Integrand size = 37, antiderivative size = 598 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2 \left (3 a d^2 (5 C d-6 c D)+b \left (72 c^2 C d-84 B c d^2+105 A d^3-64 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{315 b d^4}+\frac {2 \left (7 a d^2 D-b \left (18 c C d-21 B d^2-16 c^2 D\right )\right ) x \sqrt {c+d x} \sqrt {a-b x^2}}{105 b d^3}-\frac {2 (3 C d-5 c D) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{21 b d^2}-\frac {2 D (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}{9 b d^2}-\frac {4 \sqrt {a} \left (21 a^2 d^4 D-3 a b d^2 \left (13 c C d-21 B d^2-10 c^2 D\right )+b^2 c \left (72 c^2 C d-84 B c d^2+105 A d^3-64 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 b^{3/2} d^5 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {4 \sqrt {a} \left (b c^2-a d^2\right ) \left (3 a d^2 (5 C d-6 c D)+b \left (72 c^2 C d-84 B c d^2+105 A d^3-64 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 b^{3/2} d^5 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
2/315*(3*a*d^2*(5*C*d-6*D*c)+b*(105*A*d^3-84*B*c*d^2+72*C*c^2*d-64*D*c^3)) *(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^4+2/105*(7*a*d^2*D-b*(-21*B*d^2+18*C*c *d-16*D*c^2))*x*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^3-2/21*(3*C*d-5*D*c)*(d *x+c)^(1/2)*(-b*x^2+a)^(3/2)/b/d^2-2/9*D*(d*x+c)^(3/2)*(-b*x^2+a)^(3/2)/b/ d^2-4/315*a^(1/2)*(21*a^2*d^4*D-3*a*b*d^2*(-21*B*d^2+13*C*c*d-10*D*c^2)+b^ 2*c*(105*A*d^3-84*B*c*d^2+72*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^(3/2)/d^5/((d*x+c)/(c+a^(1/2)*d/b^(1/2 )))^(1/2)/(-b*x^2+a)^(1/2)+4/315*a^(1/2)*(-a*d^2+b*c^2)*(3*a*d^2*(5*C*d-6* D*c)+b*(105*A*d^3-84*B*c*d^2+72*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^(3/2)/d^5/( d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 31.06 (sec) , antiderivative size = 748, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {a-b x^2} \left (b (c+d x) \left (-2 a d^2 (15 C d-11 c D+7 d D x)+b \left (-64 c^3 D+24 c^2 d (3 C+2 D x)-2 c d^2 (42 B+x (27 C+20 D x))+d^3 (105 A+x (63 B+5 x (9 C+7 D x)))\right )\right )-\frac {2 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (21 a^2 d^4 D+3 a b d^2 \left (-13 c C d+21 B d^2+10 c^2 D\right )+b^2 c \left (72 c^2 C d-84 B c d^2+105 A d^3-64 c^3 D\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (21 a^2 d^4 D+3 a b d^2 \left (-13 c C d+21 B d^2+10 c^2 D\right )+b^2 c \left (72 c^2 C d-84 B c d^2+105 A d^3-64 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} \sqrt {b} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (21 a^{3/2} d^3 D-3 a \sqrt {b} d^2 (5 C d-6 c D)+3 \sqrt {a} b d \left (-18 c C d+21 B d^2+16 c^2 D\right )+b^{3/2} \left (-72 c^2 C d+84 B c d^2-105 A d^3+64 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{315 b^2 d^4 \sqrt {c+d x}} \] Input:
Integrate[(Sqrt[a - b*x^2]*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]
Output:
(2*Sqrt[a - b*x^2]*(b*(c + d*x)*(-2*a*d^2*(15*C*d - 11*c*D + 7*d*D*x) + b* (-64*c^3*D + 24*c^2*d*(3*C + 2*D*x) - 2*c*d^2*(42*B + x*(27*C + 20*D*x)) + d^3*(105*A + x*(63*B + 5*x*(9*C + 7*D*x))))) - (2*(d^2*Sqrt[-c + (Sqrt[a] *d)/Sqrt[b]]*(21*a^2*d^4*D + 3*a*b*d^2*(-13*c*C*d + 21*B*d^2 + 10*c^2*D) + b^2*c*(72*c^2*C*d - 84*B*c*d^2 + 105*A*d^3 - 64*c^3*D))*(a - b*x^2) + I*S qrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(21*a^2*d^4*D + 3*a*b*d^2*(-13*c*C*d + 21*B *d^2 + 10*c^2*D) + b^2*c*(72*c^2*C*d - 84*B*c*d^2 + 105*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)*(21*a^(3/2)*d^3*D - 3*a*Sqrt [b]*d^2*(5*C*d - 6*c*D) + 3*Sqrt[a]*b*d*(-18*c*C*d + 21*B*d^2 + 16*c^2*D) + b^(3/2)*(-72*c^2*C*d + 84*B*c*d^2 - 105*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*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))))/(315*b^2*d^4*Sqrt[c + d*x])
Time = 1.17 (sec) , antiderivative size = 582, normalized size of antiderivative = 0.97, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2185, 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.
| \(\displaystyle \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {3 \sqrt {a-b x^2} \left (b (3 C d-5 c D) x^2 d^2+(3 A b d+a c D) d^2+\left (-2 b D c^2+3 b B d^2+a d^2 D\right ) x d\right )}{2 \sqrt {c+d x}}dx}{9 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {a-b x^2} \left (b (3 C d-5 c D) x^2 d^2+(3 A b d+a c D) d^2+\left (-2 b D c^2+3 b B d^2+a d^2 D\right ) x d\right )}{\sqrt {c+d x}}dx}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {b d^3 \left (d (21 A b d+3 a C d+2 a c D)+\left (7 a d^2 D-b \left (-16 D c^2+18 C d c-21 B d^2\right )\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{7 b d^2}-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} d \int \frac {\left (d (21 A b d+3 a C d+2 a c D)+\left (7 a d^2 D-b \left (-16 D c^2+18 C d c-21 B d^2\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )+3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right )}{15 d^2}-\frac {4 \int -\frac {b \left (a d \left (3 a (5 C d+c D) d^2+b \left (-16 D c^3+18 C d c^2-21 B d^2 c+105 A d^3\right )\right )+\left (5 b c d^2 (21 A b d+3 a C d+2 a c D)-\left (4 b c^2-3 a d^2\right ) \left (7 a d^2 D-b \left (-16 D c^2+18 C d c-21 B d^2\right )\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}\right )-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {2 \int \frac {a d \left (3 a (5 C d+c D) d^2+b \left (-16 D c^3+18 C d c^2-21 B d^2 c+105 A d^3\right )\right )+\left (5 b c d^2 (21 A b d+3 a C d+2 a c D)-\left (4 b c^2-3 a d^2\right ) \left (7 a d^2 D-b \left (-16 D c^2+18 C d c-21 B d^2\right )\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )+3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right )}{15 d^2}\right )-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {2 \left (\frac {\left (5 b c d^2 (2 a c D+3 a C d+21 A b d)-\left (4 b c^2-3 a d^2\right ) \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )+3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right )}{15 d^2}\right )-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (5 b c d^2 (2 a c D+3 a C d+21 A b d)-\left (4 b c^2-3 a d^2\right ) \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )+3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right )}{15 d^2}\right )-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {2 \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (5 b c d^2 (2 a c D+3 a C d+21 A b d)-\left (4 b c^2-3 a d^2\right ) \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )+3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right )}{15 d^2}\right )-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 b c d^2 (2 a c D+3 a C d+21 A b d)-\left (4 b c^2-3 a d^2\right ) \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )+3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right )}{15 d^2}\right )-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {2 \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 b c d^2 (2 a c D+3 a C d+21 A b d)-\left (4 b c^2-3 a d^2\right ) \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )+3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right )}{15 d^2}\right )-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 b c d^2 (2 a c D+3 a C d+21 A b d)-\left (4 b c^2-3 a d^2\right ) \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )+3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right )}{15 d^2}\right )-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (5 b c d^2 (2 a c D+3 a C d+21 A b d)-\left (4 b c^2-3 a d^2\right ) \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a d^2 D-b \left (-21 B d^2-16 c^2 D+18 c C d\right )\right )+3 a d^2 (5 C d-6 c D)+b \left (105 A d^3-84 B c d^2-64 c^3 D+72 c^2 C d\right )\right )}{15 d^2}\right )-\frac {2}{7} d \left (a-b x^2\right )^{3/2} \sqrt {c+d x} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{9 b d^2}\) |
Input:
Int[(Sqrt[a - b*x^2]*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]
Output:
(-2*D*(c + d*x)^(3/2)*(a - b*x^2)^(3/2))/(9*b*d^2) + ((-2*d*(3*C*d - 5*c*D )*Sqrt[c + d*x]*(a - b*x^2)^(3/2))/7 + (d*((2*Sqrt[c + d*x]*(3*a*d^2*(5*C* d - 6*c*D) + b*(72*c^2*C*d - 84*B*c*d^2 + 105*A*d^3 - 64*c^3*D) + 3*d*(7*a *d^2*D - b*(18*c*C*d - 21*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*(21*A*b*d + 3*a*C*d + 2*a*c*D) - (4*b*c^2 - 3 *a*d^2)*(7*a*d^2*D - b*(18*c*C*d - 21*B*d^2 - 16*c^2*D)))*Sqrt[c + d*x]*Sq rt[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))/(Sq rt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(3*a*d ^2*(5*C*d - 6*c*D) + b*(72*c^2*C*d - 84*B*c*d^2 + 105*A*d^3 - 64*c^3*D))*S qrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*Ellip ticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqr t[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(15*d^2)))/7)/(3*b *d^3)
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[{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]))
Time = 3.71 (sec) , antiderivative size = 1033, normalized size of antiderivative = 1.73
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1033\) |
| default | \(\text {Expression too large to display}\) | \(4685\) |
Input:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),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)*(2/9*D/d*x^3*( -b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/7*(-C*b+8/9*D/d*b*c)/b/d*x^2*(-b*d*x^3 -b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(-B*b+2/9*D*a-6/7*(-C*b+8/9*D/d*b*c)/d*c)/b/ d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(-A*b+C*a-2/3*D/d*a*c+5/7*(-C*b +8/9*D/d*b*c)/b*a-4/5*(-B*b+2/9*D*a-6/7*(-C*b+8/9*D/d*b*c)/d*c)/d*c)/b/d*( -b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(A*a+2/5*(-B*b+2/9*D*a-6/7*(-C*b+8/9*D /d*b*c)/d*c)/b/d*a*c+1/3*(-A*b+C*a-2/3*D/d*a*c+5/7*(-C*b+8/9*D/d*b*c)/b*a- 4/5*(-B*b+2/9*D*a-6/7*(-C*b+8/9*D/d*b*c)/d*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*(B*a+4/7*(-C *b+8/9*D/d*b*c)/b/d*a*c+3/5*(-B*b+2/9*D*a-6/7*(-C*b+8/9*D/d*b*c)/d*c)/b*a- 2/3*(-A*b+C*a-2/3*D/d*a*c+5/7*(-C*b+8/9*D/d*b*c)/b*a-4/5*(-B*b+2/9*D*a-6/7 *(-C*b+8/9*D/d*b*c)/d*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-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))+1/b*(a*b)^( 1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(...
Time = 0.09 (sec) , antiderivative size = 502, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=-\frac {2 \, {\left (2 \, {\left (64 \, D b^{2} c^{5} - 72 \, C b^{2} c^{4} d - 6 \, {\left (13 \, D a b - 14 \, B b^{2}\right )} c^{3} d^{2} + 3 \, {\left (31 \, C a b - 35 \, A b^{2}\right )} c^{2} d^{3} - 6 \, {\left (2 \, D a^{2} + 21 \, B a b\right )} c d^{4} + 45 \, {\left (C a^{2} + 7 \, A a b\right )} d^{5}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 6 \, {\left (64 \, D b^{2} c^{4} d - 72 \, C b^{2} c^{3} d^{2} - 6 \, {\left (5 \, D a b - 14 \, B b^{2}\right )} c^{2} d^{3} + 3 \, {\left (13 \, C a b - 35 \, A b^{2}\right )} c d^{4} - 21 \, {\left (D a^{2} + 3 \, B a b\right )} d^{5}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (35 \, D b^{2} d^{5} x^{3} - 64 \, D b^{2} c^{3} d^{2} + 72 \, C b^{2} c^{2} d^{3} + 2 \, {\left (11 \, D a b - 42 \, B b^{2}\right )} c d^{4} - 15 \, {\left (2 \, C a b - 7 \, A b^{2}\right )} d^{5} - 5 \, {\left (8 \, D b^{2} c d^{4} - 9 \, C b^{2} d^{5}\right )} x^{2} + {\left (48 \, D b^{2} c^{2} d^{3} - 54 \, C b^{2} c d^{4} - 7 \, {\left (2 \, D a b - 9 \, B b^{2}\right )} d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{945 \, b^{2} d^{6}} \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm= "fricas")
Output:
-2/945*(2*(64*D*b^2*c^5 - 72*C*b^2*c^4*d - 6*(13*D*a*b - 14*B*b^2)*c^3*d^2 + 3*(31*C*a*b - 35*A*b^2)*c^2*d^3 - 6*(2*D*a^2 + 21*B*a*b)*c*d^4 + 45*(C* a^2 + 7*A*a*b)*d^5)*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^ 2*c^4*d - 72*C*b^2*c^3*d^2 - 6*(5*D*a*b - 14*B*b^2)*c^2*d^3 + 3*(13*C*a*b - 35*A*b^2)*c*d^4 - 21*(D*a^2 + 3*B*a*b)*d^5)*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), weierstra ssPInverse(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*(35*D*b^2*d^5*x^3 - 64*D*b^2*c^3*d^2 + 72*C*b^2 *c^2*d^3 + 2*(11*D*a*b - 42*B*b^2)*c*d^4 - 15*(2*C*a*b - 7*A*b^2)*d^5 - 5* (8*D*b^2*c*d^4 - 9*C*b^2*d^5)*x^2 + (48*D*b^2*c^2*d^3 - 54*C*b^2*c*d^4 - 7 *(2*D*a*b - 9*B*b^2)*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^2*d^6)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\int \frac {\sqrt {a - b x^{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\sqrt {c + d x}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2),x)
Output:
Integral(sqrt(a - b*x**2)*(A + B*x + C*x**2 + D*x**3)/sqrt(c + d*x), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{\sqrt {d x + c}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/sqrt(d*x + c), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{\sqrt {d x + c}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/sqrt(d*x + c), x)
Timed out. \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\int \frac {\sqrt {a-b\,x^2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{\sqrt {c+d\,x}} \,d x \] Input:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(1/2),x)
Output:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(1/2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:
int((-b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x)
Output:
( - 42*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*d**3 - 126*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b**2*d**2 + 2*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c**2*d - 28* sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c*d**2*x + 126*sqrt(c + d*x)*sqrt(a - b *x**2)*b**3*c*d*x - 12*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**3*x + 10*sqr t(c + d*x)*sqrt(a - b*x**2)*b**2*c**2*d*x**2 + 70*sqrt(c + d*x)*sqrt(a - b *x**2)*b**2*c*d**2*x**3 - 63*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a* c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*b*d**4 - 315*int((sqrt(c + d*x)*s qrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b**3*c*d**2 - 189*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b**3*d**3 + 27*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/( a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b**2*c**2*d**2 + 252*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**4*c **2*d - 24*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x* *2 - b*d*x**3),x)*b**3*c**4 + 21*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**3*d**4 + 315*int((sqrt(c + d*x)*sqrt (a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*b**2*c*d**2 + 63 *int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3), x)*a**2*b**2*d**3 + 27*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*b*c**2*d**2 - 126*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b**3*c**2*d + 12*in...