\(\int \frac {A+B x+C x^2+D x^3+F x^4}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx\) [242]

Optimal result

Integrand size = 42, antiderivative size = 517 \[ \int \frac {A+B x+C x^2+D x^3+F x^4}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \left (25 a d^2 F+b \left (35 C d^2-c (49 d D-57 c F)\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{105 b^2 d^3}-\frac {2 (7 d D-16 c F) (c+d x)^{3/2} \sqrt {a-b x^2}}{35 b d^3}-\frac {2 F (c+d x)^{5/2} \sqrt {a-b x^2}}{7 b d^3}-\frac {2 \sqrt {a} \left (a d^2 (63 d D-44 c F)-b \left (70 c C d^2-105 B d^3-56 c^2 d D+48 c^3 F\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{3/2} d^4 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (25 a^2 d^4 F+b^2 \left (70 c^2 C d^2-105 B c d^3+105 A d^4-56 c^3 d D+48 c^4 F\right )+a b d^2 \left (35 C d^2-c (49 d D-32 c F)\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{5/2} d^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-2/105*(25*a*d^2*F+b*(35*C*d^2-c*(49*D*d-57*F*c)))*(d*x+c)^(1/2)*(-b*x^2+a 
)^(1/2)/b^2/d^3-2/35*(7*D*d-16*F*c)*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b/d^3-2 
/7*F*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/b/d^3-2/105*a^(1/2)*(a*d^2*(63*D*d-44* 
F*c)-b*(-105*B*d^3+70*C*c*d^2-56*D*c^2*d+48*F*c^3))*(d*x+c)^(1/2)*((-b*x^2 
+a)/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^ 
(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(3/2)/d^4/((d*x+c)/(c+a^(1/2)*d/b^ 
(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-2/105*a^(1/2)*(25*a^2*d^4*F+b^2*(105*A*d^4- 
105*B*c*d^3+70*C*c^2*d^2-56*D*c^3*d+48*F*c^4)+a*b*d^2*(35*C*d^2-c*(49*D*d- 
32*F*c)))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellip 
ticF(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^4/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.84 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x+C x^2+D x^3+F x^4}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (a d^2 (-63 d D+44 c F)+b \left (70 c C d^2-105 B d^3-56 c^2 d D+48 c^3 F\right )-(c+d x) \left (25 a d^2 F+b \left (35 C d^2+24 c^2 F+3 d^2 x (7 D+5 F x)-2 c d (14 D+9 F x)\right )\right )+\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2 (63 d D-44 c F)+b \left (-70 c C d^2+105 B d^3+56 c^2 d D-48 c^3 F\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}-\frac {i \left (105 A b^2 d^3+25 a^2 d^3 F+a^{3/2} \sqrt {b} d^2 (-63 d D+44 c F)+\sqrt {a} b^{3/2} \left (70 c C d^2-105 B d^3-56 c^2 d D+48 c^3 F\right )+a b d \left (35 C d^2+2 c (7 d D-6 c F)\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{105 b^2 d^3 \sqrt {c+d x}} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3 + F*x^4)/(Sqrt[c + d*x]*Sqrt[a - b*x^2] 
),x]
 

Output:

(2*Sqrt[a - b*x^2]*(a*d^2*(-63*d*D + 44*c*F) + b*(70*c*C*d^2 - 105*B*d^3 - 
 56*c^2*d*D + 48*c^3*F) - (c + d*x)*(25*a*d^2*F + b*(35*C*d^2 + 24*c^2*F + 
 3*d^2*x*(7*D + 5*F*x) - 2*c*d*(14*D + 9*F*x))) + (I*Sqrt[b]*(Sqrt[b]*c - 
Sqrt[a]*d)*(a*d^2*(63*d*D - 44*c*F) + b*(-70*c*C*d^2 + 105*B*d^3 + 56*c^2* 
d*D - 48*c^3*F))*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)])/(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2)) - (I*(1 
05*A*b^2*d^3 + 25*a^2*d^3*F + a^(3/2)*Sqrt[b]*d^2*(-63*d*D + 44*c*F) + Sqr 
t[a]*b^(3/2)*(70*c*C*d^2 - 105*B*d^3 - 56*c^2*d*D + 48*c^3*F) + a*b*d*(35* 
C*d^2 + 2*c*(7*d*D - 6*c*F)))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sq 
rt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*A 
rcSinh[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))))/(105*b^2*d^3*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 2.39 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2185, 27, 2185, 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.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3+F x^4}{\sqrt {a-b x^2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {2 \int -\frac {b d^3 (7 d D-16 c F) x^3+d^2 \left (-11 b F c^2+7 b C d^2+5 a d^2 F\right ) x^2+d \left (-2 b F c^3+10 a d^2 F c+7 b B d^3\right ) x+d^2 \left (5 a F c^2+7 A b d^2\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {b d^3 (7 d D-16 c F) x^3+d^2 \left (-11 b F c^2+7 b C d^2+5 a d^2 F\right ) x^2+d \left (-2 b F c^3+10 a d^2 F c+7 b B d^3\right ) x+d^2 \left (5 a F c^2+7 A b d^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {2 \int -\frac {b \left (35 b C d^2+25 a F d^2-b c (49 d D-57 c F)\right ) x^2 d^5+b \left (35 A b d^2+a c (21 d D-23 c F)\right ) d^5+b \left (35 b B d^3+a (21 d D+2 c F) d^2-2 b c^2 (7 d D-11 c F)\right ) x d^4}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {b \left (35 b C d^2+25 a F d^2-b c (49 d D-57 c F)\right ) x^2 d^5+b \left (35 A b d^2+a c (21 d D-23 c F)\right ) d^5+b \left (35 b B d^3+a (21 d D+2 c F) d^2-2 b c^2 (7 d D-11 c F)\right ) x d^4}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {-\frac {2 \int -\frac {b d^6 \left (d \left (105 A b^2 d^2+a \left (35 b C d^2+25 a F d^2+2 b c (7 d D-6 c F)\right )\right )+b \left (a d^2 (63 d D-44 c F)-b \left (48 F c^3-56 d D c^2+70 C d^2 c-105 B d^3\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 F-b c (49 d D-57 c F)+35 b C d^2\right )}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {1}{3} d^4 \int \frac {d \left (105 A b^2 d^2+a \left (35 b C d^2+25 a F d^2+2 b c (7 d D-6 c F)\right )\right )+b \left (a d^2 (63 d D-44 c F)-b \left (48 F c^3-56 d D c^2+70 C d^2 c-105 B d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 F-b c (49 d D-57 c F)+35 b C d^2\right )}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {\frac {1}{3} d^4 \left (\frac {\left (25 a^2 d^4 F+a b d^2 \left (35 C d^2-c (49 d D-32 c F)\right )+b^2 \left (105 A d^4-105 B c d^3+48 c^4 F-56 c^3 d D+70 c^2 C d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {b \left (a d^2 (63 d D-44 c F)-b \left (-105 B d^3+48 c^3 F-56 c^2 d D+70 c C d^2\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 F-b c (49 d D-57 c F)+35 b C d^2\right )}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {\frac {1}{3} d^4 \left (\frac {\left (25 a^2 d^4 F+a b d^2 \left (35 C d^2-c (49 d D-32 c F)\right )+b^2 \left (105 A d^4-105 B c d^3+48 c^4 F-56 c^3 d D+70 c^2 C d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {b \sqrt {1-\frac {b x^2}{a}} \left (a d^2 (63 d D-44 c F)-b \left (-105 B d^3+48 c^3 F-56 c^2 d D+70 c C d^2\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )-\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 F-b c (49 d D-57 c F)+35 b C d^2\right )}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {\frac {1}{3} d^4 \left (\frac {\left (25 a^2 d^4 F+a b d^2 \left (35 C d^2-c (49 d D-32 c F)\right )+b^2 \left (105 A d^4-105 B c d^3+48 c^4 F-56 c^3 d D+70 c^2 C d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (63 d D-44 c F)-b \left (-105 B d^3+48 c^3 F-56 c^2 d D+70 c C d^2\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 F-b c (49 d D-57 c F)+35 b C d^2\right )}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\frac {1}{3} d^4 \left (\frac {\left (25 a^2 d^4 F+a b d^2 \left (35 C d^2-c (49 d D-32 c F)\right )+b^2 \left (105 A d^4-105 B c d^3+48 c^4 F-56 c^3 d D+70 c^2 C d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (63 d D-44 c F)-b \left (-105 B d^3+48 c^3 F-56 c^2 d D+70 c C d^2\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 F-b c (49 d D-57 c F)+35 b C d^2\right )}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {\frac {1}{3} d^4 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (25 a^2 d^4 F+a b d^2 \left (35 C d^2-c (49 d D-32 c F)\right )+b^2 \left (105 A d^4-105 B c d^3+48 c^4 F-56 c^3 d D+70 c^2 C d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (63 d D-44 c F)-b \left (-105 B d^3+48 c^3 F-56 c^2 d D+70 c C d^2\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 F-b c (49 d D-57 c F)+35 b C d^2\right )}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {\frac {1}{3} d^4 \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (25 a^2 d^4 F+a b d^2 \left (35 C d^2-c (49 d D-32 c F)\right )+b^2 \left (105 A d^4-105 B c d^3+48 c^4 F-56 c^3 d D+70 c^2 C d^2\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (63 d D-44 c F)-b \left (-105 B d^3+48 c^3 F-56 c^2 d D+70 c C d^2\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 F-b c (49 d D-57 c F)+35 b C d^2\right )}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {\frac {1}{3} d^4 \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (25 a^2 d^4 F+a b d^2 \left (35 C d^2-c (49 d D-32 c F)\right )+b^2 \left (105 A d^4-105 B c d^3+48 c^4 F-56 c^3 d D+70 c^2 C d^2\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (63 d D-44 c F)-b \left (-105 B d^3+48 c^3 F-56 c^2 d D+70 c C d^2\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a d^2 F-b c (49 d D-57 c F)+35 b C d^2\right )}{5 b d^3}-\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (7 d D-16 c F)}{7 b d^4}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3 + F*x^4)/(Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
 

Output:

(-2*F*(c + d*x)^(5/2)*Sqrt[a - b*x^2])/(7*b*d^3) + ((-2*d*(7*d*D - 16*c*F) 
*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/5 + ((-2*d^4*(35*b*C*d^2 + 25*a*d^2*F - 
b*c*(49*d*D - 57*c*F))*Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 + (d^4*((-2*Sqrt[a 
]*Sqrt[b]*(a*d^2*(63*d*D - 44*c*F) - b*(70*c*C*d^2 - 105*B*d^3 - 56*c^2*d* 
D + 48*c^3*F))*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] 
*(25*a^2*d^4*F + b^2*(70*c^2*C*d^2 - 105*B*c*d^3 + 105*A*d^4 - 56*c^3*d*D 
+ 48*c^4*F) + a*b*d^2*(35*C*d^2 - c*(49*d*D - 32*c*F)))*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)/(5*b*d^3))/(7*b*d^4)
 

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 2185

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]))
 

Maple [A] (verified)

Time = 3.89 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.51

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {2 F \,x^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{7 b d}-\frac {2 \left (D-\frac {6 F c}{7 d}\right ) x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b d}-\frac {2 \left (C +\frac {5 F a}{7 b}-\frac {4 \left (D-\frac {6 F c}{7 d}\right ) c}{5 d}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (A +\frac {2 \left (D-\frac {6 F c}{7 d}\right ) a c}{5 b d}+\frac {\left (C +\frac {5 F a}{7 b}-\frac {4 \left (D-\frac {6 F c}{7 d}\right ) c}{5 d}\right ) a}{3 b}\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 (B +\frac {4 F a c}{7 b d}+\frac {3 \left (D-\frac {6 F c}{7 d}\right ) a}{5 b}-\frac {2 \left (C +\frac {5 F a}{7 b}-\frac {4 \left (D-\frac {6 F c}{7 d}\right ) c}{5 d}\right ) c}{3 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}}\)	\(780\)
default	\(\text {Expression too large to display}\)	\(4063\)

Input:

int((F*x^4+D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x+C x^2+D x^3+F x^4}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left ({\left (48 \, F b^{2} c^{4} - 56 \, D b^{2} c^{3} d + 2 \, {\left (4 \, F a b + 35 \, C b^{2}\right )} c^{2} d^{2} - 21 \, {\left (D a b + 5 \, B b^{2}\right )} c d^{3} + 15 \, {\left (5 \, F a^{2} + 7 \, C a b + 21 \, A b^{2}\right )} d^{4}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (48 \, F b^{2} c^{3} d - 56 \, D b^{2} c^{2} d^{2} + 2 \, {\left (22 \, F a b + 35 \, C b^{2}\right )} c d^{3} - 21 \, {\left (3 \, D a b + 5 \, B b^{2}\right )} d^{4}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (15 \, F b^{2} d^{4} x^{2} + 24 \, F b^{2} c^{2} d^{2} - 28 \, D b^{2} c d^{3} + 5 \, {\left (5 \, F a b + 7 \, C b^{2}\right )} d^{4} - 3 \, {\left (6 \, F b^{2} c d^{3} - 7 \, D b^{2} d^{4}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{315 \, b^{3} d^{5}} \] Input:

integrate((F*x^4+D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algo 
rithm="fricas")
 

Output:

-2/315*((48*F*b^2*c^4 - 56*D*b^2*c^3*d + 2*(4*F*a*b + 35*C*b^2)*c^2*d^2 - 
21*(D*a*b + 5*B*b^2)*c*d^3 + 15*(5*F*a^2 + 7*C*a*b + 21*A*b^2)*d^4)*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*(48*F*b^2*c^3*d - 56*D*b^2*c^2*d^2 
 + 2*(22*F*a*b + 35*C*b^2)*c*d^3 - 21*(3*D*a*b + 5*B*b^2)*d^4)*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*(15*F*b^2*d^4*x^2 + 24*F*b^2*c 
^2*d^2 - 28*D*b^2*c*d^3 + 5*(5*F*a*b + 7*C*b^2)*d^4 - 3*(6*F*b^2*c*d^3 - 7 
*D*b^2*d^4)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*d^5)
 

Sympy [F]

\[ \int \frac {A+B x+C x^2+D x^3+F x^4}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {A + B x + C x^{2} + D x^{3} + F x^{4}}{\sqrt {a - b x^{2}} \sqrt {c + d x}}\, dx \] Input:

integrate((F*x**4+D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2)/(-b*x**2+a)**(1/2),x 
)
 

Output:

Integral((A + B*x + C*x**2 + D*x**3 + F*x**4)/(sqrt(a - b*x**2)*sqrt(c + d 
*x)), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2+D x^3+F x^4}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {F x^{4} + D x^{3} + C x^{2} + B x + A}{\sqrt {-b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:

integrate((F*x^4+D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algo 
rithm="maxima")
 

Output:

integrate((F*x^4 + D*x^3 + C*x^2 + B*x + A)/(sqrt(-b*x^2 + a)*sqrt(d*x + c 
)), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2+D x^3+F x^4}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {F x^{4} + D x^{3} + C x^{2} + B x + A}{\sqrt {-b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:

integrate((F*x^4+D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algo 
rithm="giac")
 

Output:

integrate((F*x^4 + D*x^3 + C*x^2 + B*x + A)/(sqrt(-b*x^2 + a)*sqrt(d*x + c 
)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3+F x^4}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {A+B\,x+C\,x^2+F\,x^4+x^3\,D}{\sqrt {a-b\,x^2}\,\sqrt {c+d\,x}} \,d x \] Input:

int((A + B*x + C*x^2 + F*x^4 + x^3*D)/((a - b*x^2)^(1/2)*(c + d*x)^(1/2)), 
x)
 

Output:

int((A + B*x + C*x^2 + F*x^4 + x^3*D)/((a - b*x^2)^(1/2)*(c + d*x)^(1/2)), 
 x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2+D x^3+F x^4}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {-4 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a c d f -42 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a \,d^{3}-70 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} d^{2}+24 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b \,c^{2} f x -28 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b c \,d^{2} x -20 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b c d f \,x^{2}+44 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b c \,d^{2} f -63 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,d^{4}-105 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{3} d^{3}+48 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{3} f +14 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{2} d^{2}+2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} c \,d^{2} f +21 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} d^{4}+70 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c \,d^{2}+35 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} d^{3}-24 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,c^{3} f +28 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,c^{2} d^{2}}{70 b^{2} c \,d^{2}} \] Input:

int((F*x^4+D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x)
 

Output:

( - 4*sqrt(c + d*x)*sqrt(a - b*x**2)*a*c*d*f - 42*sqrt(c + d*x)*sqrt(a - b 
*x**2)*a*d**3 - 70*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*d**2 + 24*sqrt(c + 
d*x)*sqrt(a - b*x**2)*b*c**2*f*x - 28*sqrt(c + d*x)*sqrt(a - b*x**2)*b*c*d 
**2*x - 20*sqrt(c + d*x)*sqrt(a - b*x**2)*b*c*d*f*x**2 + 44*int((sqrt(c + 
d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b*c*d 
**2*f - 63*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x* 
*2 - b*d*x**3),x)*a*b*d**4 - 105*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2) 
/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**3*d**3 + 48*int((sqrt(c + d*x)* 
sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**2*c**3*f 
+ 14*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b 
*d*x**3),x)*b**2*c**2*d**2 + 2*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + 
 a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*c*d**2*f + 21*int((sqrt(c + d*x)*sqr 
t(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*d**4 + 70*int(( 
sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b 
**2*c*d**2 + 35*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x* 
*2 - b*d*x**3),x)*a*b**2*d**3 - 24*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a 
*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b*c**3*f + 28*int((sqrt(c + d*x)*sq 
rt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b*c**2*d**2)/(70* 
b**2*c*d**2)