Integrand size = 37, antiderivative size = 899 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\frac {4 \left (9 a^2 d^4 (65 C d-71 c D)+3 a b d^2 \left (1274 c^2 C d-1573 B c d^2+2145 A d^3-1088 c^3 D\right )-4 b^2 c^2 \left (1040 c^2 C d-1144 B c d^2+1287 A d^3-960 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{45045 b d^6}+\frac {4 \left (231 a^2 d^4 D-a b d^2 \left (793 c C d-1001 B d^2-666 c^2 D\right )+b^2 c \left (1040 c^2 C d-1144 B c d^2+1287 A d^3-960 c^3 D\right )\right ) x \sqrt {c+d x} \sqrt {a-b x^2}}{15015 b d^5}+\frac {2 \left (3 a d^2 (39 C d-58 c D)+b \left (1040 c^2 C d-1144 B c d^2+1287 A d^3-960 c^3 D\right )\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{9009 b d^4}+\frac {2 \left (33 a d^2 D-b \left (130 c C d-143 B d^2-120 c^2 D\right )\right ) x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{1287 b d^3}-\frac {2 (13 C d-23 c D) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{143 b d^2}-\frac {2 D (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{13 b d^2}-\frac {8 \sqrt {a} \left (693 a^3 d^6 D-3 a^2 b d^4 \left (598 c C d-1001 B d^2-453 c^2 D\right )+3 a b^2 c d^2 \left (2314 c^2 C d-2717 B c d^2+3432 A d^3-2048 c^3 D\right )-4 b^3 c^3 \left (1040 c^2 C d-1144 B c d^2+1287 A d^3-960 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 )}{45045 b^{3/2} d^7 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {8 \sqrt {a} \left (b c^2-a d^2\right ) \left (9 a^2 d^4 (65 C d-71 c D)+3 a b d^2 \left (1274 c^2 C d-1573 B c d^2+2145 A d^3-1088 c^3 D\right )-4 b^2 c^2 \left (1040 c^2 C d-1144 B c d^2+1287 A d^3-960 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 )}{45045 b^{3/2} d^7 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
4/45045*(9*a^2*d^4*(65*C*d-71*D*c)+3*a*b*d^2*(2145*A*d^3-1573*B*c*d^2+1274 *C*c^2*d-1088*D*c^3)-4*b^2*c^2*(1287*A*d^3-1144*B*c*d^2+1040*C*c^2*d-960*D *c^3))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^6+4/15015*(231*a^2*d^4*D-a*b*d^2 *(-1001*B*d^2+793*C*c*d-666*D*c^2)+b^2*c*(1287*A*d^3-1144*B*c*d^2+1040*C*c ^2*d-960*D*c^3))*x*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^5+2/9009*(3*a*d^2*(3 9*C*d-58*D*c)+b*(1287*A*d^3-1144*B*c*d^2+1040*C*c^2*d-960*D*c^3))*(d*x+c)^ (1/2)*(-b*x^2+a)^(3/2)/b/d^4+2/1287*(33*a*d^2*D-b*(-143*B*d^2+130*C*c*d-12 0*D*c^2))*x*(d*x+c)^(1/2)*(-b*x^2+a)^(3/2)/b/d^3-2/143*(13*C*d-23*D*c)*(d* x+c)^(1/2)*(-b*x^2+a)^(5/2)/b/d^2-2/13*D*(d*x+c)^(3/2)*(-b*x^2+a)^(5/2)/b/ d^2-8/45045*a^(1/2)*(693*a^3*d^6*D-3*a^2*b*d^4*(-1001*B*d^2+598*C*c*d-453* D*c^2)+3*a*b^2*c*d^2*(3432*A*d^3-2717*B*c*d^2+2314*C*c^2*d-2048*D*c^3)-4*b ^3*c^3*(1287*A*d^3-1144*B*c*d^2+1040*C*c^2*d-960*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^7/((d*x+c)/(c+a^(1/2) *d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)+8/45045*a^(1/2)*(-a*d^2+b*c^2)*(9*a^2* d^4*(65*C*d-71*D*c)+3*a*b*d^2*(2145*A*d^3-1573*B*c*d^2+1274*C*c^2*d-1088*D *c^3)-4*b^2*c^2*(1287*A*d^3-1144*B*c*d^2+1040*C*c^2*d-960*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^7/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 35.36 (sec) , antiderivative size = 1257, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:
Integrate[((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]
Output:
Sqrt[c + d*x]*Sqrt[a - b*x^2]*((2*(-8320*b^2*c^4*C*d + 9152*b^2*B*c^3*d^2 - 10296*A*b^2*c^2*d^3 + 12844*a*b*c^2*C*d^3 - 15158*a*b*B*c*d^4 + 19305*a* A*b*d^5 - 2340*a^2*C*d^5 + 7680*b^2*c^5*D - 11328*a*b*c^3*d^2*D + 1632*a^2 *c*d^4*D))/(45045*b*d^6) + (2*(6240*b^2*c^3*C*d - 6864*b^2*B*c^2*d^2 + 772 2*A*b^2*c*d^3 - 9308*a*b*c*C*d^3 + 11011*a*b*B*d^4 - 5760*b^2*c^4*D + 8196 *a*b*c^2*d^2*D - 924*a^2*d^4*D)*x)/(45045*b*d^5) - (2*(1040*b*c^2*C*d - 11 44*b*B*c*d^2 + 1287*A*b*d^3 - 1521*a*C*d^3 - 960*b*c^3*D + 1338*a*c*d^2*D) *x^2)/(9009*d^4) - (2*(-130*b*c*C*d + 143*b*B*d^2 + 120*b*c^2*D - 165*a*d^ 2*D)*x^3)/(1287*d^3) - (2*b*(13*C*d - 12*c*D)*x^4)/(143*d^2) - (2*b*D*x^5) /(13*d)) + (8*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^2]*(-(Sqrt[- c + (Sqrt[a]*d)/Sqrt[b]]*(693*a^3*d^6*D + 3*a^2*b*d^4*(-598*c*C*d + 1001*B *d^2 + 453*c^2*D) + 3*a*b^2*c*d^2*(2314*c^2*C*d - 2717*B*c*d^2 + 3432*A*d^ 3 - 2048*c^3*D) + 4*b^3*c^3*(-1040*c^2*C*d + 1144*B*c*d^2 - 1287*A*d^3 + 9 60*c^3*D))*(-((a*d^2)/(c + d*x)^2) + b*(-1 + c/(c + d*x))^2)) + (I*Sqrt[b] *(Sqrt[b]*c - Sqrt[a]*d)*(693*a^3*d^6*D + 3*a^2*b*d^4*(-598*c*C*d + 1001*B *d^2 + 453*c^2*D) + 3*a*b^2*c*d^2*(2314*c^2*C*d - 2717*B*c*d^2 + 3432*A*d^ 3 - 2048*c^3*D) + 4*b^3*c^3*(-1040*c^2*C*d + 1144*B*c*d^2 - 1287*A*d^3 + 9 60*c^3*D))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[ -c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt...
Time = 1.64 (sec) , antiderivative size = 860, normalized size of antiderivative = 0.96, 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, 682, 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.
| \(\displaystyle \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {\left (a-b x^2\right )^{3/2} \left (b (13 C d-23 c D) x^2 d^2+(13 A b d+3 a c D) d^2+\left (-10 b D c^2+13 b B d^2+3 a d^2 D\right ) x d\right )}{2 \sqrt {c+d x}}dx}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (a-b x^2\right )^{3/2} \left (b (13 C d-23 c D) x^2 d^2+(13 A b d+3 a c D) d^2+\left (-10 b D c^2+13 b B d^2+3 a d^2 D\right ) x d\right )}{\sqrt {c+d x}}dx}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {b d^3 \left (d (143 A b d+13 a C d+10 a c D)+\left (33 a d^2 D-b \left (-120 D c^2+130 C d c-143 B d^2\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{2 \sqrt {c+d x}}dx}{11 b d^2}-\frac {2}{11} d \left (a-b x^2\right )^{5/2} \sqrt {c+d x} (13 C d-23 c D)}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} d \int \frac {\left (d (143 A b d+13 a C d+10 a c D)+\left (33 a d^2 D-b \left (-120 D c^2+130 C d c-143 B d^2\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}}dx-\frac {2}{11} d \left (a-b x^2\right )^{5/2} \sqrt {c+d x} (13 C d-23 c D)}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )+3 a d^2 (39 C d-58 c D)+b \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{63 d^2}-\frac {4 \int -\frac {b \left (a d \left (3 a (39 C d+19 c D) d^2+b \left (-120 D c^3+130 C d c^2-143 B d^2 c+1287 A d^3\right )\right )+\left (9 b c d^2 (143 A b d+13 a C d+10 a c D)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-120 D c^2+130 C d c-143 B d^2\right )\right )\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{21 b d^2}\right )-\frac {2}{11} d \left (a-b x^2\right )^{5/2} \sqrt {c+d x} (13 C d-23 c D)}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \int \frac {\left (a d \left (3 a (39 C d+19 c D) d^2+b \left (-120 D c^3+130 C d c^2-143 B d^2 c+1287 A d^3\right )\right )+\left (9 b c d^2 (143 A b d+13 a C d+10 a c D)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-120 D c^2+130 C d c-143 B d^2\right )\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )+3 a d^2 (39 C d-58 c D)+b \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{63 d^2}\right )-\frac {2}{11} d \left (a-b x^2\right )^{5/2} \sqrt {c+d x} (13 C d-23 c D)}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (9 a^2 d^4 (65 C d-71 c D)+3 d x \left (9 b c d^2 (10 a c D+13 a C d+143 A b d)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )\right )+3 a b d^2 \left (2145 A d^3-1573 B c d^2-1088 c^3 D+1274 c^2 C d\right )-4 b^2 c^2 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{15 d^2}-\frac {4 \int -\frac {b \left (a d \left (9 a^2 (65 C d+6 c D) d^4+3 a b \left (-422 D c^3+481 C d c^2-572 B d^2 c+2145 A d^3\right ) d^2-b^2 c^2 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right )+\left (693 a^3 D d^6-3 a^2 b \left (-453 D c^2+598 C d c-1001 B d^2\right ) d^4+3 a b^2 c \left (-2048 D c^3+2314 C d c^2-2717 B d^2 c+3432 A d^3\right ) d^2-4 b^3 c^3 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )+3 a d^2 (39 C d-58 c D)+b \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{63 d^2}\right )-\frac {2}{11} d \left (a-b x^2\right )^{5/2} \sqrt {c+d x} (13 C d-23 c D)}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \left (\frac {2 \int \frac {a d \left (9 a^2 (65 C d+6 c D) d^4+3 a b \left (-422 D c^3+481 C d c^2-572 B d^2 c+2145 A d^3\right ) d^2-b^2 c^2 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right )+\left (693 a^3 D d^6-3 a^2 b \left (-453 D c^2+598 C d c-1001 B d^2\right ) d^4+3 a b^2 c \left (-2048 D c^3+2314 C d c^2-2717 B d^2 c+3432 A d^3\right ) d^2-4 b^3 c^3 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (9 a^2 d^4 (65 C d-71 c D)+3 d x \left (9 b c d^2 (10 a c D+13 a C d+143 A b d)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )\right )+3 a b d^2 \left (2145 A d^3-1573 B c d^2-1088 c^3 D+1274 c^2 C d\right )-4 b^2 c^2 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )+3 a d^2 (39 C d-58 c D)+b \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{63 d^2}\right )-\frac {2}{11} d \left (a-b x^2\right )^{5/2} \sqrt {c+d x} (13 C d-23 c D)}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \left (\frac {2 \left (\frac {\left (693 a^3 d^6 D-3 a^2 b d^4 \left (-1001 B d^2-453 c^2 D+598 c C d\right )+3 a b^2 c d^2 \left (3432 A d^3-2717 B c d^2-2048 c^3 D+2314 c^2 C d\right )-4 b^3 c^3 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (9 a^2 d^4 (65 C d-71 c D)+3 a b d^2 \left (2145 A d^3-1573 B c d^2-1088 c^3 D+1274 c^2 C d\right )-4 b^2 c^2 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (9 a^2 d^4 (65 C d-71 c D)+3 d x \left (9 b c d^2 (10 a c D+13 a C d+143 A b d)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )\right )+3 a b d^2 \left (2145 A d^3-1573 B c d^2-1088 c^3 D+1274 c^2 C d\right )-4 b^2 c^2 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )+3 a d^2 (39 C d-58 c D)+b \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{63 d^2}\right )-\frac {2}{11} d \left (a-b x^2\right )^{5/2} \sqrt {c+d x} (13 C d-23 c D)}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \left (\frac {2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (693 a^3 d^6 D-3 a^2 b d^4 \left (-1001 B d^2-453 c^2 D+598 c C d\right )+3 a b^2 c d^2 \left (3432 A d^3-2717 B c d^2-2048 c^3 D+2314 c^2 C d\right )-4 b^3 c^3 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (9 a^2 d^4 (65 C d-71 c D)+3 a b d^2 \left (2145 A d^3-1573 B c d^2-1088 c^3 D+1274 c^2 C d\right )-4 b^2 c^2 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (9 a^2 d^4 (65 C d-71 c D)+3 d x \left (9 b c d^2 (10 a c D+13 a C d+143 A b d)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )\right )+3 a b d^2 \left (2145 A d^3-1573 B c d^2-1088 c^3 D+1274 c^2 C d\right )-4 b^2 c^2 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )+3 a d^2 (39 C d-58 c D)+b \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{63 d^2}\right )-\frac {2}{11} d \left (a-b x^2\right )^{5/2} \sqrt {c+d x} (13 C d-23 c D)}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \sqrt {c+d x} \left (3 a (39 C d-58 c D) d^2+7 \left (33 a d^2 D-b \left (-120 D c^2+130 C d c-143 B d^2\right )\right ) x d+b \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{63 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (9 a^2 (65 C d-71 c D) d^4+3 a b \left (-1088 D c^3+1274 C d c^2-1573 B d^2 c+2145 A d^3\right ) d^2+3 \left (9 b c d^2 (143 A b d+13 a C d+10 a c D)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-120 D c^2+130 C d c-143 B d^2\right )\right )\right ) x d-4 b^2 c^2 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right )}{15 d^2}+\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (9 a^2 (65 C d-71 c D) d^4+3 a b \left (-1088 D c^3+1274 C d c^2-1573 B d^2 c+2145 A d^3\right ) d^2-4 b^2 c^2 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \left (693 a^3 D d^6-3 a^2 b \left (-453 D c^2+598 C d c-1001 B d^2\right ) d^4+3 a b^2 c \left (-2048 D c^3+2314 C d c^2-2717 B d^2 c+3432 A d^3\right ) d^2-4 b^3 c^3 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{21 d^2}\right )-\frac {2}{11} d (13 C d-23 c D) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{13 b d^3}-\frac {2 D (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \left (\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (9 a^2 d^4 (65 C d-71 c D)+3 a b d^2 \left (2145 A d^3-1573 B c d^2-1088 c^3 D+1274 c^2 C d\right )-4 b^2 c^2 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (693 a^3 d^6 D-3 a^2 b d^4 \left (-1001 B d^2-453 c^2 D+598 c C d\right )+3 a b^2 c d^2 \left (3432 A d^3-2717 B c d^2-2048 c^3 D+2314 c^2 C d\right )-4 b^3 c^3 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (9 a^2 d^4 (65 C d-71 c D)+3 d x \left (9 b c d^2 (10 a c D+13 a C d+143 A b d)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )\right )+3 a b d^2 \left (2145 A d^3-1573 B c d^2-1088 c^3 D+1274 c^2 C d\right )-4 b^2 c^2 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )+3 a d^2 (39 C d-58 c D)+b \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{63 d^2}\right )-\frac {2}{11} d \left (a-b x^2\right )^{5/2} \sqrt {c+d x} (13 C d-23 c D)}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \left (\frac {2 \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (9 a^2 d^4 (65 C d-71 c D)+3 a b d^2 \left (2145 A d^3-1573 B c d^2-1088 c^3 D+1274 c^2 C d\right )-4 b^2 c^2 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (693 a^3 d^6 D-3 a^2 b d^4 \left (-1001 B d^2-453 c^2 D+598 c C d\right )+3 a b^2 c d^2 \left (3432 A d^3-2717 B c d^2-2048 c^3 D+2314 c^2 C d\right )-4 b^3 c^3 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (9 a^2 d^4 (65 C d-71 c D)+3 d x \left (9 b c d^2 (10 a c D+13 a C d+143 A b d)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )\right )+3 a b d^2 \left (2145 A d^3-1573 B c d^2-1088 c^3 D+1274 c^2 C d\right )-4 b^2 c^2 \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (33 a d^2 D-b \left (-143 B d^2-120 c^2 D+130 c C d\right )\right )+3 a d^2 (39 C d-58 c D)+b \left (1287 A d^3-1144 B c d^2-960 c^3 D+1040 c^2 C d\right )\right )}{63 d^2}\right )-\frac {2}{11} d \left (a-b x^2\right )^{5/2} \sqrt {c+d x} (13 C d-23 c D)}{13 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{3/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \sqrt {c+d x} \left (3 a (39 C d-58 c D) d^2+7 \left (33 a d^2 D-b \left (-120 D c^2+130 C d c-143 B d^2\right )\right ) x d+b \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{63 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (9 a^2 (65 C d-71 c D) d^4+3 a b \left (-1088 D c^3+1274 C d c^2-1573 B d^2 c+2145 A d^3\right ) d^2+3 \left (9 b c d^2 (143 A b d+13 a C d+10 a c D)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-120 D c^2+130 C d c-143 B d^2\right )\right )\right ) x d-4 b^2 c^2 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (9 a^2 (65 C d-71 c D) d^4+3 a b \left (-1088 D c^3+1274 C d c^2-1573 B d^2 c+2145 A d^3\right ) d^2-4 b^2 c^2 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (693 a^3 D d^6-3 a^2 b \left (-453 D c^2+598 C d c-1001 B d^2\right ) d^4+3 a b^2 c \left (-2048 D c^3+2314 C d c^2-2717 B d^2 c+3432 A d^3\right ) d^2-4 b^3 c^3 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{21 d^2}\right )-\frac {2}{11} d (13 C d-23 c D) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{13 b d^3}-\frac {2 D (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{13 b d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {1}{11} d \left (\frac {2 \sqrt {c+d x} \left (3 a (39 C d-58 c D) d^2+7 \left (33 a d^2 D-b \left (-120 D c^2+130 C d c-143 B d^2\right )\right ) x d+b \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{63 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (9 a^2 (65 C d-71 c D) d^4+3 a b \left (-1088 D c^3+1274 C d c^2-1573 B d^2 c+2145 A d^3\right ) d^2+3 \left (9 b c d^2 (143 A b d+13 a C d+10 a c D)-\left (8 b c^2-7 a d^2\right ) \left (33 a d^2 D-b \left (-120 D c^2+130 C d c-143 B d^2\right )\right )\right ) x d-4 b^2 c^2 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (9 a^2 (65 C d-71 c D) d^4+3 a b \left (-1088 D c^3+1274 C d c^2-1573 B d^2 c+2145 A d^3\right ) d^2-4 b^2 c^2 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (693 a^3 D d^6-3 a^2 b \left (-453 D c^2+598 C d c-1001 B d^2\right ) d^4+3 a b^2 c \left (-2048 D c^3+2314 C d c^2-2717 B d^2 c+3432 A d^3\right ) d^2-4 b^3 c^3 \left (-960 D c^3+1040 C d c^2-1144 B d^2 c+1287 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{21 d^2}\right )-\frac {2}{11} d (13 C d-23 c D) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{13 b d^3}-\frac {2 D (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{13 b d^2}\) |
Input:
Int[((a - b*x^2)^(3/2)*(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)^(5/2))/(13*b*d^2) + ((-2*d*(13*C*d - 23* c*D)*Sqrt[c + d*x]*(a - b*x^2)^(5/2))/11 + (d*((2*Sqrt[c + d*x]*(3*a*d^2*( 39*C*d - 58*c*D) + b*(1040*c^2*C*d - 1144*B*c*d^2 + 1287*A*d^3 - 960*c^3*D ) + 7*d*(33*a*d^2*D - b*(130*c*C*d - 143*B*d^2 - 120*c^2*D))*x)*(a - b*x^2 )^(3/2))/(63*d^2) + (2*((2*Sqrt[c + d*x]*(9*a^2*d^4*(65*C*d - 71*c*D) + 3* a*b*d^2*(1274*c^2*C*d - 1573*B*c*d^2 + 2145*A*d^3 - 1088*c^3*D) - 4*b^2*c^ 2*(1040*c^2*C*d - 1144*B*c*d^2 + 1287*A*d^3 - 960*c^3*D) + 3*d*(9*b*c*d^2* (143*A*b*d + 13*a*C*d + 10*a*c*D) - (8*b*c^2 - 7*a*d^2)*(33*a*d^2*D - b*(1 30*c*C*d - 143*B*d^2 - 120*c^2*D)))*x)*Sqrt[a - b*x^2])/(15*d^2) + (2*((-2 *Sqrt[a]*(693*a^3*d^6*D - 3*a^2*b*d^4*(598*c*C*d - 1001*B*d^2 - 453*c^2*D) + 3*a*b^2*c*d^2*(2314*c^2*C*d - 2717*B*c*d^2 + 3432*A*d^3 - 2048*c^3*D) - 4*b^3*c^3*(1040*c^2*C*d - 1144*B*c*d^2 + 1287*A*d^3 - 960*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)])/(Sqrt[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)*(9*a^2*d^4*(65*C*d - 71*c*D) + 3*a*b*d^2*(1274*c^2*C*d - 1573*B*c*d^2 + 2145*A*d^3 - 1088*c^3*D) - 4*b^2*c^2*(1040*c^2*C*d - 1144*B*c*d^2 + 1287* A*d^3 - 960*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^...
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]))
Leaf count of result is larger than twice the leaf count of optimal. \(2420\) vs. \(2(809)=1618\).
Time = 5.17 (sec) , antiderivative size = 2421, normalized size of antiderivative = 2.69
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2421\) |
| default | \(\text {Expression too large to display}\) | \(7072\) |
Input:
int((-b*x^2+a)^(3/2)*(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)*(-2/13*D*b/d*x ^5*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/11*(b^2*C-12/13*D*b^2/d*c)/b/d*x^4 *(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/9*(B*b^2-15/13*a*b*D-10/11*(b^2*C-12 /13*D*b^2/d*c)/d*c)/b/d*x^3*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/7*(b^2*A- 2*C*a*b+10/13*D*b/d*a*c+9/11*(b^2*C-12/13*D*b^2/d*c)/b*a-8/9*(B*b^2-15/13* a*b*D-10/11*(b^2*C-12/13*D*b^2/d*c)/d*c)/d*c)/b/d*x^2*(-b*d*x^3-b*c*x^2+a* d*x+a*c)^(1/2)-2/5*(-2*a*b*B+D*a^2+8/11*(b^2*C-12/13*D*b^2/d*c)/b/d*a*c+7/ 9*(B*b^2-15/13*a*b*D-10/11*(b^2*C-12/13*D*b^2/d*c)/d*c)/b*a-6/7*(b^2*A-2*C *a*b+10/13*D*b/d*a*c+9/11*(b^2*C-12/13*D*b^2/d*c)/b*a-8/9*(B*b^2-15/13*a*b *D-10/11*(b^2*C-12/13*D*b^2/d*c)/d*c)/d*c)/d*c)/b/d*x*(-b*d*x^3-b*c*x^2+a* d*x+a*c)^(1/2)-2/3*(-2*a*b*A+C*a^2+2/3*(B*b^2-15/13*a*b*D-10/11*(b^2*C-12/ 13*D*b^2/d*c)/d*c)/b/d*a*c+5/7*(b^2*A-2*C*a*b+10/13*D*b/d*a*c+9/11*(b^2*C- 12/13*D*b^2/d*c)/b*a-8/9*(B*b^2-15/13*a*b*D-10/11*(b^2*C-12/13*D*b^2/d*c)/ d*c)/d*c)/b*a-4/5*(-2*a*b*B+D*a^2+8/11*(b^2*C-12/13*D*b^2/d*c)/b/d*a*c+7/9 *(B*b^2-15/13*a*b*D-10/11*(b^2*C-12/13*D*b^2/d*c)/d*c)/b*a-6/7*(b^2*A-2*C* a*b+10/13*D*b/d*a*c+9/11*(b^2*C-12/13*D*b^2/d*c)/b*a-8/9*(B*b^2-15/13*a*b* D-10/11*(b^2*C-12/13*D*b^2/d*c)/d*c)/d*c)/d*c)/d*c)/b/d*(-b*d*x^3-b*c*x^2+ a*d*x+a*c)^(1/2)+2*(a^2*A+2/5*(-2*a*b*B+D*a^2+8/11*(b^2*C-12/13*D*b^2/d*c) /b/d*a*c+7/9*(B*b^2-15/13*a*b*D-10/11*(b^2*C-12/13*D*b^2/d*c)/d*c)/b*a-6/7 *(b^2*A-2*C*a*b+10/13*D*b/d*a*c+9/11*(b^2*C-12/13*D*b^2/d*c)/b*a-8/9*(B...
Time = 0.11 (sec) , antiderivative size = 822, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:
integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x, algorithm= "fricas")
Output:
2/135135*(4*(3840*D*b^3*c^7 - 4160*C*b^3*c^6*d - 32*(282*D*a*b^2 - 143*B*b ^3)*c^5*d^2 + 234*(43*C*a*b^2 - 22*A*b^3)*c^4*d^3 + 27*(191*D*a^2*b - 429* B*a*b^2)*c^3*d^4 - 39*(157*C*a^2*b - 363*A*a*b^2)*c^2*d^5 + 3*(177*D*a^3 + 2717*B*a^2*b)*c*d^6 - 1755*(C*a^3 + 11*A*a^2*b)*d^7)*sqrt(-b*d)*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) + 12*(3840*D*b^3*c^6*d - 4160*C*b^3*c^5*d^2 - 32*(19 2*D*a*b^2 - 143*B*b^3)*c^4*d^3 + 78*(89*C*a*b^2 - 66*A*b^3)*c^3*d^4 + 3*(4 53*D*a^2*b - 2717*B*a*b^2)*c^2*d^5 - 78*(23*C*a^2*b - 132*A*a*b^2)*c*d^6 + 231*(3*D*a^3 + 13*B*a^2*b)*d^7)*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*(3465*D*b^3*d^7*x^5 - 7680*D*b^3*c^5*d^2 + 8320*C*b^3*c^4*d^ 3 + 64*(177*D*a*b^2 - 143*B*b^3)*c^3*d^4 - 52*(247*C*a*b^2 - 198*A*b^3)*c^ 2*d^5 - 2*(816*D*a^2*b - 7579*B*a*b^2)*c*d^6 + 585*(4*C*a^2*b - 33*A*a*b^2 )*d^7 - 315*(12*D*b^3*c*d^6 - 13*C*b^3*d^7)*x^4 + 35*(120*D*b^3*c^2*d^5 - 130*C*b^3*c*d^6 - 11*(15*D*a*b^2 - 13*B*b^3)*d^7)*x^3 - 5*(960*D*b^3*c^3*d ^4 - 1040*C*b^3*c^2*d^5 - 2*(669*D*a*b^2 - 572*B*b^3)*c*d^6 + 117*(13*C*a* b^2 - 11*A*b^3)*d^7)*x^2 + (5760*D*b^3*c^4*d^3 - 6240*C*b^3*c^3*d^4 - 12*( 683*D*a*b^2 - 572*B*b^3)*c^2*d^5 + 26*(358*C*a*b^2 - 297*A*b^3)*c*d^6 + 77 *(12*D*a^2*b - 143*B*a*b^2)*d^7)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^...
\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\sqrt {c + d x}}\, dx \] Input:
integrate((-b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2),x)
Output:
Integral((a - b*x**2)**(3/2)*(A + B*x + C*x**2 + D*x**3)/sqrt(c + d*x), x)
\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c}} \,d x } \] Input:
integrate((-b*x^2+a)^(3/2)*(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)*(-b*x^2 + a)^(3/2)/sqrt(d*x + c), x)
\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c}} \,d x } \] Input:
integrate((-b*x^2+a)^(3/2)*(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)*(-b*x^2 + a)^(3/2)/sqrt(d*x + c), x)
Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{\sqrt {c+d\,x}} \,d x \] Input:
int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(1/2),x)
Output:
int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(1/2), x)
\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {c+d x}} \, dx=\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\sqrt {d x +c}}d x \] Input:
int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x)
Output:
int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x)