Integrand size = 25, antiderivative size = 601 \[ \int x^2 (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=-\frac {4 c \left (120 b^2 c^4-245 a b c^2 d^2+1629 a^2 d^4\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{45045 b^2 d^3}-\frac {4 \left (120 b^2 c^4-95 a b c^2 d^2+539 a^2 d^4\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{45045 b^2 d^3}-\frac {4 c \left (24 b c^2+23 a d^2\right ) (c+d x)^{5/2} \sqrt {a-b x^2}}{9009 b d^3}+\frac {2 (c+d x)^{7/2} \left (24 b c^2+77 a d^2-54 b c d x\right ) \sqrt {a-b x^2}}{1287 b d^3}-\frac {2 (c+d x)^{7/2} \left (a-b x^2\right )^{3/2}}{13 b d}-\frac {4 \sqrt {a} \left (120 b^3 c^6-335 a b^2 c^4 d^2+4614 a^2 b c^2 d^4+1617 a^3 d^6\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{45045 b^{5/2} d^4 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {4 \sqrt {a} c \left (120 b^3 c^6-365 a b^2 c^4 d^2+1874 a^2 b c^2 d^4-1629 a^3 d^6\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{45045 b^{5/2} d^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-4/45045*c*(1629*a^2*d^4-245*a*b*c^2*d^2+120*b^2*c^4)*(d*x+c)^(1/2)*(-b*x^ 2+a)^(1/2)/b^2/d^3-4/45045*(539*a^2*d^4-95*a*b*c^2*d^2+120*b^2*c^4)*(d*x+c )^(3/2)*(-b*x^2+a)^(1/2)/b^2/d^3-4/9009*c*(23*a*d^2+24*b*c^2)*(d*x+c)^(5/2 )*(-b*x^2+a)^(1/2)/b/d^3+2/1287*(d*x+c)^(7/2)*(-54*b*c*d*x+77*a*d^2+24*b*c ^2)*(-b*x^2+a)^(1/2)/b/d^3-2/13*(d*x+c)^(7/2)*(-b*x^2+a)^(3/2)/b/d-4/45045 *a^(1/2)*(1617*a^3*d^6+4614*a^2*b*c^2*d^4-335*a*b^2*c^4*d^2+120*b^3*c^6)*( d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2 ^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(5/2)/d^4/(b^(1/ 2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)+4/45045*a^(1/2)*c *(-1629*a^3*d^6+1874*a^2*b*c^2*d^4-365*a*b^2*c^4*d^2+120*b^3*c^6)*(b^(1/2) *(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/2)*EllipticF(1/2*(1-b ^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^ (1/2))/b^(5/2)/d^4/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.78 (sec) , antiderivative size = 753, normalized size of antiderivative = 1.25 \[ \int x^2 (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\frac {2 \sqrt {a-b x^2} \left (b (c+d x) \left (-2 a^2 d^4 (2168 c+539 d x)-10 a b d^2 \left (32 c^3+258 c^2 d x+254 c d^2 x^2+77 d^3 x^3\right )+15 b^2 \left (8 c^5-6 c^4 d x+5 c^3 d^2 x^2+371 c^2 d^3 x^3+567 c d^4 x^4+231 d^5 x^5\right )\right )-\frac {2 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (120 b^3 c^6-335 a b^2 c^4 d^2+4614 a^2 b c^2 d^4+1617 a^3 d^6\right ) \left (a-b x^2\right )+i \sqrt {b} \left (120 b^{7/2} c^7-120 \sqrt {a} b^3 c^6 d-335 a b^{5/2} c^5 d^2+335 a^{3/2} b^2 c^4 d^3+4614 a^2 b^{3/2} c^3 d^4-4614 a^{5/2} b c^2 d^5+1617 a^3 \sqrt {b} c d^6-1617 a^{7/2} d^7\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (120 b^3 c^6-30 \sqrt {a} b^{5/2} c^5 d-335 a b^2 c^4 d^2-2740 a^{3/2} b^{3/2} c^3 d^3+4614 a^2 b c^2 d^4-3246 a^{5/2} \sqrt {b} c d^5+1617 a^3 d^6\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{45045 b^3 d^3 \sqrt {c+d x}} \] Input:
Integrate[x^2*(c + d*x)^(5/2)*Sqrt[a - b*x^2],x]
Output:
(2*Sqrt[a - b*x^2]*(b*(c + d*x)*(-2*a^2*d^4*(2168*c + 539*d*x) - 10*a*b*d^ 2*(32*c^3 + 258*c^2*d*x + 254*c*d^2*x^2 + 77*d^3*x^3) + 15*b^2*(8*c^5 - 6* c^4*d*x + 5*c^3*d^2*x^2 + 371*c^2*d^3*x^3 + 567*c*d^4*x^4 + 231*d^5*x^5)) - (2*(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(120*b^3*c^6 - 335*a*b^2*c^4*d^2 + 4614*a^2*b*c^2*d^4 + 1617*a^3*d^6)*(a - b*x^2) + I*Sqrt[b]*(120*b^(7/2)* c^7 - 120*Sqrt[a]*b^3*c^6*d - 335*a*b^(5/2)*c^5*d^2 + 335*a^(3/2)*b^2*c^4* d^3 + 4614*a^2*b^(3/2)*c^3*d^4 - 4614*a^(5/2)*b*c^2*d^5 + 1617*a^3*Sqrt[b] *c*d^6 - 1617*a^(7/2)*d^7)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[ -(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcS inh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d) /(Sqrt[b]*c - Sqrt[a]*d)] + I*Sqrt[a]*Sqrt[b]*d*(120*b^3*c^6 - 30*Sqrt[a]* b^(5/2)*c^5*d - 335*a*b^2*c^4*d^2 - 2740*a^(3/2)*b^(3/2)*c^3*d^3 + 4614*a^ 2*b*c^2*d^4 - 3246*a^(5/2)*Sqrt[b]*c*d^5 + 1617*a^3*d^6)*Sqrt[(d*(Sqrt[a]/ Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)]))/(d^2*Sqrt[-c + ( Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2))))/(45045*b^3*d^3*Sqrt[c + d*x])
Time = 0.93 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {604, 27, 687, 27, 687, 27, 687, 27, 682, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int x^2 \sqrt {a-b x^2} (c+d x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 604 |
| \(\displaystyle -\frac {2 \int -\frac {1}{2} d (7 a d-6 b c x) (c+d x)^{5/2} \sqrt {a-b x^2}dx}{13 b d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (7 a d-6 b c x) (c+d x)^{5/2} \sqrt {a-b x^2}dx}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}-\frac {2 \int -\frac {1}{2} b (c+d x)^{3/2} \left (47 a c d-\left (30 b c^2-77 a d^2\right ) x\right ) \sqrt {a-b x^2}dx}{11 b}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \int (c+d x)^{3/2} \left (47 a c d-\left (30 b c^2-77 a d^2\right ) x\right ) \sqrt {a-b x^2}dx+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}-\frac {2 \int -\frac {3}{2} \sqrt {c+d x} \left (a d \left (111 b c^2+77 a d^2\right )-2 b c \left (15 b c^2-109 a d^2\right ) x\right ) \sqrt {a-b x^2}dx}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\int \sqrt {c+d x} \left (a d \left (111 b c^2+77 a d^2\right )-2 b c \left (15 b c^2-109 a d^2\right ) x\right ) \sqrt {a-b x^2}dx}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )-\frac {2 \int -\frac {b \left (a c d \left (747 b c^2+757 a d^2\right )-\left (30 b^2 c^4-995 a b d^2 c^2-539 a^2 d^4\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{7 b}}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {1}{7} \int \frac {\left (a c d \left (747 b c^2+757 a d^2\right )-\left (30 b^2 c^4-995 a b d^2 c^2-539 a^2 d^4\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx+\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {1}{7} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right )-3 d x \left (-539 a^2 d^4-995 a b c^2 d^2+30 b^2 c^4\right )\right )}{15 d^2}-\frac {4 \int -\frac {b \left (2 a c d \left (15 b^2 c^4+1370 a b d^2 c^2+1623 a^2 d^4\right )+\left (120 b^3 c^6-335 a b^2 d^2 c^4+4614 a^2 b d^4 c^2+1617 a^3 d^6\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}\right )+\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {1}{7} \left (\frac {2 \int \frac {2 a c d \left (15 b^2 c^4+1370 a b d^2 c^2+1623 a^2 d^4\right )+\left (120 b^3 c^6-335 a b^2 d^2 c^4+4614 a^2 b d^4 c^2+1617 a^3 d^6\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right )-3 d x \left (-539 a^2 d^4-995 a b c^2 d^2+30 b^2 c^4\right )\right )}{15 d^2}\right )+\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {1}{7} \left (\frac {2 \left (\frac {\left (1617 a^3 d^6+4614 a^2 b c^2 d^4-335 a b^2 c^4 d^2+120 b^3 c^6\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \left (b c^2-a d^2\right ) \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right )-3 d x \left (-539 a^2 d^4-995 a b c^2 d^2+30 b^2 c^4\right )\right )}{15 d^2}\right )+\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {1}{7} \left (\frac {2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (1617 a^3 d^6+4614 a^2 b c^2 d^4-335 a b^2 c^4 d^2+120 b^3 c^6\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {c \left (b c^2-a d^2\right ) \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right )-3 d x \left (-539 a^2 d^4-995 a b c^2 d^2+30 b^2 c^4\right )\right )}{15 d^2}\right )+\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {1}{7} \left (\frac {2 \left (-\frac {c \left (b c^2-a d^2\right ) \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (1617 a^3 d^6+4614 a^2 b c^2 d^4-335 a b^2 c^4 d^2+120 b^3 c^6\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right )-3 d x \left (-539 a^2 d^4-995 a b c^2 d^2+30 b^2 c^4\right )\right )}{15 d^2}\right )+\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {1}{7} \left (\frac {2 \left (-\frac {c \left (b c^2-a d^2\right ) \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (1617 a^3 d^6+4614 a^2 b c^2 d^4-335 a b^2 c^4 d^2+120 b^3 c^6\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right )-3 d x \left (-539 a^2 d^4-995 a b c^2 d^2+30 b^2 c^4\right )\right )}{15 d^2}\right )+\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {1}{7} \left (\frac {2 \left (-\frac {c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (1617 a^3 d^6+4614 a^2 b c^2 d^4-335 a b^2 c^4 d^2+120 b^3 c^6\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right )-3 d x \left (-539 a^2 d^4-995 a b c^2 d^2+30 b^2 c^4\right )\right )}{15 d^2}\right )+\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {1}{7} \left (\frac {2 \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (1617 a^3 d^6+4614 a^2 b c^2 d^4-335 a b^2 c^4 d^2+120 b^3 c^6\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right )-3 d x \left (-539 a^2 d^4-995 a b c^2 d^2+30 b^2 c^4\right )\right )}{15 d^2}\right )+\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {\frac {1}{7} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right )-3 d x \left (-539 a^2 d^4-995 a b c^2 d^2+30 b^2 c^4\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (1629 a^2 d^4-245 a b c^2 d^2+120 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (1617 a^3 d^6+4614 a^2 b c^2 d^4-335 a b^2 c^4 d^2+120 b^3 c^6\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}\right )+\frac {4}{7} c \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 b c^2-109 a d^2\right )}{3 b}+\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (30 b c^2-77 a d^2\right )}{9 b}\right )+\frac {12}{11} c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{13 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d}\) |
Input:
Int[x^2*(c + d*x)^(5/2)*Sqrt[a - b*x^2],x]
Output:
(-2*(c + d*x)^(7/2)*(a - b*x^2)^(3/2))/(13*b*d) + ((12*c*(c + d*x)^(5/2)*( a - b*x^2)^(3/2))/11 + ((2*(30*b*c^2 - 77*a*d^2)*(c + d*x)^(3/2)*(a - b*x^ 2)^(3/2))/(9*b) + ((4*c*(15*b*c^2 - 109*a*d^2)*Sqrt[c + d*x]*(a - b*x^2)^( 3/2))/7 + ((2*Sqrt[c + d*x]*(c*(120*b^2*c^4 - 245*a*b*c^2*d^2 + 1629*a^2*d ^4) - 3*d*(30*b^2*c^4 - 995*a*b*c^2*d^2 - 539*a^2*d^4)*x)*Sqrt[a - b*x^2]) /(15*d^2) + (2*((-2*Sqrt[a]*(120*b^3*c^6 - 335*a*b^2*c^4*d^2 + 4614*a^2*b* c^2*d^4 + 1617*a^3*d^6)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin [Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)]) /(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x ^2]) + (2*Sqrt[a]*c*(b*c^2 - a*d^2)*(120*b^2*c^4 - 245*a*b*c^2*d^2 + 1629* a^2*d^4)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2 )/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt [b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(15*d^2) )/7)/(3*b))/11)/(13*b*d)
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_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[(c + d*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*d^(m - 1)*(m + n + 2*p + 1))), x] + Simp[1/(b*d^m*(m + n + 2*p + 1)) Int[(c + d*x)^n*(a + b* x^2)^p*ExpandToSum[b*d^m*(m + n + 2*p + 1)*x^m - b*(m + n + 2*p + 1)*(c + d *x)^m - (c + d*x)^(m - 2)*(a*d^2*(m + n - 1) - b*c^2*(m + n + 2*p + 1) - 2* b*c*d*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 1] && NeQ[m + n + 2*p + 1, 0] && IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Time = 3.11 (sec) , antiderivative size = 987, normalized size of antiderivative = 1.64
| method | result | size |
| risch | \(-\frac {2 \left (-3465 b^{2} x^{5} d^{5}-8505 b^{2} c \,x^{4} d^{4}+770 a b \,d^{5} x^{3}-5565 c^{2} d^{3} x^{3} b^{2}+2540 a b c \,d^{4} x^{2}-75 b^{2} c^{3} d^{2} x^{2}+1078 a^{2} x \,d^{5}+2580 a b \,c^{2} d^{3} x +90 b^{2} c^{4} d x +4336 a^{2} c \,d^{4}+320 a \,c^{3} d^{2} b -120 c^{5} b^{2}\right ) \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{45045 d^{3} b^{2}}+\frac {2 \left (\frac {\left (1617 a^{3} d^{6}+4614 a^{2} b \,c^{2} d^{4}-335 a \,b^{2} c^{4} d^{2}+120 b^{3} c^{6}\right ) \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {3246 a^{3} c \,d^{5} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {30 b \,c^{5} d a \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2740 c^{3} d^{3} a^{2} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{45045 b^{2} d^{3} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(987\) |
| elliptic | \(\text {Expression too large to display}\) | \(1493\) |
| default | \(\text {Expression too large to display}\) | \(2227\) |
Input:
int(x^2*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/45045/d^3*(-3465*b^2*d^5*x^5-8505*b^2*c*d^4*x^4+770*a*b*d^5*x^3-5565*b^ 2*c^2*d^3*x^3+2540*a*b*c*d^4*x^2-75*b^2*c^3*d^2*x^2+1078*a^2*d^5*x+2580*a* b*c^2*d^3*x+90*b^2*c^4*d*x+4336*a^2*c*d^4+320*a*b*c^3*d^2-120*b^2*c^5)/b^2 *(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)+2/45045/b^2/d^3*((1617*a^3*d^6+4614*a^2*b* c^2*d^4-335*a*b^2*c^4*d^2+120*b^3*c^6)/b*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b) ^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x- 1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*( (c/d-1/b*(a*b)^(1/2))*EllipticE(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^( 1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))-c/d*EllipticF( 1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c /d-1/b*(a*b)^(1/2)))^(1/2)))+3246*a^3*c*d^5/b*(a*b)^(1/2)*2^(1/2)*((x+1/b* (a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(- 2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1 /2)*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b* (a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))+30*b*c^5*d*a*(a*b)^(1/2)*2^(1/2) *((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2))) ^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d* x+a*c)^(1/2)*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/ 2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))+2740*c^3*d^3*a^2*(a*b)^ (1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1...
Time = 0.19 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.70 \[ \int x^2 (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\frac {2 \, {\left (2 \, {\left (120 \, b^{3} c^{7} - 425 \, a b^{2} c^{5} d^{2} - 3606 \, a^{2} b c^{3} d^{4} - 8121 \, a^{3} c d^{6}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 6 \, {\left (120 \, b^{3} c^{6} d - 335 \, a b^{2} c^{4} d^{3} + 4614 \, a^{2} b c^{2} d^{5} + 1617 \, a^{3} d^{7}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (3465 \, b^{3} d^{7} x^{5} + 8505 \, b^{3} c d^{6} x^{4} + 120 \, b^{3} c^{5} d^{2} - 320 \, a b^{2} c^{3} d^{4} - 4336 \, a^{2} b c d^{6} + 35 \, {\left (159 \, b^{3} c^{2} d^{5} - 22 \, a b^{2} d^{7}\right )} x^{3} + 5 \, {\left (15 \, b^{3} c^{3} d^{4} - 508 \, a b^{2} c d^{6}\right )} x^{2} - 2 \, {\left (45 \, b^{3} c^{4} d^{3} + 1290 \, a b^{2} c^{2} d^{5} + 539 \, a^{2} b d^{7}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{135135 \, b^{3} d^{5}} \] Input:
integrate(x^2*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
2/135135*(2*(120*b^3*c^7 - 425*a*b^2*c^5*d^2 - 3606*a^2*b*c^3*d^4 - 8121*a ^3*c*d^6)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8 /27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 6*(120*b^3*c^6*d - 3 35*a*b^2*c^4*d^3 + 4614*a^2*b*c^2*d^5 + 1617*a^3*d^7)*sqrt(-b*d)*weierstra ssZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), w eierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2 )/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(3465*b^3*d^7*x^5 + 8505*b^3*c*d^6*x^4 + 120*b^3*c^5*d^2 - 320*a*b^2*c^3*d^4 - 4336*a^2*b*c*d^6 + 35*(159*b^3*c^2 *d^5 - 22*a*b^2*d^7)*x^3 + 5*(15*b^3*c^3*d^4 - 508*a*b^2*c*d^6)*x^2 - 2*(4 5*b^3*c^4*d^3 + 1290*a*b^2*c^2*d^5 + 539*a^2*b*d^7)*x)*sqrt(-b*x^2 + a)*sq rt(d*x + c))/(b^3*d^5)
\[ \int x^2 (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\int x^{2} \sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {5}{2}}\, dx \] Input:
integrate(x**2*(d*x+c)**(5/2)*(-b*x**2+a)**(1/2),x)
Output:
Integral(x**2*sqrt(a - b*x**2)*(c + d*x)**(5/2), x)
\[ \int x^2 (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {5}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^(5/2)*x^2, x)
\[ \int x^2 (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {5}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^(5/2)*x^2, x)
Timed out. \[ \int x^2 (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\int x^2\,\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{5/2} \,d x \] Input:
int(x^2*(a - b*x^2)^(1/2)*(c + d*x)^(5/2),x)
Output:
int(x^2*(a - b*x^2)^(1/2)*(c + d*x)^(5/2), x)
\[ \int x^2 (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\frac {-3234 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a^{3} d^{5}-17900 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a^{2} b \,c^{2} d^{3}-2156 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a^{2} b c \,d^{4} x +30 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a \,b^{2} c^{4} d -5160 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a \,b^{2} c^{3} d^{2} x -5080 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a \,b^{2} c^{2} d^{3} x^{2}-1540 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a \,b^{2} c \,d^{4} x^{3}-180 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{3} c^{5} x +150 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{3} c^{4} d \,x^{2}+11130 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{3} c^{3} d^{2} x^{3}+17010 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{3} c^{2} d^{3} x^{4}+6930 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{3} c \,d^{4} x^{5}-4851 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{3} b \,d^{6}-13842 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b^{2} c^{2} d^{4}+1005 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{3} c^{4} d^{2}-360 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{4} c^{6}+1617 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{4} d^{6}+11106 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{3} b \,c^{2} d^{4}+5145 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b^{2} c^{4} d^{2}+180 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{3} c^{6}}{45045 b^{3} c \,d^{2}} \] Input:
int(x^2*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2),x)
Output:
( - 3234*sqrt(c + d*x)*sqrt(a - b*x**2)*a**3*d**5 - 17900*sqrt(c + d*x)*sq rt(a - b*x**2)*a**2*b*c**2*d**3 - 2156*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2 *b*c*d**4*x + 30*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b**2*c**4*d - 5160*sqrt( c + d*x)*sqrt(a - b*x**2)*a*b**2*c**3*d**2*x - 5080*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b**2*c**2*d**3*x**2 - 1540*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b** 2*c*d**4*x**3 - 180*sqrt(c + d*x)*sqrt(a - b*x**2)*b**3*c**5*x + 150*sqrt( c + d*x)*sqrt(a - b*x**2)*b**3*c**4*d*x**2 + 11130*sqrt(c + d*x)*sqrt(a - b*x**2)*b**3*c**3*d**2*x**3 + 17010*sqrt(c + d*x)*sqrt(a - b*x**2)*b**3*c* *2*d**3*x**4 + 6930*sqrt(c + d*x)*sqrt(a - b*x**2)*b**3*c*d**4*x**5 - 4851 *int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x **3),x)*a**3*b*d**6 - 13842*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*b**2*c**2*d**4 + 1005*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b** 3*c**4*d**2 - 360*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**4*c**6 + 1617*int((sqrt(c + d*x)*sqrt(a - b*x* *2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**4*d**6 + 11106*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**3*b*c** 2*d**4 + 5145*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*b**2*c**4*d**2 + 180*int((sqrt(c + d*x)*sqrt(a - b*x* *2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b**3*c**6)/(45045*b**3*c*...