Integrand size = 35, antiderivative size = 450 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x} (a (b B c-A b d-a C d)+b (A b c+a c C-a B d) x)}{b^2 \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}+\frac {2 C \sqrt {c+d x} \sqrt {a-b x^2}}{3 b^2 d}+\frac {\sqrt {a} \left (a d^2 (7 c C-9 B d)-b c \left (4 c^2 C-6 B c d-3 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d^2 \left (b c^2-a d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {\sqrt {a} \left (5 a C d^2+b \left (4 c^2 C-6 B c d+3 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{5/2} d^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(d*x+c)^(1/2)*(a*(-A*b*d+B*b*c-C*a*d)+b*(A*b*c-B*a*d+C*a*c)*x)/b^2/(-a*d^2 +b*c^2)/(-b*x^2+a)^(1/2)+2/3*C*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d+1/3*a^ (1/2)*(a*d^2*(-9*B*d+7*C*c)-b*c*(-3*A*d^2-6*B*c*d+4*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^2/(-a*d^2+b*c^2)/( (d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)+1/3*a^(1/2)*(5*a*C*d ^2+b*(3*A*d^2-6*B*c*d+4*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^(5/2)/d^2/(d*x+c)^(1/2)/(-b*x^ 2+a)^(1/2)
Result contains complex when optimal does not.
Time = 27.12 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.43 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (\frac {2 C}{d}+\frac {3 \left (a^2 C d-A b^2 c x+a b (-B c+A d-c C x+B d x)\right )}{\left (b c^2-a d^2\right ) \left (-a+b x^2\right )}\right )}{b^2}-\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a d^2 (-7 c C+9 B d)+b c \left (4 c^2 C-6 B c d-3 A d^2\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2 (-7 c C+9 B d)+b c \left (4 c^2 C-6 B c d-3 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (6 A b^{3/2} c d+5 a^{3/2} C d^2+3 a \sqrt {b} d (4 c C-3 B d)+\sqrt {a} b \left (4 c^2 C-6 B c d+3 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{b^2 d^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-b c^2+a d^2\right ) \left (-a+b x^2\right )}\right )}{3 \sqrt {c+d x}} \] Input:
Integrate[(x^2*(A + B*x + C*x^2))/(Sqrt[c + d*x]*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*((2*C)/d + (3*(a^2*C*d - A*b^2*c*x + a*b*(-(B *c) + A*d - c*C*x + B*d*x)))/((b*c^2 - a*d^2)*(-a + b*x^2))))/b^2 - (d^2*S qrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a*d^2*(-7*c*C + 9*B*d) + b*c*(4*c^2*C - 6*B *c*d - 3*A*d^2))*(a - b*x^2) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(a*d^2*(- 7*c*C + 9*B*d) + b*c*(4*c^2*C - 6*B*c*d - 3*A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[ b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d* x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]] , (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*d*(Sqrt[b]*c - Sqrt [a]*d)*(6*A*b^(3/2)*c*d + 5*a^(3/2)*C*d^2 + 3*a*Sqrt[b]*d*(4*c*C - 3*B*d) + Sqrt[a]*b*(4*c^2*C - 6*B*c*d + 3*A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/ (c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)* EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b ]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(b^2*d^3*Sqrt[-c + (Sqrt[a]*d)/ Sqrt[b]]*(-(b*c^2) + a*d^2)*(-a + b*x^2))))/(3*Sqrt[c + d*x])
Time = 2.13 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2180, 27, 2185, 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.
| \(\displaystyle \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2180 |
| \(\displaystyle \frac {\int -\frac {2 a C \left (c^2-\frac {a d^2}{b}\right ) x^2+\frac {a (b c (2 B c+A d)+a d (c C-3 B d)) x}{b}+\frac {a \left (A b \left (2 b c^2-a d^2\right )-a \left (a C d^2-b c (2 c C-B d)\right )\right )}{b^2}}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\int \frac {2 a C \left (c^2-\frac {a d^2}{b}\right ) x^2+\frac {a (b c (2 B c+A d)+a d (c C-3 B d)) x}{b}+\frac {a \left (A b \left (2 b c^2-a d^2\right )-a \left (a C d^2-b c (2 c C-B d)\right )\right )}{b^2}}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {-\frac {2 \int -\frac {a d \left (d \left (3 A b \left (2 b c^2-a d^2\right )-a \left (5 a C d^2-b c (8 c C-3 B d)\right )\right )+b \left (a d^2 (7 c C-9 B d)-b c \left (4 C c^2-6 B d c-3 A d^2\right )\right ) x\right )}{2 b \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {4 a C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b^2 d}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a \int \frac {d \left (3 A b \left (2 b c^2-a d^2\right )-a \left (5 a C d^2-b c (8 c C-3 B d)\right )\right )+b \left (a d^2 (7 c C-9 B d)-b c \left (4 C c^2-6 B d c-3 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b^2 d}-\frac {4 a C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b^2 d}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (3 A d^2-6 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {b \left (a d^2 (7 c C-9 B d)-b c \left (-3 A d^2-6 B c d+4 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{3 b^2 d}-\frac {4 a C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b^2 d}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (3 A d^2-6 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {b \sqrt {1-\frac {b x^2}{a}} \left (a d^2 (7 c C-9 B d)-b c \left (-3 A d^2-6 B c d+4 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{3 b^2 d}-\frac {4 a C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b^2 d}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (3 A d^2-6 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (7 c C-9 B d)-b c \left (-3 A d^2-6 B c d+4 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b^2 d}-\frac {4 a C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b^2 d}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (3 A d^2-6 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (7 c C-9 B d)-b c \left (-3 A d^2-6 B c d+4 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b^2 d}-\frac {4 a C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b^2 d}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (3 A d^2-6 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (7 c C-9 B d)-b c \left (-3 A d^2-6 B c d+4 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b^2 d}-\frac {4 a C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b^2 d}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (5 a C d^2+b \left (3 A d^2-6 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (7 c C-9 B d)-b c \left (-3 A d^2-6 B c d+4 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b^2 d}-\frac {4 a C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b^2 d}}{2 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (5 a C d^2+b \left (3 A d^2-6 B c d+4 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (7 c C-9 B d)-b c \left (-3 A d^2-6 B c d+4 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b^2 d}-\frac {4 a C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b^2 d}}{2 a \left (b c^2-a d^2\right )}\) |
Input:
Int[(x^2*(A + B*x + C*x^2))/(Sqrt[c + d*x]*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[c + d*x]*(a*(b*B*c - A*b*d - a*C*d) + b*(A*b*c + a*c*C - a*B*d)*x))/ (b^2*(b*c^2 - a*d^2)*Sqrt[a - b*x^2]) - ((-4*a*C*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(3*b^2*d) + (a*((-2*Sqrt[a]*Sqrt[b]*(a*d^2*(7*c*C - 9*B*d) - b*c*(4*c^2*C - 6*B*c*d - 3*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)])/(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)*(5*a*C*d^2 + b*(4*c^2*C - 6* B*c*d + 3*A*d^2))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2* d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2]))) /(3*b^2*d))/(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[{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. \(800\) vs. \(2(386)=772\).
Time = 8.48 (sec) , antiderivative size = 801, normalized size of antiderivative = 1.78
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (-\frac {\left (A b c -B a d +C a c \right ) x}{2 \left (a \,d^{2}-b \,c^{2}\right ) b^{2}}+\frac {a \left (A b d -B b c +a C d \right )}{2 \left (a \,d^{2}-b \,c^{2}\right ) b^{3}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 C \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b^{2} d}+\frac {2 \left (-\frac {d a \left (A b d -B b c +a C d \right )}{2 b^{2} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {c \left (A b c -B a d +C a c \right )}{b \left (a \,d^{2}-b \,c^{2}\right )}-\frac {C a}{3 b^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (-\frac {B}{b}+\frac {d \left (A b c -B a d +C a c \right )}{2 b \left (a \,d^{2}-b \,c^{2}\right )}+\frac {2 C c}{3 b d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(801\) |
| risch | \(\text {Expression too large to display}\) | \(1503\) |
| default | \(\text {Expression too large to display}\) | \(3282\) |
Input:
int(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-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*(-b*d*x-b*c) *(-1/2*(A*b*c-B*a*d+C*a*c)/(a*d^2-b*c^2)/b^2*x+1/2*a*(A*b*d-B*b*c+C*a*d)/( a*d^2-b*c^2)/b^3)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2/3*C/b^2/d*(-b*d*x^3-b*c *x^2+a*d*x+a*c)^(1/2)+2*(-1/2*d*a*(A*b*d-B*b*c+C*a*d)/b^2/(a*d^2-b*c^2)+1/ b*c*(A*b*c-B*a*d+C*a*c)/(a*d^2-b*c^2)-1/3*C/b^2*a)*(c/d-1/b*(a*b)^(1/2))*( (x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^ (1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3 -b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2), ((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(-B/b+1/2*d*(A*b* c-B*a*d+C*a*c)/b/(a*d^2-b*c^2)+2/3*C/b/d*c)*(c/d-1/b*(a*b)^(1/2))*((x+c/d) /(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2))) ^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^ 2+a*d*x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a* b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/ b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*( a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))))
Time = 0.10 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.29 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {{\left (4 \, C a b^{2} c^{4} - 6 \, B a b^{2} c^{3} d + {\left (17 \, C a^{2} b + 15 \, A a b^{2}\right )} c^{2} d^{2} - 3 \, {\left (5 \, C a^{3} + 3 \, A a^{2} b\right )} d^{4} - {\left (4 \, C b^{3} c^{4} - 6 \, B b^{3} c^{3} d + {\left (17 \, C a b^{2} + 15 \, A b^{3}\right )} c^{2} d^{2} - 3 \, {\left (5 \, C a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (4 \, C a b^{2} c^{3} d - 6 \, B a b^{2} c^{2} d^{2} + 9 \, B a^{2} b d^{4} - {\left (7 \, C a^{2} b + 3 \, A a b^{2}\right )} c d^{3} - {\left (4 \, C b^{3} c^{3} d - 6 \, B b^{3} c^{2} d^{2} + 9 \, B a b^{2} d^{4} - {\left (7 \, C a b^{2} + 3 \, A b^{3}\right )} c d^{3}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (2 \, C a b^{2} c^{2} d^{2} + 3 \, B a b^{2} c d^{3} - {\left (5 \, C a^{2} b + 3 \, A a b^{2}\right )} d^{4} - 2 \, {\left (C b^{3} c^{2} d^{2} - C a b^{2} d^{4}\right )} x^{2} - 3 \, {\left (B a b^{2} d^{4} - {\left (C a b^{2} + A b^{3}\right )} c d^{3}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{9 \, {\left (a b^{4} c^{2} d^{3} - a^{2} b^{3} d^{5} - {\left (b^{5} c^{2} d^{3} - a b^{4} d^{5}\right )} x^{2}\right )}} \] Input:
integrate(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="f ricas")
Output:
1/9*((4*C*a*b^2*c^4 - 6*B*a*b^2*c^3*d + (17*C*a^2*b + 15*A*a*b^2)*c^2*d^2 - 3*(5*C*a^3 + 3*A*a^2*b)*d^4 - (4*C*b^3*c^4 - 6*B*b^3*c^3*d + (17*C*a*b^2 + 15*A*b^3)*c^2*d^2 - 3*(5*C*a^2*b + 3*A*a*b^2)*d^4)*x^2)*sqrt(-b*d)*weie rstrassPInverse(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*(4*C*a*b^2*c^3*d - 6*B*a*b^2*c^2*d^2 + 9*B* a^2*b*d^4 - (7*C*a^2*b + 3*A*a*b^2)*c*d^3 - (4*C*b^3*c^3*d - 6*B*b^3*c^2*d ^2 + 9*B*a*b^2*d^4 - (7*C*a*b^2 + 3*A*b^3)*c*d^3)*x^2)*sqrt(-b*d)*weierstr assZeta(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*(2*C*a*b^2*c^2*d^2 + 3*B*a*b^2*c*d^3 - (5*C*a^2*b + 3*A*a*b^2)*d^4 - 2*(C*b^3*c^2*d^2 - C*a*b^2*d^4)*x^2 - 3*(B* a*b^2*d^4 - (C*a*b^2 + A*b^3)*c*d^3)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a *b^4*c^2*d^3 - a^2*b^3*d^5 - (b^5*c^2*d^3 - a*b^4*d^5)*x^2)
\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (A + B x + C x^{2}\right )}{\left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(x**2*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(x**2*(A + B*x + C*x**2)/((a - b*x**2)**(3/2)*sqrt(c + d*x)), x)
\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{2}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="m axima")
Output:
integrate((C*x^2 + B*x + A)*x^2/((-b*x^2 + a)^(3/2)*sqrt(d*x + c)), x)
\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{2}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="g iac")
Output:
integrate((C*x^2 + B*x + A)*x^2/((-b*x^2 + a)^(3/2)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (C\,x^2+B\,x+A\right )}{{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int((x^2*(A + B*x + C*x^2))/((a - b*x^2)^(3/2)*(c + d*x)^(1/2)),x)
Output:
int((x^2*(A + B*x + C*x^2))/((a - b*x^2)^(3/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (C \,x^{2}+B x +A \right )}{\sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{\frac {3}{2}}}d x \] Input:
int(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x)
Output:
int(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x)