\(\int \frac {x^2 (A+B x+C x^2)}{\sqrt {c+d x} (a-b x^2)^{5/2}} \, dx\) [234]

Optimal result

Integrand size = 35, antiderivative size = 601 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {c+d x} (a (b B c-A b d-a C d)+b (A b c+a c C-a B d) x)}{3 b^2 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}-\frac {\sqrt {c+d x} \left (a \left (7 a^2 C d^3+b^2 c^2 (6 B c-5 A d)-a b d \left (11 c^2 C+2 B c d-A d^2\right )\right )+b \left (2 A b c \left (b c^2+a d^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) x\right )}{6 a b^2 \left (b c^2-a d^2\right )^2 \sqrt {a-b x^2}}-\frac {\left (2 A b c \left (b c^2+a d^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 \sqrt {a} b^{3/2} \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {\left (A b \left (2 b c^2-a d^2\right )+a \left (5 a C d^2-b c (4 c C+B d)\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 \sqrt {a} b^{5/2} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

1/3*(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)^(3/2)-1/6*(d*x+c)^(1/2)*(a*(7*a^2*C*d^3+b^2*c^2*(-5 
*A*d+6*B*c)-a*b*d*(-A*d^2+2*B*c*d+11*C*c^2))+b*(2*A*b*c*(a*d^2+b*c^2)+a*(b 
*c^2*(-7*B*d+8*C*c)-a*d^2*(-3*B*d+4*C*c)))*x)/a/b^2/(-a*d^2+b*c^2)^2/(-b*x 
^2+a)^(1/2)-1/6*(2*A*b*c*(a*d^2+b*c^2)+a*(b*c^2*(-7*B*d+8*C*c)-a*d^2*(-3*B 
*d+4*C*c)))*(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))/a^ 
(1/2)/b^(3/2)/(-a*d^2+b*c^2)^2/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x 
^2+a)^(1/2)+1/6*(A*b*(-a*d^2+2*b*c^2)+a*(5*a*C*d^2-b*c*(B*d+4*C*c)))*((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))/a^(1/2)/b^(5/2)/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 29.66 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.37 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (-5 a^4 C d^3+2 A b^4 c^3 x^3+a b^3 c x^2 \left (8 c^2 C x+B c (6 c-7 d x)+A d (-5 c+2 d x)\right )+a^2 b^2 \left (A d \left (3 c^2-4 c d x+d^2 x^2\right )-c C x \left (6 c^2+11 c d x+4 d^2 x^2\right )+B \left (-4 c^3+5 c^2 d x-2 c d^2 x^2+3 d^3 x^3\right )\right )+a^3 b d \left (9 c^2 C+2 c C d x+d^2 (A+x (-B+7 C x))\right )\right )}{\left (a-b x^2\right )^2}-\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (2 A b c \left (b c^2+a d^2\right )+a \left (b c^2 (8 c C-7 B d)+a d^2 (-4 c C+3 B d)\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (2 A b c \left (b c^2+a d^2\right )+a \left (b c^2 (8 c C-7 B d)+a d^2 (-4 c C+3 B d)\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (2 A b^2 c^2 d+5 a^2 C d^3-3 a^{3/2} \sqrt {b} d^2 (-3 c C+B d)-3 \sqrt {a} b^{3/2} c \left (4 c^2 C-2 B c d+A d^2\right )-a b d \left (4 c^2 C+B c d+A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{6 a b^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}} \] Input:

Integrate[(x^2*(A + B*x + C*x^2))/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
 

Output:

(Sqrt[a - b*x^2]*(((c + d*x)*(-5*a^4*C*d^3 + 2*A*b^4*c^3*x^3 + a*b^3*c*x^2 
*(8*c^2*C*x + B*c*(6*c - 7*d*x) + A*d*(-5*c + 2*d*x)) + a^2*b^2*(A*d*(3*c^ 
2 - 4*c*d*x + d^2*x^2) - c*C*x*(6*c^2 + 11*c*d*x + 4*d^2*x^2) + B*(-4*c^3 
+ 5*c^2*d*x - 2*c*d^2*x^2 + 3*d^3*x^3)) + a^3*b*d*(9*c^2*C + 2*c*C*d*x + d 
^2*(A + x*(-B + 7*C*x)))))/(a - b*x^2)^2 - (d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt 
[b]]*(2*A*b*c*(b*c^2 + a*d^2) + a*(b*c^2*(8*c*C - 7*B*d) + a*d^2*(-4*c*C + 
 3*B*d)))*(a - b*x^2) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(2*A*b*c*(b*c^2 
+ a*d^2) + a*(b*c^2*(8*c*C - 7*B*d) + a*d^2*(-4*c*C + 3*B*d)))*Sqrt[(d*(Sq 
rt[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]]/Sqr 
t[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*Sqrt[a]* 
(Sqrt[b]*c - Sqrt[a]*d)*(2*A*b^2*c^2*d + 5*a^2*C*d^3 - 3*a^(3/2)*Sqrt[b]*d 
^2*(-3*c*C + B*d) - 3*Sqrt[a]*b^(3/2)*c*(4*c^2*C - 2*B*c*d + A*d^2) - a*b* 
d*(4*c^2*C + B*c*d + A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqr 
t[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*Ar 
cSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]* 
d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2) 
)))/(6*a*b^2*(b*c^2 - a*d^2)^2*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 2.47 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2180, 27, 2180, 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.

\(\displaystyle \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a-b x^2\right )^{5/2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 2180

\(\displaystyle \frac {\int -\frac {6 a C \left (c^2-\frac {a d^2}{b}\right ) x^2+\frac {3 a (b c (2 B c-A d)-a d (c C+B d)) x}{b}+\frac {a \left (A b \left (2 b c^2-a d^2\right )-a \left (a C d^2-b c (2 c C-B d)\right )\right )}{b^2}}{2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{3 a \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\int \frac {6 a C \left (c^2-\frac {a d^2}{b}\right ) x^2+\frac {3 a (b c (2 B c-A d)-a d (c C+B d)) x}{b}+\frac {a \left (A b \left (2 b c^2-a d^2\right )-a \left (a C d^2-b c (2 c C-B d)\right )\right )}{b^2}}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 2180

\(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\int -\frac {a \left (a \left (5 a^2 C d^4-a b \left (13 C c^2-2 B d c+A d^2\right ) d^2+b^2 c^2 \left (12 C c^2-6 B d c+5 A d^2\right )\right )+b d \left (2 A b c \left (b c^2+a d^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) x\right )}{2 b^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (a \left (7 a^2 C d^3-a b d \left (-A d^2+2 B c d+11 c^2 C\right )+b^2 c^2 (6 B c-5 A d)\right )+b x \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\sqrt {c+d x} \left (a \left (7 a^2 C d^3-a b d \left (-A d^2+2 B c d+11 c^2 C\right )+b^2 c^2 (6 B c-5 A d)\right )+b x \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\int \frac {a \left (5 a^2 C d^4-a b \left (13 C c^2-2 B d c+A d^2\right ) d^2+b^2 c^2 \left (12 C c^2-6 B d c+5 A d^2\right )\right )+b d \left (2 A b c \left (b c^2+a d^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\sqrt {c+d x} \left (a \left (7 a^2 C d^3-a b d \left (-A d^2+2 B c d+11 c^2 C\right )+b^2 c^2 (6 B c-5 A d)\right )+b x \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {b \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx-\left (b c^2-a d^2\right ) \left (A b \left (2 b c^2-a d^2\right )+a \left (5 a C d^2-b c (B d+4 c C)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\sqrt {c+d x} \left (a \left (7 a^2 C d^3-a b d \left (-A d^2+2 B c d+11 c^2 C\right )+b^2 c^2 (6 B c-5 A d)\right )+b x \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {b \sqrt {1-\frac {b x^2}{a}} \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\left (b c^2-a d^2\right ) \left (A b \left (2 b c^2-a d^2\right )+a \left (5 a C d^2-b c (B d+4 c C)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\sqrt {c+d x} \left (a \left (7 a^2 C d^3-a b d \left (-A d^2+2 B c d+11 c^2 C\right )+b^2 c^2 (6 B c-5 A d)\right )+b x \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {-\left (\left (b c^2-a d^2\right ) \left (A b \left (2 b c^2-a d^2\right )+a \left (5 a C d^2-b c (B d+4 c C)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\sqrt {c+d x} \left (a \left (7 a^2 C d^3-a b d \left (-A d^2+2 B c d+11 c^2 C\right )+b^2 c^2 (6 B c-5 A d)\right )+b x \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {-\left (b c^2-a d^2\right ) \left (A b \left (2 b c^2-a d^2\right )+a \left (5 a C d^2-b c (B d+4 c C)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\sqrt {c+d x} \left (a \left (7 a^2 C d^3-a b d \left (-A d^2+2 B c d+11 c^2 C\right )+b^2 c^2 (6 B c-5 A d)\right )+b x \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (A b \left (2 b c^2-a d^2\right )+a \left (5 a C d^2-b c (B d+4 c C)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\sqrt {c+d x} \left (a \left (7 a^2 C d^3-a b d \left (-A d^2+2 B c d+11 c^2 C\right )+b^2 c^2 (6 B c-5 A d)\right )+b x \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (A b \left (2 b c^2-a d^2\right )+a \left (5 a C d^2-b c (B d+4 c C)\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\sqrt {c+d x} \left (a \left (7 a^2 C d^3-a b d \left (-A d^2+2 B c d+11 c^2 C\right )+b^2 c^2 (6 B c-5 A d)\right )+b x \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (A b \left (2 b c^2-a d^2\right )+a \left (5 a C d^2-b c (B d+4 c C)\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (2 A b c \left (a d^2+b c^2\right )+a \left (b c^2 (8 c C-7 B d)-a d^2 (4 c C-3 B d)\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

Input:

Int[(x^2*(A + B*x + C*x^2))/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),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))/ 
(3*b^2*(b*c^2 - a*d^2)*(a - b*x^2)^(3/2)) - ((Sqrt[c + d*x]*(a*(7*a^2*C*d^ 
3 + b^2*c^2*(6*B*c - 5*A*d) - a*b*d*(11*c^2*C + 2*B*c*d - A*d^2)) + b*(2*A 
*b*c*(b*c^2 + a*d^2) + a*(b*c^2*(8*c*C - 7*B*d) - a*d^2*(4*c*C - 3*B*d)))* 
x))/(b^2*(b*c^2 - a*d^2)*Sqrt[a - b*x^2]) - ((-2*Sqrt[a]*Sqrt[b]*(2*A*b*c* 
(b*c^2 + a*d^2) + a*(b*c^2*(8*c*C - 7*B*d) - a*d^2*(4*c*C - 3*B*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[(Sqrt[b]*(c + d*x))/(S 
qrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(A*b* 
(2*b*c^2 - a*d^2) + a*(5*a*C*d^2 - b*c*(4*c*C + B*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]*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/(2*b^2*(b*c^2 - a*d^2)))/(6*a*(b*c^2 - 
a*d^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

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])
 

rule 327

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]
 

rule 508

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]
 

rule 509

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]
 

rule 511

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]
 

rule 512

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]
 

rule 600

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]
 

rule 2180

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]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1125\) vs. \(2(535)=1070\).

Time = 6.51 (sec) , antiderivative size = 1126, normalized size of antiderivative = 1.87

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1126\)
default	\(\text {Expression too large to display}\)	\(7616\)

Input:

int(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),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)*((-1/3*(A*b*c-B* 
a*d+C*a*c)/b^3/(a*d^2-b*c^2)*x+1/3*(A*b*d-B*b*c+C*a*d)*a/b^4/(a*d^2-b*c^2) 
)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x^2-a/b)^2-2*(-b*d*x-b*c)*(-1/12/b^2 
*(2*A*a*b*c*d^2+2*A*b^2*c^3+3*B*a^2*d^3-7*B*a*b*c^2*d-4*C*a^2*c*d^2+8*C*a* 
b*c^3)/a/(a*d^2-b*c^2)^2*x-1/12*(A*a*b*d^3-5*A*b^2*c^2*d-2*B*a*b*c*d^2+6*B 
*b^2*c^3+7*C*a^2*d^3-11*C*a*b*c^2*d)/(a*d^2-b*c^2)^2/b^3)/((x^2-a/b)*(-b*d 
*x-b*c))^(1/2)+2*(C/b^2-1/6/b^2/(a*d^2-b*c^2)*(A*a*b*d^2-2*A*b^2*c^2+B*a*b 
*c*d+7*C*a^2*d^2-8*C*a*b*c^2)/a+1/12/b^2*d*(A*a*b*d^3-5*A*b^2*c^2*d-2*B*a* 
b*c*d^2+6*B*b^2*c^3+7*C*a^2*d^3-11*C*a*b*c^2*d)/(a*d^2-b*c^2)^2+1/6/b*c*(2 
*A*a*b*c*d^2+2*A*b^2*c^3+3*B*a^2*d^3-7*B*a*b*c^2*d-4*C*a^2*c*d^2+8*C*a*b*c 
^3)/a/(a*d^2-b*c^2)^2)*(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))+1/6*d*(2*A*a*b*c*d^2+2*A*b^2*c^3+3*B*a^2*d^3-7* 
B*a*b*c^2*d-4*C*a^2*c*d^2+8*C*a*b*c^3)/b/a/(a*d^2-b*c^2)^2*(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)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1093 vs. \(2 (539) = 1078\).

Time = 0.13 (sec) , antiderivative size = 1093, normalized size of antiderivative = 1.82 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="f 
ricas")
 

Output:

1/18*((11*B*a^3*b^2*c^3*d - 3*B*a^4*b*c*d^3 - 2*(14*C*a^3*b^2 - A*a^2*b^3) 
*c^4 + (35*C*a^4*b - 13*A*a^3*b^2)*c^2*d^2 - 3*(5*C*a^5 - A*a^4*b)*d^4 + ( 
11*B*a*b^4*c^3*d - 3*B*a^2*b^3*c*d^3 - 2*(14*C*a*b^4 - A*b^5)*c^4 + (35*C* 
a^2*b^3 - 13*A*a*b^4)*c^2*d^2 - 3*(5*C*a^3*b^2 - A*a^2*b^3)*d^4)*x^4 - 2*( 
11*B*a^2*b^3*c^3*d - 3*B*a^3*b^2*c*d^3 - 2*(14*C*a^2*b^3 - A*a*b^4)*c^4 + 
(35*C*a^3*b^2 - 13*A*a^2*b^3)*c^2*d^2 - 3*(5*C*a^4*b - A*a^3*b^2)*d^4)*x^2 
)*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*(7*B*a^3*b^2*c^2*d^2 - 3*B 
*a^4*b*d^4 - 2*(4*C*a^3*b^2 + A*a^2*b^3)*c^3*d + 2*(2*C*a^4*b - A*a^3*b^2) 
*c*d^3 + (7*B*a*b^4*c^2*d^2 - 3*B*a^2*b^3*d^4 - 2*(4*C*a*b^4 + A*b^5)*c^3* 
d + 2*(2*C*a^2*b^3 - A*a*b^4)*c*d^3)*x^4 - 2*(7*B*a^2*b^3*c^2*d^2 - 3*B*a^ 
3*b^2*d^4 - 2*(4*C*a^2*b^3 + A*a*b^4)*c^3*d + 2*(2*C*a^3*b^2 - A*a^2*b^3)* 
c*d^3)*x^2)*sqrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/2 
7*(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*(4*B*a^ 
2*b^3*c^3*d - 3*(3*C*a^3*b^2 + A*a^2*b^3)*c^2*d^2 + (5*C*a^4*b - A*a^3*b^2 
)*d^4 + (7*B*a*b^4*c^2*d^2 - 3*B*a^2*b^3*d^4 - 2*(4*C*a*b^4 + A*b^5)*c^3*d 
 + 2*(2*C*a^2*b^3 - A*a*b^4)*c*d^3)*x^3 - (6*B*a*b^4*c^3*d - 2*B*a^2*b^3*c 
*d^3 - (11*C*a^2*b^3 + 5*A*a*b^4)*c^2*d^2 + (7*C*a^3*b^2 + A*a^2*b^3)*d^4) 
*x^2 + (6*C*a^2*b^3*c^3*d - 5*B*a^2*b^3*c^2*d^2 + B*a^3*b^2*d^4 - 2*(C*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**2*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(-b*x**2+a)**(5/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{2}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="m 
axima")
 

Output:

integrate((C*x^2 + B*x + A)*x^2/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
 

Giac [F]

\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{2}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="g 
iac")
 

Output:

integrate((C*x^2 + B*x + A)*x^2/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^2\,\left (C\,x^2+B\,x+A\right )}{{\left (a-b\,x^2\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int((x^2*(A + B*x + C*x^2))/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)),x)
 

Output:

int((x^2*(A + B*x + C*x^2))/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^{2} \left (C \,x^{2}+B x +A \right )}{\sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:

int(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)
 

Output:

int(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)