Integrand size = 37, antiderivative size = 553 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=-\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a-b x^2}}{d^4 \sqrt {c+d x}}-\frac {2 \left (10 a d^2 D+b \left (84 c C d-35 B d^2-141 c^2 D\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{105 b d^4}+\frac {2 (7 C d-23 c D) (c+d x)^{3/2} \sqrt {a-b x^2}}{35 d^4}+\frac {2 D (c+d x)^{5/2} \sqrt {a-b x^2}}{7 d^4}-\frac {4 \sqrt {a} \left (a d^2 (21 C d-34 c D)-b \left (168 c^2 C d-140 B c d^2+105 A d^3-192 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 \sqrt {b} d^5 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {4 \sqrt {a} \left (5 a^2 d^4 D-a b d^2 \left (63 c C d-35 B d^2-82 c^2 D\right )+b^2 c \left (168 c^2 C d-140 B c d^2+105 A d^3-192 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{3/2} d^5 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2*(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)^(1/2)-2/105* (10*a*d^2*D+b*(-35*B*d^2+84*C*c*d-141*D*c^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/ 2)/b/d^4+2/35*(7*C*d-23*D*c)*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/d^4+2/7*D*(d*x +c)^(5/2)*(-b*x^2+a)^(1/2)/d^4-4/105*a^(1/2)*(a*d^2*(21*C*d-34*D*c)-b*(105 *A*d^3-140*B*c*d^2+168*C*c^2*d-192*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/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^( 1/2)/(-b*x^2+a)^(1/2)-4/105*a^(1/2)*(5*a^2*d^4*D-a*b*d^2*(-35*B*d^2+63*C*c *d-82*D*c^2)+b^2*c*(105*A*d^3-140*B*c*d^2+168*C*c^2*d-192*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^(3/2)/d^5/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 29.46 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (105 b \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )+\left (-10 a d^2 D+b \left (-63 c C d+35 B d^2+87 c^2 D\right )\right ) (c+d x)+3 b d (7 C d-13 c D) x (c+d x)+15 b d^2 D x^2 (c+d x)-\frac {2 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a d^2 (21 C d-34 c D)+b \left (-168 c^2 C d+140 B c d^2-105 A d^3+192 c^3 D\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2 (21 C d-34 c D)+b \left (-168 c^2 C d+140 B c d^2-105 A d^3+192 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} d \left (5 a^{3/2} d^3 D+a \sqrt {b} d^2 (-21 C d+34 c D)+\sqrt {a} b d \left (-42 c C d+35 B d^2+48 c^2 D\right )+b^{3/2} \left (168 c^2 C d-140 B c d^2+105 A d^3-192 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{105 b d^4 \sqrt {c+d x}} \] Input:
Integrate[(Sqrt[a - b*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]
Output:
(2*Sqrt[a - b*x^2]*(105*b*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D) + (-10*a* d^2*D + b*(-63*c*C*d + 35*B*d^2 + 87*c^2*D))*(c + d*x) + 3*b*d*(7*C*d - 13 *c*D)*x*(c + d*x) + 15*b*d^2*D*x^2*(c + d*x) - (2*(d^2*Sqrt[-c + (Sqrt[a]* d)/Sqrt[b]]*(a*d^2*(21*C*d - 34*c*D) + b*(-168*c^2*C*d + 140*B*c*d^2 - 105 *A*d^3 + 192*c^3*D))*(a - b*x^2) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(a*d^ 2*(21*C*d - 34*c*D) + b*(-168*c^2*C*d + 140*B*c*d^2 - 105*A*d^3 + 192*c^3* D))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]* d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d )] - I*Sqrt[a]*d*(5*a^(3/2)*d^3*D + a*Sqrt[b]*d^2*(-21*C*d + 34*c*D) + Sqr t[a]*b*d*(-42*c*C*d + 35*B*d^2 + 48*c^2*D) + b^(3/2)*(168*c^2*C*d - 140*B* c*d^2 + 105*A*d^3 - 192*c^3*D))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]* Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I *ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[ a]*d)/(Sqrt[b]*c - Sqrt[a]*d)]))/(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2))))/(105*b*d^4*Sqrt[c + d*x])
Time = 1.53 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.23, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2182, 27, 2185, 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.
| \(\displaystyle \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {a-b x^2} \left (\left (\frac {b c^2}{d}-a d\right ) D x^2-\left (a (C d-c D)+b \left (\frac {6 D c^3}{d^2}-\frac {6 C c^2}{d}+5 B c-5 A d\right )\right ) x+\frac {-a D c^2+A b d c+a C d c-a B d^2}{d}\right )}{2 \sqrt {c+d x}}dx}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {a-b x^2} \left (\left (\frac {b c^2}{d}-a d\right ) D x^2-\left (a (C d-c D)+b \left (\frac {6 D c^3}{d^2}-\frac {6 C c^2}{d}+5 B c-5 A d\right )\right ) x+A b c+a \left (-\frac {D c^2}{d}+C c-B d\right )\right )}{\sqrt {c+d x}}dx}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )-\frac {2 \int -\frac {\left (d \left (7 A b^2 c d-a \left (a d^2 D-b \left (-6 D c^2+7 C d c-7 B d^2\right )\right )\right )-b \left (a d^2 (7 C d-13 c D)-b \left (-48 D c^3+42 C d c^2-35 B d^2 c+35 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{7 b d^2}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (d \left (7 A b^2 c d-a \left (a d^2 D-b \left (-6 D c^2+7 C d c-7 B d^2\right )\right )\right )-b \left (a d^2 (7 C d-13 c D)-b \left (-48 D c^3+42 C d c^2-35 B d^2 c+35 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 b d^2}+\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {-\frac {4 \int -\frac {b \left (a d \left (b c^2-a d^2\right ) \left (5 a d^2 D-b \left (-48 D c^2+42 C d c-35 B d^2\right )\right )+b \left (b c^2-a d^2\right ) \left (a d^2 (21 C d-34 c D)-b \left (-192 D c^3+168 C d c^2-140 B d^2 c+105 A d^3\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 a^2 d^4 D+3 b d x \left (a d^2 (7 C d-13 c D)-b \left (35 A d^3-35 B c d^2-48 c^3 D+42 c^2 C d\right )\right )-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{15 d^2}}{7 b d^2}+\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {a d \left (b c^2-a d^2\right ) \left (5 a d^2 D-b \left (-48 D c^2+42 C d c-35 B d^2\right )\right )+b \left (b c^2-a d^2\right ) \left (a d^2 (21 C d-34 c D)-b \left (-192 D c^3+168 C d c^2-140 B d^2 c+105 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 a^2 d^4 D+3 b d x \left (a d^2 (7 C d-13 c D)-b \left (35 A d^3-35 B c d^2-48 c^3 D+42 c^2 C d\right )\right )-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{15 d^2}}{7 b d^2}+\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {\left (b c^2-a d^2\right ) \left (5 a^2 d^4 D-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {b \left (b c^2-a d^2\right ) \left (a d^2 (21 C d-34 c D)-b \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 a^2 d^4 D+3 b d x \left (a d^2 (7 C d-13 c D)-b \left (35 A d^3-35 B c d^2-48 c^3 D+42 c^2 C d\right )\right )-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{15 d^2}}{7 b d^2}+\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {\left (b c^2-a d^2\right ) \left (5 a^2 d^4 D-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {b \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (a d^2 (21 C d-34 c D)-b \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 a^2 d^4 D+3 b d x \left (a d^2 (7 C d-13 c D)-b \left (35 A d^3-35 B c d^2-48 c^3 D+42 c^2 C d\right )\right )-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{15 d^2}}{7 b d^2}+\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {\left (b c^2-a d^2\right ) \left (5 a^2 d^4 D-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-a d^2\right ) \left (a d^2 (21 C d-34 c D)-b \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 a^2 d^4 D+3 b d x \left (a d^2 (7 C d-13 c D)-b \left (35 A d^3-35 B c d^2-48 c^3 D+42 c^2 C d\right )\right )-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{15 d^2}}{7 b d^2}+\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {\left (b c^2-a d^2\right ) \left (5 a^2 d^4 D-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (21 C d-34 c D)-b \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 a^2 d^4 D+3 b d x \left (a d^2 (7 C d-13 c D)-b \left (35 A d^3-35 B c d^2-48 c^3 D+42 c^2 C d\right )\right )-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{15 d^2}}{7 b d^2}+\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (5 a^2 d^4 D-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (21 C d-34 c D)-b \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 a^2 d^4 D+3 b d x \left (a d^2 (7 C d-13 c D)-b \left (35 A d^3-35 B c d^2-48 c^3 D+42 c^2 C d\right )\right )-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{15 d^2}}{7 b d^2}+\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (5 a^2 d^4 D-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (21 C d-34 c D)-b \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 a^2 d^4 D+3 b d x \left (a d^2 (7 C d-13 c D)-b \left (35 A d^3-35 B c d^2-48 c^3 D+42 c^2 C d\right )\right )-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{15 d^2}}{7 b d^2}+\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 a^2 d^4 D-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (21 C d-34 c D)-b \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 a^2 d^4 D+3 b d x \left (a d^2 (7 C d-13 c D)-b \left (35 A d^3-35 B c d^2-48 c^3 D+42 c^2 C d\right )\right )-a b d^2 \left (-35 B d^2-82 c^2 D+63 c C d\right )+b^2 c \left (105 A d^3-140 B c d^2-192 c^3 D+168 c^2 C d\right )\right )}{15 d^2}}{7 b d^2}+\frac {2}{7} D \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
Input:
Int[(Sqrt[a - b*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]
Output:
(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a - b*x^2)^(3/2))/(d^2*(b*c^2 - a* d^2)*Sqrt[c + d*x]) + ((2*(a/b - c^2/d^2)*D*Sqrt[c + d*x]*(a - b*x^2)^(3/2 ))/7 + ((-2*Sqrt[c + d*x]*(5*a^2*d^4*D - a*b*d^2*(63*c*C*d - 35*B*d^2 - 82 *c^2*D) + b^2*c*(168*c^2*C*d - 140*B*c*d^2 + 105*A*d^3 - 192*c^3*D) + 3*b* d*(a*d^2*(7*C*d - 13*c*D) - b*(42*c^2*C*d - 35*B*c*d^2 + 35*A*d^3 - 48*c^3 *D))*x)*Sqrt[a - b*x^2])/(15*d^2) + (2*((-2*Sqrt[a]*Sqrt[b]*(b*c^2 - a*d^2 )*(a*d^2*(21*C*d - 34*c*D) - b*(168*c^2*C*d - 140*B*c*d^2 + 105*A*d^3 - 19 2*c^3*D))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqr t[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)*(5*a^2*d^4*D - a*b*d^2*(63*c*C*d - 35*B*d^2 - 82*c^2*D) + b^2* c*(168*c^2*C*d - 140*B*c*d^2 + 105*A*d^3 - 192*c^3*D))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[ b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(15*d^2))/(7*b*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]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(1177\) vs. \(2(477)=954\).
Time = 3.53 (sec) , antiderivative size = 1178, normalized size of antiderivative = 2.13
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1178\) |
| default | \(\text {Expression too large to display}\) | \(3797\) |
Input:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),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*(-b*d*x^2+ a*d)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^5/((x+c/d)*(-b*d*x^2+a*d))^(1/2)+2/7* D/d^2*x^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(-b/d^2*(C*d-D*c)+6/7*D/d ^2*b*c)/b/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(-1/d^3*(B*b*d^2-C*b* c*d-D*a*d^2+D*b*c^2)-5/7*D/d*a-4/5*(-b/d^2*(C*d-D*c)+6/7*D/d^2*b*c)/d*c)/b /d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*((A*b*c*d^3+B*a*d^4-B*b*c^2*d^2-C* a*c*d^3+C*b*c^3*d+D*a*c^2*d^2-D*b*c^4)/d^5-b*(A*d^3-B*c*d^2+C*c^2*d-D*c^3) /d^5*c+2/5*(-b/d^2*(C*d-D*c)+6/7*D/d^2*b*c)/b/d*a*c+1/3*(-1/d^3*(B*b*d^2-C *b*c*d-D*a*d^2+D*b*c^2)-5/7*D/d*a-4/5*(-b/d^2*(C*d-D*c)+6/7*D/d^2*b*c)/d*c )/b*a)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b *(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b *(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*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*(A*b*d^3-B*b*c*d^2-C*a*d^3+C*b*c^2*d+D*a*c*d^2-D*b*c^ 3)-b*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^4-4/7*D/d^2*a*c+3/5*(-b/d^2*(C*d-D*c) +6/7*D/d^2*b*c)/b*a-2/3*(-1/d^3*(B*b*d^2-C*b*c*d-D*a*d^2+D*b*c^2)-5/7*D/d* a-4/5*(-b/d^2*(C*d-D*c)+6/7*D/d^2*b*c)/d*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...
Time = 0.10 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (2 \, {\left (192 \, D b^{2} c^{5} - 168 \, C b^{2} c^{4} d - 2 \, {\left (89 \, D a b - 70 \, B b^{2}\right )} c^{3} d^{2} + 21 \, {\left (7 \, C a b - 5 \, A b^{2}\right )} c^{2} d^{3} - 15 \, {\left (D a^{2} + 7 \, B a b\right )} c d^{4} + {\left (192 \, D b^{2} c^{4} d - 168 \, C b^{2} c^{3} d^{2} - 2 \, {\left (89 \, D a b - 70 \, B b^{2}\right )} c^{2} d^{3} + 21 \, {\left (7 \, C a b - 5 \, A b^{2}\right )} c d^{4} - 15 \, {\left (D a^{2} + 7 \, B a b\right )} d^{5}\right )} x\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 6 \, {\left (192 \, D b^{2} c^{4} d - 168 \, C b^{2} c^{3} d^{2} - 2 \, {\left (17 \, D a b - 70 \, B b^{2}\right )} c^{2} d^{3} + 21 \, {\left (C a b - 5 \, A b^{2}\right )} c d^{4} + {\left (192 \, D b^{2} c^{3} d^{2} - 168 \, C b^{2} c^{2} d^{3} - 2 \, {\left (17 \, D a b - 70 \, B b^{2}\right )} c d^{4} + 21 \, {\left (C a b - 5 \, A b^{2}\right )} d^{5}\right )} x\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (15 \, D b^{2} d^{5} x^{3} + 192 \, D b^{2} c^{3} d^{2} - 168 \, C b^{2} c^{2} d^{3} - 105 \, A b^{2} d^{5} - 10 \, {\left (D a b - 14 \, B b^{2}\right )} c d^{4} - 3 \, {\left (8 \, D b^{2} c d^{4} - 7 \, C b^{2} d^{5}\right )} x^{2} + {\left (48 \, D b^{2} c^{2} d^{3} - 42 \, C b^{2} c d^{4} - 5 \, {\left (2 \, D a b - 7 \, B b^{2}\right )} d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{315 \, {\left (b^{2} d^{7} x + b^{2} c d^{6}\right )}} \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm= "fricas")
Output:
2/315*(2*(192*D*b^2*c^5 - 168*C*b^2*c^4*d - 2*(89*D*a*b - 70*B*b^2)*c^3*d^ 2 + 21*(7*C*a*b - 5*A*b^2)*c^2*d^3 - 15*(D*a^2 + 7*B*a*b)*c*d^4 + (192*D*b ^2*c^4*d - 168*C*b^2*c^3*d^2 - 2*(89*D*a*b - 70*B*b^2)*c^2*d^3 + 21*(7*C*a *b - 5*A*b^2)*c*d^4 - 15*(D*a^2 + 7*B*a*b)*d^5)*x)*sqrt(-b*d)*weierstrassP Inverse(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*(192*D*b^2*c^4*d - 168*C*b^2*c^3*d^2 - 2*(17*D*a*b - 70*B*b^2)*c^2*d^3 + 21*(C*a*b - 5*A*b^2)*c*d^4 + (192*D*b^2*c^3*d^2 - 16 8*C*b^2*c^2*d^3 - 2*(17*D*a*b - 70*B*b^2)*c*d^4 + 21*(C*a*b - 5*A*b^2)*d^5 )*x)*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*(15*D*b^2*d^5* x^3 + 192*D*b^2*c^3*d^2 - 168*C*b^2*c^2*d^3 - 105*A*b^2*d^5 - 10*(D*a*b - 14*B*b^2)*c*d^4 - 3*(8*D*b^2*c*d^4 - 7*C*b^2*d^5)*x^2 + (48*D*b^2*c^2*d^3 - 42*C*b^2*c*d^4 - 5*(2*D*a*b - 7*B*b^2)*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^2*d^7*x + b^2*c*d^6)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int \frac {\sqrt {a - b x^{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(3/2),x)
Output:
Integral(sqrt(a - b*x**2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**(3/2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{{\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/(d*x + c)^(3/2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{{\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/(d*x + c)^(3/2), x)
Timed out. \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int \frac {\sqrt {a-b\,x^2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(3/2),x)
Output:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(3/2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int \frac {\sqrt {-b \,x^{2}+a}\, \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{\frac {3}{2}}}d x \] Input:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x)
Output:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x)