^2) + 15*(3*sqrt(b*d)*b^6*c^5*abs(b) - 25* sqrt(b*d)*a*b^5*c^4*d*abs(b) + 150*sqrt(b*d)*a^2*b^4*c^3*d^2*abs(b) + 150* sqrt(b*d)*a^3*b^3*c^2*d^3*abs(b) - 25*sqrt(b*d)*a^4*b^2*c*d^4*abs(b) + 3*s qrt(b*d)*a^5*b*d^5*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b* x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt (-a*b*c*d)*a^2*b*c^2) - 2*(45*sqrt(b*d)*b^24*c^14*abs(b) - 810*sqrt(b*d)*a *b^23*c^13*d*abs(b) + 1871*sqrt(b*d)*a^2*b^22*c^12*d^2*abs(b) + 15580*sqrt (b*d)*a^3*b^21*c^11*d^3*abs(b) - 112635*sqrt(b*d)*a^4*b^20*c^10*d^4*abs(b) + 346890*sqrt(b*d)*a^5*b^19*c^9*d^5*abs(b) - 642945*sqrt(b*d)*a^6*b^18*c^ 8*d^6*abs(b) + 784008*sqrt(b*d)*a^7*b^17*c^7*d^7*abs(b) - 642945*sqrt(b*d) *a^8*b^16*c^6*d^8*abs(b) + 346890*sqrt(b*d)*a^9*b^15*c^5*d^9*abs(b) - 1126 35*sqrt(b*d)*a^10*b^14*c^4*d^10*abs(b) + 15580*sqrt(b*d)*a^11*b^13*c^3*d^1 1*abs(b) + 1871*sqrt(b*d)*a^12*b^12*c^2*d^12*abs(b) - 810*sqrt(b*d)*a^13*b ^11*c*d^13*abs(b) + 45*sqrt(b*d)*a^14*b^10*d^14*abs(b) - 405*sqrt(b*d)*(sq rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^22*c^13*a bs(b) + 6015*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b *d - a*b*d))^2*a*b^21*c^12*d*abs(b) - 1670*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^20*c^11*d^2*abs(b) - 12 2710*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - ...
Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^6} \,d x \]