Integrand size = 37, antiderivative size = 533 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\frac {\left (a \left (B+\frac {a D}{b}\right )+(A b+a C) x\right ) (c+d x)^{5/2}}{3 a b \left (a-b x^2\right )^{3/2}}-\frac {(c+d x)^{3/2} \left (a (A b d+7 a C d+6 a c D)-\left (4 A b^2 c-a (2 b c C+5 b B d+11 a d D)\right ) x\right )}{6 a^2 b^2 \sqrt {a-b x^2}}+\frac {d \left (4 A b^2 c-a (2 b c C+5 b B d+15 a d D)\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{6 a^2 b^3}+\frac {\left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+5 B d)+3 a d (7 C d+15 c D))\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{3/2} b^{5/2} \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c-a (2 b c C+5 b B d+15 a d D)\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{3/2} b^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
1/3*(a*(B+a*D/b)+(A*b+C*a)*x)*(d*x+c)^(5/2)/a/b/(-b*x^2+a)^(3/2)-1/6*(d*x+ c)^(3/2)*(a*(A*b*d+7*C*a*d+6*D*a*c)-(4*A*b^2*c-a*(5*B*b*d+2*C*b*c+11*D*a*d ))*x)/a^2/b^2/(-b*x^2+a)^(1/2)+1/6*d*(4*A*b^2*c-a*(5*B*b*d+2*C*b*c+15*D*a* d))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/b^3+1/6*(A*b*(-3*a*d^2+4*b*c^2)-a*( b*c*(5*B*d+2*C*c)+3*a*d*(7*C*d+15*D*c)))*(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))/a^(3/2)/b^(5/2)/((d*x+c)/(c+a^(1/2)*d/b^(1/2))) ^(1/2)/(-b*x^2+a)^(1/2)-1/6*(-a*d^2+b*c^2)*(4*A*b^2*c-a*(5*B*b*d+2*C*b*c+1 5*D*a*d))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellip ticF(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))/a^(3/2)/b^(7/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 30.70 (sec) , antiderivative size = 779, normalized size of antiderivative = 1.46 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (-15 a^4 d^2 D-4 A b^4 c^2 x^3+a b^3 x \left (c (2 c C+5 B d) x^2+A \left (6 c^2+c d x+3 d^2 x^2\right )\right )-a^3 b \left (4 c^2 D+c d (9 C+13 D x)+d^2 (5 B+7 x (C-3 D x))\right )+a^2 b^2 \left (A d (3 c-d x)+B \left (2 c^2-c d x+7 d^2 x^2\right )+x^2 \left (6 c^2 D+d^2 x (9 C-4 D x)+c d (13 C+17 D x)\right )\right )\right )}{a^2 b^3 \left (a-b x^2\right )^2}-\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+5 B d)+3 a d (7 C d+15 c D))\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+5 B d)+3 a d (7 C d+15 c D))\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (A \left (4 b^2 c+3 \sqrt {a} b^{3/2} d\right )+a \left (-b (2 c C+5 B d)-15 a d D+3 \sqrt {a} \sqrt {b} (7 C d+10 c D)\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{a^2 b^3 d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{6 \sqrt {c+d x}} \] Input:
Integrate[((c + d*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(a - b*x^2)^(5/2),x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*(-15*a^4*d^2*D - 4*A*b^4*c^2*x^3 + a*b^3*x*(c *(2*c*C + 5*B*d)*x^2 + A*(6*c^2 + c*d*x + 3*d^2*x^2)) - a^3*b*(4*c^2*D + c *d*(9*C + 13*D*x) + d^2*(5*B + 7*x*(C - 3*D*x))) + a^2*b^2*(A*d*(3*c - d*x ) + B*(2*c^2 - c*d*x + 7*d^2*x^2) + x^2*(6*c^2*D + d^2*x*(9*C - 4*D*x) + c *d*(13*C + 17*D*x)))))/(a^2*b^3*(a - b*x^2)^2) - (d^2*Sqrt[-c + (Sqrt[a]*d )/Sqrt[b]]*(A*b*(4*b*c^2 - 3*a*d^2) - a*(b*c*(2*c*C + 5*B*d) + 3*a*d*(7*C* d + 15*c*D)))*(a - b*x^2) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(A*b*(4*b*c^ 2 - 3*a*d^2) - a*(b*c*(2*c*C + 5*B*d) + 3*a*d*(7*C*d + 15*c*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]]/S qrt[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)*(A*(4*b^2*c + 3*Sqrt[a]*b^(3/2)*d) + a*(-(b*(2 *c*C + 5*B*d)) - 15*a*d*D + 3*Sqrt[a]*Sqrt[b]*(7*C*d + 10*c*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]]/S qrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(a^2*b^3* d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(6*Sqrt[c + d*x])
Time = 1.24 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2176, 27, 2176, 27, 687, 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.
| \(\displaystyle \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2176 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{3/2} \left (-6 a d D x^2-(A b d+7 a C d+6 a c D) x+4 A b c-\frac {a (2 b c C+5 b B d+5 a d D)}{b}\right )}{2 \left (a-b x^2\right )^{3/2}}dx}{3 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{3/2} \left (-6 a d D x^2-(A b d+7 a C d+6 a c D) x+4 A b c-\frac {a (2 b c C+5 b B d+5 a d D)}{b}\right )}{\left (a-b x^2\right )^{3/2}}dx}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2176 |
| \(\displaystyle \frac {\frac {\int \frac {3 d \sqrt {c+d x} \left (a (A b d+7 a C d+10 a c D)-\left (4 A b^2 c-a (2 b c C+5 b B d+15 a d D)\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 d \int \frac {\sqrt {c+d x} \left (a (A b d+7 a C d+10 a c D)-\left (4 A b^2 c-a (2 b c C+5 b B d+15 a d D)\right ) x\right )}{\sqrt {a-b x^2}}dx}{2 a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {3 d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right )}{3 b}-\frac {2 \int \frac {a \left (A b^2 c d-a \left (15 a D d^2+b \left (30 D c^2+23 C d c+5 B d^2\right )\right )\right )+b \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+5 B d)+3 a d (7 C d+15 c D))\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}\right )}{2 a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right )}{3 b}-\frac {\int \frac {a \left (A b^2 c d-a \left (15 a D d^2+b \left (30 D c^2+23 C d c+5 B d^2\right )\right )\right )+b \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+5 B d)+3 a d (7 C d+15 c D))\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}\right )}{2 a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {3 d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right )}{3 b}-\frac {\frac {b \left (A b \left (4 b c^2-3 a d^2\right )-a (3 a d (15 c D+7 C d)+b c (5 B d+2 c C))\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}\right )}{2 a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {3 d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right )}{3 b}-\frac {\frac {b \sqrt {1-\frac {b x^2}{a}} \left (A b \left (4 b c^2-3 a d^2\right )-a (3 a d (15 c D+7 C d)+b c (5 B d+2 c C))\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}\right )}{2 a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {3 d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right )}{3 b}-\frac {-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (A b \left (4 b c^2-3 a d^2\right )-a (3 a d (15 c D+7 C d)+b c (5 B d+2 c C))\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )}{2 a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {3 d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right )}{3 b}-\frac {-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (A b \left (4 b c^2-3 a d^2\right )-a (3 a d (15 c D+7 C d)+b c (5 B d+2 c C))\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )}{2 a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {3 d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right )}{3 b}-\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (A b \left (4 b c^2-3 a d^2\right )-a (3 a d (15 c D+7 C d)+b c (5 B d+2 c C))\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )}{2 a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {3 d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right )}{3 b}-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (A b \left (4 b c^2-3 a d^2\right )-a (3 a d (15 c D+7 C d)+b c (5 B d+2 c C))\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )}{2 a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {3 d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right )}{3 b}-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (4 A b^2 c-a (15 a d D+5 b B d+2 b c C)\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (A b \left (4 b c^2-3 a d^2\right )-a (3 a d (15 c D+7 C d)+b c (5 B d+2 c C))\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )}{2 a b}-\frac {(c+d x)^{3/2} \left (a (6 a c D+7 a C d+A b d)-x \left (4 A b^2 c-a (11 a d D+5 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
Input:
Int[((c + d*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(a - b*x^2)^(5/2),x]
Output:
((a*(B + (a*D)/b) + (A*b + a*C)*x)*(c + d*x)^(5/2))/(3*a*b*(a - b*x^2)^(3/ 2)) + (-(((c + d*x)^(3/2)*(a*(A*b*d + 7*a*C*d + 6*a*c*D) - (4*A*b^2*c - a* (2*b*c*C + 5*b*B*d + 11*a*d*D))*x))/(a*b*Sqrt[a - b*x^2])) + (3*d*((2*(4*A *b^2*c - a*(2*b*c*C + 5*b*B*d + 15*a*d*D))*Sqrt[c + d*x]*Sqrt[a - b*x^2])/ (3*b) - ((-2*Sqrt[a]*Sqrt[b]*(A*b*(4*b*c^2 - 3*a*d^2) - a*(b*c*(2*c*C + 5* B*d) + 3*a*d*(7*C*d + 15*c*D)))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*Elliptic E[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)*(4*A*b^2*c - a*(2*b*c*C + 5*b*B*d + 15*a *d*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]))/(3*b)))/(2* a*b))/(6*a*b)
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[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[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 , x], x, 1]}, Simp[(d + e*x)^m*(a + b*x^2)^(p + 1)*((a*S - b*R*x)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*(d + e*x)*Qx - a*e*S*m + b*d*R*(2*p + 3) + b*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && !(IGtQ[m, 0] && R ationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(1183\) vs. \(2(461)=922\).
Time = 7.16 (sec) , antiderivative size = 1184, normalized size of antiderivative = 2.22
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1184\) |
| default | \(\text {Expression too large to display}\) | \(7619\) |
Input:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(5/2),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)*((1/3*(A*a*b*d ^2+A*b^2*c^2+2*B*a*b*c*d+C*a^2*d^2+C*a*b*c^2+2*D*a^2*c*d)/b^4/a*x+1/3*(2*A *b^2*c*d+B*a*b*d^2+B*b^2*c^2+2*C*a*b*c*d+D*a^2*d^2+D*a*b*c^2)/b^5)*(-b*d*x ^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x^2-a/b)^2-2*(-b*d*x-b*c)*(-1/12*(3*A*a*b*d^2 -4*A*b^2*c^2+5*B*a*b*c*d+9*C*a^2*d^2+2*C*a*b*c^2+17*D*a^2*c*d)/b^3/a^2*x-1 /12*(A*b^2*c*d+7*B*a*b*d^2+13*C*a*b*c*d+13*D*a^2*d^2+6*D*a*b*c^2)/a/b^4)/( (x^2-a/b)*(-b*d*x-b*c))^(1/2)-2/3/b^3*d^2*D*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^( 1/2)+2*(d*(B*b*d^2+3*C*b*c*d+2*D*a*d^2+3*D*b*c^2)/b^3-1/6/b^3*(4*A*a*b^2*c *d^2-4*A*b^3*c^3+7*B*a^2*b*d^3+5*B*a*b^2*c^2*d+22*C*a^2*b*c*d^2+2*C*a*b^2* c^3+13*D*a^3*d^3+23*D*a^2*b*c^2*d)/a^2+1/12/b^3*d*(A*b^2*c*d+7*B*a*b*d^2+1 3*C*a*b*c*d+13*D*a^2*d^2+6*D*a*b*c^2)/a+1/6/b^2*c*(3*A*a*b*d^2-4*A*b^2*c^2 +5*B*a*b*c*d+9*C*a^2*d^2+2*C*a*b*c^2+17*D*a^2*c*d)/a^2+1/3/b^3*d^3*D*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*(1/b^2*d^2*(C*d+3*D*c)+1/12*(3*A*a*b*d^2-4*A*b^2*c^2+5*B*a*b*c*d+9*C*a ^2*d^2+2*C*a*b*c^2+17*D*a^2*c*d)*d/a^2/b^2-2/3/b^2*d^2*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/...
Leaf count of result is larger than twice the leaf count of optimal. 987 vs. \(2 (465) = 930\).
Time = 0.14 (sec) , antiderivative size = 987, normalized size of antiderivative = 1.85 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(5/2),x, algorithm= "fricas")
Output:
1/18*(((2*(C*a*b^4 - 2*A*b^5)*c^3 - 5*(9*D*a^2*b^3 - B*a*b^4)*c^2*d - 6*(8 *C*a^2*b^3 - A*a*b^4)*c*d^2 - 15*(3*D*a^3*b^2 + B*a^2*b^3)*d^3)*x^4 + 2*(C *a^3*b^2 - 2*A*a^2*b^3)*c^3 - 5*(9*D*a^4*b - B*a^3*b^2)*c^2*d - 6*(8*C*a^4 *b - A*a^3*b^2)*c*d^2 - 15*(3*D*a^5 + B*a^4*b)*d^3 - 2*(2*(C*a^2*b^3 - 2*A *a*b^4)*c^3 - 5*(9*D*a^3*b^2 - B*a^2*b^3)*c^2*d - 6*(8*C*a^3*b^2 - A*a^2*b ^3)*c*d^2 - 15*(3*D*a^4*b + B*a^3*b^2)*d^3)*x^2)*sqrt(-b*d)*weierstrassPIn verse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/ 3*(3*d*x + c)/d) + 3*((2*(C*a*b^4 - 2*A*b^5)*c^2*d + 5*(9*D*a^2*b^3 + B*a* b^4)*c*d^2 + 3*(7*C*a^2*b^3 + A*a*b^4)*d^3)*x^4 + 2*(C*a^3*b^2 - 2*A*a^2*b ^3)*c^2*d + 5*(9*D*a^4*b + B*a^3*b^2)*c*d^2 + 3*(7*C*a^4*b + A*a^3*b^2)*d^ 3 - 2*(2*(C*a^2*b^3 - 2*A*a*b^4)*c^2*d + 5*(9*D*a^3*b^2 + B*a^2*b^3)*c*d^2 + 3*(7*C*a^3*b^2 + A*a^2*b^3)*d^3)*x^2)*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), weierstrassPIn verse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/ 3*(3*d*x + c)/d)) - 3*(4*D*a^2*b^3*d^3*x^4 + 2*(2*D*a^3*b^2 - B*a^2*b^3)*c ^2*d + 3*(3*C*a^3*b^2 - A*a^2*b^3)*c*d^2 + 5*(3*D*a^4*b + B*a^3*b^2)*d^3 - (2*(C*a*b^4 - 2*A*b^5)*c^2*d + (17*D*a^2*b^3 + 5*B*a*b^4)*c*d^2 + 3*(3*C* a^2*b^3 + A*a*b^4)*d^3)*x^3 - (6*D*a^2*b^3*c^2*d + (13*C*a^2*b^3 + A*a*b^4 )*c*d^2 + 7*(3*D*a^3*b^2 + B*a^2*b^3)*d^3)*x^2 - (6*A*a*b^4*c^2*d - (13*D* a^3*b^2 + B*a^2*b^3)*c*d^2 - (7*C*a^3*b^2 + A*a^2*b^3)*d^3)*x)*sqrt(-b*...
Timed out. \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(5/2)*(D*x**3+C*x**2+B*x+A)/(-b*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(5/2),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^(5/2)/(-b*x^2 + a)^(5/2), x)
\[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(5/2),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^(5/2)/(-b*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a-b\,x^2\right )}^{5/2}} \,d x \] Input:
int(((c + d*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(5/2),x)
Output:
int(((c + d*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(5/2), x)
\[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\int \frac {\left (d x +c \right )^{\frac {5}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (-b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(5/2),x)
Output:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(5/2),x)