Integrand size = 37, antiderivative size = 754 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=-\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a-b x^2}}{5 d^4 (c+d x)^{5/2}}-\frac {2 \left (5 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b c \left (12 c^2 C d-7 B c d^2+2 A d^3-17 c^3 D\right )\right ) \sqrt {a-b x^2}}{15 d^4 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {2 \left (15 a^2 d^4 (C d-3 c D)+b^2 c^2 \left (33 c^2 C d-8 B c d^2-2 A d^3-73 c^3 D\right )-2 a b d^2 \left (28 c^2 C d-8 B c d^2+3 A d^3-63 c^3 D\right )\right ) \sqrt {a-b x^2}}{15 d^4 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 D \sqrt {c+d x} \sqrt {a-b x^2}}{3 d^4}+\frac {4 \sqrt {a} \sqrt {b} \left (5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (43 c^2 C d-8 B c d^2+3 A d^3-118 c^3 D\right )+b^2 c^2 \left (24 c^2 C d-4 B c d^2-A d^3-64 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 d^5 \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {4 \sqrt {a} \left (5 a^2 d^4 D+5 a b d^2 \left (5 c C d-B d^2-14 c^2 D\right )-b^2 c \left (24 c^2 C d-4 B c d^2-A d^3-64 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 \sqrt {b} d^5 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/5*(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)^(5/2)-2/15 *(5*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)-b*c*(2*A*d^3-7*B*c*d^2+12*C*c^2*d-17*D* c^3))*(-b*x^2+a)^(1/2)/d^4/(-a*d^2+b*c^2)/(d*x+c)^(3/2)-2/15*(15*a^2*d^4*( C*d-3*D*c)+b^2*c^2*(-2*A*d^3-8*B*c*d^2+33*C*c^2*d-73*D*c^3)-2*a*b*d^2*(3*A *d^3-8*B*c*d^2+28*C*c^2*d-63*D*c^3))*(-b*x^2+a)^(1/2)/d^4/(-a*d^2+b*c^2)^2 /(d*x+c)^(1/2)+2/3*D*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/d^4+4/15*a^(1/2)*b^(1/ 2)*(5*a^2*d^4*(3*C*d-10*D*c)-a*b*d^2*(3*A*d^3-8*B*c*d^2+43*C*c^2*d-118*D*c ^3)+b^2*c^2*(-A*d^3-4*B*c*d^2+24*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))/d^5/(-a*d^2+b*c^2)^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)*(5*a^2*d^4*D+5*a*b*d^ 2*(-B*d^2+5*C*c*d-14*D*c^2)-b^2*c*(-A*d^3-4*B*c*d^2+24*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/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 33.30 (sec) , antiderivative size = 1099, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(Sqrt[a - b*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(7/2),x]
Output:
Sqrt[c + d*x]*Sqrt[a - b*x^2]*((2*D)/(3*d^4) + (2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(5*d^4*(c + d*x)^3) - (2*(12*b*c^3*C*d - 7*b*B*c^2*d^2 + 2 *A*b*c*d^3 - 10*a*c*C*d^3 + 5*a*B*d^4 - 17*b*c^4*D + 15*a*c^2*d^2*D))/(15* d^4*(-(b*c^2) + a*d^2)*(c + d*x)^2) - (2*(33*b^2*c^4*C*d - 8*b^2*B*c^3*d^2 - 2*A*b^2*c^2*d^3 - 56*a*b*c^2*C*d^3 + 16*a*b*B*c*d^4 - 6*a*A*b*d^5 + 15* a^2*C*d^5 - 73*b^2*c^5*D + 126*a*b*c^3*d^2*D - 45*a^2*c*d^4*D))/(15*d^4*(- (b*c^2) + a*d^2)^2*(c + d*x))) + (4*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d *x))^2)/d^2]*(-(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(5*a^2*d^4*(-3*C*d + 10*c*D ) + a*b*d^2*(43*c^2*C*d - 8*B*c*d^2 + 3*A*d^3 - 118*c^3*D) + b^2*c^2*(-24* c^2*C*d + 4*B*c*d^2 + A*d^3 + 64*c^3*D))*(-((a*d^2)/(c + d*x)^2) + b*(-1 + c/(c + d*x))^2)) + (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(5*a^2*d^4*(-3*C*d + 10*c*D) + a*b*d^2*(43*c^2*C*d - 8*B*c*d^2 + 3*A*d^3 - 118*c^3*D) + b^2*c ^2*(-24*c^2*C*d + 4*B*c*d^2 + A*d^3 + 64*c^3*D))*Sqrt[1 - c/(c + d*x) - (S qrt[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]*d*(Sqrt[b]*c - Sqrt[a]*d)*(5*a^2*d^4*D + 15*a^(3/2)*Sqrt[b]*d^3*( C*d - 3*c*D) - 5*a*b*d^2*(-5*c*C*d + B*d^2 + 14*c^2*D) + 3*Sqrt[a]*b^(3/2) *d*(-6*c^2*C*d + B*c*d^2 - A*d^3 + 16*c^3*D) + b^2*c*(-24*c^2*C*d + 4*B*c* d^2 + A*d^3 + 64*c^3*D))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/(Sqrt[b]*(c...
Time = 1.61 (sec) , antiderivative size = 822, normalized size of antiderivative = 1.09, 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, 681, 25, 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.
| \(\displaystyle \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {a-b x^2} \left (5 \left (\frac {b c^2}{d}-a d\right ) D x^2-\left (a (5 C d-5 c D)+b \left (\frac {6 D c^3}{d^2}-\frac {6 C c^2}{d}+B c-A d\right )\right ) x+\frac {5 \left (A b c d+a \left (-D c^2+C d c-B d^2\right )\right )}{d}\right )}{2 (c+d x)^{5/2}}dx}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {a-b x^2} \left (5 \left (\frac {b c^2}{d}-a d\right ) D x^2-\left (a (5 C d-5 c D)+b \left (\frac {6 D c^3}{d^2}-\frac {6 C c^2}{d}+B c-A d\right )\right ) x+5 \left (A b c+a \left (-\frac {D c^2}{d}+C c-B d\right )\right )\right )}{(c+d x)^{5/2}}dx}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {\frac {2 \int \frac {3 \left (d \left (A b d \left (5 b c^2-a d^2\right )+a \left (5 a d^2 (C d-2 c D)-b c \left (-6 D c^2+C d c+4 B d^2\right )\right )\right )+\left (5 a^2 D d^4+5 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-16 D c^3+6 C d c^2-B d^2 c-4 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{2 d^2 (c+d x)^{3/2}}dx}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-B c d^2-11 c^3 D+6 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (d \left (A b d \left (5 b c^2-a d^2\right )+a \left (5 a d^2 (C d-2 c D)-b c \left (-6 D c^2+C d c+4 B d^2\right )\right )\right )+\left (5 a^2 D d^4+5 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-16 D c^3+6 C d c^2-B d^2 c-4 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{(c+d x)^{3/2}}dx}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-B c d^2-11 c^3 D+6 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 681 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {a d \left (5 a^2 D d^4+5 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-16 D c^3+6 C d c^2-B d^2 c-4 A d^3\right )\right )-b \left (5 a^2 (3 C d-10 c D) d^4-a b \left (-118 D c^3+43 C d c^2-8 B d^2 c+3 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+24 C d c^2-4 B d^2 c-A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 d^2}-\frac {2 \sqrt {a-b x^2} \left (-d x \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-5 c^2 D+2 c C d\right )-b^2 c \left (-4 A d^3-B c d^2-16 c^3 D+6 c^2 C d\right )\right )+5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right )}{3 d^2 \sqrt {c+d x}}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-B c d^2-11 c^3 D+6 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {a d \left (5 a^2 D d^4+5 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-16 D c^3+6 C d c^2-B d^2 c-4 A d^3\right )\right )-b \left (5 a^2 (3 C d-10 c D) d^4-a b \left (-118 D c^3+43 C d c^2-8 B d^2 c+3 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+24 C d c^2-4 B d^2 c-A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 d^2}-\frac {2 \sqrt {a-b x^2} \left (-d x \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-5 c^2 D+2 c C d\right )-b^2 c \left (-4 A d^3-B c d^2-16 c^3 D+6 c^2 C d\right )\right )+5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right )}{3 d^2 \sqrt {c+d x}}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-B c d^2-11 c^3 D+6 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-14 c^2 D+5 c C d\right )-b^2 c \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \left (5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{3 d^2}-\frac {2 \sqrt {a-b x^2} \left (-d x \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-5 c^2 D+2 c C d\right )-b^2 c \left (-4 A d^3-B c d^2-16 c^3 D+6 c^2 C d\right )\right )+5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right )}{3 d^2 \sqrt {c+d x}}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-B c d^2-11 c^3 D+6 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-14 c^2 D+5 c C d\right )-b^2 c \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \sqrt {1-\frac {b x^2}{a}} \left (5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{3 d^2}-\frac {2 \sqrt {a-b x^2} \left (-d x \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-5 c^2 D+2 c C d\right )-b^2 c \left (-4 A d^3-B c d^2-16 c^3 D+6 c^2 C d\right )\right )+5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right )}{3 d^2 \sqrt {c+d x}}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-B c d^2-11 c^3 D+6 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-14 c^2 D+5 c C d\right )-b^2 c \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{3 d^2}-\frac {2 \sqrt {a-b x^2} \left (-d x \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-5 c^2 D+2 c C d\right )-b^2 c \left (-4 A d^3-B c d^2-16 c^3 D+6 c^2 C d\right )\right )+5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right )}{3 d^2 \sqrt {c+d x}}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-B c d^2-11 c^3 D+6 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-14 c^2 D+5 c C d\right )-b^2 c \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{3 d^2}-\frac {2 \sqrt {a-b x^2} \left (-d x \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-5 c^2 D+2 c C d\right )-b^2 c \left (-4 A d^3-B c d^2-16 c^3 D+6 c^2 C d\right )\right )+5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right )}{3 d^2 \sqrt {c+d x}}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-B c d^2-11 c^3 D+6 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-14 c^2 D+5 c C d\right )-b^2 c \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{3 d^2}-\frac {2 \sqrt {a-b x^2} \left (-d x \left (5 a^2 d^4 D+5 a b d^2 \left (-B d^2-5 c^2 D+2 c C d\right )-b^2 c \left (-4 A d^3-B c d^2-16 c^3 D+6 c^2 C d\right )\right )+5 a^2 d^4 (3 C d-10 c D)-a b d^2 \left (3 A d^3-8 B c d^2-118 c^3 D+43 c^2 C d\right )+b^2 c^2 \left (-A d^3-4 B c d^2-64 c^3 D+24 c^2 C d\right )\right )}{3 d^2 \sqrt {c+d x}}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-B c d^2-11 c^3 D+6 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{3/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (5 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-11 D c^3+6 C d c^2-B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \left (5 a^2 (3 C d-10 c D) d^4-a b \left (-118 D c^3+43 C d c^2-8 B d^2 c+3 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+24 C d c^2-4 B d^2 c-A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (5 a^2 D d^4+5 a b \left (-14 D c^2+5 C d c-B d^2\right ) d^2-b^2 c \left (-64 D c^3+24 C d c^2-4 B d^2 c-A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{3 d^2}-\frac {2 \left (5 a^2 (3 C d-10 c D) d^4-a b \left (-118 D c^3+43 C d c^2-8 B d^2 c+3 A d^3\right ) d^2-\left (5 a^2 D d^4+5 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-16 D c^3+6 C d c^2-B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-64 D c^3+24 C d c^2-4 B d^2 c-A d^3\right )\right ) \sqrt {a-b x^2}}{3 d^2 \sqrt {c+d x}}}{d^2 \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{3/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (5 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-11 D c^3+6 C d c^2-B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \left (5 a^2 (3 C d-10 c D) d^4-a b \left (-118 D c^3+43 C d c^2-8 B d^2 c+3 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+24 C d c^2-4 B d^2 c-A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (5 a^2 D d^4+5 a b \left (-14 D c^2+5 C d c-B d^2\right ) d^2-b^2 c \left (-64 D c^3+24 C d c^2-4 B d^2 c-A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{3 d^2}-\frac {2 \left (5 a^2 (3 C d-10 c D) d^4-a b \left (-118 D c^3+43 C d c^2-8 B d^2 c+3 A d^3\right ) d^2-\left (5 a^2 D d^4+5 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-16 D c^3+6 C d c^2-B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-64 D c^3+24 C d c^2-4 B d^2 c-A d^3\right )\right ) \sqrt {a-b x^2}}{3 d^2 \sqrt {c+d x}}}{d^2 \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}\) |
Input:
Int[(Sqrt[a - b*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(7/2),x]
Output:
(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a - b*x^2)^(3/2))/(5*d^2*(b*c^2 - a*d^2)*(c + d*x)^(5/2)) + ((2*(5*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) - b*c*( 6*c^2*C*d - B*c*d^2 - 4*A*d^3 - 11*c^3*D))*(a - b*x^2)^(3/2))/(3*d^2*(b*c^ 2 - a*d^2)*(c + d*x)^(3/2)) + ((-2*(5*a^2*d^4*(3*C*d - 10*c*D) - a*b*d^2*( 43*c^2*C*d - 8*B*c*d^2 + 3*A*d^3 - 118*c^3*D) + b^2*c^2*(24*c^2*C*d - 4*B* c*d^2 - A*d^3 - 64*c^3*D) - d*(5*a^2*d^4*D + 5*a*b*d^2*(2*c*C*d - B*d^2 - 5*c^2*D) - b^2*c*(6*c^2*C*d - B*c*d^2 - 4*A*d^3 - 16*c^3*D))*x)*Sqrt[a - b *x^2])/(3*d^2*Sqrt[c + d*x]) + (2*((2*Sqrt[a]*Sqrt[b]*(5*a^2*d^4*(3*C*d - 10*c*D) - a*b*d^2*(43*c^2*C*d - 8*B*c*d^2 + 3*A*d^3 - 118*c^3*D) + b^2*c^2 *(24*c^2*C*d - 4*B*c*d^2 - A*d^3 - 64*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)/((Sqr t[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)*(5*a^2*d^4*D + 5*a*b*d^2*( 5*c*C*d - B*d^2 - 14*c^2*D) - b^2*c*(24*c^2*C*d - 4*B*c*d^2 - 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])))/(3*d^2))/( d^2*(b*c^2 - a*d^2)))/(5*(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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ (e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] , x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ[m + 2 *p + 1, 0] && (IntegerQ[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]
Time = 5.98 (sec) , antiderivative size = 1281, normalized size of antiderivative = 1.70
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1281\) |
| default | \(\text {Expression too large to display}\) | \(16906\) |
Input:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),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)*(-2/5*(A*d^3-B *c*d^2+C*c^2*d-D*c^3)/d^7*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^3-2/1 5*(2*A*b*c*d^3+5*B*a*d^4-7*B*b*c^2*d^2-10*C*a*c*d^3+12*C*b*c^3*d+15*D*a*c^ 2*d^2-17*D*b*c^4)/d^6/(a*d^2-b*c^2)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+ c/d)^2+2/15*(-b*d*x^2+a*d)/d^5/(a*d^2-b*c^2)^2*(6*A*a*b*d^5+2*A*b^2*c^2*d^ 3-16*B*a*b*c*d^4+8*B*b^2*c^3*d^2-15*C*a^2*d^5+56*C*a*b*c^2*d^3-33*C*b^2*c^ 4*d+45*D*a^2*c*d^4-126*D*a*b*c^3*d^2+73*D*b^2*c^5)/((x+c/d)*(-b*d*x^2+a*d) )^(1/2)+2/3/d^4*D*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-(B*b*d^2-3*C*b*c* d-D*a*d^2+6*D*b*c^2)/d^5+1/15*b*(2*A*b*c*d^3+5*B*a*d^4-7*B*b*c^2*d^2-10*C* a*c*d^3+12*C*b*c^3*d+15*D*a*c^2*d^2-17*D*b*c^4)/d^5/(a*d^2-b*c^2)+1/15*b/d ^5*c*(6*A*a*b*d^5+2*A*b^2*c^2*d^3-16*B*a*b*c*d^4+8*B*b^2*c^3*d^2-15*C*a^2* d^5+56*C*a*b*c^2*d^3-33*C*b^2*c^4*d+45*D*a^2*c*d^4-126*D*a*b*c^3*d^2+73*D* b^2*c^5)/(a*d^2-b*c^2)^2-1/3/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*(-(C*d-3*D*c)*b/d^4+1/15*b/d^4 *(6*A*a*b*d^5+2*A*b^2*c^2*d^3-16*B*a*b*c*d^4+8*B*b^2*c^3*d^2-15*C*a^2*d^5+ 56*C*a*b*c^2*d^3-33*C*b^2*c^4*d+45*D*a^2*c*d^4-126*D*a*b*c^3*d^2+73*D*b^2* c^5)/(a*d^2-b*c^2)^2+2/3/d^4*D*b*c)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d...
Leaf count of result is larger than twice the leaf count of optimal. 1722 vs. \(2 (676) = 1352\).
Time = 0.17 (sec) , antiderivative size = 1722, normalized size of antiderivative = 2.28 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algorithm= "fricas")
Output:
2/45*(2*(64*D*b^3*c^9 - 24*C*b^3*c^8*d - 2*(83*D*a*b^2 - 2*B*b^3)*c^7*d^2 + (61*C*a*b^2 + A*b^3)*c^6*d^3 + (125*D*a^2*b - 11*B*a*b^2)*c^5*d^4 - 9*(5 *C*a^2*b + A*a*b^2)*c^4*d^5 - 15*(D*a^3 - B*a^2*b)*c^3*d^6 + (64*D*b^3*c^6 *d^3 - 24*C*b^3*c^5*d^4 - 2*(83*D*a*b^2 - 2*B*b^3)*c^4*d^5 + (61*C*a*b^2 + A*b^3)*c^3*d^6 + (125*D*a^2*b - 11*B*a*b^2)*c^2*d^7 - 9*(5*C*a^2*b + A*a* b^2)*c*d^8 - 15*(D*a^3 - B*a^2*b)*d^9)*x^3 + 3*(64*D*b^3*c^7*d^2 - 24*C*b^ 3*c^6*d^3 - 2*(83*D*a*b^2 - 2*B*b^3)*c^5*d^4 + (61*C*a*b^2 + A*b^3)*c^4*d^ 5 + (125*D*a^2*b - 11*B*a*b^2)*c^3*d^6 - 9*(5*C*a^2*b + A*a*b^2)*c^2*d^7 - 15*(D*a^3 - B*a^2*b)*c*d^8)*x^2 + 3*(64*D*b^3*c^8*d - 24*C*b^3*c^7*d^2 - 2*(83*D*a*b^2 - 2*B*b^3)*c^6*d^3 + (61*C*a*b^2 + A*b^3)*c^5*d^4 + (125*D*a ^2*b - 11*B*a*b^2)*c^4*d^5 - 9*(5*C*a^2*b + A*a*b^2)*c^3*d^6 - 15*(D*a^3 - B*a^2*b)*c^2*d^7)*x)*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^8*d - 24*C*b^3*c^7*d^2 - 2*(59*D*a*b^2 - 2*B*b^3)*c^6*d^3 + (43*C*a* b^2 + A*b^3)*c^5*d^4 + 2*(25*D*a^2*b - 4*B*a*b^2)*c^4*d^5 - 3*(5*C*a^2*b - A*a*b^2)*c^3*d^6 + (64*D*b^3*c^5*d^4 - 24*C*b^3*c^4*d^5 - 2*(59*D*a*b^2 - 2*B*b^3)*c^3*d^6 + (43*C*a*b^2 + A*b^3)*c^2*d^7 + 2*(25*D*a^2*b - 4*B*a*b ^2)*c*d^8 - 3*(5*C*a^2*b - A*a*b^2)*d^9)*x^3 + 3*(64*D*b^3*c^6*d^3 - 24*C* b^3*c^5*d^4 - 2*(59*D*a*b^2 - 2*B*b^3)*c^4*d^5 + (43*C*a*b^2 + A*b^3)*c^3* d^6 + 2*(25*D*a^2*b - 4*B*a*b^2)*c^2*d^7 - 3*(5*C*a^2*b - A*a*b^2)*c*d^...
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int \frac {\sqrt {a - b x^{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(7/2),x)
Output:
Integral(sqrt(a - b*x**2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**(7/2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/(d*x + c)^(7/2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/(d*x + c)^(7/2), x)
Timed out. \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int \frac {\sqrt {a-b\,x^2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{7/2}} \,d x \] Input:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(7/2),x)
Output:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(7/2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int \frac {\sqrt {-b \,x^{2}+a}\, \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{\frac {7}{2}}}d x \] Input:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x)
Output:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x)