Integrand size = 26, antiderivative size = 386 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=-\frac {11 b d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{30 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {b d^2 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{160 a^4 \left (\frac {d}{x}\right )^{3/2}}-\frac {7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{1280 a^6 \sqrt {\frac {d}{x}}}+\frac {\left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{640 a^5}-\frac {\left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{240 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3}{3 a}-\frac {\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{512 a^{13/2}} \]
-1/512*(-231*b^6*d^3+1260*a*b^4*c*d^2-1680*a^2*b^2*c^2*d+320*a^3*c^3)*arct anh(1/2*(2*a+b*(d/x)^(1/2))/a^(1/2)/(a+c/x+b*(d/x)^(1/2))^(1/2))/a^(13/2)- 11/30*b*d^3*(a+c/x+b*(d/x)^(1/2))^(1/2)/a^2/(d/x)^(5/2)+1/160*b*d^2*(-77*b ^2*d+156*a*c)*(a+c/x+b*(d/x)^(1/2))^(1/2)/a^4/(d/x)^(3/2)+1/640*(385*b^4*d ^2-1176*a*b^2*c*d+400*a^2*c^2)*x*(a+c/x+b*(d/x)^(1/2))^(1/2)/a^5-1/240*(-9 9*b^2*d+100*a*c)*x^2*(a+c/x+b*(d/x)^(1/2))^(1/2)/a^3+1/3*x^3*(a+c/x+b*(d/x )^(1/2))^(1/2)/a-7/1280*b*d*(165*b^4*d^2-680*a*b^2*c*d+528*a^2*c^2)*(a+c/x +b*(d/x)^(1/2))^(1/2)/a^6/(d/x)^(1/2)
Time = 2.69 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.06 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\frac {\sqrt {a} d \left (-3465 b^5 d^2 \left (b d+c \sqrt {\frac {d}{x}}\right )+1280 a^6 x^3-64 a^5 x^2 \left (5 c+2 b \sqrt {\frac {d}{x}} x\right )+16 a^4 x \left (50 c^2+11 b^2 d x+46 b c \sqrt {\frac {d}{x}} x\right )-105 a b^3 d \left (-158 b c d-136 c^2 \sqrt {\frac {d}{x}}+11 b^2 d \sqrt {\frac {d}{x}} x\right )+42 a^2 b \left (-432 b c^2 d-264 c^3 \sqrt {\frac {d}{x}}+11 b^3 d^2 x+128 b^2 c d \sqrt {\frac {d}{x}} x\right )+24 a^3 \left (100 c^3-72 b^2 c d x-206 b c^2 \sqrt {\frac {d}{x}} x-11 b^3 \left (\frac {d}{x}\right )^{3/2} x^3\right )\right )-15 \sqrt {d} \left (-320 a^3 c^3+1680 a^2 b^2 c^2 d-1260 a b^4 c d^2+231 b^6 d^3\right ) \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{3840 a^{13/2} d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \]
(Sqrt[a]*d*(-3465*b^5*d^2*(b*d + c*Sqrt[d/x]) + 1280*a^6*x^3 - 64*a^5*x^2* (5*c + 2*b*Sqrt[d/x]*x) + 16*a^4*x*(50*c^2 + 11*b^2*d*x + 46*b*c*Sqrt[d/x] *x) - 105*a*b^3*d*(-158*b*c*d - 136*c^2*Sqrt[d/x] + 11*b^2*d*Sqrt[d/x]*x) + 42*a^2*b*(-432*b*c^2*d - 264*c^3*Sqrt[d/x] + 11*b^3*d^2*x + 128*b^2*c*d* Sqrt[d/x]*x) + 24*a^3*(100*c^3 - 72*b^2*c*d*x - 206*b*c^2*Sqrt[d/x]*x - 11 *b^3*(d/x)^(3/2)*x^3)) - 15*Sqrt[d]*(-320*a^3*c^3 + 1680*a^2*b^2*c^2*d - 1 260*a*b^4*c*d^2 + 231*b^6*d^3)*Sqrt[(d*(c + (a + b*Sqrt[d/x])*x))/x]*ArcTa nh[(Sqrt[c]*Sqrt[d/x] - Sqrt[(d*(c + a*x + b*Sqrt[d/x]*x))/x])/(Sqrt[a]*Sq rt[d])])/(3840*a^(13/2)*d*Sqrt[a + b*Sqrt[d/x] + c/x])
Time = 0.83 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {2066, 1693, 1167, 27, 1237, 27, 1237, 27, 1237, 27, 1237, 27, 1228, 1154, 219}
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.
| \(\displaystyle \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx\) |
| \(\Big \downarrow \) 2066 |
| \(\displaystyle -d^3 \int \frac {x^4}{d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}d\frac {d}{x}\) |
| \(\Big \downarrow \) 1693 |
| \(\displaystyle -2 d^3 \int \frac {x^7}{d^7 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}\) |
| \(\Big \downarrow \) 1167 |
| \(\displaystyle -2 d^3 \left (-\frac {\int \frac {\left (10 \sqrt {\frac {d}{x}} c+11 b d\right ) x^6}{2 d^7 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{6 a}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 d^3 \left (-\frac {\int \frac {\left (10 \sqrt {\frac {d}{x}} c+11 b d\right ) x^6}{d^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -2 d^3 \left (-\frac {-\frac {\int -\frac {\left (-99 d b^2-88 c \sqrt {\frac {d}{x}} b+100 a c\right ) x^5}{2 d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{5 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 d^3 \left (-\frac {\frac {\int \frac {\left (-99 d b^2-88 c \sqrt {\frac {d}{x}} b+100 a c\right ) x^5}{d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{10 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {\int \frac {3 \left (2 c \sqrt {\frac {d}{x}} \left (100 a c-99 b^2 d\right )+3 b d \left (156 a c-77 b^2 d\right )\right ) x^4}{2 d^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{4 a}-\frac {x^4 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}}{10 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {3 \int \frac {\left (2 c \sqrt {\frac {d}{x}} \left (100 a c-99 b^2 d\right )+3 b d \left (156 a c-77 b^2 d\right )\right ) x^4}{d^4 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{8 a d}-\frac {x^4 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}}{10 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {3 \left (-\frac {\int -\frac {3 \left (385 d^2 b^4-1176 a c d b^2-4 c \left (156 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}} b+400 a^2 c^2\right ) x^3}{2 d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{3 a}-\frac {b x^3 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a d^2}\right )}{8 a d}-\frac {x^4 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}}{10 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {3 \left (\frac {\int \frac {\left (385 d^2 b^4-1176 a c d b^2-4 c \left (156 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}} b+400 a^2 c^2\right ) x^3}{d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{2 a}-\frac {b x^3 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a d^2}\right )}{8 a d}-\frac {x^4 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}}{10 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {3 \left (\frac {-\frac {\int \frac {\left (7 b d \left (165 d^2 b^4-680 a c d b^2+528 a^2 c^2\right )+2 c \left (385 d^2 b^4-1176 a c d b^2+400 a^2 c^2\right ) \sqrt {\frac {d}{x}}\right ) x^2}{2 d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{2 a}-\frac {x^2 \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{2 a}-\frac {b x^3 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a d^2}\right )}{8 a d}-\frac {x^4 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}}{10 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {3 \left (\frac {-\frac {\int \frac {\left (7 b d \left (165 d^2 b^4-680 a c d b^2+528 a^2 c^2\right )+2 c \left (385 d^2 b^4-1176 a c d b^2+400 a^2 c^2\right ) \sqrt {\frac {d}{x}}\right ) x^2}{d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{4 a d}-\frac {x^2 \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{2 a}-\frac {b x^3 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a d^2}\right )}{8 a d}-\frac {x^4 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}}{10 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {3 \left (\frac {-\frac {\frac {5 \left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \int \frac {x}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{2 a}-\frac {7 b x \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a}}{4 a d}-\frac {x^2 \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{2 a}-\frac {b x^3 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a d^2}\right )}{8 a d}-\frac {x^4 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}}{10 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {3 \left (\frac {-\frac {-\frac {5 \left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \int \frac {1}{4 a-\frac {d^2}{x^2}}d\frac {2 a+b \sqrt {\frac {d}{x}}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}}{a}-\frac {7 b x \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a}}{4 a d}-\frac {x^2 \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}}{2 a}-\frac {b x^3 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a d^2}\right )}{8 a d}-\frac {x^4 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}}{10 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {3 \left (\frac {-\frac {x^2 \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{2 a d^2}-\frac {-\frac {7 b x \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a}-\frac {5 \left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}\right )}{2 a^{3/2}}}{4 a d}}{2 a}-\frac {b x^3 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{a d^2}\right )}{8 a d}-\frac {x^4 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^4}}{10 a}-\frac {11 b x^5 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{5 a d^4}}{12 a d}-\frac {x^6 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{6 a d^6}\right )\) |
-2*d^3*(-1/6*(Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x^6)/(a*d^6) - ((-11*b*Sqr t[a + b*Sqrt[d/x] + (c*d)/x^2]*x^5)/(5*a*d^4) + (-1/4*((100*a*c - 99*b^2*d )*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x^4)/(a*d^4) - (3*(-((b*(156*a*c - 77* b^2*d)*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x^3)/(a*d^2)) + (-1/2*((400*a^2*c ^2 - 1176*a*b^2*c*d + 385*b^4*d^2)*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2]*x^2)/ (a*d^2) - ((-7*b*(528*a^2*c^2 - 680*a*b^2*c*d + 165*b^4*d^2)*Sqrt[a + b*Sq rt[d/x] + (c*d)/x^2]*x)/a - (5*(320*a^3*c^3 - 1680*a^2*b^2*c^2*d + 1260*a* b^4*c*d^2 - 231*b^6*d^3)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b *Sqrt[d/x] + (c*d)/x^2])])/(2*a^(3/2)))/(4*a*d))/(2*a)))/(8*a*d))/(10*a))/ (12*a*d))
3.31.61.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d ^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[ (d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m , -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ [Simplify[(m + 1)/n]]
Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_), x _Symbol] :> Simp[-d^(m + 1) Subst[Int[(a + b*x^n + (c/d^(2*n))*x^(2*n))^p /x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, -2*n ] && IntegerQ[2*n] && IntegerQ[m]
Time = 0.26 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \sqrt {x}\, \left (2560 x^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {13}{2}}-6930 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{\frac {5}{2}} b^{5}+3168 d \,a^{\frac {9}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, x^{\frac {3}{2}} b^{2}-3696 a^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{\frac {5}{2}} b^{3}+3465 d^{3} \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a \,b^{6}+4620 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {5}{2}} d^{2} \sqrt {x}\, b^{4}-2816 a^{\frac {11}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x^{\frac {5}{2}} b -3200 a^{\frac {11}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c \,x^{\frac {3}{2}}+28560 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {5}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3} c -14112 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} d \sqrt {x}\, b^{2} c +7488 a^{\frac {9}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x^{\frac {3}{2}} b c -18900 d^{2} \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{2} b^{4} c +4800 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {9}{2}} c^{2} \sqrt {x}-22176 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} \sqrt {\frac {d}{x}}\, \sqrt {x}\, b \,c^{2}+25200 d \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{3} b^{2} c^{2}-4800 \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{4} c^{3}\right )}{7680 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {15}{2}}}\) | \(655\) |
1/7680*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(2560*x^(5/2)*(b*(d/x)^(1 /2)*x+a*x+c)^(1/2)*a^(13/2)-6930*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(3/2)*(d/ x)^(5/2)*x^(5/2)*b^5+3168*d*a^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^(3/2)* b^2-3696*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^(5/2)*b^3+346 5*d^3*ln(1/2*((d/x)^(1/2)*x^(1/2)*b+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2 )+2*a*x^(1/2))/a^(1/2))*a*b^6+4620*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(5/2)*d ^2*x^(1/2)*b^4-2816*a^(11/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^( 5/2)*b-3200*a^(11/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*c*x^(3/2)+28560*(b*(d/x )^(1/2)*x+a*x+c)^(1/2)*a^(5/2)*(d/x)^(3/2)*x^(3/2)*b^3*c-14112*(b*(d/x)^(1 /2)*x+a*x+c)^(1/2)*a^(7/2)*d*x^(1/2)*b^2*c+7488*a^(9/2)*(b*(d/x)^(1/2)*x+a *x+c)^(1/2)*(d/x)^(1/2)*x^(3/2)*b*c-18900*d^2*ln(1/2*((d/x)^(1/2)*x^(1/2)* b+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^2*b^4*c+ 4800*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(9/2)*c^2*x^(1/2)-22176*(b*(d/x)^(1/2 )*x+a*x+c)^(1/2)*a^(7/2)*(d/x)^(1/2)*x^(1/2)*b*c^2+25200*d*ln(1/2*((d/x)^( 1/2)*x^(1/2)*b+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2 ))*a^3*b^2*c^2-4800*ln(1/2*((d/x)^(1/2)*x^(1/2)*b+2*(b*(d/x)^(1/2)*x+a*x+c )^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^4*c^3)/(b*(d/x)^(1/2)*x+a*x+c)^(1/ 2)/a^(15/2)
Timed out. \[ \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\int \frac {x^{2}}{\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \]
\[ \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\int { \frac {x^{2}}{\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}} \,d x } \]
Time = 0.43 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.40 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=-\frac {2 \, \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}} {\left (2 \, \sqrt {d x} {\left (4 \, \sqrt {d x} {\left (2 \, \sqrt {d x} {\left (8 \, \sqrt {d x} {\left (\frac {11 \, b}{a^{2} d^{2}} - \frac {10 \, \sqrt {d x}}{a d^{3}}\right )} - \frac {99 \, a^{3} b^{2} d^{2} - 100 \, a^{4} c d}{a^{6} d^{3}}\right )} + \frac {3 \, {\left (77 \, a^{2} b^{3} d^{3} - 156 \, a^{3} b c d^{2}\right )}}{a^{6} d^{3}}\right )} - \frac {3 \, {\left (385 \, a b^{4} d^{4} - 1176 \, a^{2} b^{2} c d^{3} + 400 \, a^{3} c^{2} d^{2}\right )}}{a^{6} d^{3}}\right )} + \frac {21 \, {\left (165 \, b^{5} d^{5} - 680 \, a b^{3} c d^{4} + 528 \, a^{2} b c^{2} d^{3}\right )}}{a^{6} d^{3}}\right )} + \frac {15 \, {\left (231 \, b^{6} d^{4} - 1260 \, a b^{4} c d^{3} + 1680 \, a^{2} b^{2} c^{2} d^{2} - 320 \, a^{3} c^{3} d\right )} \log \left ({\left | -b d^{2} - 2 \, \sqrt {a d} {\left (\sqrt {a d} \sqrt {d x} - \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}}\right )} \right |}\right )}{\sqrt {a d} a^{6}} - \frac {3 \, {\left (1155 \, b^{6} d^{4} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) - 6300 \, a b^{4} c d^{3} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 8400 \, a^{2} b^{2} c^{2} d^{2} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 2310 \, \sqrt {c d^{2}} \sqrt {a d} b^{5} d^{2} - 1600 \, a^{3} c^{3} d \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) - 9520 \, \sqrt {c d^{2}} \sqrt {a d} a b^{3} c d + 7392 \, \sqrt {c d^{2}} \sqrt {a d} a^{2} b c^{2}\right )}}{\sqrt {a d} a^{6}}}{7680 \, \sqrt {d} \mathrm {sgn}\left (x\right )} \]
-1/7680*(2*sqrt(a*d^2*x + sqrt(d*x)*b*d^2 + c*d^2)*(2*sqrt(d*x)*(4*sqrt(d* x)*(2*sqrt(d*x)*(8*sqrt(d*x)*(11*b/(a^2*d^2) - 10*sqrt(d*x)/(a*d^3)) - (99 *a^3*b^2*d^2 - 100*a^4*c*d)/(a^6*d^3)) + 3*(77*a^2*b^3*d^3 - 156*a^3*b*c*d ^2)/(a^6*d^3)) - 3*(385*a*b^4*d^4 - 1176*a^2*b^2*c*d^3 + 400*a^3*c^2*d^2)/ (a^6*d^3)) + 21*(165*b^5*d^5 - 680*a*b^3*c*d^4 + 528*a^2*b*c^2*d^3)/(a^6*d ^3)) + 15*(231*b^6*d^4 - 1260*a*b^4*c*d^3 + 1680*a^2*b^2*c^2*d^2 - 320*a^3 *c^3*d)*log(abs(-b*d^2 - 2*sqrt(a*d)*(sqrt(a*d)*sqrt(d*x) - sqrt(a*d^2*x + sqrt(d*x)*b*d^2 + c*d^2))))/(sqrt(a*d)*a^6) - 3*(1155*b^6*d^4*log(abs(-b* d^2 + 2*sqrt(c*d^2)*sqrt(a*d))) - 6300*a*b^4*c*d^3*log(abs(-b*d^2 + 2*sqrt (c*d^2)*sqrt(a*d))) + 8400*a^2*b^2*c^2*d^2*log(abs(-b*d^2 + 2*sqrt(c*d^2)* sqrt(a*d))) + 2310*sqrt(c*d^2)*sqrt(a*d)*b^5*d^2 - 1600*a^3*c^3*d*log(abs( -b*d^2 + 2*sqrt(c*d^2)*sqrt(a*d))) - 9520*sqrt(c*d^2)*sqrt(a*d)*a*b^3*c*d + 7392*sqrt(c*d^2)*sqrt(a*d)*a^2*b*c^2)/(sqrt(a*d)*a^6))/(sqrt(d)*sgn(x))
Timed out. \[ \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx=\int \frac {x^2}{\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \]