Integrand size = 37, antiderivative size = 477 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=-\frac {2 \left (25 a D+b \left (35 B-\frac {2 c (7 C d-4 c D)}{d^2}\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{105 b^2}-\frac {2 (7 C d-9 c D) (c+d x)^{3/2} \sqrt {a-b x^2}}{35 b d^2}-\frac {2 D (c+d x)^{5/2} \sqrt {a-b x^2}}{7 b d^2}-\frac {2 \sqrt {a} \left (a d^2 (63 C d+19 c D)-b \left (14 c^2 C d-35 B c d^2-105 A d^3-8 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{3/2} d^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (25 a d^2 D-b \left (14 c C d-35 B d^2-8 c^2 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{5/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/105*(25*D*a+b*(35*B-2*c*(7*C*d-4*D*c)/d^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(1 /2)/b^2-2/35*(7*C*d-9*D*c)*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b/d^2-2/7*D*(d*x +c)^(5/2)*(-b*x^2+a)^(1/2)/b/d^2-2/105*a^(1/2)*(a*d^2*(63*C*d+19*D*c)-b*(- 105*A*d^3-35*B*c*d^2+14*C*c^2*d-8*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^3/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1 /2)/(-b*x^2+a)^(1/2)+2/105*a^(1/2)*(-a*d^2+b*c^2)*(25*a*d^2*D-b*(-35*B*d^2 +14*C*c*d-8*D*c^2))*((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^(5/2)/d^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 28.35 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (-a d^2 (63 C d+19 c D)-b \left (-14 c^2 C d+35 B c d^2+105 A d^3+8 c^3 D\right )-(c+d x) \left (25 a d^2 D+b \left (-4 c^2 D+c d (7 C+3 D x)+d^2 (35 B+3 x (7 C+5 D x))\right )\right )+\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2 (63 C d+19 c D)+b \left (-14 c^2 C d+35 B c d^2+105 A d^3+8 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}-\frac {i \left (\sqrt {b} c-\sqrt {a} d\right ) \left (105 A b^{3/2} d^2-25 a^{3/2} d^2 D+3 a \sqrt {b} d (21 C d-2 c D)+\sqrt {a} b \left (14 c C d-35 B d^2-8 c^2 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{105 b^2 d^2 \sqrt {c+d x}} \] Input:
Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2 + D*x^3))/Sqrt[a - b*x^2],x]
Output:
(2*Sqrt[a - b*x^2]*(-(a*d^2*(63*C*d + 19*c*D)) - b*(-14*c^2*C*d + 35*B*c*d ^2 + 105*A*d^3 + 8*c^3*D) - (c + d*x)*(25*a*d^2*D + b*(-4*c^2*D + c*d*(7*C + 3*D*x) + d^2*(35*B + 3*x*(7*C + 5*D*x)))) + (I*Sqrt[b]*(Sqrt[b]*c - Sqr t[a]*d)*(a*d^2*(63*C*d + 19*c*D) + b*(-14*c^2*C*d + 35*B*c*d^2 + 105*A*d^3 + 8*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)])/(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2)) - (I*(Sqrt[ b]*c - Sqrt[a]*d)*(105*A*b^(3/2)*d^2 - 25*a^(3/2)*d^2*D + 3*a*Sqrt[b]*d*(2 1*C*d - 2*c*D) + Sqrt[a]*b*(14*c*C*d - 35*B*d^2 - 8*c^2*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*Sqrt[-c + ( Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(105*b^2*d^2*Sqrt[c + d*x])
Time = 1.06 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.04, 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, 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.
\(\displaystyle \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx\) |
\(\Big \downarrow \) 2185 |
\(\displaystyle -\frac {2 \int -\frac {\sqrt {c+d x} \left (b (7 C d-9 c D) x^2 d^2+(7 A b d+5 a c D) d^2+\left (-2 b D c^2+7 b B d^2+5 a d^2 D\right ) x d\right )}{2 \sqrt {a-b x^2}}dx}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sqrt {c+d x} \left (b (7 C d-9 c D) x^2 d^2+(7 A b d+5 a c D) d^2+\left (-2 b D c^2+7 b B d^2+5 a d^2 D\right ) x d\right )}{\sqrt {a-b x^2}}dx}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 2185 |
\(\displaystyle \frac {-\frac {2 \int -\frac {b d^3 \sqrt {c+d x} \left (d (35 A b d+21 a C d-2 a c D)+\left (25 a d^2 D-b \left (-8 D c^2+14 C d c-35 B d^2\right )\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{5 b d^2}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{5} d \int \frac {\sqrt {c+d x} \left (d (35 A b d+21 a C d-2 a c D)+\left (25 a d^2 D-b \left (-8 D c^2+14 C d c-35 B d^2\right )\right ) x\right )}{\sqrt {a-b x^2}}dx-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 687 |
\(\displaystyle \frac {\frac {1}{5} d \left (-\frac {2 \int -\frac {d \left (105 A c d b^2+a \left (25 a D d^2+b \left (2 D c^2+49 C d c+35 B d^2\right )\right )\right )+b \left (a d^2 (63 C d+19 c D)-b \left (-8 D c^3+14 C d c^2-35 B d^2 c-105 A d^3\right )\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right )}{3 b}\right )-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{5} d \left (\frac {\int \frac {d \left (105 A c d b^2+a \left (25 a D d^2+b \left (2 D c^2+49 C d c+35 B d^2\right )\right )\right )+b \left (a d^2 (63 C d+19 c D)-b \left (-8 D c^3+14 C d c^2-35 B d^2 c-105 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right )}{3 b}\right )-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 600 |
\(\displaystyle \frac {\frac {1}{5} d \left (\frac {\frac {b \left (a d^2 (19 c D+63 C d)-b \left (-105 A d^3-35 B c d^2-8 c^3 D+14 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right )}{3 b}\right )-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 509 |
\(\displaystyle \frac {\frac {1}{5} d \left (\frac {\frac {b \sqrt {1-\frac {b x^2}{a}} \left (a d^2 (19 c D+63 C d)-b \left (-105 A d^3-35 B c d^2-8 c^3 D+14 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right )}{3 b}\right )-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 508 |
\(\displaystyle \frac {\frac {1}{5} d \left (\frac {-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (19 c D+63 C d)-b \left (-105 A d^3-35 B c d^2-8 c^3 D+14 c^2 C d\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right )}{3 b}\right )-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\frac {1}{5} d \left (\frac {-\frac {\left (b c^2-a d^2\right ) \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (19 c D+63 C d)-b \left (-105 A d^3-35 B c d^2-8 c^3 D+14 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right )}{3 b}\right )-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 512 |
\(\displaystyle \frac {\frac {1}{5} d \left (\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (19 c D+63 C d)-b \left (-105 A d^3-35 B c d^2-8 c^3 D+14 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right )}{3 b}\right )-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 511 |
\(\displaystyle \frac {\frac {1}{5} d \left (\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (19 c D+63 C d)-b \left (-105 A d^3-35 B c d^2-8 c^3 D+14 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right )}{3 b}\right )-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {\frac {1}{5} d \left (\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (19 c D+63 C d)-b \left (-105 A d^3-35 B c d^2-8 c^3 D+14 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 D-b \left (-35 B d^2-8 c^2 D+14 c C d\right )\right )}{3 b}\right )-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 C d-9 c D)}{7 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^2}\) |
Input:
Int[(Sqrt[c + d*x]*(A + B*x + C*x^2 + D*x^3))/Sqrt[a - b*x^2],x]
Output:
(-2*D*(c + d*x)^(5/2)*Sqrt[a - b*x^2])/(7*b*d^2) + ((-2*d*(7*C*d - 9*c*D)* (c + d*x)^(3/2)*Sqrt[a - b*x^2])/5 + (d*((-2*(25*a*d^2*D - b*(14*c*C*d - 3 5*B*d^2 - 8*c^2*D))*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(3*b) + ((-2*Sqrt[a]*Sq rt[b]*(a*d^2*(63*C*d + 19*c*D) - b*(14*c^2*C*d - 35*B*c*d^2 - 105*A*d^3 - 8*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)*(25*a*d^2*D - b*(14*c*C*d - 35*B*d^2 - 8*c^2*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*b)))/5)/(7*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[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[{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 = 3.20 (sec) , antiderivative size = 793, normalized size of antiderivative = 1.66
method | result | size |
elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 D x^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{7 b}-\frac {2 \left (C d +\frac {D c}{7}\right ) x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b d}-\frac {2 \left (B d +C c +\frac {5 D a d}{7 b}-\frac {4 \left (C d +\frac {D c}{7}\right ) c}{5 d}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (A c +\frac {2 \left (C d +\frac {D c}{7}\right ) a c}{5 b d}+\frac {\left (B d +C c +\frac {5 D a d}{7 b}-\frac {4 \left (C d +\frac {D c}{7}\right ) c}{5 d}\right ) a}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (A d +B c +\frac {4 D a c}{7 b}+\frac {3 \left (C d +\frac {D c}{7}\right ) a}{5 b}-\frac {2 \left (B d +C c +\frac {5 D a d}{7 b}-\frac {4 \left (C d +\frac {D c}{7}\right ) c}{5 d}\right ) c}{3 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(793\) |
default | \(\text {Expression too large to display}\) | \(3628\) |
Input:
int((d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(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)*(-2/7*D/b*x^2* (-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(C*d+1/7*D*c)/b/d*x*(-b*d*x^3-b*c*x ^2+a*d*x+a*c)^(1/2)-2/3*(B*d+C*c+5/7*D/b*a*d-4/5*(C*d+1/7*D*c)/d*c)/b/d*(- b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(A*c+2/5*(C*d+1/7*D*c)/b/d*a*c+1/3*(B*d +C*c+5/7*D/b*a*d-4/5*(C*d+1/7*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*(A*d+B*c+4/7*D/b*a*c+3/ 5*(C*d+1/7*D*c)/b*a-2/3*(B*d+C*c+5/7*D/b*a*d-4/5*(C*d+1/7*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))*El lipticE(((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)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))))
Time = 0.10 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\frac {2 \, {\left ({\left (8 \, D b^{2} c^{4} - 14 \, C b^{2} c^{3} d + {\left (13 \, D a b + 35 \, B b^{2}\right )} c^{2} d^{2} - 42 \, {\left (2 \, C a b + 5 \, A b^{2}\right )} c d^{3} - 15 \, {\left (5 \, D a^{2} + 7 \, B a b\right )} d^{4}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (8 \, D b^{2} c^{3} d - 14 \, C b^{2} c^{2} d^{2} + {\left (19 \, D a b + 35 \, B b^{2}\right )} c d^{3} + 21 \, {\left (3 \, C a b + 5 \, A b^{2}\right )} d^{4}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (15 \, D b^{2} d^{4} x^{2} - 4 \, D b^{2} c^{2} d^{2} + 7 \, C b^{2} c d^{3} + 5 \, {\left (5 \, D a b + 7 \, B b^{2}\right )} d^{4} + 3 \, {\left (D b^{2} c d^{3} + 7 \, C b^{2} d^{4}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{315 \, b^{3} d^{4}} \] Input:
integrate((d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(1/2),x, algorithm= "fricas")
Output:
2/315*((8*D*b^2*c^4 - 14*C*b^2*c^3*d + (13*D*a*b + 35*B*b^2)*c^2*d^2 - 42* (2*C*a*b + 5*A*b^2)*c*d^3 - 15*(5*D*a^2 + 7*B*a*b)*d^4)*sqrt(-b*d)*weierst rassPInverse(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*(8*D*b^2*c^3*d - 14*C*b^2*c^2*d^2 + (19*D*a*b + 35*B*b^2)*c*d^3 + 21*(3*C*a*b + 5*A*b^2)*d^4)*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), weierst rassPInverse(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^4*x^2 - 4*D*b^2*c^2*d^2 + 7*C*b^2 *c*d^3 + 5*(5*D*a*b + 7*B*b^2)*d^4 + 3*(D*b^2*c*d^3 + 7*C*b^2*d^4)*x)*sqrt (-b*x^2 + a)*sqrt(d*x + c))/(b^3*d^4)
\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int \frac {\sqrt {c + d x} \left (A + B x + C x^{2} + D x^{3}\right )}{\sqrt {a - b x^{2}}}\, dx \] Input:
integrate((d*x+c)**(1/2)*(D*x**3+C*x**2+B*x+A)/(-b*x**2+a)**(1/2),x)
Output:
Integral(sqrt(c + d*x)*(A + B*x + C*x**2 + D*x**3)/sqrt(a - b*x**2), x)
\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {d x + c}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate((d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(1/2),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(d*x + c)/sqrt(-b*x^2 + a), x)
\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {d x + c}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate((d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(1/2),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(d*x + c)/sqrt(-b*x^2 + a), x)
Timed out. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int \frac {\sqrt {c+d\,x}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{\sqrt {a-b\,x^2}} \,d x \] Input:
int(((c + d*x)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(1/2),x)
Output:
int(((c + d*x)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(1/2), x)
\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\frac {-70 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b d -88 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a c d -70 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c -32 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b \,c^{2} x -20 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b c d \,x^{2}-105 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} d^{2}-82 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b c \,d^{2}-35 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{3} c d +6 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{3}+35 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,d^{2}+44 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} c \,d^{2}+70 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{2}+35 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c d +32 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,c^{3}}{70 b^{2} c} \] Input:
int((d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(1/2),x)
Output:
( - 70*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*d - 88*sqrt(c + d*x)*sqrt(a - b* x**2)*a*c*d - 70*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c - 32*sqrt(c + d*x)* sqrt(a - b*x**2)*b*c**2*x - 20*sqrt(c + d*x)*sqrt(a - b*x**2)*b*c*d*x**2 - 105*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*d**2 - 82*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*c*d**2 - 35*int((sqrt(c + d*x)*sqr t(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**3*c*d + 6*in t((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**2*c**3 + 35*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*d**2 + 44*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*c*d**2 + 70*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**2*c**2 + 35*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**2*c*d + 32*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*c**3)/(70*b**2*c)