Integrand size = 37, antiderivative size = 736 \[ \int \sqrt {c+d x} \sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \left (75 a^2 d^4 D-3 a b d^2 \left (11 c C d-55 B d^2-2 c^2 D\right )+b^2 c \left (88 c^2 C d-132 B c d^2+231 A d^3-64 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{3465 b^2 d^4}+\frac {2 \left (a d^2 (77 C d+c D)-b \left (22 c^2 C d-33 B c d^2-231 A d^3-16 c^3 D\right )\right ) x \sqrt {c+d x} \sqrt {a-b x^2}}{1155 b d^3}-\frac {2 \left (15 a d^2 D-b \left (22 c C d-33 B d^2-16 c^2 D\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{231 b^2 d^2}-\frac {2 (11 C d-17 c D) (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}{99 b d^2}-\frac {2 D (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}{11 b d^2}-\frac {4 \sqrt {a} \left (3 a^2 d^4 (77 C d+26 c D)+b^2 c^2 \left (88 c^2 C d-132 B c d^2+231 A d^3-64 c^3 D\right )-3 a b d^2 \left (33 c^2 C d-88 B c d^2-231 A d^3-18 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3465 b^{3/2} d^5 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {4 \sqrt {a} \left (b c^2-a d^2\right ) \left (75 a^2 d^4 D-3 a b d^2 \left (11 c C d-55 B d^2-2 c^2 D\right )+b^2 c \left (88 c^2 C d-132 B c d^2+231 A d^3-64 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3465 b^{5/2} d^5 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
2/3465*(75*a^2*d^4*D-3*a*b*d^2*(-55*B*d^2+11*C*c*d-2*D*c^2)+b^2*c*(231*A*d ^3-132*B*c*d^2+88*C*c^2*d-64*D*c^3))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d^ 4+2/1155*(a*d^2*(77*C*d+D*c)-b*(-231*A*d^3-33*B*c*d^2+22*C*c^2*d-16*D*c^3) )*x*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^3-2/231*(15*a*d^2*D-b*(-33*B*d^2+22 *C*c*d-16*D*c^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(3/2)/b^2/d^2-2/99*(11*C*d-17*D *c)*(d*x+c)^(3/2)*(-b*x^2+a)^(3/2)/b/d^2-2/11*D*(d*x+c)^(5/2)*(-b*x^2+a)^( 3/2)/b/d^2-4/3465*a^(1/2)*(3*a^2*d^4*(77*C*d+26*D*c)+b^2*c^2*(231*A*d^3-13 2*B*c*d^2+88*C*c^2*d-64*D*c^3)-3*a*b*d^2*(-231*A*d^3-88*B*c*d^2+33*C*c^2*d -18*D*c^3))*(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^5/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)+4/3465*a^ (1/2)*(-a*d^2+b*c^2)*(75*a^2*d^4*D-3*a*b*d^2*(-55*B*d^2+11*C*c*d-2*D*c^2)+ b^2*c*(231*A*d^3-132*B*c*d^2+88*C*c^2*d-64*D*c^3))*((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 = 33.33 (sec) , antiderivative size = 881, normalized size of antiderivative = 1.20 \[ \int \sqrt {c+d x} \sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2 \sqrt {a-b x^2} \left ((c+d x) \left (-150 a^2 d^4 D-2 a b d^2 \left (-23 c^2 D+4 c d (11 C+4 D x)+d^2 \left (165 B+77 C x+45 D x^2\right )\right )+b^2 \left (-64 c^4 D+8 c^3 d (11 C+6 D x)-2 c^2 d^2 (66 B+x (33 C+20 D x))+c d^3 \left (231 A+x \left (99 B+55 C x+35 D x^2\right )\right )+d^4 x \left (693 A+5 x \left (99 B+77 C x+63 D x^2\right )\right )\right )\right )-\frac {2 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (3 a^2 d^4 (77 C d+26 c D)+b^2 c^2 \left (88 c^2 C d-132 B c d^2+231 A d^3-64 c^3 D\right )+3 a b d^2 \left (-33 c^2 C d+88 B c d^2+231 A d^3+18 c^3 D\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 a^2 d^4 (77 C d+26 c D)+b^2 c^2 \left (88 c^2 C d-132 B c d^2+231 A d^3-64 c^3 D\right )+3 a b d^2 \left (-33 c^2 C d+88 B c d^2+231 A d^3+18 c^3 D\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 )+i \sqrt {a} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (75 a^2 d^4 D-3 a^{3/2} \sqrt {b} d^3 (77 C d+c D)+3 a b d^2 \left (-11 c C d+55 B d^2+2 c^2 D\right )+b^2 c \left (88 c^2 C d-132 B c d^2+231 A d^3-64 c^3 D\right )-3 \sqrt {a} b^{3/2} d \left (-22 c^2 C d+33 B c d^2+231 A d^3+16 c^3 D\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 )\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{3465 b^2 d^4 \sqrt {c+d x}} \] Input:
Integrate[Sqrt[c + d*x]*Sqrt[a - b*x^2]*(A + B*x + C*x^2 + D*x^3),x]
Output:
(2*Sqrt[a - b*x^2]*((c + d*x)*(-150*a^2*d^4*D - 2*a*b*d^2*(-23*c^2*D + 4*c *d*(11*C + 4*D*x) + d^2*(165*B + 77*C*x + 45*D*x^2)) + b^2*(-64*c^4*D + 8* c^3*d*(11*C + 6*D*x) - 2*c^2*d^2*(66*B + x*(33*C + 20*D*x)) + c*d^3*(231*A + x*(99*B + 55*C*x + 35*D*x^2)) + d^4*x*(693*A + 5*x*(99*B + 77*C*x + 63* D*x^2)))) - (2*(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(3*a^2*d^4*(77*C*d + 26 *c*D) + b^2*c^2*(88*c^2*C*d - 132*B*c*d^2 + 231*A*d^3 - 64*c^3*D) + 3*a*b* d^2*(-33*c^2*C*d + 88*B*c*d^2 + 231*A*d^3 + 18*c^3*D))*(a - b*x^2) + I*Sqr t[b]*(Sqrt[b]*c - Sqrt[a]*d)*(3*a^2*d^4*(77*C*d + 26*c*D) + b^2*c^2*(88*c^ 2*C*d - 132*B*c*d^2 + 231*A*d^3 - 64*c^3*D) + 3*a*b*d^2*(-33*c^2*C*d + 88* B*c*d^2 + 231*A*d^3 + 18*c^3*D))*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]*d*(Sqrt[b]*c - Sqrt[a]*d)*(75* a^2*d^4*D - 3*a^(3/2)*Sqrt[b]*d^3*(77*C*d + c*D) + 3*a*b*d^2*(-11*c*C*d + 55*B*d^2 + 2*c^2*D) + b^2*c*(88*c^2*C*d - 132*B*c*d^2 + 231*A*d^3 - 64*c^3 *D) - 3*Sqrt[a]*b^(3/2)*d*(-22*c^2*C*d + 33*B*c*d^2 + 231*A*d^3 + 16*c^3*D ))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d )/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d) ]))/(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2))))/(3465*b^2*d^4*Sq...
Time = 1.47 (sec) , antiderivative size = 725, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {2185, 27, 2185, 27, 687, 27, 682, 27, 600, 509, 508, 327, 512, 511, 321}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {a-b x^2} \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {1}{2} \sqrt {c+d x} \sqrt {a-b x^2} \left (b (11 C d-17 c D) x^2 d^2+(11 A b d+5 a c D) d^2+\left (-6 b D c^2+11 b B d^2+5 a d^2 D\right ) x d\right )dx}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {c+d x} \sqrt {a-b x^2} \left (b (11 C d-17 c D) x^2 d^2+(11 A b d+5 a c D) d^2+\left (-6 b D c^2+11 b B d^2+5 a d^2 D\right ) x d\right )dx}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {3}{2} b d^3 \sqrt {c+d x} \left (d (33 A b d+11 a C d-2 a c D)+\left (15 a d^2 D-b \left (-16 D c^2+22 C d c-33 B d^2\right )\right ) x\right ) \sqrt {a-b x^2}dx}{9 b d^2}-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} d \int \sqrt {c+d x} \left (d (33 A b d+11 a C d-2 a c D)+\left (15 a d^2 D-b \left (-16 D c^2+22 C d c-33 B d^2\right )\right ) x\right ) \sqrt {a-b x^2}dx-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{3} d \left (-\frac {2 \int -\frac {\left (d \left (231 A c d b^2+a \left (15 a D d^2+b \left (2 D c^2+55 C d c+33 B d^2\right )\right )\right )+b \left (a d^2 (77 C d+c D)-b \left (-16 D c^3+22 C d c^2-33 B d^2 c-231 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\int \frac {\left (d \left (231 A c d b^2+a \left (15 a D d^2+b \left (2 D c^2+55 C d c+33 B d^2\right )\right )\right )+b \left (a d^2 (77 C d+c D)-b \left (-16 D c^3+22 C d c^2-33 B d^2 c-231 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (75 a^2 d^4 D+3 b d x \left (a d^2 (c D+77 C d)-b \left (-231 A d^3-33 B c d^2-16 c^3 D+22 c^2 C d\right )\right )-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{15 d^2}-\frac {4 \int -\frac {b \left (a d \left (75 a^2 D d^4+3 a b \left (3 D c^2+66 C d c+55 B d^2\right ) d^2+b^2 c \left (-16 D c^3+22 C d c^2-33 B d^2 c+924 A d^3\right )\right )+b \left (3 a^2 (77 C d+26 c D) d^4-3 a b \left (-18 D c^3+33 C d c^2-88 B d^2 c-231 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+88 C d c^2-132 B d^2 c+231 A d^3\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\frac {2 \int \frac {a d \left (75 a^2 D d^4+3 a b \left (3 D c^2+66 C d c+55 B d^2\right ) d^2+b^2 c \left (-16 D c^3+22 C d c^2-33 B d^2 c+924 A d^3\right )\right )+b \left (3 a^2 (77 C d+26 c D) d^4-3 a b \left (-18 D c^3+33 C d c^2-88 B d^2 c-231 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+88 C d c^2-132 B d^2 c+231 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (75 a^2 d^4 D+3 b d x \left (a d^2 (c D+77 C d)-b \left (-231 A d^3-33 B c d^2-16 c^3 D+22 c^2 C d\right )\right )-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\frac {2 \left (\frac {b \left (3 a^2 d^4 (26 c D+77 C d)-3 a b d^2 \left (-231 A d^3-88 B c d^2-18 c^3 D+33 c^2 C d\right )+b^2 c^2 \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (75 a^2 d^4 D-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (75 a^2 d^4 D+3 b d x \left (a d^2 (c D+77 C d)-b \left (-231 A d^3-33 B c d^2-16 c^3 D+22 c^2 C d\right )\right )-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\frac {2 \left (\frac {b \sqrt {1-\frac {b x^2}{a}} \left (3 a^2 d^4 (26 c D+77 C d)-3 a b d^2 \left (-231 A d^3-88 B c d^2-18 c^3 D+33 c^2 C d\right )+b^2 c^2 \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (75 a^2 d^4 D-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (75 a^2 d^4 D+3 b d x \left (a d^2 (c D+77 C d)-b \left (-231 A d^3-33 B c d^2-16 c^3 D+22 c^2 C d\right )\right )-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (75 a^2 d^4 D-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (26 c D+77 C d)-3 a b d^2 \left (-231 A d^3-88 B c d^2-18 c^3 D+33 c^2 C d\right )+b^2 c^2 \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (75 a^2 d^4 D+3 b d x \left (a d^2 (c D+77 C d)-b \left (-231 A d^3-33 B c d^2-16 c^3 D+22 c^2 C d\right )\right )-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (75 a^2 d^4 D-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a^2 d^4 (26 c D+77 C d)-3 a b d^2 \left (-231 A d^3-88 B c d^2-18 c^3 D+33 c^2 C d\right )+b^2 c^2 \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (75 a^2 d^4 D+3 b d x \left (a d^2 (c D+77 C d)-b \left (-231 A d^3-33 B c d^2-16 c^3 D+22 c^2 C d\right )\right )-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\frac {2 \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (75 a^2 d^4 D-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a^2 d^4 (26 c D+77 C d)-3 a b d^2 \left (-231 A d^3-88 B c d^2-18 c^3 D+33 c^2 C d\right )+b^2 c^2 \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (75 a^2 d^4 D+3 b d x \left (a d^2 (c D+77 C d)-b \left (-231 A d^3-33 B c d^2-16 c^3 D+22 c^2 C d\right )\right )-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (75 a^2 d^4 D-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a^2 d^4 (26 c D+77 C d)-3 a b d^2 \left (-231 A d^3-88 B c d^2-18 c^3 D+33 c^2 C d\right )+b^2 c^2 \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (75 a^2 d^4 D+3 b d x \left (a d^2 (c D+77 C d)-b \left (-231 A d^3-33 B c d^2-16 c^3 D+22 c^2 C d\right )\right )-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (75 a^2 d^4 D-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a^2 d^4 (26 c D+77 C d)-3 a b d^2 \left (-231 A d^3-88 B c d^2-18 c^3 D+33 c^2 C d\right )+b^2 c^2 \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (75 a^2 d^4 D+3 b d x \left (a d^2 (c D+77 C d)-b \left (-231 A d^3-33 B c d^2-16 c^3 D+22 c^2 C d\right )\right )-3 a b d^2 \left (-55 B d^2-2 c^2 D+11 c C d\right )+b^2 c \left (231 A d^3-132 B c d^2-64 c^3 D+88 c^2 C d\right )\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (15 a d^2 D-b \left (-33 B d^2-16 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \left (a-b x^2\right )^{3/2} (c+d x)^{3/2} (11 C d-17 c D)}{11 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d^2}\) |
Input:
Int[Sqrt[c + d*x]*Sqrt[a - b*x^2]*(A + B*x + C*x^2 + D*x^3),x]
Output:
(-2*D*(c + d*x)^(5/2)*(a - b*x^2)^(3/2))/(11*b*d^2) + ((-2*d*(11*C*d - 17* c*D)*(c + d*x)^(3/2)*(a - b*x^2)^(3/2))/9 + (d*((-2*(15*a*d^2*D - b*(22*c* C*d - 33*B*d^2 - 16*c^2*D))*Sqrt[c + d*x]*(a - b*x^2)^(3/2))/(7*b) + ((2*S qrt[c + d*x]*(75*a^2*d^4*D - 3*a*b*d^2*(11*c*C*d - 55*B*d^2 - 2*c^2*D) + b ^2*c*(88*c^2*C*d - 132*B*c*d^2 + 231*A*d^3 - 64*c^3*D) + 3*b*d*(a*d^2*(77* C*d + c*D) - b*(22*c^2*C*d - 33*B*c*d^2 - 231*A*d^3 - 16*c^3*D))*x)*Sqrt[a - b*x^2])/(15*d^2) + (2*((-2*Sqrt[a]*Sqrt[b]*(3*a^2*d^4*(77*C*d + 26*c*D) + b^2*c^2*(88*c^2*C*d - 132*B*c*d^2 + 231*A*d^3 - 64*c^3*D) - 3*a*b*d^2*( 33*c^2*C*d - 88*B*c*d^2 - 231*A*d^3 - 18*c^3*D))*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)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(75*a^2*d^4*D - 3*a*b*d ^2*(11*c*C*d - 55*B*d^2 - 2*c^2*D) + b^2*c*(88*c^2*C*d - 132*B*c*d^2 + 231 *A*d^3 - 64*c^3*D))*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*b)))/3)/(11*b*d^3)
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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
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. \(1526\) vs. \(2(652)=1304\).
Time = 2.78 (sec) , antiderivative size = 1527, normalized size of antiderivative = 2.07
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1527\) |
| default | \(\text {Expression too large to display}\) | \(5953\) |
Input:
int((d*x+c)^(1/2)*(-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVER BOSE)
Output:
1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((d*x+c)*(-b*x^2+a))^(1/2)*(2/11*D*x^4*(- b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/9*(-b*d*C-1/11*b*c*D)/b/d*x^3*(-b*d*x^3 -b*c*x^2+a*d*x+a*c)^(1/2)-2/7*(-B*b*d-b*c*C+2/11*D*a*d-8/9*(-b*d*C-1/11*b* c*D)/d*c)/b/d*x^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(-A*b*d-B*b*c+C*a *d+3/11*D*a*c+7/9*(-b*d*C-1/11*b*c*D)/b*a-6/7*(-B*b*d-b*c*C+2/11*D*a*d-8/9 *(-b*d*C-1/11*b*c*D)/d*c)/d*c)/b/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/ 3*(-A*b*c+B*a*d+a*c*C+2/3*(-b*d*C-1/11*b*c*D)/b/d*a*c+5/7*(-B*b*d-b*c*C+2/ 11*D*a*d-8/9*(-b*d*C-1/11*b*c*D)/d*c)/b*a-4/5*(-A*b*d-B*b*c+C*a*d+3/11*D*a *c+7/9*(-b*d*C-1/11*b*c*D)/b*a-6/7*(-B*b*d-b*c*C+2/11*D*a*d-8/9*(-b*d*C-1/ 11*b*c*D)/d*c)/d*c)/d*c)/b/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(A*a*c+2 /5*(-A*b*d-B*b*c+C*a*d+3/11*D*a*c+7/9*(-b*d*C-1/11*b*c*D)/b*a-6/7*(-B*b*d- b*c*C+2/11*D*a*d-8/9*(-b*d*C-1/11*b*c*D)/d*c)/d*c)/b/d*a*c+1/3*(-A*b*c+B*a *d+a*c*C+2/3*(-b*d*C-1/11*b*c*D)/b/d*a*c+5/7*(-B*b*d-b*c*C+2/11*D*a*d-8/9* (-b*d*C-1/11*b*c*D)/d*c)/b*a-4/5*(-A*b*d-B*b*c+C*a*d+3/11*D*a*c+7/9*(-b*d* C-1/11*b*c*D)/b*a-6/7*(-B*b*d-b*c*C+2/11*D*a*d-8/9*(-b*d*C-1/11*b*c*D)/d*c )/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)*Elliptic F(((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*(A*a*d+a*B*c+4/7*(-B*b*d-b*c*C+2/11*D*a*d-8/9*(-...
Time = 0.09 (sec) , antiderivative size = 662, normalized size of antiderivative = 0.90 \[ \int \sqrt {c+d x} \sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(1/2)*(-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm= "fricas")
Output:
-2/10395*(2*(64*D*b^3*c^6 - 88*C*b^3*c^5*d - 6*(17*D*a*b^2 - 22*B*b^3)*c^4 *d^2 + 33*(5*C*a*b^2 - 7*A*b^3)*c^3*d^3 - 3*(17*D*a^2*b + 121*B*a*b^2)*c^2 *d^4 + 33*(11*C*a^2*b + 63*A*a*b^2)*c*d^5 + 45*(5*D*a^3 + 11*B*a^2*b)*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*(64*D*b^3*c^5*d - 88*C*b^3* c^4*d^2 - 6*(9*D*a*b^2 - 22*B*b^3)*c^3*d^3 + 33*(3*C*a*b^2 - 7*A*b^3)*c^2* d^4 - 6*(13*D*a^2*b + 44*B*a*b^2)*c*d^5 - 231*(C*a^2*b + 3*A*a*b^2)*d^6)*s qrt(-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*(315*D*b^3*d^6*x^4 - 64*D*b^3*c^4*d^2 + 88*C*b^3*c^3*d^3 + 2*(23*D*a*b^2 - 66*B*b^3)*c^2*d^4 - 11*(8*C*a*b^2 - 21*A*b^3)*c*d^5 - 30*(5*D*a^2*b + 11*B*a*b^2)*d^6 + 35*(D *b^3*c*d^5 + 11*C*b^3*d^6)*x^3 - 5*(8*D*b^3*c^2*d^4 - 11*C*b^3*c*d^5 + 9*( 2*D*a*b^2 - 11*B*b^3)*d^6)*x^2 + (48*D*b^3*c^3*d^3 - 66*C*b^3*c^2*d^4 - (3 2*D*a*b^2 - 99*B*b^3)*c*d^5 - 77*(2*C*a*b^2 - 9*A*b^3)*d^6)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*d^6)
\[ \int \sqrt {c+d x} \sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int \sqrt {a - b x^{2}} \sqrt {c + d x} \left (A + B x + C x^{2} + D x^{3}\right )\, dx \] Input:
integrate((d*x+c)**(1/2)*(-b*x**2+a)**(1/2)*(D*x**3+C*x**2+B*x+A),x)
Output:
Integral(sqrt(a - b*x**2)*sqrt(c + d*x)*(A + B*x + C*x**2 + D*x**3), x)
\[ \int \sqrt {c+d x} \sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int { {\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c} \,d x } \] Input:
integrate((d*x+c)^(1/2)*(-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(-b*x^2 + a)*sqrt(d*x + c), x)
\[ \int \sqrt {c+d x} \sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int { {\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c} \,d x } \] Input:
integrate((d*x+c)^(1/2)*(-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(-b*x^2 + a)*sqrt(d*x + c), x)
Timed out. \[ \int \sqrt {c+d x} \sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int \sqrt {a-b\,x^2}\,\sqrt {c+d\,x}\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \] Input:
int((a - b*x^2)^(1/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2 + x^3*D),x)
Output:
int((a - b*x^2)^(1/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2 + x^3*D), x)
\[ \int \sqrt {c+d x} \sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:
int((d*x+c)^(1/2)*(-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x)
Output:
( - 462*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*b*d**3 - 306*sqrt(c + d*x)*sqr t(a - b*x**2)*a**2*c*d**3 + 462*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b**2*c*d* *2*x - 396*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b**2*c*d**2 + 2*sqrt(c + d*x)* sqrt(a - b*x**2)*a*b*c**3*d - 124*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c**2* d**2*x - 60*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c*d**3*x**2 + 66*sqrt(c + d *x)*sqrt(a - b*x**2)*b**3*c**2*d*x + 330*sqrt(c + d*x)*sqrt(a - b*x**2)*b* *3*c*d**2*x**2 - 12*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**4*x + 10*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**3*d*x**2 + 280*sqrt(c + d*x)*sqrt(a - b*x **2)*b**2*c**2*d**2*x**3 + 210*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c*d**3* x**4 - 693*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 - 309*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 - 231*int((sqr t(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 - 264*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*d**3 + 45*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 + 132*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 - 24*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 + 231*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 + 153*...