Integrand size = 37, antiderivative size = 594 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=-\frac {2 \left (3 a d^2 (25 C d+13 c D)-b \left (18 c^2 C d-63 B c d^2-105 A d^3-8 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{315 b^2 d^2}-\frac {2 \left (49 a D+b \left (63 B-\frac {2 c (9 C d-4 c D)}{d^2}\right )\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{315 b^2}-\frac {2 (9 C d-11 c D) (c+d x)^{5/2} \sqrt {a-b x^2}}{63 b d^2}-\frac {2 D (c+d x)^{7/2} \sqrt {a-b x^2}}{9 b d^2}-\frac {2 \sqrt {a} \left (147 a^2 d^4 D+3 a b d^2 \left (82 c C d+63 B d^2+11 c^2 D\right )-b^2 c \left (18 c^2 C d-63 B c d^2-420 A d^3-8 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 b^{5/2} d^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (3 a d^2 (25 C d+13 c D)-b \left (18 c^2 C d-63 B c d^2-105 A d^3-8 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 b^{5/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/315*(3*a*d^2*(25*C*d+13*D*c)-b*(-105*A*d^3-63*B*c*d^2+18*C*c^2*d-8*D*c^ 3))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d^2-2/315*(49*D*a+b*(63*B-2*c*(9*C* d-4*D*c)/d^2))*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b^2-2/63*(9*C*d-11*D*c)*(d*x +c)^(5/2)*(-b*x^2+a)^(1/2)/b/d^2-2/9*D*(d*x+c)^(7/2)*(-b*x^2+a)^(1/2)/b/d^ 2-2/315*a^(1/2)*(147*a^2*d^4*D+3*a*b*d^2*(63*B*d^2+82*C*c*d+11*D*c^2)-b^2* c*(-420*A*d^3-63*B*c*d^2+18*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^(5/2)/d^3/((d*x+c)/(c+a^(1/2)*d/b^(1/2)) )^(1/2)/(-b*x^2+a)^(1/2)+2/315*a^(1/2)*(-a*d^2+b*c^2)*(3*a*d^2*(25*C*d+13* D*c)-b*(-105*A*d^3-63*B*c*d^2+18*C*c^2*d-8*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^(5/2)/d^3/( d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 31.10 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.27 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (-147 a^2 d^4 D-3 a b d^2 \left (82 c C d+63 B d^2+11 c^2 D\right )-b^2 c \left (-18 c^2 C d+63 B c d^2+420 A d^3+8 c^3 D\right )-b (c+d x) \left (a d^2 (75 C d+88 c D+49 d D x)+b \left (-4 c^3 D+3 c^2 d (3 C+D x)+2 c d^2 (63 B+x (36 C+25 D x))+d^3 (105 A+x (63 B+5 x (9 C+7 D x)))\right )\right )+\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (147 a^2 d^4 D+3 a b d^2 \left (82 c C d+63 B d^2+11 c^2 D\right )+b^2 c \left (-18 c^2 C d+63 B c d^2+420 A d^3+8 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}-\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (315 A b^2 c d^2+147 a^2 d^3 D-3 a^{3/2} \sqrt {b} d^2 (25 C d+13 c D)+3 a b d \left (57 c C d+63 B d^2-2 c^2 D\right )-\sqrt {a} b^{3/2} \left (-18 c^2 C d+63 B c d^2+105 A d^3+8 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{315 b^3 d^2 \sqrt {c+d x}} \] Input:
Integrate[((c + d*x)^(3/2)*(A + B*x + C*x^2 + D*x^3))/Sqrt[a - b*x^2],x]
Output:
(2*Sqrt[a - b*x^2]*(-147*a^2*d^4*D - 3*a*b*d^2*(82*c*C*d + 63*B*d^2 + 11*c ^2*D) - b^2*c*(-18*c^2*C*d + 63*B*c*d^2 + 420*A*d^3 + 8*c^3*D) - b*(c + d* x)*(a*d^2*(75*C*d + 88*c*D + 49*d*D*x) + b*(-4*c^3*D + 3*c^2*d*(3*C + D*x) + 2*c*d^2*(63*B + x*(36*C + 25*D*x)) + d^3*(105*A + x*(63*B + 5*x*(9*C + 7*D*x))))) + (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(147*a^2*d^4*D + 3*a*b*d^2 *(82*c*C*d + 63*B*d^2 + 11*c^2*D) + b^2*c*(-18*c^2*C*d + 63*B*c*d^2 + 420* 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*S qrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(315*A*b^2*c*d^2 + 147*a^2*d^3*D - 3*a^(3/2 )*Sqrt[b]*d^2*(25*C*d + 13*c*D) + 3*a*b*d*(57*c*C*d + 63*B*d^2 - 2*c^2*D) - Sqrt[a]*b^(3/2)*(-18*c^2*C*d + 63*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)*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*Sqr t[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(315*b^3*d^2*Sqrt[c + d*x])
Time = 1.32 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {2185, 27, 2185, 27, 687, 27, 687, 27, 600, 509, 508, 327, 512, 511, 321}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {(c+d x)^{3/2} \left (b (9 C d-11 c D) x^2 d^2+(9 A b d+7 a c D) d^2+\left (-2 b D c^2+9 b B d^2+7 a d^2 D\right ) x d\right )}{2 \sqrt {a-b x^2}}dx}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{3/2} \left (b (9 C d-11 c D) x^2 d^2+(9 A b d+7 a c D) d^2+\left (-2 b D c^2+9 b B d^2+7 a d^2 D\right ) x d\right )}{\sqrt {a-b x^2}}dx}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {b d^3 (c+d x)^{3/2} \left (3 d (21 A b d+15 a C d-2 a c D)+\left (49 a d^2 D-b \left (-8 D c^2+18 C d c-63 B d^2\right )\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{7 b d^2}-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} d \int \frac {(c+d x)^{3/2} \left (3 d (21 A b d+15 a C d-2 a c D)+\left (49 a d^2 D-b \left (-8 D c^2+18 C d c-63 B d^2\right )\right ) x\right )}{\sqrt {a-b x^2}}dx-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{7} d \left (-\frac {2 \int -\frac {3 \sqrt {c+d x} \left (d \left (105 A c d b^2+a \left (49 a D d^2+b \left (-2 D c^2+57 C d c+63 B d^2\right )\right )\right )+b \left (3 a d^2 (25 C d+13 c D)-b \left (-8 D c^3+18 C d c^2-63 B d^2 c-105 A d^3\right )\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {3 \int \frac {\sqrt {c+d x} \left (d \left (105 A c d b^2+a \left (49 a D d^2+b \left (-2 D c^2+57 C d c+63 B d^2\right )\right )\right )+b \left (3 a d^2 (25 C d+13 c D)-b \left (-8 D c^3+18 C d c^2-63 B d^2 c-105 A d^3\right )\right ) x\right )}{\sqrt {a-b x^2}}dx}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {3 \left (-\frac {2 \int -\frac {b \left (d \left (105 A b d \left (3 b c^2+a d^2\right )+a \left (3 a (25 C d+62 c D) d^2+b c \left (2 D c^2+153 C d c+252 B d^2\right )\right )\right )+\left (147 a^2 D d^4+3 a b \left (11 D c^2+82 C d c+63 B d^2\right ) d^2-b^2 c \left (-8 D c^3+18 C d c^2-63 B d^2 c-420 A d^3\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )\right )}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {3 \left (\frac {1}{3} \int \frac {d \left (105 A b d \left (3 b c^2+a d^2\right )+a \left (3 a (25 C d+62 c D) d^2+b c \left (2 D c^2+153 C d c+252 B d^2\right )\right )\right )+\left (147 a^2 D d^4+3 a b \left (11 D c^2+82 C d c+63 B d^2\right ) d^2-b^2 c \left (-8 D c^3+18 C d c^2-63 B d^2 c-420 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )\right )}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {3 \left (\frac {1}{3} \left (\frac {\left (147 a^2 d^4 D+3 a b d^2 \left (63 B d^2+11 c^2 D+82 c C d\right )-b^2 c \left (-420 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )\right )}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {3 \left (\frac {1}{3} \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (147 a^2 d^4 D+3 a b d^2 \left (63 B d^2+11 c^2 D+82 c C d\right )-b^2 c \left (-420 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )\right )}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {3 \left (\frac {1}{3} \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (147 a^2 d^4 D+3 a b d^2 \left (63 B d^2+11 c^2 D+82 c C d\right )-b^2 c \left (-420 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )\right )}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {3 \left (\frac {1}{3} \left (-\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (147 a^2 d^4 D+3 a b d^2 \left (63 B d^2+11 c^2 D+82 c C d\right )-b^2 c \left (-420 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )\right )}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {3 \left (\frac {1}{3} \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (147 a^2 d^4 D+3 a b d^2 \left (63 B d^2+11 c^2 D+82 c C d\right )-b^2 c \left (-420 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )\right )}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (147 a^2 d^4 D+3 a b d^2 \left (63 B d^2+11 c^2 D+82 c C d\right )-b^2 c \left (-420 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )\right )}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {1}{7} d \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (147 a^2 d^4 D+3 a b d^2 \left (63 B d^2+11 c^2 D+82 c C d\right )-b^2 c \left (-420 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2 (13 c D+25 C d)-b \left (-105 A d^3-63 B c d^2-8 c^3 D+18 c^2 C d\right )\right )\right )}{5 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (49 a d^2 D-b \left (-63 B d^2-8 c^2 D+18 c C d\right )\right )}{5 b}\right )-\frac {2}{7} d \sqrt {a-b x^2} (c+d x)^{5/2} (9 C d-11 c D)}{9 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d^2}\) |
Input:
Int[((c + d*x)^(3/2)*(A + B*x + C*x^2 + D*x^3))/Sqrt[a - b*x^2],x]
Output:
(-2*D*(c + d*x)^(7/2)*Sqrt[a - b*x^2])/(9*b*d^2) + ((-2*d*(9*C*d - 11*c*D) *(c + d*x)^(5/2)*Sqrt[a - b*x^2])/7 + (d*((-2*(49*a*d^2*D - b*(18*c*C*d - 63*B*d^2 - 8*c^2*D))*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5*b) + (3*((-2*(3*a *d^2*(25*C*d + 13*c*D) - b*(18*c^2*C*d - 63*B*c*d^2 - 105*A*d^3 - 8*c^3*D) )*Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 + ((-2*Sqrt[a]*(147*a^2*d^4*D + 3*a*b*d ^2*(82*c*C*d + 63*B*d^2 + 11*c^2*D) - b^2*c*(18*c^2*C*d - 63*B*c*d^2 - 420 *A*d^3 - 8*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)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqr t[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(3*a*d^2*(25*C*d + 13*c*D) - b*(18*c^2*C*d - 63*B*c*d^2 - 105*A*d^3 - 8*c^3*D))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + S qrt[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]*S qrt[a - b*x^2]))/3))/(5*b)))/7)/(9*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]))
Leaf count of result is larger than twice the leaf count of optimal. \(1149\) vs. \(2(516)=1032\).
Time = 5.14 (sec) , antiderivative size = 1150, normalized size of antiderivative = 1.94
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1150\) |
| default | \(\text {Expression too large to display}\) | \(4971\) |
Input:
int((d*x+c)^(3/2)*(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)*(-2/9*D/b*d*x^ 3*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/7*(C*d^2+10/9*D*c*d)/b/d*x^2*(-b*d* x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(B*d^2+2*C*c*d+D*c^2+7/9*D/b*d^2*a-6/7*(C *d^2+10/9*D*c*d)/d*c)/b/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(A*d^2+ 2*B*c*d+C*c^2+2/3*D/b*d*a*c+5/7*(C*d^2+10/9*D*c*d)/b*a-4/5*(B*d^2+2*C*c*d+ D*c^2+7/9*D/b*d^2*a-6/7*(C*d^2+10/9*D*c*d)/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+2/5*(B*d^2+2*C*c*d+D*c^2+7/9*D/b*d^2*a-6/7*(C*d ^2+10/9*D*c*d)/d*c)/b/d*a*c+1/3*(A*d^2+2*B*c*d+C*c^2+2/3*D/b*d*a*c+5/7*(C* d^2+10/9*D*c*d)/b*a-4/5*(B*d^2+2*C*c*d+D*c^2+7/9*D/b*d^2*a-6/7*(C*d^2+10/9 *D*c*d)/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)*El lipticF(((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*(2*A*c*d+B*c^2+4/7*(C*d^2+10/9*D*c*d)/b/d*a*c +3/5*(B*d^2+2*C*c*d+D*c^2+7/9*D/b*d^2*a-6/7*(C*d^2+10/9*D*c*d)/d*c)/b*a-2/ 3*(A*d^2+2*B*c*d+C*c^2+2/3*D/b*d*a*c+5/7*(C*d^2+10/9*D*c*d)/b*a-4/5*(B*d^2 +2*C*c*d+D*c^2+7/9*D/b*d^2*a-6/7*(C*d^2+10/9*D*c*d)/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))*Ellip...
Time = 0.09 (sec) , antiderivative size = 503, normalized size of antiderivative = 0.85 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\frac {2 \, {\left ({\left (8 \, D b^{2} c^{5} - 18 \, C b^{2} c^{4} d + 9 \, {\left (3 \, D a b + 7 \, B b^{2}\right )} c^{3} d^{2} - 3 \, {\left (71 \, C a b + 175 \, A b^{2}\right )} c^{2} d^{3} - 3 \, {\left (137 \, D a^{2} + 189 \, B a b\right )} c d^{4} - 45 \, {\left (5 \, C a^{2} + 7 \, A a b\right )} d^{5}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (8 \, D b^{2} c^{4} d - 18 \, C b^{2} c^{3} d^{2} + 3 \, {\left (11 \, D a b + 21 \, B b^{2}\right )} c^{2} d^{3} + 6 \, {\left (41 \, C a b + 70 \, A b^{2}\right )} c d^{4} + 21 \, {\left (7 \, D a^{2} + 9 \, B a b\right )} d^{5}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (35 \, D b^{2} d^{5} x^{3} - 4 \, D b^{2} c^{3} d^{2} + 9 \, C b^{2} c^{2} d^{3} + 2 \, {\left (44 \, D a b + 63 \, B b^{2}\right )} c d^{4} + 15 \, {\left (5 \, C a b + 7 \, A b^{2}\right )} d^{5} + 5 \, {\left (10 \, D b^{2} c d^{4} + 9 \, C b^{2} d^{5}\right )} x^{2} + {\left (3 \, D b^{2} c^{2} d^{3} + 72 \, C b^{2} c d^{4} + 7 \, {\left (7 \, D a b + 9 \, B b^{2}\right )} d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{945 \, b^{3} d^{4}} \] Input:
integrate((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(1/2),x, algorithm= "fricas")
Output:
2/945*((8*D*b^2*c^5 - 18*C*b^2*c^4*d + 9*(3*D*a*b + 7*B*b^2)*c^3*d^2 - 3*( 71*C*a*b + 175*A*b^2)*c^2*d^3 - 3*(137*D*a^2 + 189*B*a*b)*c*d^4 - 45*(5*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) + 3*(8*D*b^2 *c^4*d - 18*C*b^2*c^3*d^2 + 3*(11*D*a*b + 21*B*b^2)*c^2*d^3 + 6*(41*C*a*b + 70*A*b^2)*c*d^4 + 21*(7*D*a^2 + 9*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), 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*(35*D*b^2*d^5*x^3 - 4*D*b^2*c^3*d^2 + 9*C*b^2 *c^2*d^3 + 2*(44*D*a*b + 63*B*b^2)*c*d^4 + 15*(5*C*a*b + 7*A*b^2)*d^5 + 5* (10*D*b^2*c*d^4 + 9*C*b^2*d^5)*x^2 + (3*D*b^2*c^2*d^3 + 72*C*b^2*c*d^4 + 7 *(7*D*a*b + 9*B*b^2)*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*d^4)
\[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\sqrt {a - b x^{2}}}\, dx \] Input:
integrate((d*x+c)**(3/2)*(D*x**3+C*x**2+B*x+A)/(-b*x**2+a)**(1/2),x)
Output:
Integral((c + d*x)**(3/2)*(A + B*x + C*x**2 + D*x**3)/sqrt(a - b*x**2), x)
\[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {3}{2}}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate((d*x+c)^(3/2)*(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)*(d*x + c)^(3/2)/sqrt(-b*x^2 + a), x)
\[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {3}{2}}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate((d*x+c)^(3/2)*(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)*(d*x + c)^(3/2)/sqrt(-b*x^2 + a), x)
Timed out. \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{\sqrt {a-b\,x^2}} \,d x \] Input:
int(((c + d*x)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(1/2),x)
Output:
int(((c + d*x)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(1/2), x)
\[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(1/2),x)
Output:
( - 294*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*d**3 - 1260*sqrt(c + d*x)*sqrt (a - b*x**2)*a*b**2*c*d - 378*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b**2*d**2 - 1210*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c**2*d - 196*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c*d**2*x - 630*sqrt(c + d*x)*sqrt(a - b*x**2)*b**3*c**2 - 2 52*sqrt(c + d*x)*sqrt(a - b*x**2)*b**3*c*d*x - 300*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**3*x - 380*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**2*d*x**2 - 140*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c*d**2*x**3 - 441*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 - 1260*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*c*d**2 - 567*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 - 837*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 - 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)*b**4*c**2*d + 30*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 + 147*int((sq rt(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 + 630*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 + 189*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 + 801*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**...