Integrand size = 37, antiderivative size = 662 \[ \int \frac {(c+d x)^{7/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\frac {\left (a \left (B+\frac {a D}{b}\right )+(A b+a C) x\right ) (c+d x)^{7/2}}{3 a b \left (a-b x^2\right )^{3/2}}-\frac {(c+d x)^{5/2} \left (3 a (A b d+3 a C d+2 a c D)-\left (4 A b^2 c-a (2 b c C+7 b B d+13 a d D)\right ) x\right )}{6 a^2 b^2 \sqrt {a-b x^2}}+\frac {d \left (5 A b \left (4 b c^2-5 a d^2\right )-a (5 b c (2 c C+7 B d)+3 a d (25 C d+49 c D))\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{30 a^2 b^3}+\frac {d \left (20 A b^2 c-10 a b c C-35 a b B d-77 a^2 d D\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{30 a^2 b^3}+\frac {\left (20 A b^2 c \left (b c^2-2 a d^2\right )-a \left (5 b^2 c^2 (2 c C+7 B d)+231 a^2 d^3 D+3 a b d \left (110 c C d+35 B d^2+119 c^2 D\right )\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{30 a^{3/2} b^{7/2} \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (5 A b \left (4 b c^2-5 a d^2\right )-a (5 b c (2 c C+7 B d)+3 a d (25 C d+49 c D))\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{30 a^{3/2} b^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
1/3*(a*(B+a*D/b)+(A*b+C*a)*x)*(d*x+c)^(7/2)/a/b/(-b*x^2+a)^(3/2)-1/6*(d*x+ c)^(5/2)*(3*a*(A*b*d+3*C*a*d+2*D*a*c)-(4*A*b^2*c-a*(7*B*b*d+2*C*b*c+13*D*a *d))*x)/a^2/b^2/(-b*x^2+a)^(1/2)+1/30*d*(5*A*b*(-5*a*d^2+4*b*c^2)-a*(5*b*c *(7*B*d+2*C*c)+3*a*d*(25*C*d+49*D*c)))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/ b^3+1/30*d*(20*A*b^2*c-35*B*a*b*d-10*C*a*b*c-77*D*a^2*d)*(d*x+c)^(3/2)*(-b *x^2+a)^(1/2)/a^2/b^3+1/30*(20*A*b^2*c*(-2*a*d^2+b*c^2)-a*(5*b^2*c^2*(7*B* d+2*C*c)+231*a^2*d^3*D+3*a*b*d*(35*B*d^2+110*C*c*d+119*D*c^2)))*(d*x+c)^(1 /2)*((-b*x^2+a)/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2) ,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(3/2)/b^(7/2)/((d*x+c) /(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-1/30*(-a*d^2+b*c^2)*(5*A*b* (-5*a*d^2+4*b*c^2)-a*(5*b*c*(7*B*d+2*C*c)+3*a*d*(25*C*d+49*D*c)))*((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) )/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 = 34.02 (sec) , antiderivative size = 1191, normalized size of antiderivative = 1.80 \[ \int \frac {(c+d x)^{7/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
Integrate[((c + d*x)^(7/2)*(A + B*x + C*x^2 + D*x^3))/(a - b*x^2)^(5/2),x]
Output:
Sqrt[c + d*x]*Sqrt[a - b*x^2]*((-2*d^2*(5*C*d + 16*c*D))/(15*b^3) - (2*d^3 *D*x)/(5*b^3) + (a*b^2*B*c^3 + 3*a*A*b^2*c^2*d + 3*a^2*b*c^2*C*d + 3*a^2*b *B*c*d^2 + a^2*A*b*d^3 + a^3*C*d^3 + a^2*b*c^3*D + 3*a^3*c*d^2*D + A*b^3*c ^3*x + a*b^2*c^3*C*x + 3*a*b^2*B*c^2*d*x + 3*a*A*b^2*c*d^2*x + 3*a^2*b*c*C *d^2*x + a^2*b*B*d^3*x + 3*a^2*b*c^2*d*D*x + a^3*d^3*D*x)/(3*a*b^3*(-a + b *x^2)^2) + (a*A*b^2*c^2*d + 19*a^2*b*c^2*C*d + 20*a^2*b*B*c*d^2 + 7*a^2*A* b*d^3 + 13*a^3*C*d^3 + 6*a^2*b*c^3*D + 38*a^3*c*d^2*D - 4*A*b^3*c^3*x + 2* a*b^2*c^3*C*x + 7*a*b^2*B*c^2*d*x + 8*a*A*b^2*c*d^2*x + 26*a^2*b*c*C*d^2*x + 9*a^2*b*B*d^3*x + 25*a^2*b*c^2*d*D*x + 15*a^3*d^3*D*x)/(6*a^2*b^3*(-a + b*x^2))) + (d*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^2]*(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(20*A*b^2*c*(b*c^2 - 2*a*d^2) - a*(5*b^2*c^2*(2*c* C + 7*B*d) + 231*a^2*d^3*D + 3*a*b*d*(110*c*C*d + 35*B*d^2 + 119*c^2*D)))* (-((a*d^2)/(c + d*x)^2) + b*(-1 + c/(c + d*x))^2) - (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(20*A*b^2*c*(b*c^2 - 2*a*d^2) - a*(5*b^2*c^2*(2*c*C + 7*B*d) + 231*a^2*d^3*D + 3*a*b*d*(110*c*C*d + 35*B*d^2 + 119*c^2*D)))*Sqrt[1 - c/ (c + d*x) - (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (Sqrt[ a]*d)/(Sqrt[b]*(c + d*x))]*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)])/Sqrt [c + d*x] - (I*Sqrt[a]*Sqrt[b]*d*(Sqrt[b]*c - Sqrt[a]*d)*(5*A*b^(3/2)*(4*b *c^2 + 3*Sqrt[a]*Sqrt[b]*c*d - 5*a*d^2) + a*(-5*b^(3/2)*c*(2*c*C + 7*B*...
Time = 1.56 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {2176, 27, 2176, 27, 687, 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.
| \(\displaystyle \int \frac {(c+d x)^{7/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2176 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{5/2} \left (-6 a d D x^2-3 (A b d+3 a C d+2 a c D) x+4 A b c-\frac {a (2 b c C+7 b B d+7 a d D)}{b}\right )}{2 \left (a-b x^2\right )^{3/2}}dx}{3 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{5/2} \left (-6 a d D x^2-3 (A b d+3 a C d+2 a c D) x+4 A b c-\frac {a (2 b c C+7 b B d+7 a d D)}{b}\right )}{\left (a-b x^2\right )^{3/2}}dx}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2176 |
| \(\displaystyle \frac {\frac {\int \frac {d (c+d x)^{3/2} \left (3 a (5 A b d+15 a C d+14 a c D)-\left (-77 d D a^2-10 b c C a-35 b B d a+20 A b^2 c\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \int \frac {(c+d x)^{3/2} \left (3 a (5 A b d+15 a C d+14 a c D)-\left (20 A b^2 c-a (5 b (2 c C+7 B d)+77 a d D)\right ) x\right )}{\sqrt {a-b x^2}}dx}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {d \left (\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}-\frac {2 \int -\frac {3 \sqrt {c+d x} \left (a \left (5 A c d b^2+a \left (77 a D d^2+5 b \left (14 D c^2+17 C d c+7 B d^2\right )\right )\right )-b \left (5 A b \left (4 b c^2-5 a d^2\right )-a (5 b c (2 c C+7 B d)+3 a d (25 C d+49 c D))\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \left (\frac {3 \int \frac {\sqrt {c+d x} \left (a \left (5 A c d b^2+a \left (77 a D d^2+5 b \left (14 D c^2+17 C d c+7 B d^2\right )\right )\right )-b \left (5 A b \left (4 b c^2-5 a d^2\right )-a (5 b c (2 c C+7 B d)+3 a d (25 C d+49 c D))\right ) x\right )}{\sqrt {a-b x^2}}dx}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {d \left (\frac {3 \left (\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right )-\frac {2 \int \frac {b \left (a \left (5 A b d \left (b c^2-5 a d^2\right )-a \left (3 a (25 C d+126 c D) d^2+5 b c \left (42 D c^2+53 C d c+28 B d^2\right )\right )\right )+\left (20 A b^2 c \left (b c^2-2 a d^2\right )-a \left (231 a^2 D d^3+3 a b \left (119 D c^2+110 C d c+35 B d^2\right ) d+5 b^2 c^2 (2 c C+7 B d)\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \left (\frac {3 \left (\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right )-\frac {1}{3} \int \frac {a \left (5 A b d \left (b c^2-5 a d^2\right )-a \left (3 a (25 C d+126 c D) d^2+5 b c \left (42 D c^2+53 C d c+28 B d^2\right )\right )\right )+\left (20 A b^2 c \left (b c^2-2 a d^2\right )-a \left (231 a^2 D d^3+3 a b \left (119 D c^2+110 C d c+35 B d^2\right ) d+5 b^2 c^2 (2 c C+7 B d)\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {d \left (\frac {3 \left (\frac {1}{3} \left (\frac {\left (b c^2-a d^2\right ) \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\left (20 A b^2 c \left (b c^2-2 a d^2\right )-a \left (231 a^2 d^3 D+3 a b d \left (35 B d^2+119 c^2 D+110 c C d\right )+5 b^2 c^2 (7 B d+2 c C)\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )+\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {d \left (\frac {3 \left (\frac {1}{3} \left (\frac {\left (b c^2-a d^2\right ) \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (20 A b^2 c \left (b c^2-2 a d^2\right )-a \left (231 a^2 d^3 D+3 a b d \left (35 B d^2+119 c^2 D+110 c C d\right )+5 b^2 c^2 (7 B d+2 c C)\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )+\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {d \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (20 A b^2 c \left (b c^2-2 a d^2\right )-a \left (231 a^2 d^3 D+3 a b d \left (35 B d^2+119 c^2 D+110 c C d\right )+5 b^2 c^2 (7 B d+2 c C)\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {\left (b c^2-a d^2\right ) \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )+\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {d \left (\frac {3 \left (\frac {1}{3} \left (\frac {\left (b c^2-a d^2\right ) \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (20 A b^2 c \left (b c^2-2 a d^2\right )-a \left (231 a^2 d^3 D+3 a b d \left (35 B d^2+119 c^2 D+110 c C d\right )+5 b^2 c^2 (7 B d+2 c C)\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {d \left (\frac {3 \left (\frac {1}{3} \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (20 A b^2 c \left (b c^2-2 a d^2\right )-a \left (231 a^2 d^3 D+3 a b d \left (35 B d^2+119 c^2 D+110 c C d\right )+5 b^2 c^2 (7 B d+2 c C)\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {d \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (20 A b^2 c \left (b c^2-2 a d^2\right )-a \left (231 a^2 d^3 D+3 a b d \left (35 B d^2+119 c^2 D+110 c C d\right )+5 b^2 c^2 (7 B d+2 c C)\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )+\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {d \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (20 A b^2 c \left (b c^2-2 a d^2\right )-a \left (231 a^2 d^3 D+3 a b d \left (35 B d^2+119 c^2 D+110 c C d\right )+5 b^2 c^2 (7 B d+2 c C)\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )+\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (5 A b \left (4 b c^2-5 a d^2\right )-a (3 a d (49 c D+25 C d)+5 b c (7 B d+2 c C))\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 A b^2 c-a (77 a d D+5 b (7 B d+2 c C))\right )}{5 b}\right )}{2 a b}-\frac {(c+d x)^{5/2} \left (3 a (2 a c D+3 a C d+A b d)-x \left (4 A b^2 c-a (13 a d D+7 b B d+2 b c C)\right )\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{7/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\) |
Input:
Int[((c + d*x)^(7/2)*(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)^(7/2))/(3*a*b*(a - b*x^2)^(3/ 2)) + (-(((c + d*x)^(5/2)*(3*a*(A*b*d + 3*a*C*d + 2*a*c*D) - (4*A*b^2*c - a*(2*b*c*C + 7*b*B*d + 13*a*d*D))*x))/(a*b*Sqrt[a - b*x^2])) + (d*((2*(20* A*b^2*c - a*(5*b*(2*c*C + 7*B*d) + 77*a*d*D))*(c + d*x)^(3/2)*Sqrt[a - b*x ^2])/(5*b) + (3*((2*(5*A*b*(4*b*c^2 - 5*a*d^2) - a*(5*b*c*(2*c*C + 7*B*d) + 3*a*d*(25*C*d + 49*c*D)))*Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 + ((2*Sqrt[a] *(20*A*b^2*c*(b*c^2 - 2*a*d^2) - a*(5*b^2*c^2*(2*c*C + 7*B*d) + 231*a^2*d^ 3*D + 3*a*b*d*(110*c*C*d + 35*B*d^2 + 119*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)/ ((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]*(b*c^2 - a*d^2)*(5*A*b*(4*b*c ^2 - 5*a*d^2) - a*(5*b*c*(2*c*C + 7*B*d) + 3*a*d*(25*C*d + 49*c*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))/(5*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. \(1570\) vs. \(2(584)=1168\).
Time = 7.58 (sec) , antiderivative size = 1571, normalized size of antiderivative = 2.37
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1571\) |
| default | \(\text {Expression too large to display}\) | \(9889\) |
Input:
int((d*x+c)^(7/2)*(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*(3*A*a*b ^2*c*d^2+A*b^3*c^3+B*a^2*b*d^3+3*B*a*b^2*c^2*d+3*C*a^2*b*c*d^2+C*a*b^2*c^3 +D*a^3*d^3+3*D*a^2*b*c^2*d)/a/b^5*x+1/3*(A*a*b*d^3+3*A*b^2*c^2*d+3*B*a*b*c *d^2+B*b^2*c^3+C*a^2*d^3+3*C*a*b*c^2*d+3*D*a^2*c*d^2+D*a*b*c^3)/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*(8*A*a*b^2 *c*d^2-4*A*b^3*c^3+9*B*a^2*b*d^3+7*B*a*b^2*c^2*d+26*C*a^2*b*c*d^2+2*C*a*b^ 2*c^3+15*D*a^3*d^3+25*D*a^2*b*c^2*d)/a^2/b^4*x-1/12*(7*A*a*b*d^3+A*b^2*c^2 *d+20*B*a*b*c*d^2+13*C*a^2*d^3+19*C*a*b*c^2*d+38*D*a^2*c*d^2+6*D*a*b*c^3)/ a/b^4)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)-2/5*D*d^3/b^3*x*(-b*d*x^3-b*c*x^2+a* d*x+a*c)^(1/2)-2/3*(1/b^2*d^3*(C*d+4*D*c)-4/5*D*d^3/b^2*c)/b/d*(-b*d*x^3-b *c*x^2+a*d*x+a*c)^(1/2)+2*(d*(A*b*d^3+4*B*b*c*d^2+2*C*a*d^3+6*C*b*c^2*d+8* D*a*c*d^2+4*D*b*c^3)/b^3-1/6/b^3*(7*A*a^2*b*d^4+9*A*a*b^2*c^2*d^2-4*A*b^3* c^4+29*B*a^2*b*c*d^3+7*B*a*b^2*c^3*d+13*C*a^3*d^4+45*C*a^2*b*c^2*d^2+2*C*a *b^2*c^4+53*D*a^3*c*d^3+31*D*a^2*b*c^3*d)/a^2+1/12/b^3*d*(7*A*a*b*d^3+A*b^ 2*c^2*d+20*B*a*b*c*d^2+13*C*a^2*d^3+19*C*a*b*c^2*d+38*D*a^2*c*d^2+6*D*a*b* c^3)/a+1/6/b^3*c*(8*A*a*b^2*c*d^2-4*A*b^3*c^3+9*B*a^2*b*d^3+7*B*a*b^2*c^2* d+26*C*a^2*b*c*d^2+2*C*a*b^2*c^3+15*D*a^3*d^3+25*D*a^2*b*c^2*d)/a^2+2/5*D* d^3/b^3*a*c+1/3*(1/b^2*d^3*(C*d+4*D*c)-4/5*D*d^3/b^2*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...
Leaf count of result is larger than twice the leaf count of optimal. 1279 vs. \(2 (588) = 1176\).
Time = 0.12 (sec) , antiderivative size = 1279, normalized size of antiderivative = 1.93 \[ \int \frac {(c+d x)^{7/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(7/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(5/2),x, algorithm= "fricas")
Output:
1/90*((10*(C*a^3*b^2 - 2*A*a^2*b^3)*c^4 - 7*(39*D*a^4*b - 5*B*a^3*b^2)*c^3 *d - 5*(93*C*a^4*b - 11*A*a^3*b^2)*c^2*d^2 - 21*(43*D*a^5 + 15*B*a^4*b)*c* d^3 - 75*(3*C*a^5 + A*a^4*b)*d^4 + (10*(C*a*b^4 - 2*A*b^5)*c^4 - 7*(39*D*a ^2*b^3 - 5*B*a*b^4)*c^3*d - 5*(93*C*a^2*b^3 - 11*A*a*b^4)*c^2*d^2 - 21*(43 *D*a^3*b^2 + 15*B*a^2*b^3)*c*d^3 - 75*(3*C*a^3*b^2 + A*a^2*b^3)*d^4)*x^4 - 2*(10*(C*a^2*b^3 - 2*A*a*b^4)*c^4 - 7*(39*D*a^3*b^2 - 5*B*a^2*b^3)*c^3*d - 5*(93*C*a^3*b^2 - 11*A*a^2*b^3)*c^2*d^2 - 21*(43*D*a^4*b + 15*B*a^3*b^2) *c*d^3 - 75*(3*C*a^4*b + A*a^3*b^2)*d^4)*x^2)*sqrt(-b*d)*weierstrassPInver se(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*(10*(C*a^3*b^2 - 2*A*a^2*b^3)*c^3*d + 7*(51*D*a^4*b + 5* B*a^3*b^2)*c^2*d^2 + 10*(33*C*a^4*b + 4*A*a^3*b^2)*c*d^3 + 21*(11*D*a^5 + 5*B*a^4*b)*d^4 + (10*(C*a*b^4 - 2*A*b^5)*c^3*d + 7*(51*D*a^2*b^3 + 5*B*a*b ^4)*c^2*d^2 + 10*(33*C*a^2*b^3 + 4*A*a*b^4)*c*d^3 + 21*(11*D*a^3*b^2 + 5*B *a^2*b^3)*d^4)*x^4 - 2*(10*(C*a^2*b^3 - 2*A*a*b^4)*c^3*d + 7*(51*D*a^3*b^2 + 5*B*a^2*b^3)*c^2*d^2 + 10*(33*C*a^3*b^2 + 4*A*a^2*b^3)*c*d^3 + 21*(11*D *a^4*b + 5*B*a^3*b^2)*d^4)*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), 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*(12*D*a^2*b^3*d^4*x^5 + 10*(2*D*a^3*b^2 - B*a^2*b^3)*c^3*d + 5*(13*C*a^3*b^2 - 5*A*a^2*b^3)*c^2*d^2 + 14*(16*D*a^4*b + 5*B*a^3*b^2)*...
Timed out. \[ \int \frac {(c+d x)^{7/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(7/2)*(D*x**3+C*x**2+B*x+A)/(-b*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {(c+d x)^{7/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {7}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)^(7/2)*(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)^(7/2)/(-b*x^2 + a)^(5/2), x)
\[ \int \frac {(c+d x)^{7/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {7}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)^(7/2)*(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)^(7/2)/(-b*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {(c+d x)^{7/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{7/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a-b\,x^2\right )}^{5/2}} \,d x \] Input:
int(((c + d*x)^(7/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(5/2),x)
Output:
int(((c + d*x)^(7/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(5/2), x)
\[ \int \frac {(c+d x)^{7/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\int \frac {\left (d x +c \right )^{\frac {7}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (-b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:
int((d*x+c)^(7/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(5/2),x)
Output:
int((d*x+c)^(7/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(5/2),x)