Integrand size = 35, antiderivative size = 831 \[ \int \frac {x^3 \sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=-\frac {2 \left (3 a^2 d^4 (432 c C-325 B d)+64 b^2 c^3 \left (120 c^2 C-130 B c d+143 A d^2\right )+2 a b c d^2 \left (1312 c^2 C-1326 B c d+1287 A d^2\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{45045 b^2 d^6}+\frac {2 \left (539 a^2 C d^4+16 b^2 c^2 \left (120 c^2 C-130 B c d+143 A d^2\right )+a b d^2 \left (956 c^2 C-988 B c d+1001 A d^2\right )\right ) x \sqrt {c+d x} \sqrt {a-b x^2}}{15015 b^2 d^5}+\frac {2 \left (a d^2 (367 c C-195 B d)+b c \left (1083 c^2 C-923 B c d+715 A d^2\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{3003 b^2 d^4}-\frac {2 \left (77 a C d^2+b \left (633 c^2 C-364 B c d+143 A d^2\right )\right ) (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}{1287 b^2 d^4}+\frac {2 (45 c C-13 B d) (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}{143 b d^4}-\frac {2 C (c+d x)^{7/2} \left (a-b x^2\right )^{3/2}}{13 b d^4}-\frac {4 \sqrt {a} \left (1617 a^3 C d^6-64 b^3 c^4 \left (120 c^2 C-130 B c d+143 A d^2\right )+3 a^2 b d^4 \left (524 c^2 C-663 B c d+1001 A d^2\right )+2 a b^2 c^2 d^2 \left (1568 c^2 C-1794 B c d+2145 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{45045 b^{5/2} d^7 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {4 \sqrt {a} \left (b c^2-a d^2\right ) \left (3 a^2 d^4 (432 c C-325 B d)+64 b^2 c^3 \left (120 c^2 C-130 B c d+143 A d^2\right )+2 a b c d^2 \left (1312 c^2 C-1326 B c d+1287 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{45045 b^{5/2} d^7 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/45045*(3*a^2*d^4*(-325*B*d+432*C*c)+64*b^2*c^3*(143*A*d^2-130*B*c*d+120 *C*c^2)+2*a*b*c*d^2*(1287*A*d^2-1326*B*c*d+1312*C*c^2))*(d*x+c)^(1/2)*(-b* x^2+a)^(1/2)/b^2/d^6+2/15015*(539*a^2*C*d^4+16*b^2*c^2*(143*A*d^2-130*B*c* d+120*C*c^2)+a*b*d^2*(1001*A*d^2-988*B*c*d+956*C*c^2))*x*(d*x+c)^(1/2)*(-b *x^2+a)^(1/2)/b^2/d^5+2/3003*(a*d^2*(-195*B*d+367*C*c)+b*c*(715*A*d^2-923* B*c*d+1083*C*c^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(3/2)/b^2/d^4-2/1287*(77*a*C*d ^2+b*(143*A*d^2-364*B*c*d+633*C*c^2))*(d*x+c)^(3/2)*(-b*x^2+a)^(3/2)/b^2/d ^4+2/143*(-13*B*d+45*C*c)*(d*x+c)^(5/2)*(-b*x^2+a)^(3/2)/b/d^4-2/13*C*(d*x +c)^(7/2)*(-b*x^2+a)^(3/2)/b/d^4-4/45045*a^(1/2)*(1617*a^3*C*d^6-64*b^3*c^ 4*(143*A*d^2-130*B*c*d+120*C*c^2)+3*a^2*b*d^4*(1001*A*d^2-663*B*c*d+524*C* c^2)+2*a*b^2*c^2*d^2*(2145*A*d^2-1794*B*c*d+1568*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^(5/2)/d^7/((d*x+c)/(c+a^(1/2) *d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-4/45045*a^(1/2)*(-a*d^2+b*c^2)*(3*a^2* d^4*(-325*B*d+432*C*c)+64*b^2*c^3*(143*A*d^2-130*B*c*d+120*C*c^2)+2*a*b*c* d^2*(1287*A*d^2-1326*B*c*d+1312*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/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(5/2)/d^7/(d*x+c)^(1/2 )/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 32.63 (sec) , antiderivative size = 982, normalized size of antiderivative = 1.18 \[ \int \frac {x^3 \sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:
Integrate[(x^3*Sqrt[a - b*x^2]*(A + B*x + C*x^2))/Sqrt[c + d*x],x]
Output:
(2*Sqrt[a - b*x^2]*(-(b*(c + d*x)*(2*a^2*d^4*(-757*c*C + 975*B*d + 539*C*d *x) - 2*a*b*d^2*(1088*c^3*C - 2*c^2*d*(637*B + 333*C*x) - d^3*x*(1001*A + 585*B*x + 385*C*x^2) + c*d^2*(1573*A + 793*B*x + 485*C*x^2)) + b^2*(7680*c ^5*C - 640*c^4*d*(13*B + 9*C*x) + 32*c^3*d^2*(286*A + 15*x*(13*B + 10*C*x) ) - 35*d^5*x^3*(143*A + 9*x*(13*B + 11*C*x)) - 8*c^2*d^3*x*(858*A + 25*x*( 26*B + 21*C*x)) + 10*c*d^4*x^2*(572*A + 7*x*(65*B + 54*C*x))))) - (2*(d^2* Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(1617*a^3*C*d^6 - 64*b^3*c^4*(120*c^2*C - 1 30*B*c*d + 143*A*d^2) + 3*a^2*b*d^4*(524*c^2*C - 663*B*c*d + 1001*A*d^2) + 2*a*b^2*c^2*d^2*(1568*c^2*C - 1794*B*c*d + 2145*A*d^2))*(a - b*x^2) + I*S qrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(1617*a^3*C*d^6 - 64*b^3*c^4*(120*c^2*C - 1 30*B*c*d + 143*A*d^2) + 3*a^2*b*d^4*(524*c^2*C - 663*B*c*d + 1001*A*d^2) + 2*a*b^2*c^2*d^2*(1568*c^2*C - 1794*B*c*d + 2145*A*d^2))*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*ArcSinh[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*(Sqrt[b]*c - Sqrt[a]*d)*(1617*a^(5/2)*C*d^5 + 3*a^2*Sqrt[b]*d^4*(432*c *C - 325*B*d) + 64*b^(5/2)*c^3*(120*c^2*C - 130*B*c*d + 143*A*d^2) + 48*Sq rt[a]*b^2*c^2*d*(120*c^2*C - 130*B*c*d + 143*A*d^2) + 3*a^(3/2)*b*d^3*(956 *c^2*C - 988*B*c*d + 1001*A*d^2) + 2*a*b^(3/2)*c*d^2*(1312*c^2*C - 1326*B* c*d + 1287*A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((S...
Time = 4.07 (sec) , antiderivative size = 834, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {2185, 27, 2185, 27, 2185, 27, 2185, 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.
| \(\displaystyle \int \frac {x^3 \sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {\sqrt {a-b x^2} \left (-b d^4 (45 c C-13 B d) x^4-d^3 \left (57 b C c^2-13 A b d^2-7 a C d^2\right ) x^3-c C d^2 \left (31 b c^2-21 a d^2\right ) x^2-3 c^2 C d \left (2 b c^2-7 a d^2\right ) x+7 a c^3 C d^2\right )}{2 \sqrt {c+d x}}dx}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {a-b x^2} \left (-b d^4 (45 c C-13 B d) x^4-d^3 \left (57 b C c^2-13 A b d^2-7 a C d^2\right ) x^3-c C d^2 \left (31 b c^2-21 a d^2\right ) x^2-3 c^2 C d \left (2 b c^2-7 a d^2\right ) x+7 a c^3 C d^2\right )}{\sqrt {c+d x}}dx}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {2 \int \frac {\sqrt {a-b x^2} \left (-b \left (77 a C d^2+b \left (633 C c^2-364 B d c+143 A d^2\right )\right ) x^3 d^7-b \left (b (694 c C-299 B d) c^2+a d^2 (6 c C+65 B d)\right ) x^2 d^6+a b c^2 (148 c C-65 B d) d^6+b c \left (a d^2 (219 c C-130 B d)-6 b c^2 (34 c C-13 B d)\right ) x d^5\right )}{2 \sqrt {c+d x}}dx}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\int \frac {\sqrt {a-b x^2} \left (-b \left (77 a C d^2+b \left (633 C c^2-364 B d c+143 A d^2\right )\right ) x^3 d^7-b \left (b (694 c C-299 B d) c^2+a d^2 (6 c C+65 B d)\right ) x^2 d^6+a b c^2 (148 c C-65 B d) d^6+b c \left (a d^2 (219 c C-130 B d)-6 b c^2 (34 c C-13 B d)\right ) x d^5\right )}{\sqrt {c+d x}}dx}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {2 \int \frac {3 \sqrt {a-b x^2} \left (-b^2 \left (a (367 c C-195 B d) d^2+b c \left (1083 C c^2-923 B d c+715 A d^2\right )\right ) x^2 d^9+a b c \left (77 a C d^2+b \left (189 C c^2-169 B d c+143 A d^2\right )\right ) d^9+b \left (77 a^2 C d^4-a b \left (178 C c^2-26 B d c-143 A d^2\right ) d^2-2 b^2 c^2 \left (327 C c^2-247 B d c+143 A d^2\right )\right ) x d^8\right )}{2 \sqrt {c+d x}}dx}{9 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {\int \frac {\sqrt {a-b x^2} \left (-b^2 \left (a (367 c C-195 B d) d^2+b c \left (1083 C c^2-923 B d c+715 A d^2\right )\right ) x^2 d^9+a b c \left (77 a C d^2+b \left (189 C c^2-169 B d c+143 A d^2\right )\right ) d^9+b \left (77 a^2 C d^4-a b \left (178 C c^2-26 B d c-143 A d^2\right ) d^2-2 b^2 c^2 \left (327 C c^2-247 B d c+143 A d^2\right )\right ) x d^8\right )}{\sqrt {c+d x}}dx}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {\frac {2}{7} b d^8 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (a d^2 (367 c C-195 B d)+b c \left (715 A d^2-923 B c d+1083 c^2 C\right )\right )-\frac {2 \int -\frac {b^2 d^{10} \left (a d \left (a (172 c C+195 B d) d^2+2 b c \left (120 C c^2-130 B d c+143 A d^2\right )\right )+\left (539 a^2 C d^4+a b \left (956 C c^2-988 B d c+1001 A d^2\right ) d^2+16 b^2 c^2 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{7 b d^2}}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {\frac {1}{7} b d^8 \int \frac {\left (a d \left (a (172 c C+195 B d) d^2+2 b c \left (120 C c^2-130 B d c+143 A d^2\right )\right )+\left (539 a^2 C d^4+a b \left (956 C c^2-988 B d c+1001 A d^2\right ) d^2+16 b^2 c^2 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx+\frac {2}{7} b d^8 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (a d^2 (367 c C-195 B d)+b c \left (715 A d^2-923 B c d+1083 c^2 C\right )\right )}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {\frac {1}{7} b d^8 \left (-\frac {4 \int -\frac {b \left (a d \left (3 a^2 (107 c C+325 B d) d^4+a b c \left (244 C c^2-312 B d c+429 A d^2\right ) d^2-16 b^2 c^3 \left (120 C c^2-130 B d c+143 A d^2\right )\right )+\left (1617 a^3 C d^6+3 a^2 b \left (524 C c^2-663 B d c+1001 A d^2\right ) d^4+2 a b^2 c^2 \left (1568 C c^2-1794 B d c+2145 A d^2\right ) d^2-64 b^3 c^4 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-3 d x \left (539 a^2 C d^4+a b d^2 \left (1001 A d^2-988 B c d+956 c^2 C\right )+16 b^2 c^2 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )+3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )}{15 d^2}\right )+\frac {2}{7} b d^8 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (a d^2 (367 c C-195 B d)+b c \left (715 A d^2-923 B c d+1083 c^2 C\right )\right )}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {\frac {1}{7} b d^8 \left (\frac {2 \int \frac {a d \left (3 a^2 (107 c C+325 B d) d^4+a b c \left (244 C c^2-312 B d c+429 A d^2\right ) d^2-16 b^2 c^3 \left (120 C c^2-130 B d c+143 A d^2\right )\right )+\left (1617 a^3 C d^6+3 a^2 b \left (524 C c^2-663 B d c+1001 A d^2\right ) d^4+2 a b^2 c^2 \left (1568 C c^2-1794 B d c+2145 A d^2\right ) d^2-64 b^3 c^4 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-3 d x \left (539 a^2 C d^4+a b d^2 \left (1001 A d^2-988 B c d+956 c^2 C\right )+16 b^2 c^2 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )+3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )}{15 d^2}\right )+\frac {2}{7} b d^8 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (a d^2 (367 c C-195 B d)+b c \left (715 A d^2-923 B c d+1083 c^2 C\right )\right )}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {\frac {1}{7} b d^8 \left (\frac {2 \left (\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {\left (1617 a^3 C d^6+3 a^2 b d^4 \left (1001 A d^2-663 B c d+524 c^2 C\right )+2 a b^2 c^2 d^2 \left (2145 A d^2-1794 B c d+1568 c^2 C\right )-64 b^3 c^4 \left (143 A d^2-130 B c d+120 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-3 d x \left (539 a^2 C d^4+a b d^2 \left (1001 A d^2-988 B c d+956 c^2 C\right )+16 b^2 c^2 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )+3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )}{15 d^2}\right )+\frac {2}{7} b d^8 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (a d^2 (367 c C-195 B d)+b c \left (715 A d^2-923 B c d+1083 c^2 C\right )\right )}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {\frac {1}{7} b d^8 \left (\frac {2 \left (\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {\sqrt {1-\frac {b x^2}{a}} \left (1617 a^3 C d^6+3 a^2 b d^4 \left (1001 A d^2-663 B c d+524 c^2 C\right )+2 a b^2 c^2 d^2 \left (2145 A d^2-1794 B c d+1568 c^2 C\right )-64 b^3 c^4 \left (143 A d^2-130 B c d+120 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-3 d x \left (539 a^2 C d^4+a b d^2 \left (1001 A d^2-988 B c d+956 c^2 C\right )+16 b^2 c^2 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )+3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )}{15 d^2}\right )+\frac {2}{7} b d^8 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (a d^2 (367 c C-195 B d)+b c \left (715 A d^2-923 B c d+1083 c^2 C\right )\right )}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {\frac {1}{7} b d^8 \left (\frac {2 \left (\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (1617 a^3 C d^6+3 a^2 b d^4 \left (1001 A d^2-663 B c d+524 c^2 C\right )+2 a b^2 c^2 d^2 \left (2145 A d^2-1794 B c d+1568 c^2 C\right )-64 b^3 c^4 \left (143 A d^2-130 B c d+120 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-3 d x \left (539 a^2 C d^4+a b d^2 \left (1001 A d^2-988 B c d+956 c^2 C\right )+16 b^2 c^2 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )+3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )}{15 d^2}\right )+\frac {2}{7} b d^8 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (a d^2 (367 c C-195 B d)+b c \left (715 A d^2-923 B c d+1083 c^2 C\right )\right )}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {\frac {1}{7} b d^8 \left (\frac {2 \left (\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (1617 a^3 C d^6+3 a^2 b d^4 \left (1001 A d^2-663 B c d+524 c^2 C\right )+2 a b^2 c^2 d^2 \left (2145 A d^2-1794 B c d+1568 c^2 C\right )-64 b^3 c^4 \left (143 A d^2-130 B c d+120 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-3 d x \left (539 a^2 C d^4+a b d^2 \left (1001 A d^2-988 B c d+956 c^2 C\right )+16 b^2 c^2 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )+3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )}{15 d^2}\right )+\frac {2}{7} b d^8 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (a d^2 (367 c C-195 B d)+b c \left (715 A d^2-923 B c d+1083 c^2 C\right )\right )}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {2}{11} d \left (a-b x^2\right )^{3/2} (c+d x)^{5/2} (45 c C-13 B d)-\frac {\frac {2}{9} d^5 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} \left (77 a C d^2+b \left (143 A d^2-364 B c d+633 c^2 C\right )\right )-\frac {\frac {1}{7} b d^8 \left (\frac {2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (1617 a^3 C d^6+3 a^2 b d^4 \left (1001 A d^2-663 B c d+524 c^2 C\right )+2 a b^2 c^2 d^2 \left (2145 A d^2-1794 B c d+1568 c^2 C\right )-64 b^3 c^4 \left (143 A d^2-130 B c d+120 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-3 d x \left (539 a^2 C d^4+a b d^2 \left (1001 A d^2-988 B c d+956 c^2 C\right )+16 b^2 c^2 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )+3 a^2 d^4 (432 c C-325 B d)+2 a b c d^2 \left (1287 A d^2-1326 B c d+1312 c^2 C\right )+64 b^2 c^3 \left (143 A d^2-130 B c d+120 c^2 C\right )\right )}{15 d^2}\right )+\frac {2}{7} b d^8 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (a d^2 (367 c C-195 B d)+b c \left (715 A d^2-923 B c d+1083 c^2 C\right )\right )}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C \left (a-b x^2\right )^{3/2} (c+d x)^{7/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {2}{11} d (45 c C-13 B d) (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}-\frac {\frac {2}{9} d^5 \left (77 a C d^2+b \left (633 C c^2-364 B d c+143 A d^2\right )\right ) (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}-\frac {\frac {2}{7} b \left (a (367 c C-195 B d) d^2+b c \left (1083 C c^2-923 B d c+715 A d^2\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2} d^8+\frac {1}{7} b \left (\frac {2 \left (-\frac {2 \sqrt {a} \left (1617 a^3 C d^6+3 a^2 b \left (524 C c^2-663 B d c+1001 A d^2\right ) d^4+2 a b^2 c^2 \left (1568 C c^2-1794 B d c+2145 A d^2\right ) d^2-64 b^3 c^4 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (3 a^2 (432 c C-325 B d) d^4+2 a b c \left (1312 C c^2-1326 B d c+1287 A d^2\right ) d^2+64 b^2 c^3 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{15 d^2}-\frac {2 \sqrt {c+d x} \left (3 a^2 (432 c C-325 B d) d^4+2 a b c \left (1312 C c^2-1326 B d c+1287 A d^2\right ) d^2-3 \left (539 a^2 C d^4+a b \left (956 C c^2-988 B d c+1001 A d^2\right ) d^2+16 b^2 c^2 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) x d+64 b^2 c^3 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) \sqrt {a-b x^2}}{15 d^2}\right ) d^8}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C (c+d x)^{7/2} \left (a-b x^2\right )^{3/2}}{13 b d^4}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {2}{11} d (45 c C-13 B d) (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}-\frac {\frac {2}{9} d^5 \left (77 a C d^2+b \left (633 C c^2-364 B d c+143 A d^2\right )\right ) (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}-\frac {\frac {2}{7} b \left (a (367 c C-195 B d) d^2+b c \left (1083 C c^2-923 B d c+715 A d^2\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2} d^8+\frac {1}{7} b \left (\frac {2 \left (-\frac {2 \sqrt {a} \left (1617 a^3 C d^6+3 a^2 b \left (524 C c^2-663 B d c+1001 A d^2\right ) d^4+2 a b^2 c^2 \left (1568 C c^2-1794 B d c+2145 A d^2\right ) d^2-64 b^3 c^4 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (3 a^2 (432 c C-325 B d) d^4+2 a b c \left (1312 C c^2-1326 B d c+1287 A d^2\right ) d^2+64 b^2 c^3 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{15 d^2}-\frac {2 \sqrt {c+d x} \left (3 a^2 (432 c C-325 B d) d^4+2 a b c \left (1312 C c^2-1326 B d c+1287 A d^2\right ) d^2-3 \left (539 a^2 C d^4+a b \left (956 C c^2-988 B d c+1001 A d^2\right ) d^2+16 b^2 c^2 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) x d+64 b^2 c^3 \left (120 C c^2-130 B d c+143 A d^2\right )\right ) \sqrt {a-b x^2}}{15 d^2}\right ) d^8}{3 b d^3}}{11 b d^4}}{13 b d^5}-\frac {2 C (c+d x)^{7/2} \left (a-b x^2\right )^{3/2}}{13 b d^4}\) |
Input:
Int[(x^3*Sqrt[a - b*x^2]*(A + B*x + C*x^2))/Sqrt[c + d*x],x]
Output:
(-2*C*(c + d*x)^(7/2)*(a - b*x^2)^(3/2))/(13*b*d^4) + ((2*d*(45*c*C - 13*B *d)*(c + d*x)^(5/2)*(a - b*x^2)^(3/2))/11 - ((2*d^5*(77*a*C*d^2 + b*(633*c ^2*C - 364*B*c*d + 143*A*d^2))*(c + d*x)^(3/2)*(a - b*x^2)^(3/2))/9 - ((2* b*d^8*(a*d^2*(367*c*C - 195*B*d) + b*c*(1083*c^2*C - 923*B*c*d + 715*A*d^2 ))*Sqrt[c + d*x]*(a - b*x^2)^(3/2))/7 + (b*d^8*((-2*Sqrt[c + d*x]*(3*a^2*d ^4*(432*c*C - 325*B*d) + 64*b^2*c^3*(120*c^2*C - 130*B*c*d + 143*A*d^2) + 2*a*b*c*d^2*(1312*c^2*C - 1326*B*c*d + 1287*A*d^2) - 3*d*(539*a^2*C*d^4 + 16*b^2*c^2*(120*c^2*C - 130*B*c*d + 143*A*d^2) + a*b*d^2*(956*c^2*C - 988* B*c*d + 1001*A*d^2))*x)*Sqrt[a - b*x^2])/(15*d^2) + (2*((-2*Sqrt[a]*(1617* a^3*C*d^6 - 64*b^3*c^4*(120*c^2*C - 130*B*c*d + 143*A*d^2) + 3*a^2*b*d^4*( 524*c^2*C - 663*B*c*d + 1001*A*d^2) + 2*a*b^2*c^2*d^2*(1568*c^2*C - 1794*B *c*d + 2145*A*d^2))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqr t[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sq rt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*(b*c^2 - a*d^2)*(3*a^2*d^4*(432*c*C - 325*B*d) + 64*b^2*c^3* (120*c^2*C - 130*B*c*d + 143*A*d^2) + 2*a*b*c*d^2*(1312*c^2*C - 1326*B*c*d + 1287*A*d^2))*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*d^3))/(11*b*d^4))/(13*b*d^5)
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. \(2090\) vs. \(2(741)=1482\).
Time = 9.38 (sec) , antiderivative size = 2091, normalized size of antiderivative = 2.52
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2091\) |
| risch | \(\text {Expression too large to display}\) | \(2093\) |
| default | \(\text {Expression too large to display}\) | \(5767\) |
Input:
int(x^3*(-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x,method=_RETURNVERBO SE)
Output:
((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(2/13*C/d*x^5*(- b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/11*(-B*b+12/13*C/d*b*c)/b/d*x^4*(-b*d*x ^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/9*(-A*b+2/13*a*C-10/11*(-B*b+12/13*C/d*b*c)/ d*c)/b/d*x^3*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/7*(B*a-10/13*C/d*a*c+9/1 1*(-B*b+12/13*C/d*b*c)/b*a-8/9*(-A*b+2/13*a*C-10/11*(-B*b+12/13*C/d*b*c)/d *c)/d*c)/b/d*x^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(A*a+8/11*(-B*b+12 /13*C/d*b*c)/b/d*a*c+7/9*(-A*b+2/13*a*C-10/11*(-B*b+12/13*C/d*b*c)/d*c)/b* a-6/7*(B*a-10/13*C/d*a*c+9/11*(-B*b+12/13*C/d*b*c)/b*a-8/9*(-A*b+2/13*a*C- 10/11*(-B*b+12/13*C/d*b*c)/d*c)/d*c)/d*c)/b/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a* c)^(1/2)-2/3*(2/3*(-A*b+2/13*a*C-10/11*(-B*b+12/13*C/d*b*c)/d*c)/b/d*a*c+5 /7*(B*a-10/13*C/d*a*c+9/11*(-B*b+12/13*C/d*b*c)/b*a-8/9*(-A*b+2/13*a*C-10/ 11*(-B*b+12/13*C/d*b*c)/d*c)/d*c)/b*a-4/5*(A*a+8/11*(-B*b+12/13*C/d*b*c)/b /d*a*c+7/9*(-A*b+2/13*a*C-10/11*(-B*b+12/13*C/d*b*c)/d*c)/b*a-6/7*(B*a-10/ 13*C/d*a*c+9/11*(-B*b+12/13*C/d*b*c)/b*a-8/9*(-A*b+2/13*a*C-10/11*(-B*b+12 /13*C/d*b*c)/d*c)/d*c)/d*c)/d*c)/b/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2* (2/5*(A*a+8/11*(-B*b+12/13*C/d*b*c)/b/d*a*c+7/9*(-A*b+2/13*a*C-10/11*(-B*b +12/13*C/d*b*c)/d*c)/b*a-6/7*(B*a-10/13*C/d*a*c+9/11*(-B*b+12/13*C/d*b*c)/ b*a-8/9*(-A*b+2/13*a*C-10/11*(-B*b+12/13*C/d*b*c)/d*c)/d*c)/d*c)/b/d*a*c+1 /3*(2/3*(-A*b+2/13*a*C-10/11*(-B*b+12/13*C/d*b*c)/d*c)/b/d*a*c+5/7*(B*a-10 /13*C/d*a*c+9/11*(-B*b+12/13*C/d*b*c)/b*a-8/9*(-A*b+2/13*a*C-10/11*(-B*...
Time = 0.10 (sec) , antiderivative size = 739, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:
integrate(x^3*(-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="f ricas")
Output:
-2/135135*(2*(7680*C*b^3*c^7 - 8320*B*b^3*c^6*d + 9828*B*a*b^2*c^4*d^3 + 1 053*B*a^2*b*c^2*d^5 + 2925*B*a^3*d^7 - 64*(139*C*a*b^2 - 143*A*b^3)*c^5*d^ 2 - 6*(140*C*a^2*b + 1859*A*a*b^2)*c^3*d^4 - 6*(109*C*a^3 + 286*A*a^2*b)*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*(7680*C*b^3*c^6*d - 83 20*B*b^3*c^5*d^2 + 3588*B*a*b^2*c^3*d^4 + 1989*B*a^2*b*c*d^6 - 64*(49*C*a* b^2 - 143*A*b^3)*c^4*d^3 - 6*(262*C*a^2*b + 715*A*a*b^2)*c^2*d^5 - 231*(7* C*a^3 + 13*A*a^2*b)*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*(3465*C*b^3*d^7*x^5 - 7680*C*b^3*c^5*d^2 + 8320*B*b^3*c^4*d^3 - 2548 *B*a*b^2*c^2*d^5 - 1950*B*a^2*b*d^7 + 64*(34*C*a*b^2 - 143*A*b^3)*c^3*d^4 + 2*(757*C*a^2*b + 1573*A*a*b^2)*c*d^6 - 315*(12*C*b^3*c*d^6 - 13*B*b^3*d^ 7)*x^4 + 35*(120*C*b^3*c^2*d^5 - 130*B*b^3*c*d^6 - 11*(2*C*a*b^2 - 13*A*b^ 3)*d^7)*x^3 - 10*(480*C*b^3*c^3*d^4 - 520*B*b^3*c^2*d^5 + 117*B*a*b^2*d^7 - (97*C*a*b^2 - 572*A*b^3)*c*d^6)*x^2 + 2*(2880*C*b^3*c^4*d^3 - 3120*B*b^3 *c^3*d^4 + 793*B*a*b^2*c*d^6 - 6*(111*C*a*b^2 - 572*A*b^3)*c^2*d^5 - 77*(7 *C*a^2*b + 13*A*a*b^2)*d^7)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*d^8)
\[ \int \frac {x^3 \sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\int \frac {x^{3} \sqrt {a - b x^{2}} \left (A + B x + C x^{2}\right )}{\sqrt {c + d x}}\, dx \] Input:
integrate(x**3*(-b*x**2+a)**(1/2)*(C*x**2+B*x+A)/(d*x+c)**(1/2),x)
Output:
Integral(x**3*sqrt(a - b*x**2)*(A + B*x + C*x**2)/sqrt(c + d*x), x)
\[ \int \frac {x^3 \sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a} x^{3}}{\sqrt {d x + c}} \,d x } \] Input:
integrate(x^3*(-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="m axima")
Output:
integrate((C*x^2 + B*x + A)*sqrt(-b*x^2 + a)*x^3/sqrt(d*x + c), x)
\[ \int \frac {x^3 \sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a} x^{3}}{\sqrt {d x + c}} \,d x } \] Input:
integrate(x^3*(-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm="g iac")
Output:
integrate((C*x^2 + B*x + A)*sqrt(-b*x^2 + a)*x^3/sqrt(d*x + c), x)
Timed out. \[ \int \frac {x^3 \sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\int \frac {x^3\,\sqrt {a-b\,x^2}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {c+d\,x}} \,d x \] Input:
int((x^3*(a - b*x^2)^(1/2)*(A + B*x + C*x^2))/(c + d*x)^(1/2),x)
Output:
int((x^3*(a - b*x^2)^(1/2)*(A + B*x + C*x^2))/(c + d*x)^(1/2), x)
\[ \int \frac {x^3 \sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\int \frac {x^{3} \sqrt {-b \,x^{2}+a}\, \left (C \,x^{2}+B x +A \right )}{\sqrt {d x +c}}d x \] Input:
int(x^3*(-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x)
Output:
int(x^3*(-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2),x)