Integrand size = 35, antiderivative size = 767 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {a (A b c+a c C-a B d+(b B c-A b d-a C d) x)}{b^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {\left (3 a b c d^3 (B c-2 A d)-3 a^2 d^4 (2 c C-B d)-2 b^2 c^3 \left (c^2 C-B c d+A d^2\right )\right ) \sqrt {a-b x^2}}{3 b^2 d^2 \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}+\frac {\left (3 a^3 C d^6-2 b^3 c^4 \left (5 c^2 C-2 B c d-A d^2\right )+3 a^2 b d^4 \left (3 c^2 C-3 B c d+A d^2\right )+3 a b^2 c^2 d^2 \left (10 c^2 C-9 B c d+9 A d^2\right )\right ) \sqrt {a-b x^2}}{3 b^2 d^2 \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}-\frac {\sqrt {a} \left (9 a^3 C d^6-2 b^3 c^4 \left (8 c^2 C-2 B c d-A d^2\right )-3 a^2 b d^4 \left (3 c^2 C+3 B c d-A d^2\right )+3 a b^2 c^2 d^2 \left (16 c^2 C-9 B c d+9 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d^3 \left (b c^2-a d^2\right )^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\sqrt {a} \left (3 a^2 d^4 (4 c C-B d)+2 b^2 c^3 \left (8 c^2 C-2 B c d-A d^2\right )-3 a b c d^2 \left (12 c^2 C-5 B c d+2 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d^3 \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
a*(A*b*c+C*a*c-B*a*d+(-A*b*d+B*b*c-C*a*d)*x)/b^2/(-a*d^2+b*c^2)/(d*x+c)^(3 /2)/(-b*x^2+a)^(1/2)-1/3*(3*a*b*c*d^3*(-2*A*d+B*c)-3*a^2*d^4*(-B*d+2*C*c)- 2*b^2*c^3*(A*d^2-B*c*d+C*c^2))*(-b*x^2+a)^(1/2)/b^2/d^2/(-a*d^2+b*c^2)^2/( d*x+c)^(3/2)+1/3*(3*a^3*C*d^6-2*b^3*c^4*(-A*d^2-2*B*c*d+5*C*c^2)+3*a^2*b*d ^4*(A*d^2-3*B*c*d+3*C*c^2)+3*a*b^2*c^2*d^2*(9*A*d^2-9*B*c*d+10*C*c^2))*(-b *x^2+a)^(1/2)/b^2/d^2/(-a*d^2+b*c^2)^3/(d*x+c)^(1/2)-1/3*a^(1/2)*(9*a^3*C* d^6-2*b^3*c^4*(-A*d^2-2*B*c*d+8*C*c^2)-3*a^2*b*d^4*(-A*d^2+3*B*c*d+3*C*c^2 )+3*a*b^2*c^2*d^2*(9*A*d^2-9*B*c*d+16*C*c^2))*(d*x+c)^(1/2)*((-b*x^2+a)/a) ^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)* d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(3/2)/d^3/(-a*d^2+b*c^2)^3/((d*x+c)/(c+a ^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-1/3*a^(1/2)*(3*a^2*d^4*(-B*d+4*C *c)+2*b^2*c^3*(-A*d^2-2*B*c*d+8*C*c^2)-3*a*b*c*d^2*(2*A*d^2-5*B*c*d+12*C*c ^2))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF( 1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1 /2)*d))^(1/2))/b^(3/2)/d^3/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 33.06 (sec) , antiderivative size = 1167, normalized size of antiderivative = 1.52 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(x^3*(A + B*x + C*x^2))/((c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
Output:
Sqrt[c + d*x]*Sqrt[a - b*x^2]*((2*c^3*(c^2*C - B*c*d + A*d^2))/(3*d^2*(-(b *c^2) + a*d^2)^2*(c + d*x)^2) + (2*c^2*(5*b*c^4*C - 2*b*B*c^3*d - A*b*c^2* d^2 - 15*a*c^2*C*d^2 + 12*a*B*c*d^3 - 9*a*A*d^4))/(3*d^2*(-(b*c^2) + a*d^2 )^3*(c + d*x)) + (-(a*A*b^2*c^3) - a^2*b*c^3*C + 3*a^2*b*B*c^2*d - 3*a^2*A *b*c*d^2 - 3*a^3*c*C*d^2 + a^3*B*d^3 - a*b^2*B*c^3*x + 3*a*A*b^2*c^2*d*x + 3*a^2*b*c^2*C*d*x - 3*a^2*b*B*c*d^2*x + a^2*A*b*d^3*x + a^3*C*d^3*x)/(b*( b*c^2 - a*d^2)^3*(-a + b*x^2))) - (Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d* x))^2)/d^2]*(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-9*a^3*C*d^6 + 2*b^3*c^4*(8*c ^2*C - 2*B*c*d - A*d^2) + 3*a^2*b*d^4*(3*c^2*C + 3*B*c*d - A*d^2) - 3*a*b^ 2*c^2*d^2*(16*c^2*C - 9*B*c*d + 9*A*d^2))*(-((a*d^2)/(c + d*x)^2) + b*(-1 + c/(c + d*x))^2) - (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(-9*a^3*C*d^6 + 2*b ^3*c^4*(8*c^2*C - 2*B*c*d - A*d^2) + 3*a^2*b*d^4*(3*c^2*C + 3*B*c*d - A*d^ 2) - 3*a*b^2*c^2*d^2*(16*c^2*C - 9*B*c*d + 9*A*d^2))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (Sqrt[a]*d)/(Sqr t[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[b]*c - Sqrt[a]*d)])/Sqrt[c + d*x] + (I*Sqrt[a]*Sqrt[b]*d*(Sqrt[b]*c - Sqrt[a]*d)*(-9*a^(5/2)*C*d^5 + 3*a^2*S qrt[b]*d^4*(-4*c*C + B*d) - 3*a^(3/2)*b*d^3*(c^2*C - 4*B*c*d + A*d^2) + 2* b^(5/2)*c^3*(-8*c^2*C + 2*B*c*d + A*d^2) + 3*a*b^(3/2)*c*d^2*(12*c^2*C - 5 *B*c*d + 2*A*d^2) - 3*Sqrt[a]*b^2*c^2*d*(4*c^2*C - 4*B*c*d + 7*A*d^2))*...
Time = 3.70 (sec) , antiderivative size = 810, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2180, 27, 2182, 27, 2182, 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.
| \(\displaystyle \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a-b x^2\right )^{3/2} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2180 |
| \(\displaystyle \frac {\int -\frac {2 a C \left (c^2-\frac {a d^2}{b}\right ) x^3+2 a B \left (c^2-\frac {a d^2}{b}\right ) x^2+\frac {a \left (A b \left (2 b c^2+a d^2\right )+a \left (a C d^2+b c (2 c C-3 B d)\right )\right ) x}{b^2}+\frac {a^2 (b c (2 B c-5 A d)-a d (5 c C-3 B d))}{b^2}}{2 (c+d x)^{5/2} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}+\frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\int \frac {2 a C \left (c^2-\frac {a d^2}{b}\right ) x^3+2 a B \left (c^2-\frac {a d^2}{b}\right ) x^2+\frac {a \left (A b \left (2 b c^2+a d^2\right )+a \left (a C d^2+b c (2 c C-3 B d)\right )\right ) x}{b^2}+\frac {a^2 (b c (2 B c-5 A d)-a d (5 c C-3 B d))}{b^2}}{(c+d x)^{5/2} \sqrt {a-b x^2}}dx}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 \int -\frac {\frac {3 \left (a^2 C d^4+a b \left (5 C c^2-4 B d c+A d^2\right ) d^2+b^2 c^2 \left (2 C c^2-4 B d c+7 A d^2\right )\right ) a^2}{b^2 d}-\frac {6 C \left (b c^2-a d^2\right )^2 x^2 a}{b d}-\frac {\left (3 a^2 B d^5+3 a b c \left (4 C c^2-5 B d c+2 A d^2\right ) d^2-2 b^2 c^3 \left (2 C c^2-2 B d c-A d^2\right )\right ) x a}{b d^2}}{2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (-3 a^2 d^4 (2 c C-B d)+3 a b c d^3 (B c-2 A d)-2 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (-3 a^2 d^4 (2 c C-B d)+3 a b c d^3 (B c-2 A d)-2 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\int \frac {\frac {3 \left (a^2 C d^4+a b \left (5 C c^2-4 B d c+A d^2\right ) d^2+b^2 c^2 \left (2 C c^2-4 B d c+7 A d^2\right )\right ) a^2}{b^2 d}-\frac {6 C \left (b c^2-a d^2\right )^2 x^2 a}{b d}-\frac {\left (3 a^2 B d^5+3 a b c \left (4 C c^2-5 B d c+2 A d^2\right ) d^2-2 b^2 c^3 \left (2 C c^2-2 B d c-A d^2\right )\right ) x a}{b d^2}}{(c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (-3 a^2 d^4 (2 c C-B d)+3 a b c d^3 (B c-2 A d)-2 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 \int -\frac {a \left (a d \left (3 a^2 (c C-B d) d^4-3 a b c \left (13 C c^2-9 B d c+3 A d^2\right ) d^2+b^2 c^3 \left (4 C c^2+8 B d c-23 A d^2\right )\right )-\left (9 a^3 C d^6-3 a^2 b \left (3 C c^2+3 B d c-A d^2\right ) d^4+3 a b^2 c^2 \left (16 C c^2-9 B d c+9 A d^2\right ) d^2-2 b^3 c^4 \left (8 C c^2-2 B d c-A d^2\right )\right ) x\right )}{2 b d^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 a \sqrt {a-b x^2} \left (3 a^3 C d^6+3 a^2 b d^4 \left (A d^2-3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+10 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+5 c^2 C\right )\right )}{b^2 d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (-3 a^2 d^4 (2 c C-B d)+3 a b c d^3 (B c-2 A d)-2 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (3 a^3 C d^6+3 a^2 b d^4 \left (A d^2-3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+10 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+5 c^2 C\right )\right )}{b^2 d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {a \int \frac {a d \left (3 a^2 (c C-B d) d^4-3 a b c \left (13 C c^2-9 B d c+3 A d^2\right ) d^2+b^2 c^3 \left (4 C c^2+8 B d c-23 A d^2\right )\right )-\left (9 a^3 C d^6-3 a^2 b \left (3 C c^2+3 B d c-A d^2\right ) d^4+3 a b^2 c^2 \left (16 C c^2-9 B d c+9 A d^2\right ) d^2-2 b^3 c^4 \left (8 C c^2-2 B d c-A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (-3 a^2 d^4 (2 c C-B d)+3 a b c d^3 (B c-2 A d)-2 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (3 a^3 C d^6+3 a^2 b d^4 \left (A d^2-3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+10 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+5 c^2 C\right )\right )}{b^2 d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {a \left (-\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (4 c C-B d)-3 a b c d^2 \left (2 A d^2-5 B c d+12 c^2 C\right )+2 b^2 c^3 \left (-A d^2-2 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\left (9 a^3 C d^6-3 a^2 b d^4 \left (-A d^2+3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+16 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+8 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{b d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (-3 a^2 d^4 (2 c C-B d)+3 a b c d^3 (B c-2 A d)-2 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (3 a^3 C d^6+3 a^2 b d^4 \left (A d^2-3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+10 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+5 c^2 C\right )\right )}{b^2 d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {a \left (-\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (4 c C-B d)-3 a b c d^2 \left (2 A d^2-5 B c d+12 c^2 C\right )+2 b^2 c^3 \left (-A d^2-2 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (9 a^3 C d^6-3 a^2 b d^4 \left (-A d^2+3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+16 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+8 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{b d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (-3 a^2 d^4 (2 c C-B d)+3 a b c d^3 (B c-2 A d)-2 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (3 a^3 C d^6+3 a^2 b d^4 \left (A d^2-3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+10 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+5 c^2 C\right )\right )}{b^2 d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {a \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a^3 C d^6-3 a^2 b d^4 \left (-A d^2+3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+16 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+8 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (4 c C-B d)-3 a b c d^2 \left (2 A d^2-5 B c d+12 c^2 C\right )+2 b^2 c^3 \left (-A d^2-2 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (-3 a^2 d^4 (2 c C-B d)+3 a b c d^3 (B c-2 A d)-2 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (3 a^3 C d^6+3 a^2 b d^4 \left (A d^2-3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+10 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+5 c^2 C\right )\right )}{b^2 d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {a \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a^3 C d^6-3 a^2 b d^4 \left (-A d^2+3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+16 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+8 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (4 c C-B d)-3 a b c d^2 \left (2 A d^2-5 B c d+12 c^2 C\right )+2 b^2 c^3 \left (-A d^2-2 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {a (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (-3 a^2 d^4 (2 c C-B d)+3 a b c d^3 (B c-2 A d)-2 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 a \sqrt {a-b x^2} \left (3 a^3 C d^6+3 a^2 b d^4 \left (A d^2-3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+10 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+5 c^2 C\right )\right )}{b^2 d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {a \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a^3 C d^6-3 a^2 b d^4 \left (-A d^2+3 B c d+3 c^2 C\right )+3 a b^2 c^2 d^2 \left (9 A d^2-9 B c d+16 c^2 C\right )-2 b^3 c^4 \left (-A d^2-2 B c d+8 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 d^4 (4 c C-B d)-3 a b c d^2 \left (2 A d^2-5 B c d+12 c^2 C\right )+2 b^2 c^3 \left (-A d^2-2 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{b d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {a (A b c+a C c-a B d+(b B c-A b d-a C d) x)}{b^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {\frac {2 a \left (-3 a^2 (2 c C-B d) d^4+3 a b c (B c-2 A d) d^3-2 b^2 c^3 \left (C c^2-B d c+A d^2\right )\right ) \sqrt {a-b x^2}}{3 b^2 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {\frac {2 a \left (3 a^3 C d^6+3 a^2 b \left (3 C c^2-3 B d c+A d^2\right ) d^4+3 a b^2 c^2 \left (10 C c^2-9 B d c+9 A d^2\right ) d^2-2 b^3 c^4 \left (5 C c^2-2 B d c-A d^2\right )\right ) \sqrt {a-b x^2}}{b^2 d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {a \left (\frac {2 \sqrt {a} \left (9 a^3 C d^6-3 a^2 b \left (3 C c^2+3 B d c-A d^2\right ) d^4+3 a b^2 c^2 \left (16 C c^2-9 B d c+9 A d^2\right ) d^2-2 b^3 c^4 \left (8 C c^2-2 B d c-A d^2\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (3 a^2 (4 c C-B d) d^4-3 a b c \left (12 C c^2-5 B d c+2 A d^2\right ) d^2+2 b^2 c^3 \left (8 C c^2-2 B d c-A d^2\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{b d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {a (A b c+a C c-a B d+(b B c-A b d-a C d) x)}{b^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {\frac {2 a \left (-3 a^2 (2 c C-B d) d^4+3 a b c (B c-2 A d) d^3-2 b^2 c^3 \left (C c^2-B d c+A d^2\right )\right ) \sqrt {a-b x^2}}{3 b^2 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {\frac {2 a \left (3 a^3 C d^6+3 a^2 b \left (3 C c^2-3 B d c+A d^2\right ) d^4+3 a b^2 c^2 \left (10 C c^2-9 B d c+9 A d^2\right ) d^2-2 b^3 c^4 \left (5 C c^2-2 B d c-A d^2\right )\right ) \sqrt {a-b x^2}}{b^2 d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {a \left (\frac {2 \sqrt {a} \left (9 a^3 C d^6-3 a^2 b \left (3 C c^2+3 B d c-A d^2\right ) d^4+3 a b^2 c^2 \left (16 C c^2-9 B d c+9 A d^2\right ) d^2-2 b^3 c^4 \left (8 C c^2-2 B d c-A d^2\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (3 a^2 (4 c C-B d) d^4-3 a b c \left (12 C c^2-5 B d c+2 A d^2\right ) d^2+2 b^2 c^3 \left (8 C c^2-2 B d c-A d^2\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{b d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\) |
Input:
Int[(x^3*(A + B*x + C*x^2))/((c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
Output:
(a*(A*b*c + a*c*C - a*B*d + (b*B*c - A*b*d - a*C*d)*x))/(b^2*(b*c^2 - a*d^ 2)*(c + d*x)^(3/2)*Sqrt[a - b*x^2]) - ((2*a*(3*a*b*c*d^3*(B*c - 2*A*d) - 3 *a^2*d^4*(2*c*C - B*d) - 2*b^2*c^3*(c^2*C - B*c*d + A*d^2))*Sqrt[a - b*x^2 ])/(3*b^2*d^2*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) - ((2*a*(3*a^3*C*d^6 - 2*b^ 3*c^4*(5*c^2*C - 2*B*c*d - A*d^2) + 3*a^2*b*d^4*(3*c^2*C - 3*B*c*d + A*d^2 ) + 3*a*b^2*c^2*d^2*(10*c^2*C - 9*B*c*d + 9*A*d^2))*Sqrt[a - b*x^2])/(b^2* d^2*(b*c^2 - a*d^2)*Sqrt[c + d*x]) - (a*((2*Sqrt[a]*(9*a^3*C*d^6 - 2*b^3*c ^4*(8*c^2*C - 2*B*c*d - A*d^2) - 3*a^2*b*d^4*(3*c^2*C + 3*B*c*d - A*d^2) + 3*a*b^2*c^2*d^2*(16*c^2*C - 9*B*c*d + 9*A*d^2))*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)*(3*a^2*d^4*(4*c *C - B*d) + 2*b^2*c^3*(8*c^2*C - 2*B*c*d - A*d^2) - 3*a*b*c*d^2*(12*c^2*C - 5*B*c*d + 2*A*d^2))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sq rt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2 ])))/(b*d^2*(b*c^2 - a*d^2)))/(3*(b*c^2 - a*d^2)))/(2*a*(b*c^2 - a*d^2))
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[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 , x], x, 1]}, Simp[(-(d + e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((a*(e*R - d*S) + (b*d*R + a*e*S)*x)/(2*a*(p + 1)*(b*d^2 + a*e^2))), x] + Simp[1/(2*a*(p + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a* (p + 1)*(b*d^2 + a*e^2)*Qx + b*d^2*R*(2*p + 3) - a*e*(d*S*m - e*R*(m + 2*p + 3)) + e*(b*d*R + a*e*S)*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, d, e , m}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && !(IGtQ[ m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Time = 9.10 (sec) , antiderivative size = 1376, normalized size of antiderivative = 1.79
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1376\) |
| default | \(\text {Expression too large to display}\) | \(10264\) |
Input:
int(x^3*(C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBO SE)
Output:
((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(2/3/d^4/(a*d^2- b*c^2)^2*c^3*(A*d^2-B*c*d+C*c^2)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d )^2-2/3*(-b*d*x^2+a*d)/d^3/(a*d^2-b*c^2)^3*c^2*(9*A*a*d^4+A*b*c^2*d^2-12*B *a*c*d^3+2*B*b*c^3*d+15*C*a*c^2*d^2-5*C*b*c^4)/((x+c/d)*(-b*d*x^2+a*d))^(1 /2)-2*(-b*d*x-b*c)*(1/2*a*(A*a*b*d^3+3*A*b^2*c^2*d-3*B*a*b*c*d^2-B*b^2*c^3 +C*a^2*d^3+3*C*a*b*c^2*d)/(a*d^2-b*c^2)^3/b^2*x-1/2*a*(3*A*a*b*c*d^2+A*b^2 *c^3-B*a^2*d^3-3*B*a*b*c^2*d+3*C*a^2*c*d^2+C*a*b*c^3)/(a*d^2-b*c^2)^3/b^2) /((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2*(-1/b/d^3*(B*d-2*C*c)-1/3*b/d^3*c^3*(A*d ^2-B*c*d+C*c^2)/(a*d^2-b*c^2)^2-1/3*b/d^3*c^3*(9*A*a*d^4+A*b*c^2*d^2-12*B* a*c*d^3+2*B*b*c^3*d+15*C*a*c^2*d^2-5*C*b*c^4)/(a*d^2-b*c^2)^3-a*(2*A*b*c*d -B*a*d^2-B*b*c^2+2*C*a*c*d)/(a*d^2-b*c^2)^2/b+1/2*a*d*(3*A*a*b*c*d^2+A*b^2 *c^3-B*a^2*d^3-3*B*a*b*c^2*d+3*C*a^2*c*d^2+C*a*b*c^3)/b/(a*d^2-b*c^2)^3-1/ b*c*a*(A*a*b*d^3+3*A*b^2*c^2*d-3*B*a*b*c*d^2-B*b^2*c^3+C*a^2*d^3+3*C*a*b*c ^2*d)/(a*d^2-b*c^2)^3)*(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*(-C/d^2/b-1/3*b*c^2/d^2*(9*A*a*d^4+A*b*c^2*d^ 2-12*B*a*c*d^3+2*B*b*c^3*d+15*C*a*c^2*d^2-5*C*b*c^4)/(a*d^2-b*c^2)^3-1/2*d *a*(A*a*b*d^3+3*A*b^2*c^2*d-3*B*a*b*c*d^2-B*b^2*c^3+C*a^2*d^3+3*C*a*b*c...
Leaf count of result is larger than twice the leaf count of optimal. 2290 vs. \(2 (701) = 1402\).
Time = 0.24 (sec) , antiderivative size = 2290, normalized size of antiderivative = 2.99 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(x^3*(C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="f ricas")
Output:
-1/9*((16*C*a*b^3*c^9 - 4*B*a*b^3*c^8*d + 3*B*a^2*b^2*c^6*d^3 - 72*B*a^3*b *c^4*d^5 + 9*B*a^4*c^2*d^7 - 2*(30*C*a^2*b^2 + A*a*b^3)*c^7*d^2 + 42*(3*C* a^3*b + A*a^2*b^2)*c^5*d^4 - 6*(3*C*a^4 - 4*A*a^3*b)*c^3*d^6 - (16*C*b^4*c ^7*d^2 - 4*B*b^4*c^6*d^3 + 3*B*a*b^3*c^4*d^5 - 72*B*a^2*b^2*c^2*d^7 + 9*B* a^3*b*d^9 - 2*(30*C*a*b^3 + A*b^4)*c^5*d^4 + 42*(3*C*a^2*b^2 + A*a*b^3)*c^ 3*d^6 - 6*(3*C*a^3*b - 4*A*a^2*b^2)*c*d^8)*x^4 - 2*(16*C*b^4*c^8*d - 4*B*b ^4*c^7*d^2 + 3*B*a*b^3*c^5*d^4 - 72*B*a^2*b^2*c^3*d^6 + 9*B*a^3*b*c*d^8 - 2*(30*C*a*b^3 + A*b^4)*c^6*d^3 + 42*(3*C*a^2*b^2 + A*a*b^3)*c^4*d^5 - 6*(3 *C*a^3*b - 4*A*a^2*b^2)*c^2*d^7)*x^3 - (16*C*b^4*c^9 - 4*B*b^4*c^8*d + 7*B *a*b^3*c^6*d^3 - 75*B*a^2*b^2*c^4*d^5 + 81*B*a^3*b*c^2*d^7 - 9*B*a^4*d^9 - 2*(38*C*a*b^3 + A*b^4)*c^7*d^2 + 2*(93*C*a^2*b^2 + 22*A*a*b^3)*c^5*d^4 - 18*(8*C*a^3*b + A*a^2*b^2)*c^3*d^6 + 6*(3*C*a^4 - 4*A*a^3*b)*c*d^8)*x^2 + 2*(16*C*a*b^3*c^8*d - 4*B*a*b^3*c^7*d^2 + 3*B*a^2*b^2*c^5*d^4 - 72*B*a^3*b *c^3*d^6 + 9*B*a^4*c*d^8 - 2*(30*C*a^2*b^2 + A*a*b^3)*c^6*d^3 + 42*(3*C*a^ 3*b + A*a^2*b^2)*c^4*d^5 - 6*(3*C*a^4 - 4*A*a^3*b)*c^2*d^7)*x)*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*(16*C*a*b^3*c^8*d - 4*B*a*b^3*c^7*d^2 + 27*B*a^2*b^2*c^5*d^4 + 9*B*a^3*b*c^3*d^6 - 2*(24*C*a^2*b^2 + A*a*b^3)*c^6 *d^3 + 9*(C*a^3*b - 3*A*a^2*b^2)*c^4*d^5 - 3*(3*C*a^4 + A*a^3*b)*c^2*d^7 - (16*C*b^4*c^6*d^3 - 4*B*b^4*c^5*d^4 + 27*B*a*b^3*c^3*d^6 + 9*B*a^2*b^2...
Timed out. \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x**3*(C*x**2+B*x+A)/(d*x+c)**(5/2)/(-b*x**2+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{3}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^3*(C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="m axima")
Output:
integrate((C*x^2 + B*x + A)*x^3/((-b*x^2 + a)^(3/2)*(d*x + c)^(5/2)), x)
\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{3}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^3*(C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="g iac")
Output:
integrate((C*x^2 + B*x + A)*x^3/((-b*x^2 + a)^(3/2)*(d*x + c)^(5/2)), x)
Timed out. \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^3\,\left (C\,x^2+B\,x+A\right )}{{\left (a-b\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int((x^3*(A + B*x + C*x^2))/((a - b*x^2)^(3/2)*(c + d*x)^(5/2)),x)
Output:
int((x^3*(A + B*x + C*x^2))/((a - b*x^2)^(3/2)*(c + d*x)^(5/2)), x)
\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \left (C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{\frac {5}{2}} \left (-b \,x^{2}+a \right )^{\frac {3}{2}}}d x \] Input:
int(x^3*(C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)
Output:
int(x^3*(C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)