Integrand size = 37, antiderivative size = 739 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=-\frac {2 \left (675 a^2 d^4 D+3 a b d^2 \left (418 c C d+275 B d^2+95 c^2 D\right )-b^2 c \left (110 c^2 C d-495 B c d^2-1848 A d^3-40 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{3465 b^3 d^2}-\frac {2 \left (a d^2 (539 C d+335 c D)-b \left (110 c^2 C d-495 B c d^2-693 A d^3-40 c^3 D\right )\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{3465 b^2 d^2}-\frac {2 \left (81 a d^2 D-b \left (22 c C d-99 B d^2-8 c^2 D\right )\right ) (c+d x)^{5/2} \sqrt {a-b x^2}}{693 b^2 d^2}-\frac {2 (11 C d-13 c D) (c+d x)^{7/2} \sqrt {a-b x^2}}{99 b d^2}-\frac {2 D (c+d x)^{9/2} \sqrt {a-b x^2}}{11 b d^2}-\frac {2 \sqrt {a} \left (3 a^2 d^4 (539 C d+1235 c D)-b^2 c^2 \left (110 c^2 C d-495 B c d^2-5313 A d^3-40 c^3 D\right )+3 a b d^2 \left (1023 c^2 C d+1595 B c d^2+693 A d^3+85 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3465 b^{5/2} d^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (675 a^2 d^4 D+3 a b d^2 \left (418 c C d+275 B d^2+95 c^2 D\right )-b^2 c \left (110 c^2 C d-495 B c d^2-1848 A d^3-40 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3465 b^{7/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/3465*(675*a^2*d^4*D+3*a*b*d^2*(275*B*d^2+418*C*c*d+95*D*c^2)-b^2*c*(-18 48*A*d^3-495*B*c*d^2+110*C*c^2*d-40*D*c^3))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2) /b^3/d^2-2/3465*(a*d^2*(539*C*d+335*D*c)-b*(-693*A*d^3-495*B*c*d^2+110*C*c ^2*d-40*D*c^3))*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b^2/d^2-2/693*(81*a*d^2*D-b *(-99*B*d^2+22*C*c*d-8*D*c^2))*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/b^2/d^2-2/99 *(11*C*d-13*D*c)*(d*x+c)^(7/2)*(-b*x^2+a)^(1/2)/b/d^2-2/11*D*(d*x+c)^(9/2) *(-b*x^2+a)^(1/2)/b/d^2-2/3465*a^(1/2)*(3*a^2*d^4*(539*C*d+1235*D*c)-b^2*c ^2*(-5313*A*d^3-495*B*c*d^2+110*C*c^2*d-40*D*c^3)+3*a*b*d^2*(693*A*d^3+159 5*B*c*d^2+1023*C*c^2*d+85*D*c^3))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellip ticE(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)+2/3465*a^(1/2)*(-a*d^2+b*c^2)*(675*a^2*d^4*D+3*a*b*d^2*(275*B *d^2+418*C*c*d+95*D*c^2)-b^2*c*(-1848*A*d^3-495*B*c*d^2+110*C*c^2*d-40*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^(7/2)/d^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 33.16 (sec) , antiderivative size = 896, normalized size of antiderivative = 1.21 \[ \int \frac {(c+d x)^{5/2} \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 (-3 a^2 d^4 (539 C d+1235 c D)-b^2 c^2 \left (-110 c^2 C d+495 B c d^2+5313 A d^3+40 c^3 D\right )-3 a b d^2 \left (1023 c^2 C d+1595 B c d^2+693 A d^3+85 c^3 D\right )-(c+d x) \left (675 a^2 d^4 D+a b d^2 \left (1025 c^2 D+c d (1793 C+1145 D x)+d^2 \left (825 B+539 C x+405 D x^2\right )\right )+b^2 \left (-20 c^4 D+5 c^3 d (11 C+3 D x)+5 c^2 d^2 (297 B+x (165 C+113 D x))+d^4 x \left (693 A+5 x \left (99 B+77 C x+63 D x^2\right )\right )+c d^3 \left (2541 A+5 x \left (297 B+209 C x+161 D x^2\right )\right )\right )\right )+\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 a^2 d^4 (539 C d+1235 c D)+b^2 c^2 \left (-110 c^2 C d+495 B c d^2+5313 A d^3+40 c^3 D\right )+3 a b d^2 \left (1023 c^2 C d+1595 B c d^2+693 A d^3+85 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 (3465 A b^{5/2} c^2 d^2-675 a^{5/2} d^4 D+3 a^2 \sqrt {b} d^3 (539 C d+1010 c D)-3 a^{3/2} b d^2 \left (418 c C d+275 B d^2+95 c^2 D\right )+3 a b^{3/2} d \left (605 c^2 C d+1320 B c d^2+693 A d^3-10 c^3 D\right )-\sqrt {a} b^2 c \left (-110 c^2 C d+495 B c d^2+1848 A d^3+40 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 )}{3465 b^3 d^2 \sqrt {c+d x}} \] Input:
Integrate[((c + d*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/Sqrt[a - b*x^2],x]
Output:
(2*Sqrt[a - b*x^2]*(-3*a^2*d^4*(539*C*d + 1235*c*D) - b^2*c^2*(-110*c^2*C* d + 495*B*c*d^2 + 5313*A*d^3 + 40*c^3*D) - 3*a*b*d^2*(1023*c^2*C*d + 1595* B*c*d^2 + 693*A*d^3 + 85*c^3*D) - (c + d*x)*(675*a^2*d^4*D + a*b*d^2*(1025 *c^2*D + c*d*(1793*C + 1145*D*x) + d^2*(825*B + 539*C*x + 405*D*x^2)) + b^ 2*(-20*c^4*D + 5*c^3*d*(11*C + 3*D*x) + 5*c^2*d^2*(297*B + x*(165*C + 113* D*x)) + d^4*x*(693*A + 5*x*(99*B + 77*C*x + 63*D*x^2)) + c*d^3*(2541*A + 5 *x*(297*B + 209*C*x + 161*D*x^2)))) + (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*( 3*a^2*d^4*(539*C*d + 1235*c*D) + b^2*c^2*(-110*c^2*C*d + 495*B*c*d^2 + 531 3*A*d^3 + 40*c^3*D) + 3*a*b*d^2*(1023*c^2*C*d + 1595*B*c*d^2 + 693*A*d^3 + 85*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 - S qrt[a]*d)])/(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2)) - (I*(Sqrt[b ]*c - Sqrt[a]*d)*(3465*A*b^(5/2)*c^2*d^2 - 675*a^(5/2)*d^4*D + 3*a^2*Sqrt[ b]*d^3*(539*C*d + 1010*c*D) - 3*a^(3/2)*b*d^2*(418*c*C*d + 275*B*d^2 + 95* c^2*D) + 3*a*b^(3/2)*d*(605*c^2*C*d + 1320*B*c*d^2 + 693*A*d^3 - 10*c^3*D) - Sqrt[a]*b^2*c*(-110*c^2*C*d + 495*B*c*d^2 + 1848*A*d^3 + 40*c^3*D))*Sqr t[(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)])/...
Time = 1.55 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.459, Rules used = {2185, 27, 2185, 27, 687, 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.
| \(\displaystyle \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {(c+d x)^{5/2} \left (b (11 C d-13 c D) x^2 d^2+(11 A b d+9 a c D) d^2+\left (-2 b D c^2+11 b B d^2+9 a d^2 D\right ) x d\right )}{2 \sqrt {a-b x^2}}dx}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{5/2} \left (b (11 C d-13 c D) x^2 d^2+(11 A b d+9 a c D) d^2+\left (-2 b D c^2+11 b B d^2+9 a d^2 D\right ) x d\right )}{\sqrt {a-b x^2}}dx}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {b d^3 (c+d x)^{5/2} \left (d (99 A b d+77 a C d-10 a c D)+\left (81 a d^2 D-b \left (-8 D c^2+22 C d c-99 B d^2\right )\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{9 b d^2}-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} d \int \frac {(c+d x)^{5/2} \left (d (99 A b d+77 a C d-10 a c D)+\left (81 a d^2 D-b \left (-8 D c^2+22 C d c-99 B d^2\right )\right ) x\right )}{\sqrt {a-b x^2}}dx-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{9} d \left (-\frac {2 \int -\frac {(c+d x)^{3/2} \left (3 d \left (231 A c d b^2+a \left (135 a D d^2+b \left (-10 D c^2+143 C d c+165 B d^2\right )\right )\right )+b \left (a d^2 (539 C d+335 c D)-b \left (-40 D c^3+110 C d c^2-495 B d^2 c-693 A d^3\right )\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\int \frac {(c+d x)^{3/2} \left (3 d \left (231 A c d b^2+a \left (135 a D d^2+b \left (-10 D c^2+143 C d c+165 B d^2\right )\right )\right )+b \left (a d^2 (539 C d+335 c D)-b \left (-40 D c^3+110 C d c^2-495 B d^2 c-693 A d^3\right )\right ) x\right )}{\sqrt {a-b x^2}}dx}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {-\frac {2 \int -\frac {3 b \sqrt {c+d x} \left (d \left (231 A b d \left (5 b c^2+3 a d^2\right )+a \left (a (539 C d+1010 c D) d^2+5 b c \left (-2 D c^2+121 C d c+264 B d^2\right )\right )\right )+\left (675 a^2 D d^4+3 a b \left (95 D c^2+418 C d c+275 B d^2\right ) d^2-b^2 c \left (-40 D c^3+110 C d c^2-495 B d^2 c-1848 A d^3\right )\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{5 b}-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} \left (a d^2 (335 c D+539 C d)-b \left (-693 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {3}{5} \int \frac {\sqrt {c+d x} \left (d \left (231 A b d \left (5 b c^2+3 a d^2\right )+a \left (a (539 C d+1010 c D) d^2+5 b c \left (-2 D c^2+121 C d c+264 B d^2\right )\right )\right )+\left (675 a^2 D d^4+3 a b \left (95 D c^2+418 C d c+275 B d^2\right ) d^2-b^2 c \left (-40 D c^3+110 C d c^2-495 B d^2 c-1848 A d^3\right )\right ) x\right )}{\sqrt {a-b x^2}}dx-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} \left (a d^2 (335 c D+539 C d)-b \left (-693 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {3}{5} \left (-\frac {2 \int -\frac {d \left (231 A c d \left (15 b c^2+17 a d^2\right ) b^2+a \left (675 a^2 D d^4+3 a b \left (1105 D c^2+957 C d c+275 B d^2\right ) d^2+5 b^2 c^2 \left (2 D c^2+341 C d c+891 B d^2\right )\right )\right )+b \left (3 a^2 (539 C d+1235 c D) d^4+3 a b \left (85 D c^3+1023 C d c^2+1595 B d^2 c+693 A d^3\right ) d^2-b^2 c^2 \left (-40 D c^3+110 C d c^2-495 B d^2 c-5313 A d^3\right )\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} \left (a d^2 (335 c D+539 C d)-b \left (-693 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {3}{5} \left (\frac {\int \frac {d \left (231 A c d \left (15 b c^2+17 a d^2\right ) b^2+a \left (675 a^2 D d^4+3 a b \left (1105 D c^2+957 C d c+275 B d^2\right ) d^2+5 b^2 c^2 \left (2 D c^2+341 C d c+891 B d^2\right )\right )\right )+b \left (3 a^2 (539 C d+1235 c D) d^4+3 a b \left (85 D c^3+1023 C d c^2+1595 B d^2 c+693 A d^3\right ) d^2-b^2 c^2 \left (-40 D c^3+110 C d c^2-495 B d^2 c-5313 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} \left (a d^2 (335 c D+539 C d)-b \left (-693 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {3}{5} \left (\frac {\frac {b \left (3 a^2 d^4 (1235 c D+539 C d)+3 a b d^2 \left (693 A d^3+1595 B c d^2+85 c^3 D+1023 c^2 C d\right )-b^2 c^2 \left (-5313 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} \left (a d^2 (335 c D+539 C d)-b \left (-693 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {3}{5} \left (\frac {\frac {b \sqrt {1-\frac {b x^2}{a}} \left (3 a^2 d^4 (1235 c D+539 C d)+3 a b d^2 \left (693 A d^3+1595 B c d^2+85 c^3 D+1023 c^2 C d\right )-b^2 c^2 \left (-5313 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} \left (a d^2 (335 c D+539 C d)-b \left (-693 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {3}{5} \left (\frac {-\frac {\left (b c^2-a d^2\right ) \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (1235 c D+539 C d)+3 a b d^2 \left (693 A d^3+1595 B c d^2+85 c^3 D+1023 c^2 C d\right )-b^2 c^2 \left (-5313 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} \left (a d^2 (335 c D+539 C d)-b \left (-693 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {3}{5} \left (\frac {-\frac {\left (b c^2-a d^2\right ) \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a^2 d^4 (1235 c D+539 C d)+3 a b d^2 \left (693 A d^3+1595 B c d^2+85 c^3 D+1023 c^2 C d\right )-b^2 c^2 \left (-5313 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} \left (a d^2 (335 c D+539 C d)-b \left (-693 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {3}{5} \left (\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a^2 d^4 (1235 c D+539 C d)+3 a b d^2 \left (693 A d^3+1595 B c d^2+85 c^3 D+1023 c^2 C d\right )-b^2 c^2 \left (-5313 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} \left (a d^2 (335 c D+539 C d)-b \left (-693 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {3}{5} \left (\frac {\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (675 a^2 D d^4+3 a b \left (95 D c^2+418 C d c+275 B d^2\right ) d^2-b^2 c \left (-40 D c^3+110 C d c^2-495 B d^2 c-1848 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^2 (539 C d+1235 c D) d^4+3 a b \left (85 D c^3+1023 C d c^2+1595 B d^2 c+693 A d^3\right ) d^2-b^2 c^2 \left (-40 D c^3+110 C d c^2-495 B d^2 c-5313 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}}{3 b}-\frac {2 \left (675 a^2 D d^4+3 a b \left (95 D c^2+418 C d c+275 B d^2\right ) d^2-b^2 c \left (-40 D c^3+110 C d c^2-495 B d^2 c-1848 A d^3\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{3 b}\right )-\frac {2}{5} \left (a d^2 (539 C d+335 c D)-b \left (-40 D c^3+110 C d c^2-495 B d^2 c-693 A d^3\right )\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{7 b}-\frac {2 \left (81 a d^2 D-b \left (-8 D c^2+22 C d c-99 B d^2\right )\right ) (c+d x)^{5/2} \sqrt {a-b x^2}}{7 b}\right )-\frac {2}{9} d (11 C d-13 c D) (c+d x)^{7/2} \sqrt {a-b x^2}}{11 b d^3}-\frac {2 D (c+d x)^{9/2} \sqrt {a-b x^2}}{11 b d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {3}{5} \left (\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a^2 d^4 (1235 c D+539 C d)+3 a b d^2 \left (693 A d^3+1595 B c d^2+85 c^3 D+1023 c^2 C d\right )-b^2 c^2 \left (-5313 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (675 a^2 d^4 D+3 a b d^2 \left (275 B d^2+95 c^2 D+418 c C d\right )-b^2 c \left (-1848 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} \left (a d^2 (335 c D+539 C d)-b \left (-693 A d^3-495 B c d^2-40 c^3 D+110 c^2 C d\right )\right )}{7 b}-\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2 D-b \left (-99 B d^2-8 c^2 D+22 c C d\right )\right )}{7 b}\right )-\frac {2}{9} d \sqrt {a-b x^2} (c+d x)^{7/2} (11 C d-13 c D)}{11 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
Input:
Int[((c + d*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/Sqrt[a - b*x^2],x]
Output:
(-2*D*(c + d*x)^(9/2)*Sqrt[a - b*x^2])/(11*b*d^2) + ((-2*d*(11*C*d - 13*c* D)*(c + d*x)^(7/2)*Sqrt[a - b*x^2])/9 + (d*((-2*(81*a*d^2*D - b*(22*c*C*d - 99*B*d^2 - 8*c^2*D))*(c + d*x)^(5/2)*Sqrt[a - b*x^2])/(7*b) + ((-2*(a*d^ 2*(539*C*d + 335*c*D) - b*(110*c^2*C*d - 495*B*c*d^2 - 693*A*d^3 - 40*c^3* D))*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/5 + (3*((-2*(675*a^2*d^4*D + 3*a*b*d^ 2*(418*c*C*d + 275*B*d^2 + 95*c^2*D) - b^2*c*(110*c^2*C*d - 495*B*c*d^2 - 1848*A*d^3 - 40*c^3*D))*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(3*b) + ((-2*Sqrt[a ]*Sqrt[b]*(3*a^2*d^4*(539*C*d + 1235*c*D) - b^2*c^2*(110*c^2*C*d - 495*B*c *d^2 - 5313*A*d^3 - 40*c^3*D) + 3*a*b*d^2*(1023*c^2*C*d + 1595*B*c*d^2 + 6 93*A*d^3 + 85*c^3*D))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[S qrt[1 - (Sqrt[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)*(675*a^2*d^4*D + 3*a*b*d^2*(418*c*C*d + 275*B*d^2 + 95*c^2*D) - b^2*c*(110*c^2*C*d - 495*B*c*d^2 - 1848*A*d^3 - 40*c^3*D))*S qrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*Ellip ticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqr t[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/(3*b)))/5)/(7*b)))/ 9)/(11*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. \(1847\) vs. \(2(655)=1310\).
Time = 6.42 (sec) , antiderivative size = 1848, normalized size of antiderivative = 2.50
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1848\) |
| default | \(\text {Expression too large to display}\) | \(6097\) |
Input:
int((d*x+c)^(5/2)*(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)*(-2/11*D*d^2/b *x^4*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/9*(C*d^3+23/11*d^2*c*D)/b/d*x^3* (-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/7*(B*d^3+3*C*c*d^2+3*D*c^2*d+9/11*D*d ^3/b*a-8/9*(C*d^3+23/11*d^2*c*D)/d*c)/b/d*x^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c) ^(1/2)-2/5*(A*d^3+3*B*c*d^2+3*C*c^2*d+D*c^3+8/11*D*d^2/b*a*c+7/9*(C*d^3+23 /11*d^2*c*D)/b*a-6/7*(B*d^3+3*C*c*d^2+3*D*c^2*d+9/11*D*d^3/b*a-8/9*(C*d^3+ 23/11*d^2*c*D)/d*c)/d*c)/b/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(3*A *c*d^2+3*B*c^2*d+C*c^3+2/3*(C*d^3+23/11*d^2*c*D)/b/d*a*c+5/7*(B*d^3+3*C*c* d^2+3*D*c^2*d+9/11*D*d^3/b*a-8/9*(C*d^3+23/11*d^2*c*D)/d*c)/b*a-4/5*(A*d^3 +3*B*c*d^2+3*C*c^2*d+D*c^3+8/11*D*d^2/b*a*c+7/9*(C*d^3+23/11*d^2*c*D)/b*a- 6/7*(B*d^3+3*C*c*d^2+3*D*c^2*d+9/11*D*d^3/b*a-8/9*(C*d^3+23/11*d^2*c*D)/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*c^3+2/5*(A*d^3+3* B*c*d^2+3*C*c^2*d+D*c^3+8/11*D*d^2/b*a*c+7/9*(C*d^3+23/11*d^2*c*D)/b*a-6/7 *(B*d^3+3*C*c*d^2+3*D*c^2*d+9/11*D*d^3/b*a-8/9*(C*d^3+23/11*d^2*c*D)/d*c)/ d*c)/b/d*a*c+1/3*(3*A*c*d^2+3*B*c^2*d+C*c^3+2/3*(C*d^3+23/11*d^2*c*D)/b/d* a*c+5/7*(B*d^3+3*C*c*d^2+3*D*c^2*d+9/11*D*d^3/b*a-8/9*(C*d^3+23/11*d^2*c*D )/d*c)/b*a-4/5*(A*d^3+3*B*c*d^2+3*C*c^2*d+D*c^3+8/11*D*d^2/b*a*c+7/9*(C*d^ 3+23/11*d^2*c*D)/b*a-6/7*(B*d^3+3*C*c*d^2+3*D*c^2*d+9/11*D*d^3/b*a-8/9*(C* d^3+23/11*d^2*c*D)/d*c)/d*c)/d*c)/b*a)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d -1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(...
Time = 0.14 (sec) , antiderivative size = 663, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(1/2),x, algorithm= "fricas")
Output:
2/10395*((40*D*b^3*c^6 - 110*C*b^3*c^5*d + 45*(5*D*a*b^2 + 11*B*b^3)*c^4*d ^2 - 66*(31*C*a*b^2 + 77*A*b^3)*c^3*d^3 - 780*(8*D*a^2*b + 11*B*a*b^2)*c^2 *d^4 - 66*(106*C*a^2*b + 147*A*a*b^2)*c*d^5 - 225*(9*D*a^3 + 11*B*a^2*b)*d ^6)*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*(40*D*b^3*c^5*d - 110*C* b^3*c^4*d^2 + 15*(17*D*a*b^2 + 33*B*b^3)*c^3*d^3 + 33*(93*C*a*b^2 + 161*A* b^3)*c^2*d^4 + 15*(247*D*a^2*b + 319*B*a*b^2)*c*d^5 + 231*(7*C*a^2*b + 9*A *a*b^2)*d^6)*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*(315*D *b^3*d^6*x^4 - 20*D*b^3*c^4*d^2 + 55*C*b^3*c^3*d^3 + 5*(205*D*a*b^2 + 297* B*b^3)*c^2*d^4 + 11*(163*C*a*b^2 + 231*A*b^3)*c*d^5 + 75*(9*D*a^2*b + 11*B *a*b^2)*d^6 + 35*(23*D*b^3*c*d^5 + 11*C*b^3*d^6)*x^3 + 5*(113*D*b^3*c^2*d^ 4 + 209*C*b^3*c*d^5 + 9*(9*D*a*b^2 + 11*B*b^3)*d^6)*x^2 + (15*D*b^3*c^3*d^ 3 + 825*C*b^3*c^2*d^4 + 5*(229*D*a*b^2 + 297*B*b^3)*c*d^5 + 77*(7*C*a*b^2 + 9*A*b^3)*d^6)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^4*d^4)
\[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\sqrt {a - b x^{2}}}\, dx \] Input:
integrate((d*x+c)**(5/2)*(D*x**3+C*x**2+B*x+A)/(-b*x**2+a)**(1/2),x)
Output:
Integral((c + d*x)**(5/2)*(A + B*x + C*x**2 + D*x**3)/sqrt(a - b*x**2), x)
\[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {5}{2}}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate((d*x+c)^(5/2)*(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)^(5/2)/sqrt(-b*x^2 + a), x)
\[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {5}{2}}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate((d*x+c)^(5/2)*(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)^(5/2)/sqrt(-b*x^2 + a), x)
Timed out. \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{\sqrt {a-b\,x^2}} \,d x \] Input:
int(((c + d*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(1/2),x)
Output:
int(((c + d*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(1/2), x)
\[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a-b x^2}} \, dx=\int \frac {\left (d x +c \right )^{\frac {5}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\sqrt {-b \,x^{2}+a}}d x \] Input:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(1/2),x)
Output:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(1/2),x)