Integrand size = 35, antiderivative size = 594 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 \left (3 a d^2 (37 c C-25 B d)+b c \left (187 c^2 C-171 B c d+147 A d^2\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{315 b^2 d^4}-\frac {2 \left (49 a C d^2+b \left (233 c^2 C-144 B c d+63 A d^2\right )\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{315 b^2 d^4}+\frac {2 (29 c C-9 B d) (c+d x)^{5/2} \sqrt {a-b x^2}}{63 b d^4}-\frac {2 C (c+d x)^{7/2} \sqrt {a-b x^2}}{9 b d^4}-\frac {2 \sqrt {a} \left (147 a^2 C d^4+8 b^2 c^2 \left (16 c^2 C-18 B c d+21 A d^2\right )+3 a b d^2 \left (36 c^2 C-44 B c d+63 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 b^{5/2} d^5 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (3 a^2 d^4 (37 c C-25 B d)+8 b^2 c^3 \left (16 c^2 C-18 B c d+21 A d^2\right )+a b c d^2 \left (76 c^2 C-96 B c d+147 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 b^{5/2} d^5 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
2/315*(3*a*d^2*(-25*B*d+37*C*c)+b*c*(147*A*d^2-171*B*c*d+187*C*c^2))*(d*x+ c)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d^4-2/315*(49*a*C*d^2+b*(63*A*d^2-144*B*c*d+ 233*C*c^2))*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b^2/d^4+2/63*(-9*B*d+29*C*c)*(d *x+c)^(5/2)*(-b*x^2+a)^(1/2)/b/d^4-2/9*C*(d*x+c)^(7/2)*(-b*x^2+a)^(1/2)/b/ d^4-2/315*a^(1/2)*(147*a^2*C*d^4+8*b^2*c^2*(21*A*d^2-18*B*c*d+16*C*c^2)+3* a*b*d^2*(63*A*d^2-44*B*c*d+36*C*c^2))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*E llipticE(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^5/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/ (-b*x^2+a)^(1/2)+2/315*a^(1/2)*(3*a^2*d^4*(-25*B*d+37*C*c)+8*b^2*c^3*(21*A *d^2-18*B*c*d+16*C*c^2)+a*b*c*d^2*(147*A*d^2-96*B*c*d+76*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^5/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 28.57 (sec) , antiderivative size = 738, normalized size of antiderivative = 1.24 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (-147 a^2 C d^4-8 b^2 c^2 \left (16 c^2 C-18 B c d+21 A d^2\right )-3 a b d^2 \left (36 c^2 C-44 B c d+63 A d^2\right )+b (c+d x) \left (a d^2 (62 c C-75 B d-49 C d x)+b \left (64 c^3 C-24 c^2 d (3 B+2 C x)-d^3 x (63 A+5 x (9 B+7 C x))+2 c d^2 (42 A+x (27 B+20 C x))\right )\right )+\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (147 a^2 C d^4+8 b^2 c^2 \left (16 c^2 C-18 B c d+21 A d^2\right )+3 a b d^2 \left (36 c^2 C-44 B c d+63 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}+\frac {i \sqrt {a} \sqrt {b} \left (147 a^2 C d^4-3 a^{3/2} \sqrt {b} d^3 (12 c C+25 B d)+8 b^2 c^2 \left (16 c^2 C-18 B c d+21 A d^2\right )-2 \sqrt {a} b^{3/2} c d \left (16 c^2 C-18 B c d+21 A d^2\right )+3 a b d^2 \left (36 c^2 C-44 B c d+63 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{315 b^3 d^4 \sqrt {c+d x}} \] Input:
Integrate[(x^3*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
Output:
(2*Sqrt[a - b*x^2]*(-147*a^2*C*d^4 - 8*b^2*c^2*(16*c^2*C - 18*B*c*d + 21*A *d^2) - 3*a*b*d^2*(36*c^2*C - 44*B*c*d + 63*A*d^2) + b*(c + d*x)*(a*d^2*(6 2*c*C - 75*B*d - 49*C*d*x) + b*(64*c^3*C - 24*c^2*d*(3*B + 2*C*x) - d^3*x* (63*A + 5*x*(9*B + 7*C*x)) + 2*c*d^2*(42*A + x*(27*B + 20*C*x)))) + (I*Sqr t[b]*(Sqrt[b]*c - Sqrt[a]*d)*(147*a^2*C*d^4 + 8*b^2*c^2*(16*c^2*C - 18*B*c *d + 21*A*d^2) + 3*a*b*d^2*(36*c^2*C - 44*B*c*d + 63*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)])/(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2)) + (I*Sqrt[a]*Sqrt[b]*(147*a^2*C*d^4 - 3*a^(3/2)*Sqrt[b]*d^3*(12*c*C + 25*B*d) + 8*b^2*c^2*(16*c^2*C - 18*B*c*d + 21*A*d^2) - 2*Sqrt[a]*b^(3/2)*c*d*(16*c^2*C - 18*B*c*d + 21*A*d^2) + 3*a *b*d^2*(36*c^2*C - 44*B*c*d + 63*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)*El lipticF[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*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] *(-a + b*x^2))))/(315*b^3*d^4*Sqrt[c + d*x])
Time = 3.67 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2185, 27, 2185, 27, 2185, 27, 2185, 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.
| \(\displaystyle \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {-b d^4 (29 c C-9 B d) x^4-d^3 \left (33 b C c^2-9 A b d^2-7 a C d^2\right ) x^3-3 c C d^2 \left (5 b c^2-7 a d^2\right ) x^2-c^2 C d \left (2 b c^2-21 a d^2\right ) x+7 a c^3 C d^2}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-b d^4 (29 c C-9 B d) x^4-d^3 \left (33 b C c^2-9 A b d^2-7 a C d^2\right ) x^3-3 c C d^2 \left (5 b c^2-7 a d^2\right ) x^2-c^2 C d \left (2 b c^2-21 a d^2\right ) x+7 a c^3 C d^2}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {2 \int \frac {-b \left (49 a C d^2+b \left (233 C c^2-144 B d c+63 A d^2\right )\right ) x^3 d^7-b \left (b (214 c C-99 B d) c^2+a d^2 (2 c C+45 B d)\right ) x^2 d^6+3 a b c^2 (32 c C-15 B d) d^6+b c \left (a d^2 (143 c C-90 B d)-2 b c^2 (22 c C-9 B d)\right ) x d^5}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\int \frac {-b \left (49 a C d^2+b \left (233 C c^2-144 B d c+63 A d^2\right )\right ) x^3 d^7-b \left (b (214 c C-99 B d) c^2+a d^2 (2 c C+45 B d)\right ) x^2 d^6+3 a b c^2 (32 c C-15 B d) d^6+b c \left (a d^2 (143 c C-90 B d)-2 b c^2 (22 c C-9 B d)\right ) x d^5}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {2 \int \frac {3 \left (-b^2 \left (3 a (37 c C-25 B d) d^2+b c \left (187 C c^2-171 B d c+147 A d^2\right )\right ) x^2 d^9+a b c \left (49 a C d^2+b \left (73 C c^2-69 B d c+63 A d^2\right )\right ) d^9+b \left (49 a^2 C d^4-a b \left (38 C c^2-6 B d c-63 A d^2\right ) d^2-2 b^2 c^2 \left (41 C c^2-33 B d c+21 A d^2\right )\right ) x d^8\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {3 \int \frac {-b^2 \left (3 a (37 c C-25 B d) d^2+b c \left (187 C c^2-171 B d c+147 A d^2\right )\right ) x^2 d^9+a b c \left (49 a C d^2+b \left (73 C c^2-69 B d c+63 A d^2\right )\right ) d^9+b \left (49 a^2 C d^4-a b \left (38 C c^2-6 B d c-63 A d^2\right ) d^2-2 b^2 c^2 \left (41 C c^2-33 B d c+21 A d^2\right )\right ) x d^8}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {3 \left (\frac {2}{3} b d^8 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (37 c C-25 B d)+b c \left (147 A d^2-171 B c d+187 c^2 C\right )\right )-\frac {2 \int -\frac {b^2 d^{10} \left (a d \left (3 a (12 c C+25 B d) d^2+2 b c \left (16 C c^2-18 B d c+21 A d^2\right )\right )+\left (147 a^2 C d^4+3 a b \left (36 C c^2-44 B d c+63 A d^2\right ) d^2+8 b^2 c^2 \left (16 C c^2-18 B d c+21 A d^2\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}\right )}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {3 \left (\frac {1}{3} b d^8 \int \frac {a d \left (3 a (12 c C+25 B d) d^2+2 b c \left (16 C c^2-18 B d c+21 A d^2\right )\right )+\left (147 a^2 C d^4+3 a b \left (36 C c^2-44 B d c+63 A d^2\right ) d^2+8 b^2 c^2 \left (16 C c^2-18 B d c+21 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2}{3} b d^8 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (37 c C-25 B d)+b c \left (147 A d^2-171 B c d+187 c^2 C\right )\right )\right )}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {3 \left (\frac {1}{3} b d^8 \left (\frac {\left (147 a^2 C d^4+3 a b d^2 \left (63 A d^2-44 B c d+36 c^2 C\right )+8 b^2 c^2 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (3 a^2 d^4 (37 c C-25 B d)+a b c d^2 \left (147 A d^2-96 B c d+76 c^2 C\right )+8 b^2 c^3 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )+\frac {2}{3} b d^8 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (37 c C-25 B d)+b c \left (147 A d^2-171 B c d+187 c^2 C\right )\right )\right )}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {3 \left (\frac {1}{3} b d^8 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (147 a^2 C d^4+3 a b d^2 \left (63 A d^2-44 B c d+36 c^2 C\right )+8 b^2 c^2 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (3 a^2 d^4 (37 c C-25 B d)+a b c d^2 \left (147 A d^2-96 B c d+76 c^2 C\right )+8 b^2 c^3 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )+\frac {2}{3} b d^8 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (37 c C-25 B d)+b c \left (147 A d^2-171 B c d+187 c^2 C\right )\right )\right )}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {3 \left (\frac {1}{3} b d^8 \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (147 a^2 C d^4+3 a b d^2 \left (63 A d^2-44 B c d+36 c^2 C\right )+8 b^2 c^2 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (3 a^2 d^4 (37 c C-25 B d)+a b c d^2 \left (147 A d^2-96 B c d+76 c^2 C\right )+8 b^2 c^3 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )+\frac {2}{3} b d^8 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (37 c C-25 B d)+b c \left (147 A d^2-171 B c d+187 c^2 C\right )\right )\right )}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {3 \left (\frac {1}{3} b d^8 \left (-\frac {\left (3 a^2 d^4 (37 c C-25 B d)+a b c d^2 \left (147 A d^2-96 B c d+76 c^2 C\right )+8 b^2 c^3 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (147 a^2 C d^4+3 a b d^2 \left (63 A d^2-44 B c d+36 c^2 C\right )+8 b^2 c^2 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} b d^8 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (37 c C-25 B d)+b c \left (147 A d^2-171 B c d+187 c^2 C\right )\right )\right )}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {3 \left (\frac {1}{3} b d^8 \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (3 a^2 d^4 (37 c C-25 B d)+a b c d^2 \left (147 A d^2-96 B c d+76 c^2 C\right )+8 b^2 c^3 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (147 a^2 C d^4+3 a b d^2 \left (63 A d^2-44 B c d+36 c^2 C\right )+8 b^2 c^2 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} b d^8 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (37 c C-25 B d)+b c \left (147 A d^2-171 B c d+187 c^2 C\right )\right )\right )}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {3 \left (\frac {1}{3} b d^8 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (3 a^2 d^4 (37 c C-25 B d)+a b c d^2 \left (147 A d^2-96 B c d+76 c^2 C\right )+8 b^2 c^3 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (147 a^2 C d^4+3 a b d^2 \left (63 A d^2-44 B c d+36 c^2 C\right )+8 b^2 c^2 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} b d^8 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (37 c C-25 B d)+b c \left (147 A d^2-171 B c d+187 c^2 C\right )\right )\right )}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (29 c C-9 B d)-\frac {\frac {2}{5} d^5 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a C d^2+b \left (63 A d^2-144 B c d+233 c^2 C\right )\right )-\frac {3 \left (\frac {1}{3} b d^8 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (3 a^2 d^4 (37 c C-25 B d)+a b c d^2 \left (147 A d^2-96 B c d+76 c^2 C\right )+8 b^2 c^3 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (147 a^2 C d^4+3 a b d^2 \left (63 A d^2-44 B c d+36 c^2 C\right )+8 b^2 c^2 \left (21 A d^2-18 B c d+16 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} b d^8 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (37 c C-25 B d)+b c \left (147 A d^2-171 B c d+187 c^2 C\right )\right )\right )}{5 b d^3}}{7 b d^4}}{9 b d^5}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^4}\) |
Input:
Int[(x^3*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
Output:
(-2*C*(c + d*x)^(7/2)*Sqrt[a - b*x^2])/(9*b*d^4) + ((2*d*(29*c*C - 9*B*d)* (c + d*x)^(5/2)*Sqrt[a - b*x^2])/7 - ((2*d^5*(49*a*C*d^2 + b*(233*c^2*C - 144*B*c*d + 63*A*d^2))*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/5 - (3*((2*b*d^8*( 3*a*d^2*(37*c*C - 25*B*d) + b*c*(187*c^2*C - 171*B*c*d + 147*A*d^2))*Sqrt[ c + d*x]*Sqrt[a - b*x^2])/3 + (b*d^8*((-2*Sqrt[a]*(147*a^2*C*d^4 + 8*b^2*c ^2*(16*c^2*C - 18*B*c*d + 21*A*d^2) + 3*a*b*d^2*(36*c^2*C - 44*B*c*d + 63* 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)])/(Sqrt[b]*d*Sqrt [(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a ]*(3*a^2*d^4*(37*c*C - 25*B*d) + 8*b^2*c^3*(16*c^2*C - 18*B*c*d + 21*A*d^2 ) + a*b*c*d^2*(76*c^2*C - 96*B*c*d + 147*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 - (Sq rt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*S qrt[c + d*x]*Sqrt[a - b*x^2])))/3))/(5*b*d^3))/(7*b*d^4))/(9*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[(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]))
Time = 6.40 (sec) , antiderivative size = 974, normalized size of antiderivative = 1.64
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {2 C \,x^{3} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{9 b d}-\frac {2 \left (B -\frac {8 c C}{9 d}\right ) x^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{7 b d}-\frac {2 \left (A +\frac {7 a C}{9 b}-\frac {6 \left (B -\frac {8 c C}{9 d}\right ) c}{7 d}\right ) x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b d}-\frac {2 \left (\frac {2 C a c}{3 b d}+\frac {5 \left (B -\frac {8 c C}{9 d}\right ) a}{7 b}-\frac {4 \left (A +\frac {7 a C}{9 b}-\frac {6 \left (B -\frac {8 c C}{9 d}\right ) c}{7 d}\right ) c}{5 d}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (\frac {2 \left (A +\frac {7 a C}{9 b}-\frac {6 \left (B -\frac {8 c C}{9 d}\right ) c}{7 d}\right ) a c}{5 b d}+\frac {\left (\frac {2 C a c}{3 b d}+\frac {5 \left (B -\frac {8 c C}{9 d}\right ) a}{7 b}-\frac {4 \left (A +\frac {7 a C}{9 b}-\frac {6 \left (B -\frac {8 c C}{9 d}\right ) c}{7 d}\right ) c}{5 d}\right ) a}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {4 \left (B -\frac {8 c C}{9 d}\right ) a c}{7 b d}+\frac {3 \left (A +\frac {7 a C}{9 b}-\frac {6 \left (B -\frac {8 c C}{9 d}\right ) c}{7 d}\right ) a}{5 b}-\frac {2 \left (\frac {2 C a c}{3 b d}+\frac {5 \left (B -\frac {8 c C}{9 d}\right ) a}{7 b}-\frac {4 \left (A +\frac {7 a C}{9 b}-\frac {6 \left (B -\frac {8 c C}{9 d}\right ) c}{7 d}\right ) c}{5 d}\right ) c}{3 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(974\) |
| risch | \(\text {Expression too large to display}\) | \(1325\) |
| default | \(\text {Expression too large to display}\) | \(4057\) |
Input:
int(x^3*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(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)*(-2/9*C/b/d*x^3* (-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/7*(B-8/9*c*C/d)/b/d*x^2*(-b*d*x^3-b*c *x^2+a*d*x+a*c)^(1/2)-2/5*(A+7/9*a*C/b-6/7*(B-8/9*c*C/d)/d*c)/b/d*x*(-b*d* x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(2/3*C/b/d*a*c+5/7*(B-8/9*c*C/d)/b*a-4/5* (A+7/9*a*C/b-6/7*(B-8/9*c*C/d)/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+7/9*a*C/b-6/7*(B-8/9*c*C/d)/d*c)/b/d*a*c+1/3*(2/3*C/b/d*a* c+5/7*(B-8/9*c*C/d)/b*a-4/5*(A+7/9*a*C/b-6/7*(B-8/9*c*C/d)/d*c)/d*c)/b*a)* (c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^ (1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^ (1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1 /b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/ 2))+2*(4/7*(B-8/9*c*C/d)/b/d*a*c+3/5*(A+7/9*a*C/b-6/7*(B-8/9*c*C/d)/d*c)/b *a-2/3*(2/3*C/b/d*a*c+5/7*(B-8/9*c*C/d)/b*a-4/5*(A+7/9*a*C/b-6/7*(B-8/9*c* C/d)/d*c)/d*c)/d*c)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^ (1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/ 2))/(-c/d+1/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(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+ 1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF( ((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a *b)^(1/2)))^(1/2))))
Time = 0.09 (sec) , antiderivative size = 467, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 \, {\left ({\left (128 \, C b^{2} c^{5} - 144 \, B b^{2} c^{4} d - 24 \, B a b c^{2} d^{3} - 225 \, B a^{2} d^{5} + 12 \, {\left (C a b + 14 \, A b^{2}\right )} c^{3} d^{2} + 3 \, {\left (13 \, C a^{2} + 21 \, A a b\right )} c d^{4}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (128 \, C b^{2} c^{4} d - 144 \, B b^{2} c^{3} d^{2} - 132 \, B a b c d^{4} + 12 \, {\left (9 \, C a b + 14 \, A b^{2}\right )} c^{2} d^{3} + 21 \, {\left (7 \, C a^{2} + 9 \, A a b\right )} d^{5}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (35 \, C b^{2} d^{5} x^{3} - 64 \, C b^{2} c^{3} d^{2} + 72 \, B b^{2} c^{2} d^{3} + 75 \, B a b d^{5} - 2 \, {\left (31 \, C a b + 42 \, A b^{2}\right )} c d^{4} - 5 \, {\left (8 \, C b^{2} c d^{4} - 9 \, B b^{2} d^{5}\right )} x^{2} + {\left (48 \, C b^{2} c^{2} d^{3} - 54 \, B b^{2} c d^{4} + 7 \, {\left (7 \, C a b + 9 \, A b^{2}\right )} d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{945 \, b^{3} d^{6}} \] Input:
integrate(x^3*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="f ricas")
Output:
2/945*((128*C*b^2*c^5 - 144*B*b^2*c^4*d - 24*B*a*b*c^2*d^3 - 225*B*a^2*d^5 + 12*(C*a*b + 14*A*b^2)*c^3*d^2 + 3*(13*C*a^2 + 21*A*a*b)*c*d^4)*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) + 3*(128*C*b^2*c^4*d - 144*B*b^2*c^3*d^2 - 132*B*a*b*c*d^4 + 12*(9*C*a*b + 14*A*b^2)*c^2*d^3 + 21*(7*C*a^2 + 9*A*a *b)*d^5)*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*(35*C*b^2* d^5*x^3 - 64*C*b^2*c^3*d^2 + 72*B*b^2*c^2*d^3 + 75*B*a*b*d^5 - 2*(31*C*a*b + 42*A*b^2)*c*d^4 - 5*(8*C*b^2*c*d^4 - 9*B*b^2*d^5)*x^2 + (48*C*b^2*c^2*d ^3 - 54*B*b^2*c*d^4 + 7*(7*C*a*b + 9*A*b^2)*d^5)*x)*sqrt(-b*x^2 + a)*sqrt( d*x + c))/(b^3*d^6)
\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {x^{3} \left (A + B x + C x^{2}\right )}{\sqrt {a - b x^{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(x**3*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral(x**3*(A + B*x + C*x**2)/(sqrt(a - b*x**2)*sqrt(c + d*x)), x)
\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{3}}{\sqrt {-b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^3*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="m axima")
Output:
integrate((C*x^2 + B*x + A)*x^3/(sqrt(-b*x^2 + a)*sqrt(d*x + c)), x)
\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{3}}{\sqrt {-b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^3*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="g iac")
Output:
integrate((C*x^2 + B*x + A)*x^3/(sqrt(-b*x^2 + a)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {x^3\,\left (C\,x^2+B\,x+A\right )}{\sqrt {a-b\,x^2}\,\sqrt {c+d\,x}} \,d x \] Input:
int((x^3*(A + B*x + C*x^2))/((a - b*x^2)^(1/2)*(c + d*x)^(1/2)),x)
Output:
int((x^3*(A + B*x + C*x^2))/((a - b*x^2)^(1/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:
int(x^3*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x)
Output:
( - 378*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*b*d**3 - 294*sqrt(c + d*x)*sqr t(a - b*x**2)*a**2*c*d**3 - 252*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b**2*c*d* *2*x - 36*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b**2*c*d**2 + 32*sqrt(c + d*x)* sqrt(a - b*x**2)*a*b*c**3*d - 196*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c**2* d**2*x + 216*sqrt(c + d*x)*sqrt(a - b*x**2)*b**3*c**2*d*x - 180*sqrt(c + d *x)*sqrt(a - b*x**2)*b**3*c*d**2*x**2 - 192*sqrt(c + d*x)*sqrt(a - b*x**2) *b**2*c**4*x + 160*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**3*d*x**2 - 140*s qrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**2*d**2*x**3 - 567*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*d **4 - 441*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*c*d**4 - 504*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**2*d**2 + 396*int((sq rt(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*d**3 - 324*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**2*c**3*d**2 + 432*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**3*d - 384 *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**3*c**5 + 189*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*d**4 + 147*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*c*d**4 + 252*int((sq...