Integrand size = 37, antiderivative size = 608 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=-\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a-b x^2}}{3 d^4 (c+d x)^{3/2}}-\frac {2 \left (3 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b c \left (8 c^2 C d-5 B c d^2+2 A d^3-11 c^3 D\right )\right ) \sqrt {a-b x^2}}{3 d^4 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {2 (5 C d-17 c D) \sqrt {c+d x} \sqrt {a-b x^2}}{15 d^4}+\frac {2 D (c+d x)^{3/2} \sqrt {a-b x^2}}{5 d^4}+\frac {4 \sqrt {a} \left (3 a^2 d^4 D+a b d^2 \left (35 c C d-15 B d^2-62 c^2 D\right )-b^2 c \left (40 c^2 C d-20 B c d^2+5 A d^3-64 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 \sqrt {b} d^5 \left (b c^2-a d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {4 \sqrt {a} \left (a d^2 (5 C d-14 c D)-b \left (40 c^2 C d-20 B c d^2+5 A d^3-64 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 \sqrt {b} d^5 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-b*x^2+a)^(1/2)/d^4/(d*x+c)^(3/2)-2/3* (3*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)-b*c*(2*A*d^3-5*B*c*d^2+8*C*c^2*d-11*D*c^ 3))*(-b*x^2+a)^(1/2)/d^4/(-a*d^2+b*c^2)/(d*x+c)^(1/2)+2/15*(5*C*d-17*D*c)* (d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/d^4+2/5*D*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/d^ 4+4/15*a^(1/2)*(3*a^2*d^4*D+a*b*d^2*(-15*B*d^2+35*C*c*d-62*D*c^2)-b^2*c*(5 *A*d^3-20*B*c*d^2+40*C*c^2*d-64*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^(1/2)/d^5/(-a*d^2+b*c^2)/((d*x+c)/(c+a^(1/2)*d /b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-4/15*a^(1/2)*(a*d^2*(5*C*d-14*D*c)-b*(5* A*d^3-20*B*c*d^2+40*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^(1/2)/d^5/(d*x+c)^(1/2) /(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 31.83 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {2 (c+d x) \left (5 C d-14 c D+3 d D x+\frac {5 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )}{(c+d x)^2}+\frac {5 \left (3 a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b c \left (8 c^2 C d-5 B c d^2+2 A d^3-11 c^3 D\right )\right )}{\left (b c^2-a d^2\right ) (c+d x)}\right )}{d^4}+\frac {4 \left (-d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (3 a^2 d^4 D+a b d^2 \left (35 c C d-15 B d^2-62 c^2 D\right )+b^2 c \left (-40 c^2 C d+20 B c d^2-5 A d^3+64 c^3 D\right )\right ) \left (a-b x^2\right )-i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 a^2 d^4 D+a b d^2 \left (35 c C d-15 B d^2-62 c^2 D\right )+b^2 c \left (-40 c^2 C d+20 B c d^2-5 A d^3+64 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 a^{3/2} d^3 D+a \sqrt {b} d^2 (-5 C d+14 c D)-3 \sqrt {a} b d \left (-10 c C d+5 B d^2+16 c^2 D\right )+b^{3/2} \left (40 c^2 C d-20 B c d^2+5 A d^3-64 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{b d^6 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2-a d^2\right ) \left (-a+b x^2\right )}\right )}{15 \sqrt {c+d x}} \] Input:
Integrate[(Sqrt[a - b*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]
Output:
(Sqrt[a - b*x^2]*((2*(c + d*x)*(5*C*d - 14*c*D + 3*d*D*x + (5*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(c + d*x)^2 + (5*(3*a*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*c*(8*c^2*C*d - 5*B*c*d^2 + 2*A*d^3 - 11*c^3*D)))/((b*c^2 - a* d^2)*(c + d*x))))/d^4 + (4*(-(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(3*a^2*d^ 4*D + a*b*d^2*(35*c*C*d - 15*B*d^2 - 62*c^2*D) + b^2*c*(-40*c^2*C*d + 20*B *c*d^2 - 5*A*d^3 + 64*c^3*D))*(a - b*x^2)) - I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a ]*d)*(3*a^2*d^4*D + a*b*d^2*(35*c*C*d - 15*B*d^2 - 62*c^2*D) + b^2*c*(-40* c^2*C*d + 20*B*c*d^2 - 5*A*d^3 + 64*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]*Sqrt[b]*d*(Sqrt[b]* c - Sqrt[a]*d)*(3*a^(3/2)*d^3*D + a*Sqrt[b]*d^2*(-5*C*d + 14*c*D) - 3*Sqrt [a]*b*d*(-10*c*C*d + 5*B*d^2 + 16*c^2*D) + b^(3/2)*(40*c^2*C*d - 20*B*c*d^ 2 + 5*A*d^3 - 64*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*ArcSin h[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/( Sqrt[b]*c - Sqrt[a]*d)]))/(b*d^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(b*c^2 - a *d^2)*(-a + b*x^2))))/(15*Sqrt[c + d*x])
Time = 1.48 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2182, 27, 2182, 27, 682, 27, 600, 509, 508, 327, 512, 511, 321}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {2 \int \frac {3 \sqrt {a-b x^2} \left (\left (\frac {b c^2}{d}-a d\right ) D x^2-\left (a (C d-c D)+b \left (\frac {2 D c^3}{d^2}-\frac {2 C c^2}{d}+B c-A d\right )\right ) x+\frac {A b c d+a \left (-D c^2+C d c-B d^2\right )}{d}\right )}{2 (c+d x)^{3/2}}dx}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {a-b x^2} \left (\left (\frac {b c^2}{d}-a d\right ) D x^2-\left (a (C d-c D)+b \left (\frac {2 D c^3}{d^2}-\frac {2 C c^2}{d}+B c-A d\right )\right ) x+A b c+a \left (-\frac {D c^2}{d}+C c-B d\right )\right )}{(c+d x)^{3/2}}dx}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {\frac {2 \int \frac {\left (b c^2-a d^2\right ) \left (d (A b d-a C d+2 a c D)-\left (a D d^2+b \left (-16 D c^2+10 C d c-5 B d^2\right )\right ) x\right ) \sqrt {a-b x^2}}{2 d^2 \sqrt {c+d x}}dx}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\left (a-\frac {b c^2}{d^2}\right ) \int \frac {\left (d (A b d-a C d+2 a c D)-\left (a D d^2+b \left (-16 D c^2+10 C d c-5 B d^2\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {-\frac {\left (a-\frac {b c^2}{d^2}\right ) \left (-\frac {4 \int \frac {b \left (a d \left (a d^2 (5 C d-11 c D)-b \left (-16 D c^3+10 C d c^2-5 B d^2 c+5 A d^3\right )\right )-\left (5 b c (A b d-a C d+2 a c D) d^2+\left (4 b c^2-3 a d^2\right ) \left (a D d^2+b \left (-16 D c^2+10 C d c-5 B d^2\right )\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )+a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right )}{15 d^2}\right )}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\left (a-\frac {b c^2}{d^2}\right ) \left (-\frac {2 \int \frac {a d \left (a d^2 (5 C d-11 c D)-b \left (-16 D c^3+10 C d c^2-5 B d^2 c+5 A d^3\right )\right )-\left (5 b c (A b d-a C d+2 a c D) d^2+\left (4 b c^2-3 a d^2\right ) \left (a D d^2+b \left (-16 D c^2+10 C d c-5 B d^2\right )\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )+a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right )}{15 d^2}\right )}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {-\frac {\left (a-\frac {b c^2}{d^2}\right ) \left (-\frac {2 \left (-\frac {\left (5 b c d^2 (2 a c D-a C d+A b d)+\left (4 b c^2-3 a d^2\right ) \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )+a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right )}{15 d^2}\right )}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {-\frac {\left (a-\frac {b c^2}{d^2}\right ) \left (-\frac {2 \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (5 b c d^2 (2 a c D-a C d+A b d)+\left (4 b c^2-3 a d^2\right ) \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )+a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right )}{15 d^2}\right )}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {-\frac {\left (a-\frac {b c^2}{d^2}\right ) \left (-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (5 b c d^2 (2 a c D-a C d+A b d)+\left (4 b c^2-3 a d^2\right ) \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )+a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right )}{15 d^2}\right )}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {\left (a-\frac {b c^2}{d^2}\right ) \left (-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 b c d^2 (2 a c D-a C d+A b d)+\left (4 b c^2-3 a d^2\right ) \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )+a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right )}{15 d^2}\right )}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {-\frac {\left (a-\frac {b c^2}{d^2}\right ) \left (-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 b c d^2 (2 a c D-a C d+A b d)+\left (4 b c^2-3 a d^2\right ) \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )+a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right )}{15 d^2}\right )}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {-\frac {\left (a-\frac {b c^2}{d^2}\right ) \left (-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 b c d^2 (2 a c D-a C d+A b d)+\left (4 b c^2-3 a d^2\right ) \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )+a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right )}{15 d^2}\right )}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {-\frac {\left (a-\frac {b c^2}{d^2}\right ) \left (-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 b c d^2 (2 a c D-a C d+A b d)+\left (4 b c^2-3 a d^2\right ) \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (a d^2 D+b \left (-5 B d^2-16 c^2 D+10 c C d\right )\right )+a d^2 (5 C d-14 c D)-b \left (5 A d^3-20 B c d^2-64 c^3 D+40 c^2 C d\right )\right )}{15 d^2}\right )}{b c^2-a d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
Input:
Int[(Sqrt[a - b*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]
Output:
(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a - b*x^2)^(3/2))/(3*d^2*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) + ((-2*(2*c*C*d - B*d^2 - 3*c^2*D)*(a - b*x^2)^(3/ 2))/(d^2*Sqrt[c + d*x]) - ((a - (b*c^2)/d^2)*((-2*Sqrt[c + d*x]*(a*d^2*(5* C*d - 14*c*D) - b*(40*c^2*C*d - 20*B*c*d^2 + 5*A*d^3 - 64*c^3*D) + 3*d*(a* d^2*D + b*(10*c*C*d - 5*B*d^2 - 16*c^2*D))*x)*Sqrt[a - b*x^2])/(15*d^2) - (2*((2*Sqrt[a]*(5*b*c*d^2*(A*b*d - a*C*d + 2*a*c*D) + (4*b*c^2 - 3*a*d^2)* (a*d^2*D + b*(10*c*C*d - 5*B*d^2 - 16*c^2*D)))*Sqrt[c + d*x]*Sqrt[1 - (b*x ^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sq rt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + S qrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(a*d^2*(5*C*d - 1 4*c*D) - b*(40*c^2*C*d - 20*B*c*d^2 + 5*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[Sq rt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(S qrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(15*d^2)))/(b*c^2 - a*d^2))/(b*c ^2 - a*d^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1142\) vs. \(2(530)=1060\).
Time = 4.16 (sec) , antiderivative size = 1143, normalized size of antiderivative = 1.88
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1143\) |
| default | \(\text {Expression too large to display}\) | \(9026\) |
Input:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(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)*(-2/3*(A*d^3-B *c*d^2+C*c^2*d-D*c^3)/d^6*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2-2/3 *(-b*d*x^2+a*d)/d^5/(a*d^2-b*c^2)*(2*A*b*c*d^3+3*B*a*d^4-5*B*b*c^2*d^2-6*C *a*c*d^3+8*C*b*c^3*d+9*D*a*c^2*d^2-11*D*b*c^4)/((x+c/d)*(-b*d*x^2+a*d))^(1 /2)+2/5*D/d^3*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(-b/d^3*(C*d-2*D*c) +4/5*D/d^3*b*c)/b/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-(A*b*d^3-2*B*b* c*d^2-C*a*d^3+3*C*b*c^2*d+2*D*a*c*d^2-4*D*b*c^3)/d^5+1/3*b*(A*d^3-B*c*d^2+ C*c^2*d-D*c^3)/d^5-1/3*b/d^5*c*(2*A*b*c*d^3+3*B*a*d^4-5*B*b*c^2*d^2-6*C*a* c*d^3+8*C*b*c^3*d+9*D*a*c^2*d^2-11*D*b*c^4)/(a*d^2-b*c^2)-2/5*D/d^3*a*c+1/ 3*(-b/d^3*(C*d-2*D*c)+4/5*D/d^3*b*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*(-1/d^4*(B*b*d^2-2*C*b*c*d -D*a*d^2+3*D*b*c^2)-1/3*b/d^4*(2*A*b*c*d^3+3*B*a*d^4-5*B*b*c^2*d^2-6*C*a*c *d^3+8*C*b*c^3*d+9*D*a*c^2*d^2-11*D*b*c^4)/(a*d^2-b*c^2)-3/5*D/d^2*a-2/3*( -b/d^3*(C*d-2*D*c)+4/5*D/d^3*b*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)...
Time = 0.11 (sec) , antiderivative size = 1035, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm= "fricas")
Output:
-2/45*(2*(64*D*b^2*c^7 - 40*C*b^2*c^6*d - 10*(11*D*a*b - 2*B*b^2)*c^5*d^2 + 5*(13*C*a*b - A*b^2)*c^4*d^3 + 6*(6*D*a^2 - 5*B*a*b)*c^3*d^4 - 15*(C*a^2 - A*a*b)*c^2*d^5 + (64*D*b^2*c^5*d^2 - 40*C*b^2*c^4*d^3 - 10*(11*D*a*b - 2*B*b^2)*c^3*d^4 + 5*(13*C*a*b - A*b^2)*c^2*d^5 + 6*(6*D*a^2 - 5*B*a*b)*c* d^6 - 15*(C*a^2 - A*a*b)*d^7)*x^2 + 2*(64*D*b^2*c^6*d - 40*C*b^2*c^5*d^2 - 10*(11*D*a*b - 2*B*b^2)*c^4*d^3 + 5*(13*C*a*b - A*b^2)*c^3*d^4 + 6*(6*D*a ^2 - 5*B*a*b)*c^2*d^5 - 15*(C*a^2 - A*a*b)*c*d^6)*x)*sqrt(-b*d)*weierstras sPInverse(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^2*c^6*d - 40*C*b^2*c^5*d^2 - 2*(31*D*a*b - 10*B*b^2)*c^4*d^3 + 5*(7*C*a*b - A*b^2)*c^3*d^4 + 3*(D*a^2 - 5*B*a*b)*c^ 2*d^5 + (64*D*b^2*c^4*d^3 - 40*C*b^2*c^3*d^4 - 2*(31*D*a*b - 10*B*b^2)*c^2 *d^5 + 5*(7*C*a*b - A*b^2)*c*d^6 + 3*(D*a^2 - 5*B*a*b)*d^7)*x^2 + 2*(64*D* b^2*c^5*d^2 - 40*C*b^2*c^4*d^3 - 2*(31*D*a*b - 10*B*b^2)*c^3*d^4 + 5*(7*C* a*b - A*b^2)*c^2*d^5 + 3*(D*a^2 - 5*B*a*b)*c*d^6)*x)*sqrt(-b*d)*weierstras sZeta(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), we ierstrassPInverse(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*(64*D*b^2*c^5*d^2 - 40*C*b^2*c^4*d^3 - 1 0*B*a*b*c*d^6 - 5*A*a*b*d^7 - 2*(27*D*a*b - 10*B*b^2)*c^3*d^4 + 5*(6*C*a*b - A*b^2)*c^2*d^5 - 3*(D*b^2*c^2*d^5 - D*a*b*d^7)*x^3 + (8*D*b^2*c^3*d^4 - 5*C*b^2*c^2*d^5 - 8*D*a*b*c*d^6 + 5*C*a*b*d^7)*x^2 + 5*(16*D*b^2*c^4*d...
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a - b x^{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)
Output:
Integral(sqrt(a - b*x**2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**(5/2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/(d*x + c)^(5/2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/(d*x + c)^(5/2), x)
Timed out. \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a-b\,x^2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(5/2),x)
Output:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(5/2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {\sqrt {-b \,x^{2}+a}\, \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{\frac {5}{2}}}d x \] Input:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)
Output:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)