Integrand size = 35, antiderivative size = 990 \[ \int \frac {x^2 \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx =\text {Too large to display} \] Output:
4/45045*(195*a^3*C*d^6+9*a^2*b*d^4*(65*A*d^2-71*B*c*d+68*C*c^2)-64*b^3*c^4 *(65*A*d^2-60*B*c*d+56*C*c^2)+6*a*b^2*c^2*d^2*(637*A*d^2-544*B*c*d+480*C*c ^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d^7-4/15015*(3*a^2*d^4*(-77*B*d+71 *C*c)-16*b^2*c^3*(65*A*d^2-60*B*c*d+56*C*c^2)+a*b*c*d^2*(793*A*d^2-666*B*c *d+580*C*c^2))*x*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^6+2/9009*(39*a^2*C*d^4 +3*a*b*d^2*(39*A*d^2-58*B*c*d+68*C*c^2)+16*b^2*c^2*(65*A*d^2-60*B*c*d+56*C *c^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(3/2)/b^2/d^5-2/1287*(3*a*d^2*(-11*B*d+12* C*c)+2*b*c*(65*A*d^2-60*B*c*d+56*C*c^2))*x*(d*x+c)^(1/2)*(-b*x^2+a)^(3/2)/ b/d^4-2/429*(13*a*C*d^2+b*(39*A*d^2-69*B*c*d+93*C*c^2))*(d*x+c)^(1/2)*(-b* x^2+a)^(5/2)/b^2/d^3+2/39*(-3*B*d+8*C*c)*(d*x+c)^(3/2)*(-b*x^2+a)^(5/2)/b/ d^3-2/15*C*(d*x+c)^(5/2)*(-b*x^2+a)^(5/2)/b/d^3+8/45045*a^(1/2)*(3*a^3*d^6 *(-231*B*d+148*C*c)+64*b^3*c^5*(65*A*d^2-60*B*c*d+56*C*c^2)+3*a^2*b*c*d^4* (598*A*d^2-453*B*c*d+376*C*c^2)-6*a*b^2*c^3*d^2*(1157*A*d^2-1024*B*c*d+928 *C*c^2))*(d*x+c)^(1/2)*((-b*x^2+a)/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^(3/ 2)/d^8/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)+8/45045*a^(1 /2)*(-a*d^2+b*c^2)*(195*a^3*C*d^6+9*a^2*b*d^4*(65*A*d^2-71*B*c*d+68*C*c^2) -64*b^3*c^4*(65*A*d^2-60*B*c*d+56*C*c^2)+6*a*b^2*c^2*d^2*(637*A*d^2-544*B* c*d+480*C*c^2))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/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/(...
Result contains complex when optimal does not.
Time = 33.82 (sec) , antiderivative size = 1335, normalized size of antiderivative = 1.35 \[ \int \frac {x^2 \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:
Integrate[(x^2*(a - b*x^2)^(3/2)*(A + B*x + C*x^2))/Sqrt[c + d*x],x]
Output:
Sqrt[c + d*x]*Sqrt[a - b*x^2]*((-8*(1792*b^3*c^6*C - 1920*b^3*B*c^5*d + 20 80*A*b^3*c^4*d^2 - 2560*a*b^2*c^4*C*d^2 + 2832*a*b^2*B*c^3*d^3 - 3211*a*A* b^2*c^2*d^4 + 321*a^2*b*c^2*C*d^4 - 408*a^2*b*B*c*d^5 + 585*a^2*A*b*d^6 + 195*a^3*C*d^6))/(45045*b^2*d^7) - (8*(-1344*b^2*c^5*C + 1440*b^2*B*c^4*d - 1560*A*b^2*c^3*d^2 + 1850*a*b*c^3*C*d^2 - 2049*a*b*B*c^2*d^3 + 2327*a*A*b *c*d^4 - 174*a^2*c*C*d^4 + 231*a^2*B*d^5)*x)/(45045*b*d^6) + (2*(-4480*b^2 *c^4*C + 4800*b^2*B*c^3*d - 5200*A*b^2*c^2*d^2 + 6036*a*b*c^2*C*d^2 - 6690 *a*b*B*c*d^3 + 7605*a*A*b*d^4 - 468*a^2*C*d^4)*x^2)/(45045*b*d^5) + (2*(56 0*b*c^3*C - 600*b*B*c^2*d + 650*A*b*c*d^2 - 744*a*c*C*d^2 + 825*a*B*d^3)*x ^3)/(6435*d^4) - (2*(168*b*c^2*C - 180*b*B*c*d + 195*A*b*d^2 - 221*a*C*d^2 )*x^4)/(2145*d^3) - (2*b*(-14*c*C + 15*B*d)*x^5)/(195*d^2) - (2*b*C*x^6)/( 15*d)) + (8*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^2]*(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(3*a^3*d^6*(148*c*C - 231*B*d) + 64*b^3*c^5*(56*c^2*C - 60*B*c*d + 65*A*d^2) + 3*a^2*b*c*d^4*(376*c^2*C - 453*B*c*d + 598*A*d^2 ) - 6*a*b^2*c^3*d^2*(928*c^2*C - 1024*B*c*d + 1157*A*d^2))*(-((a*d^2)/(c + d*x)^2) + b*(-1 + c/(c + d*x))^2) - (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(3 *a^3*d^6*(148*c*C - 231*B*d) + 64*b^3*c^5*(56*c^2*C - 60*B*c*d + 65*A*d^2) + 3*a^2*b*c*d^4*(376*c^2*C - 453*B*c*d + 598*A*d^2) - 6*a*b^2*c^3*d^2*(92 8*c^2*C - 1024*B*c*d + 1157*A*d^2))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/(Sq rt[b]*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (Sqrt[a]*d)/(Sqrt[b]*(c + d*x)...
Time = 3.62 (sec) , antiderivative size = 962, normalized size of antiderivative = 0.97, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {2185, 27, 2185, 27, 2185, 27, 682, 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.
| \(\displaystyle \int \frac {x^2 \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {5 \left (a-b x^2\right )^{3/2} \left (-b d^3 (8 c C-3 B d) x^3-d^2 \left (7 b C c^2-3 A b d^2-a C d^2\right ) x^2-2 c C d \left (b c^2-a d^2\right ) x+a c^2 C d^2\right )}{2 \sqrt {c+d x}}dx}{15 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (a-b x^2\right )^{3/2} \left (-b d^3 (8 c C-3 B d) x^3-d^2 \left (7 b C c^2-3 A b d^2-a C d^2\right ) x^2-2 c C d \left (b c^2-a d^2\right ) x+a c^2 C d^2\right )}{\sqrt {c+d x}}dx}{3 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (8 c C-3 B d)-\frac {2 \int \frac {\left (a-b x^2\right )^{3/2} \left (-b \left (13 a C d^2+b \left (93 C c^2-69 B d c+39 A d^2\right )\right ) x^2 d^5+a b c (11 c C-9 B d) d^5-b \left (6 b (9 c C-5 B d) c^2+a d^2 (2 c C+9 B d)\right ) x d^4\right )}{2 \sqrt {c+d x}}dx}{13 b d^3}}{3 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (8 c C-3 B d)-\frac {\int \frac {\left (a-b x^2\right )^{3/2} \left (-b \left (13 a C d^2+b \left (93 C c^2-69 B d c+39 A d^2\right )\right ) x^2 d^5+a b c (11 c C-9 B d) d^5-b \left (6 b (9 c C-5 B d) c^2+a d^2 (2 c C+9 B d)\right ) x d^4\right )}{\sqrt {c+d x}}dx}{13 b d^3}}{3 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (8 c C-3 B d)-\frac {\frac {2}{11} d^4 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (13 a C d^2+b \left (39 A d^2-69 B c d+93 c^2 C\right )\right )-\frac {2 \int \frac {b d^6 \left (a d \left (13 a C d^2-b \left (28 C c^2-30 B d c-39 A d^2\right )\right )-3 b \left (3 a (12 c C-11 B d) d^2+2 b c \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{2 \sqrt {c+d x}}dx}{11 b d^2}}{13 b d^3}}{3 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (8 c C-3 B d)-\frac {\frac {2}{11} d^4 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (13 a C d^2+b \left (39 A d^2-69 B c d+93 c^2 C\right )\right )-\frac {1}{11} d^4 \int \frac {\left (a d \left (13 a C d^2-b \left (28 C c^2-30 B d c-39 A d^2\right )\right )-3 b \left (3 a (12 c C-11 B d) d^2+2 b c \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}}dx}{13 b d^3}}{3 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (8 c C-3 B d)-\frac {\frac {2}{11} d^4 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (13 a C d^2+b \left (39 A d^2-69 B c d+93 c^2 C\right )\right )-\frac {1}{11} d^4 \left (\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (39 a^2 C d^4-7 b d x \left (3 a d^2 (12 c C-11 B d)+2 b c \left (65 A d^2-60 B c d+56 c^2 C\right )\right )+3 a b d^2 \left (39 A d^2-58 B c d+68 c^2 C\right )+16 b^2 c^2 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )}{21 d^2}-\frac {4 \int -\frac {3 b \left (a d \left (39 a^2 C d^4-3 a b \left (16 C c^2-19 B d c-39 A d^2\right ) d^2+2 b^2 c^2 \left (56 C c^2-60 B d c+65 A d^2\right )\right )-b \left (3 a^2 (71 c C-77 B d) d^4+a b c \left (580 C c^2-666 B d c+793 A d^2\right ) d^2-16 b^2 c^3 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{21 b d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (8 c C-3 B d)-\frac {\frac {2}{11} d^4 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (13 a C d^2+b \left (39 A d^2-69 B c d+93 c^2 C\right )\right )-\frac {1}{11} d^4 \left (\frac {2 \int \frac {\left (a d \left (39 a^2 C d^4-3 a b \left (16 C c^2-19 B d c-39 A d^2\right ) d^2+2 b^2 c^2 \left (56 C c^2-60 B d c+65 A d^2\right )\right )-b \left (3 a^2 (71 c C-77 B d) d^4+a b c \left (580 C c^2-666 B d c+793 A d^2\right ) d^2-16 b^2 c^3 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (39 a^2 C d^4-7 b d x \left (3 a d^2 (12 c C-11 B d)+2 b c \left (65 A d^2-60 B c d+56 c^2 C\right )\right )+3 a b d^2 \left (39 A d^2-58 B c d+68 c^2 C\right )+16 b^2 c^2 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )}{21 d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (8 c C-3 B d)-\frac {\frac {2}{11} d^4 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (13 a C d^2+b \left (39 A d^2-69 B c d+93 c^2 C\right )\right )-\frac {1}{11} d^4 \left (\frac {2 \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (195 a^3 C d^6-3 b d x \left (3 a^2 d^4 (71 c C-77 B d)+a b c d^2 \left (793 A d^2-666 B c d+580 c^2 C\right )-16 b^2 c^3 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )+9 a^2 b d^4 \left (65 A d^2-71 B c d+68 c^2 C\right )+6 a b^2 c^2 d^2 \left (637 A d^2-544 B c d+480 c^2 C\right )-64 b^3 c^4 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )}{15 d^2}-\frac {4 \int -\frac {b \left (a d \left (195 a^3 C d^6-9 a^2 b \left (3 C c^2-6 B d c-65 A d^2\right ) d^4+3 a b^2 c^2 \left (380 C c^2-422 B d c+481 A d^2\right ) d^2-16 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right )-b \left (3 a^3 (148 c C-231 B d) d^6+3 a^2 b c \left (376 C c^2-453 B d c+598 A d^2\right ) d^4-6 a b^2 c^3 \left (928 C c^2-1024 B d c+1157 A d^2\right ) d^2+64 b^3 c^5 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}\right )}{7 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (39 a^2 C d^4-7 b d x \left (3 a d^2 (12 c C-11 B d)+2 b c \left (65 A d^2-60 B c d+56 c^2 C\right )\right )+3 a b d^2 \left (39 A d^2-58 B c d+68 c^2 C\right )+16 b^2 c^2 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )}{21 d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (8 c C-3 B d)-\frac {\frac {2}{11} d^4 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (13 a C d^2+b \left (39 A d^2-69 B c d+93 c^2 C\right )\right )-\frac {1}{11} d^4 \left (\frac {2 \left (\frac {2 \int \frac {a d \left (195 a^3 C d^6-9 a^2 b \left (3 C c^2-6 B d c-65 A d^2\right ) d^4+3 a b^2 c^2 \left (380 C c^2-422 B d c+481 A d^2\right ) d^2-16 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right )-b \left (3 a^3 (148 c C-231 B d) d^6+3 a^2 b c \left (376 C c^2-453 B d c+598 A d^2\right ) d^4-6 a b^2 c^3 \left (928 C c^2-1024 B d c+1157 A d^2\right ) d^2+64 b^3 c^5 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (195 a^3 C d^6-3 b d x \left (3 a^2 d^4 (71 c C-77 B d)+a b c d^2 \left (793 A d^2-666 B c d+580 c^2 C\right )-16 b^2 c^3 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )+9 a^2 b d^4 \left (65 A d^2-71 B c d+68 c^2 C\right )+6 a b^2 c^2 d^2 \left (637 A d^2-544 B c d+480 c^2 C\right )-64 b^3 c^4 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )}{15 d^2}\right )}{7 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (39 a^2 C d^4-7 b d x \left (3 a d^2 (12 c C-11 B d)+2 b c \left (65 A d^2-60 B c d+56 c^2 C\right )\right )+3 a b d^2 \left (39 A d^2-58 B c d+68 c^2 C\right )+16 b^2 c^2 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )}{21 d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (8 c C-3 B d)-\frac {\frac {2}{11} d^4 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (13 a C d^2+b \left (39 A d^2-69 B c d+93 c^2 C\right )\right )-\frac {1}{11} d^4 \left (\frac {2 \left (\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (195 a^3 C d^6+9 a^2 b d^4 \left (65 A d^2-71 B c d+68 c^2 C\right )+6 a b^2 c^2 d^2 \left (637 A d^2-544 B c d+480 c^2 C\right )-64 b^3 c^4 \left (65 A d^2-60 B c d+56 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \left (3 a^3 d^6 (148 c C-231 B d)+3 a^2 b c d^4 \left (598 A d^2-453 B c d+376 c^2 C\right )-6 a b^2 c^3 d^2 \left (1157 A d^2-1024 B c d+928 c^2 C\right )+64 b^3 c^5 \left (65 A d^2-60 B c d+56 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (195 a^3 C d^6-3 b d x \left (3 a^2 d^4 (71 c C-77 B d)+a b c d^2 \left (793 A d^2-666 B c d+580 c^2 C\right )-16 b^2 c^3 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )+9 a^2 b d^4 \left (65 A d^2-71 B c d+68 c^2 C\right )+6 a b^2 c^2 d^2 \left (637 A d^2-544 B c d+480 c^2 C\right )-64 b^3 c^4 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )}{15 d^2}\right )}{7 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (39 a^2 C d^4-7 b d x \left (3 a d^2 (12 c C-11 B d)+2 b c \left (65 A d^2-60 B c d+56 c^2 C\right )\right )+3 a b d^2 \left (39 A d^2-58 B c d+68 c^2 C\right )+16 b^2 c^2 \left (65 A d^2-60 B c d+56 c^2 C\right )\right )}{21 d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {2}{13} d (8 c C-3 B d) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}-\frac {\frac {2}{11} d^4 \left (13 a C d^2+b \left (93 C c^2-69 B d c+39 A d^2\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}-\frac {1}{11} d^4 \left (\frac {2 \sqrt {c+d x} \left (39 a^2 C d^4+3 a b \left (68 C c^2-58 B d c+39 A d^2\right ) d^2-7 b \left (3 a (12 c C-11 B d) d^2+2 b c \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d+16 b^2 c^2 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-3 b \left (3 a^2 (71 c C-77 B d) d^4+a b c \left (580 C c^2-666 B d c+793 A d^2\right ) d^2-16 b^2 c^3 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right )}{15 d^2}+\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \left (3 a^3 (148 c C-231 B d) d^6+3 a^2 b c \left (376 C c^2-453 B d c+598 A d^2\right ) d^4-6 a b^2 c^3 \left (928 C c^2-1024 B d c+1157 A d^2\right ) d^2+64 b^3 c^5 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {2}{13} d (8 c C-3 B d) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}-\frac {\frac {2}{11} d^4 \left (13 a C d^2+b \left (93 C c^2-69 B d c+39 A d^2\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}-\frac {1}{11} d^4 \left (\frac {2 \sqrt {c+d x} \left (39 a^2 C d^4+3 a b \left (68 C c^2-58 B d c+39 A d^2\right ) d^2-7 b \left (3 a (12 c C-11 B d) d^2+2 b c \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d+16 b^2 c^2 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-3 b \left (3 a^2 (71 c C-77 B d) d^4+a b c \left (580 C c^2-666 B d c+793 A d^2\right ) d^2-16 b^2 c^3 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \left (3 a^3 (148 c C-231 B d) d^6+3 a^2 b c \left (376 C c^2-453 B d c+598 A d^2\right ) d^4-6 a b^2 c^3 \left (928 C c^2-1024 B d c+1157 A d^2\right ) d^2+64 b^3 c^5 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} \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}}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}\right )}{7 d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {2}{13} d (8 c C-3 B d) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}-\frac {\frac {2}{11} d^4 \left (13 a C d^2+b \left (93 C c^2-69 B d c+39 A d^2\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}-\frac {1}{11} d^4 \left (\frac {2 \sqrt {c+d x} \left (39 a^2 C d^4+3 a b \left (68 C c^2-58 B d c+39 A d^2\right ) d^2-7 b \left (3 a (12 c C-11 B d) d^2+2 b c \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d+16 b^2 c^2 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-3 b \left (3 a^2 (71 c C-77 B d) d^4+a b c \left (580 C c^2-666 B d c+793 A d^2\right ) d^2-16 b^2 c^3 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \left (3 a^3 (148 c C-231 B d) d^6+3 a^2 b c \left (376 C c^2-453 B d c+598 A d^2\right ) d^4-6 a b^2 c^3 \left (928 C c^2-1024 B d c+1157 A d^2\right ) d^2+64 b^3 c^5 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}\right )}{7 d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {2}{13} d (8 c C-3 B d) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}-\frac {\frac {2}{11} d^4 \left (13 a C d^2+b \left (93 C c^2-69 B d c+39 A d^2\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}-\frac {1}{11} d^4 \left (\frac {2 \sqrt {c+d x} \left (39 a^2 C d^4+3 a b \left (68 C c^2-58 B d c+39 A d^2\right ) d^2-7 b \left (3 a (12 c C-11 B d) d^2+2 b c \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d+16 b^2 c^2 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-3 b \left (3 a^2 (71 c C-77 B d) d^4+a b c \left (580 C c^2-666 B d c+793 A d^2\right ) d^2-16 b^2 c^3 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \left (3 a^3 (148 c C-231 B d) d^6+3 a^2 b c \left (376 C c^2-453 B d c+598 A d^2\right ) d^4-6 a b^2 c^3 \left (928 C c^2-1024 B d c+1157 A d^2\right ) d^2+64 b^3 c^5 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {2}{13} d (8 c C-3 B d) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}-\frac {\frac {2}{11} d^4 \left (13 a C d^2+b \left (93 C c^2-69 B d c+39 A d^2\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}-\frac {1}{11} d^4 \left (\frac {2 \sqrt {c+d x} \left (39 a^2 C d^4+3 a b \left (68 C c^2-58 B d c+39 A d^2\right ) d^2-7 b \left (3 a (12 c C-11 B d) d^2+2 b c \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d+16 b^2 c^2 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-3 b \left (3 a^2 (71 c C-77 B d) d^4+a b c \left (580 C c^2-666 B d c+793 A d^2\right ) d^2-16 b^2 c^3 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \left (3 a^3 (148 c C-231 B d) d^6+3 a^2 b c \left (376 C c^2-453 B d c+598 A d^2\right ) d^4-6 a b^2 c^3 \left (928 C c^2-1024 B d c+1157 A d^2\right ) d^2+64 b^3 c^5 \left (56 C c^2-60 B d c+65 A d^2\right )\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 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \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 {c+d x} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^3}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {2}{13} d (8 c C-3 B d) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}-\frac {\frac {2}{11} d^4 \left (13 a C d^2+b \left (93 C c^2-69 B d c+39 A d^2\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}-\frac {1}{11} d^4 \left (\frac {2 \sqrt {c+d x} \left (39 a^2 C d^4+3 a b \left (68 C c^2-58 B d c+39 A d^2\right ) d^2-7 b \left (3 a (12 c C-11 B d) d^2+2 b c \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d+16 b^2 c^2 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-3 b \left (3 a^2 (71 c C-77 B d) d^4+a b c \left (580 C c^2-666 B d c+793 A d^2\right ) d^2-16 b^2 c^3 \left (56 C c^2-60 B d c+65 A d^2\right )\right ) x d-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \left (3 a^3 (148 c C-231 B d) d^6+3 a^2 b c \left (376 C c^2-453 B d c+598 A d^2\right ) d^4-6 a b^2 c^3 \left (928 C c^2-1024 B d c+1157 A d^2\right ) d^2+64 b^3 c^5 \left (56 C c^2-60 B d c+65 A d^2\right )\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 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (195 a^3 C d^6+9 a^2 b \left (68 C c^2-71 B d c+65 A d^2\right ) d^4+6 a b^2 c^2 \left (480 C c^2-544 B d c+637 A d^2\right ) d^2-64 b^3 c^4 \left (56 C c^2-60 B d c+65 A d^2\right )\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 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}\right )}{13 b d^3}}{3 b d^4}-\frac {2 C (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^3}\) |
Input:
Int[(x^2*(a - b*x^2)^(3/2)*(A + B*x + C*x^2))/Sqrt[c + d*x],x]
Output:
(-2*C*(c + d*x)^(5/2)*(a - b*x^2)^(5/2))/(15*b*d^3) + ((2*d*(8*c*C - 3*B*d )*(c + d*x)^(3/2)*(a - b*x^2)^(5/2))/13 - ((2*d^4*(13*a*C*d^2 + b*(93*c^2* C - 69*B*c*d + 39*A*d^2))*Sqrt[c + d*x]*(a - b*x^2)^(5/2))/11 - (d^4*((2*S qrt[c + d*x]*(39*a^2*C*d^4 + 3*a*b*d^2*(68*c^2*C - 58*B*c*d + 39*A*d^2) + 16*b^2*c^2*(56*c^2*C - 60*B*c*d + 65*A*d^2) - 7*b*d*(3*a*d^2*(12*c*C - 11* B*d) + 2*b*c*(56*c^2*C - 60*B*c*d + 65*A*d^2))*x)*(a - b*x^2)^(3/2))/(21*d ^2) + (2*((2*Sqrt[c + d*x]*(195*a^3*C*d^6 + 9*a^2*b*d^4*(68*c^2*C - 71*B*c *d + 65*A*d^2) - 64*b^3*c^4*(56*c^2*C - 60*B*c*d + 65*A*d^2) + 6*a*b^2*c^2 *d^2*(480*c^2*C - 544*B*c*d + 637*A*d^2) - 3*b*d*(3*a^2*d^4*(71*c*C - 77*B *d) - 16*b^2*c^3*(56*c^2*C - 60*B*c*d + 65*A*d^2) + a*b*c*d^2*(580*c^2*C - 666*B*c*d + 793*A*d^2))*x)*Sqrt[a - b*x^2])/(15*d^2) + (2*((2*Sqrt[a]*Sqr t[b]*(3*a^3*d^6*(148*c*C - 231*B*d) + 64*b^3*c^5*(56*c^2*C - 60*B*c*d + 65 *A*d^2) + 3*a^2*b*c*d^4*(376*c^2*C - 453*B*c*d + 598*A*d^2) - 6*a*b^2*c^3* d^2*(928*c^2*C - 1024*B*c*d + 1157*A*d^2))*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)])/(d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*S qrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(195*a^3*C*d^6 + 9*a^2*b*d^4* (68*c^2*C - 71*B*c*d + 65*A*d^2) - 64*b^3*c^4*(56*c^2*C - 60*B*c*d + 65*A* d^2) + 6*a*b^2*c^2*d^2*(480*c^2*C - 544*B*c*d + 637*A*d^2))*Sqrt[(Sqrt[b]* (c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSi...
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[((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[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(2609\) vs. \(2(894)=1788\).
Time = 9.05 (sec) , antiderivative size = 2610, normalized size of antiderivative = 2.64
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2610\) |
| elliptic | \(\text {Expression too large to display}\) | \(3813\) |
| default | \(\text {Expression too large to display}\) | \(6620\) |
Input:
int(x^2*(-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x,method=_RETURNVERBO SE)
Output:
-2/45045/b^2*(3003*C*b^3*d^6*x^6+3465*B*b^3*d^6*x^5-3234*C*b^3*c*d^5*x^5+4 095*A*b^3*d^6*x^4-3780*B*b^3*c*d^5*x^4-4641*C*a*b^2*d^6*x^4+3528*C*b^3*c^2 *d^4*x^4-4550*A*b^3*c*d^5*x^3-5775*B*a*b^2*d^6*x^3+4200*B*b^3*c^2*d^4*x^3+ 5208*C*a*b^2*c*d^5*x^3-3920*C*b^3*c^3*d^3*x^3-7605*A*a*b^2*d^6*x^2+5200*A* b^3*c^2*d^4*x^2+6690*B*a*b^2*c*d^5*x^2-4800*B*b^3*c^3*d^3*x^2+468*C*a^2*b* d^6*x^2-6036*C*a*b^2*c^2*d^4*x^2+4480*C*b^3*c^4*d^2*x^2+9308*A*a*b^2*c*d^5 *x-6240*A*b^3*c^3*d^3*x+924*B*a^2*b*d^6*x-8196*B*a*b^2*c^2*d^4*x+5760*B*b^ 3*c^4*d^2*x-696*C*a^2*b*c*d^5*x+7400*C*a*b^2*c^3*d^3*x-5376*C*b^3*c^5*d*x+ 2340*A*a^2*b*d^6-12844*A*a*b^2*c^2*d^4+8320*A*b^3*c^4*d^2-1632*B*a^2*b*c*d ^5+11328*B*a*b^2*c^3*d^3-7680*B*b^3*c^5*d+780*C*a^3*d^6+1284*C*a^2*b*c^2*d ^4-10240*C*a*b^2*c^4*d^2+7168*C*b^3*c^6)/d^7*(-b*x^2+a)^(1/2)*(d*x+c)^(1/2 )+4/45045/b^2/d^7*(-(1794*A*a^2*b*c*d^6-6942*A*a*b^2*c^3*d^4+4160*A*b^3*c^ 5*d^2-693*B*a^3*d^7-1359*B*a^2*b*c^2*d^5+6144*B*a*b^2*c^4*d^3-3840*B*b^3*c ^6*d+444*C*a^3*c*d^6+1128*C*a^2*b*c^3*d^4-5568*C*a*b^2*c^5*d^2+3584*C*b^3* c^7)*(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)))+195*...
Time = 0.10 (sec) , antiderivative size = 916, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:
integrate(x^2*(-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="f ricas")
Output:
-2/135135*(4*(3584*C*b^4*c^8 - 3840*B*b^4*c^7*d + 9024*B*a*b^3*c^5*d^3 - 5 157*B*a^2*b^2*c^3*d^5 - 531*B*a^3*b*c*d^7 - 64*(129*C*a*b^3 - 65*A*b^4)*c^ 6*d^2 + 6*(758*C*a^2*b^2 - 1677*A*a*b^3)*c^4*d^4 + 3*(121*C*a^3*b + 2041*A *a^2*b^2)*c^2*d^6 + 585*(C*a^4 + 3*A*a^3*b)*d^8)*sqrt(-b*d)*weierstrassPIn verse(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) + 12*(3584*C*b^4*c^7*d - 3840*B*b^4*c^6*d^2 + 6144*B*a*b^ 3*c^4*d^4 - 1359*B*a^2*b^2*c^2*d^6 - 693*B*a^3*b*d^8 - 64*(87*C*a*b^3 - 65 *A*b^4)*c^5*d^3 + 6*(188*C*a^2*b^2 - 1157*A*a*b^3)*c^3*d^5 + 6*(74*C*a^3*b + 299*A*a^2*b^2)*c*d^7)*sqrt(-b*d)*weierstrassZeta(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), 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) ) + 3*(3003*C*b^4*d^8*x^6 + 7168*C*b^4*c^6*d^2 - 7680*B*b^4*c^5*d^3 + 1132 8*B*a*b^3*c^3*d^5 - 1632*B*a^2*b^2*c*d^7 - 640*(16*C*a*b^3 - 13*A*b^4)*c^4 *d^4 + 4*(321*C*a^2*b^2 - 3211*A*a*b^3)*c^2*d^6 + 780*(C*a^3*b + 3*A*a^2*b ^2)*d^8 - 231*(14*C*b^4*c*d^7 - 15*B*b^4*d^8)*x^5 + 21*(168*C*b^4*c^2*d^6 - 180*B*b^4*c*d^7 - 13*(17*C*a*b^3 - 15*A*b^4)*d^8)*x^4 - 7*(560*C*b^4*c^3 *d^5 - 600*B*b^4*c^2*d^6 + 825*B*a*b^3*d^8 - 2*(372*C*a*b^3 - 325*A*b^4)*c *d^7)*x^3 + (4480*C*b^4*c^4*d^4 - 4800*B*b^4*c^3*d^5 + 6690*B*a*b^3*c*d^7 - 4*(1509*C*a*b^3 - 1300*A*b^4)*c^2*d^6 + 117*(4*C*a^2*b^2 - 65*A*a*b^3)*d ^8)*x^2 - 4*(1344*C*b^4*c^5*d^3 - 1440*B*b^4*c^4*d^4 + 2049*B*a*b^3*c^2...
\[ \int \frac {x^2 \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\int \frac {x^{2} \left (a - b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2}\right )}{\sqrt {c + d x}}\, dx \] Input:
integrate(x**2*(-b*x**2+a)**(3/2)*(C*x**2+B*x+A)/(d*x+c)**(1/2),x)
Output:
Integral(x**2*(a - b*x**2)**(3/2)*(A + B*x + C*x**2)/sqrt(c + d*x), x)
\[ \int \frac {x^2 \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{2}}{\sqrt {d x + c}} \,d x } \] Input:
integrate(x^2*(-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="m axima")
Output:
integrate((C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)*x^2/sqrt(d*x + c), x)
\[ \int \frac {x^2 \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{2}}{\sqrt {d x + c}} \,d x } \] Input:
integrate(x^2*(-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="g iac")
Output:
integrate((C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)*x^2/sqrt(d*x + c), x)
Timed out. \[ \int \frac {x^2 \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\int \frac {x^2\,{\left (a-b\,x^2\right )}^{3/2}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {c+d\,x}} \,d x \] Input:
int((x^2*(a - b*x^2)^(3/2)*(A + B*x + C*x^2))/(c + d*x)^(1/2),x)
Output:
int((x^2*(a - b*x^2)^(3/2)*(A + B*x + C*x^2))/(c + d*x)^(1/2), x)
\[ \int \frac {x^2 \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\int \frac {x^{2} \left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (C \,x^{2}+B x +A \right )}{\sqrt {d x +c}}d x \] Input:
int(x^2*(-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x)
Output:
int(x^2*(-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x)