\(\int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx\) [43]

Optimal result

Integrand size = 43, antiderivative size = 783 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx=\frac {2 D \sqrt {c+d x} \sqrt {a+b x^2}}{3 b d f}+\frac {2 \left (\sqrt {b} c-\sqrt {-a} d\right ) \sqrt {\sqrt {b} c+\sqrt {-a} d} (3 d D e-3 C d f+2 c D f) \sqrt {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}} \sqrt {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}} E\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {-a} d}}\right )|\frac {\sqrt {b} c+\sqrt {-a} d}{\sqrt {b} c-\sqrt {-a} d}\right )}{3 b^{3/4} d^3 f^2 \sqrt {a+b x^2}}-\frac {2 \sqrt {\sqrt {b} c+\sqrt {-a} d} \left (a d D f^2-\sqrt {-a} \sqrt {b} f (3 d D e-3 C d f+2 c D f)-3 b d \left (D e^2-f (C e-B f)\right )\right ) \sqrt {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}} \sqrt {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {-a} d}}\right ),\frac {\sqrt {b} c+\sqrt {-a} d}{\sqrt {b} c-\sqrt {-a} d}\right )}{3 b^{5/4} d^2 f^3 \sqrt {a+b x^2}}-\frac {2 \sqrt {\sqrt {b} c+\sqrt {-a} d} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \sqrt {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}} \sqrt {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}} \operatorname {EllipticPi}\left (-\frac {\left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) f}{d e-c f},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {-a} d}}\right ),\frac {\sqrt {b} c+\sqrt {-a} d}{\sqrt {b} c-\sqrt {-a} d}\right )}{\sqrt [4]{b} f^3 (d e-c f) \sqrt {a+b x^2}} \] Output:

2/3*D*(d*x+c)^(1/2)*(b*x^2+a)^(1/2)/b/d/f+2/3*(b^(1/2)*c-(-a)^(1/2)*d)*(b^ 
(1/2)*c+(-a)^(1/2)*d)^(1/2)*(-3*C*d*f+2*D*c*f+3*D*d*e)*(1-b^(1/2)*(d*x+c)/ 
(b^(1/2)*c-(-a)^(1/2)*d))^(1/2)*(1-b^(1/2)*(d*x+c)/(b^(1/2)*c+(-a)^(1/2)*d 
))^(1/2)*EllipticE(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+(-a)^(1/2)*d)^(1/2),(( 
b^(1/2)*c+(-a)^(1/2)*d)/(b^(1/2)*c-(-a)^(1/2)*d))^(1/2))/b^(3/4)/d^3/f^2/( 
b*x^2+a)^(1/2)-2/3*(b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*(a*d*D*f^2-(-a)^(1/2)*b^ 
(1/2)*f*(-3*C*d*f+2*D*c*f+3*D*d*e)-3*b*d*(D*e^2-f*(-B*f+C*e)))*(1-b^(1/2)* 
(d*x+c)/(b^(1/2)*c-(-a)^(1/2)*d))^(1/2)*(1-b^(1/2)*(d*x+c)/(b^(1/2)*c+(-a) 
^(1/2)*d))^(1/2)*EllipticF(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+(-a)^(1/2)*d)^ 
(1/2),((b^(1/2)*c+(-a)^(1/2)*d)/(b^(1/2)*c-(-a)^(1/2)*d))^(1/2))/b^(5/4)/d 
^2/f^3/(b*x^2+a)^(1/2)-2*(b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*(D*e^3-f*(C*e^2-f* 
(-A*f+B*e)))*(1-b^(1/2)*(d*x+c)/(b^(1/2)*c-(-a)^(1/2)*d))^(1/2)*(1-b^(1/2) 
*(d*x+c)/(b^(1/2)*c+(-a)^(1/2)*d))^(1/2)*EllipticPi(b^(1/4)*(d*x+c)^(1/2)/ 
(b^(1/2)*c+(-a)^(1/2)*d)^(1/2),-(c+(-a)^(1/2)*d/b^(1/2))*f/(-c*f+d*e),((b^ 
(1/2)*c+(-a)^(1/2)*d)/(b^(1/2)*c-(-a)^(1/2)*d))^(1/2))/b^(1/4)/f^3/(-c*f+d 
*e)/(b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.19 (sec) , antiderivative size = 2091, normalized size of antiderivative = 2.67 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

(2*(-3*b*c^2*d^2*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*D*e^2*f - 3*a*d^4*Sqrt[- 
c - (I*Sqrt[a]*d)/Sqrt[b]]*D*e^2*f + 3*b*c^2*C*d^2*Sqrt[-c - (I*Sqrt[a]*d) 
/Sqrt[b]]*e*f^2 + 3*a*C*d^4*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*e*f^2 + b*c^3 
*d*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*D*e*f^2 + a*c*d^3*Sqrt[-c - (I*Sqrt[a] 
*d)/Sqrt[b]]*D*e*f^2 - 3*b*c^3*C*d*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*f^3 - 
3*a*c*C*d^3*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*f^3 + 2*b*c^4*Sqrt[-c - (I*Sq 
rt[a]*d)/Sqrt[b]]*D*f^3 + 2*a*c^2*d^2*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*D*f 
^3 + 6*b*c*d^2*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*D*e^2*f*(c + d*x) - 6*b*c* 
C*d^2*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*e*f^2*(c + d*x) - 2*b*c^2*d*Sqrt[-c 
 - (I*Sqrt[a]*d)/Sqrt[b]]*D*e*f^2*(c + d*x) + 6*b*c^2*C*d*Sqrt[-c - (I*Sqr 
t[a]*d)/Sqrt[b]]*f^3*(c + d*x) - 4*b*c^3*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]* 
D*f^3*(c + d*x) - 3*b*d^2*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*D*e^2*f*(c + d* 
x)^2 + 3*b*C*d^2*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*e*f^2*(c + d*x)^2 + b*c* 
d*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*D*e*f^2*(c + d*x)^2 - 3*b*c*C*d*Sqrt[-c 
 - (I*Sqrt[a]*d)/Sqrt[b]]*f^3*(c + d*x)^2 + 2*b*c^2*Sqrt[-c - (I*Sqrt[a]*d 
)/Sqrt[b]]*D*f^3*(c + d*x)^2 + d^2*Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]*D*f^2* 
(d*e - c*f)*(c + d*x)*(a + b*x^2) + Sqrt[b]*((-I)*Sqrt[b]*c + Sqrt[a]*d)*f 
*(-(d*e) + c*f)*(3*d*D*e - 3*C*d*f + 2*c*D*f)*Sqrt[(d*((I*Sqrt[a])/Sqrt[b] 
 + x))/(c + d*x)]*Sqrt[-(((I*Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d* 
x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c - (I*Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + ...
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1619\) vs. \(2(783)=1566\).

Time = 5.85 (sec) , antiderivative size = 1619, normalized size of antiderivative = 2.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2349, 729, 1540, 1416, 2185, 27, 599, 1511, 1416, 1509, 2220}

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.

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

\(\Big \downarrow \) 2349

\(\displaystyle \int \frac {\frac {D e^2}{f^3}-\frac {C e}{f^2}+\frac {D x^2}{f}+\left (\frac {C}{f}-\frac {D e}{f^2}\right ) x+\frac {B}{f}}{\sqrt {c+d x} \sqrt {b x^2+a}}dx-\frac {\left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \int \frac {1}{\sqrt {c+d x} (e+f x) \sqrt {b x^2+a}}dx}{f^3}\)

\(\Big \downarrow \) 729

\(\displaystyle \int \frac {\frac {D e^2}{f^3}-\frac {C e}{f^2}+\frac {D x^2}{f}+\left (\frac {C}{f}-\frac {D e}{f^2}\right ) x+\frac {B}{f}}{\sqrt {c+d x} \sqrt {b x^2+a}}dx-\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \int \frac {1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{f^3}\)

\(\Big \downarrow \) 1540

\(\displaystyle \int \frac {\frac {D e^2}{f^3}-\frac {C e}{f^2}+\frac {D x^2}{f}+\left (\frac {C}{f}-\frac {D e}{f^2}\right ) x+\frac {B}{f}}{\sqrt {c+d x} \sqrt {b x^2+a}}dx-\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {f \sqrt {a d^2+b c^2} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \int \frac {\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt {b} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \int \frac {1}{\sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}\right )}{f^3}\)

\(\Big \downarrow \) 1416

\(\displaystyle \int \frac {\frac {D e^2}{f^3}-\frac {C e}{f^2}+\frac {D x^2}{f}+\left (\frac {C}{f}-\frac {D e}{f^2}\right ) x+\frac {B}{f}}{\sqrt {c+d x} \sqrt {b x^2+a}}dx-\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {f \sqrt {a d^2+b c^2} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \int \frac {\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt [4]{b} \sqrt [4]{a d^2+b c^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right ) \sqrt {\frac {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}{\left (a+\frac {b c^2}{d^2}\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right )^2}} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 d \sqrt {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}} \left (a d f^2-b e (d e-2 c f)\right )}\right )}{f^3}\)

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {f \sqrt {a d^2+b c^2} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \int \frac {\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt [4]{b} \sqrt [4]{a d^2+b c^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right ) \sqrt {\frac {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}{\left (a+\frac {b c^2}{d^2}\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right )^2}} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 d \sqrt {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}} \left (a d f^2-b e (d e-2 c f)\right )}\right )}{f^3}+\frac {2 \int -\frac {d \left (d \left (a D f^2-3 b \left (D e^2-f (C e-B f)\right )\right )+b f (3 d D e-3 C d f+2 c D f) x\right )}{2 f^3 \sqrt {c+d x} \sqrt {b x^2+a}}dx}{3 b d^2}+\frac {2 D \sqrt {a+b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {f \sqrt {a d^2+b c^2} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \int \frac {\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt [4]{b} \sqrt [4]{a d^2+b c^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right ) \sqrt {\frac {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}{\left (a+\frac {b c^2}{d^2}\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right )^2}} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 d \sqrt {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}} \left (a d f^2-b e (d e-2 c f)\right )}\right )}{f^3}-\frac {\int \frac {d \left (a D f^2-3 b \left (D e^2-f (C e-B f)\right )\right )+b f (3 d D e-3 C d f+2 c D f) x}{\sqrt {c+d x} \sqrt {b x^2+a}}dx}{3 b d f^3}+\frac {2 D \sqrt {a+b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 599

\(\displaystyle -\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {f \sqrt {a d^2+b c^2} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \int \frac {\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt [4]{b} \sqrt [4]{a d^2+b c^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right ) \sqrt {\frac {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}{\left (a+\frac {b c^2}{d^2}\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right )^2}} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 d \sqrt {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}} \left (a d f^2-b e (d e-2 c f)\right )}\right )}{f^3}+\frac {2 \int \frac {-\left (\left (a D f^2-3 b \left (D e^2-f (C e-B f)\right )\right ) d^2\right )+b c f (3 d D e-3 C d f+2 c D f)-b f (3 d D e-3 C d f+2 c D f) (c+d x)}{\sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{3 b d^3 f^3}+\frac {2 D \sqrt {a+b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {2 \sqrt {c+d x} \sqrt {b x^2+a} D}{3 b d f}+\frac {2 \left (\left (-\left (\left (a D f^2-3 b \left (D e^2-f (C e-B f)\right )\right ) d^2\right )+b c f (3 d D e-3 C d f+2 c D f)-\sqrt {b} \sqrt {b c^2+a d^2} f (3 d D e-3 C d f+2 c D f)\right ) \int \frac {1}{\sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}+\sqrt {b} \sqrt {b c^2+a d^2} f (3 d D e-3 C d f+2 c D f) \int \frac {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}}{\sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}\right )}{3 b d^3 f^3}-\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {\sqrt {b c^2+a d^2} f \left (\sqrt {b c^2+a d^2} f+\sqrt {b} (d e-c f)\right ) \int \frac {\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt [4]{b} \sqrt [4]{b c^2+a d^2} \left (\sqrt {b c^2+a d^2} f+\sqrt {b} (d e-c f)\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 d \left (a d f^2-b e (d e-2 c f)\right ) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )}{f^3}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {2 \sqrt {c+d x} \sqrt {b x^2+a} D}{3 b d f}+\frac {2 \left (\frac {\sqrt [4]{b c^2+a d^2} \left (-\left (\left (a D f^2-3 b \left (D e^2-f (C e-B f)\right )\right ) d^2\right )+b c f (3 d D e-3 C d f+2 c D f)-\sqrt {b} \sqrt {b c^2+a d^2} f (3 d D e-3 C d f+2 c D f)\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 \sqrt [4]{b} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}+\sqrt {b} \sqrt {b c^2+a d^2} f (3 d D e-3 C d f+2 c D f) \int \frac {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}}{\sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}\right )}{3 b d^3 f^3}-\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {\sqrt {b c^2+a d^2} f \left (\sqrt {b c^2+a d^2} f+\sqrt {b} (d e-c f)\right ) \int \frac {\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt [4]{b} \sqrt [4]{b c^2+a d^2} \left (\sqrt {b c^2+a d^2} f+\sqrt {b} (d e-c f)\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 d \left (a d f^2-b e (d e-2 c f)\right ) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )}{f^3}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {2 \sqrt {c+d x} \sqrt {b x^2+a} D}{3 b d f}+\frac {2 \left (\sqrt {b} \sqrt {b c^2+a d^2} f (3 d D e-3 C d f+2 c D f) \left (\frac {\sqrt [4]{b c^2+a d^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{\sqrt [4]{b} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}-\frac {\sqrt {c+d x} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )}\right )+\frac {\sqrt [4]{b c^2+a d^2} \left (-\left (\left (a D f^2-3 b \left (D e^2-f (C e-B f)\right )\right ) d^2\right )+b c f (3 d D e-3 C d f+2 c D f)-\sqrt {b} \sqrt {b c^2+a d^2} f (3 d D e-3 C d f+2 c D f)\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 \sqrt [4]{b} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )}{3 b d^3 f^3}-\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {\sqrt {b c^2+a d^2} f \left (\sqrt {b c^2+a d^2} f+\sqrt {b} (d e-c f)\right ) \int \frac {\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt [4]{b} \sqrt [4]{b c^2+a d^2} \left (\sqrt {b c^2+a d^2} f+\sqrt {b} (d e-c f)\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 d \left (a d f^2-b e (d e-2 c f)\right ) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )}{f^3}\)

\(\Big \downarrow \) 2220

\(\displaystyle \frac {2 \sqrt {c+d x} \sqrt {b x^2+a} D}{3 b d f}+\frac {2 \left (\sqrt {b} \sqrt {b c^2+a d^2} f (3 d D e-3 C d f+2 c D f) \left (\frac {\sqrt [4]{b c^2+a d^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{\sqrt [4]{b} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}-\frac {\sqrt {c+d x} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )}\right )+\frac {\sqrt [4]{b c^2+a d^2} \left (-\left (\left (a D f^2-3 b \left (D e^2-f (C e-B f)\right )\right ) d^2\right )+b c f (3 d D e-3 C d f+2 c D f)-\sqrt {b} \sqrt {b c^2+a d^2} f (3 d D e-3 C d f+2 c D f)\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 \sqrt [4]{b} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )}{3 b d^3 f^3}-\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {\sqrt {b c^2+a d^2} f \left (\sqrt {b c^2+a d^2} f+\sqrt {b} (d e-c f)\right ) \left (\frac {\left (f-\frac {\sqrt {b} (d e-c f)}{\sqrt {b c^2+a d^2}}\right ) \arctan \left (\frac {\sqrt {b e^2+a f^2} \sqrt {c+d x}}{\sqrt {f} \sqrt {d e-c f} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )}{2 \sqrt {f} \sqrt {d e-c f} \sqrt {b e^2+a f^2}}+\frac {\left (\frac {\sqrt {b}}{f}+\frac {\sqrt {b c^2+a d^2}}{d e-c f}\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b c^2+a d^2} f-\sqrt {b} (d e-c f)\right )^2}{4 \sqrt {b} \sqrt {b c^2+a d^2} f (d e-c f)},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{4 \sqrt [4]{b} \sqrt [4]{b c^2+a d^2} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt [4]{b} \sqrt [4]{b c^2+a d^2} \left (\sqrt {b c^2+a d^2} f+\sqrt {b} (d e-c f)\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 d \left (a d f^2-b e (d e-2 c f)\right ) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )}{f^3}\)

Input:

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

Output:

(2*D*Sqrt[c + d*x]*Sqrt[a + b*x^2])/(3*b*d*f) + (2*(Sqrt[b]*Sqrt[b*c^2 + a 
*d^2]*f*(3*d*D*e - 3*C*d*f + 2*c*D*f)*(-((Sqrt[c + d*x]*Sqrt[a + (b*c^2)/d 
^2 - (2*b*c*(c + d*x))/d^2 + (b*(c + d*x)^2)/d^2])/((a + (b*c^2)/d^2)*(1 + 
 (Sqrt[b]*(c + d*x))/Sqrt[b*c^2 + a*d^2]))) + ((b*c^2 + a*d^2)^(1/4)*(1 + 
(Sqrt[b]*(c + d*x))/Sqrt[b*c^2 + a*d^2])*Sqrt[(a + (b*c^2)/d^2 - (2*b*c*(c 
 + d*x))/d^2 + (b*(c + d*x)^2)/d^2)/((a + (b*c^2)/d^2)*(1 + (Sqrt[b]*(c + 
d*x))/Sqrt[b*c^2 + a*d^2])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[c + d*x])/ 
(b*c^2 + a*d^2)^(1/4)], (1 + (Sqrt[b]*c)/Sqrt[b*c^2 + a*d^2])/2])/(b^(1/4) 
*Sqrt[a + (b*c^2)/d^2 - (2*b*c*(c + d*x))/d^2 + (b*(c + d*x)^2)/d^2])) + ( 
(b*c^2 + a*d^2)^(1/4)*(b*c*f*(3*d*D*e - 3*C*d*f + 2*c*D*f) - Sqrt[b]*Sqrt[ 
b*c^2 + a*d^2]*f*(3*d*D*e - 3*C*d*f + 2*c*D*f) - d^2*(a*D*f^2 - 3*b*(D*e^2 
 - f*(C*e - B*f))))*(1 + (Sqrt[b]*(c + d*x))/Sqrt[b*c^2 + a*d^2])*Sqrt[(a 
+ (b*c^2)/d^2 - (2*b*c*(c + d*x))/d^2 + (b*(c + d*x)^2)/d^2)/((a + (b*c^2) 
/d^2)*(1 + (Sqrt[b]*(c + d*x))/Sqrt[b*c^2 + a*d^2])^2)]*EllipticF[2*ArcTan 
[(b^(1/4)*Sqrt[c + d*x])/(b*c^2 + a*d^2)^(1/4)], (1 + (Sqrt[b]*c)/Sqrt[b*c 
^2 + a*d^2])/2])/(2*b^(1/4)*Sqrt[a + (b*c^2)/d^2 - (2*b*c*(c + d*x))/d^2 + 
 (b*(c + d*x)^2)/d^2])))/(3*b*d^3*f^3) - (2*(D*e^3 - f*(C*e^2 - f*(B*e - A 
*f)))*(-1/2*(b^(1/4)*(b*c^2 + a*d^2)^(1/4)*(Sqrt[b*c^2 + a*d^2]*f + Sqrt[b 
]*(d*e - c*f))*(1 + (Sqrt[b]*(c + d*x))/Sqrt[b*c^2 + a*d^2])*Sqrt[(a + (b* 
c^2)/d^2 - (2*b*c*(c + d*x))/d^2 + (b*(c + d*x)^2)/d^2)/((a + (b*c^2)/d...
 

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 599

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[-2/d^2   Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a 
*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr 
eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
 

rule 729

Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))*Sqrt[(a_) + (b_.)*(x_) 
^2]), x_Symbol] :> Simp[2   Subst[Int[1/((d*e - c*f + f*x^2)*Sqrt[(b*c^2 + 
a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; 
FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

rule 1540

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S 
ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2)   Int[1 
/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2)   I 
nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, 
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
NeQ[c*d^2 - a*e^2, 0] && PosQ[c/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]))
 

rule 2220

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a 
+ b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*El 
lipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
 /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] & 
& EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[-b + c*(d/e) + a*(e/d)]
 

rule 2349

Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. 
)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d 
*x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c 
+ d*x, x]   Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, n, p}, x] && PolynomialQ[Px, x] && LtQ[m, 0] &&  !IntegerQ[n 
] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [A] (verified)

Time = 4.95 (sec) , antiderivative size = 926, normalized size of antiderivative = 1.18

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (b \,x^{2}+a \right )}\, \left (\frac {2 D \sqrt {b d \,x^{3}+b c \,x^{2}+a d x +a c}}{3 f b d}+\frac {2 \left (\frac {B \,f^{2}-C e f +D e^{2}}{f^{3}}-\frac {D a}{3 f b}\right ) \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right ) \sqrt {\frac {x +\frac {c}{d}}{-\frac {\sqrt {-a b}}{b}+\frac {c}{d}}}\, \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 {\sqrt {-a b}}{b}+\frac {c}{d}}}, \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 {C f -D e}{f^{2}}-\frac {2 D c}{3 f d}\right ) \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right ) \sqrt {\frac {x +\frac {c}{d}}{-\frac {\sqrt {-a b}}{b}+\frac {c}{d}}}\, \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 {\sqrt {-a b}}{b}+\frac {c}{d}}}, \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 {\sqrt {-a b}}{b}+\frac {c}{d}}}, \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}}+\frac {2 \left (A \,f^{3}-B e \,f^{2}+C \,e^{2} f -D e^{3}\right ) \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right ) \sqrt {\frac {x +\frac {c}{d}}{-\frac {\sqrt {-a b}}{b}+\frac {c}{d}}}\, \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 {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{-\frac {\sqrt {-a b}}{b}+\frac {c}{d}}}, \frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}+\frac {e}{f}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{b}}}\right )}{f^{4} \sqrt {b d \,x^{3}+b c \,x^{2}+a d x +a c}\, \left (-\frac {c}{d}+\frac {e}{f}\right )}\right )}{\sqrt {d x +c}\, \sqrt {b \,x^{2}+a}}\)	\(926\)
default	\(\text {Expression too large to display}\)	\(6128\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)/(b*x^2+a)^(1/2),x,method=_RE 
TURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

((d*x+c)*(b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(b*x^2+a)^(1/2)*(2/3*D/f/b/d*(b*d* 
x^3+b*c*x^2+a*d*x+a*c)^(1/2)+2*((B*f^2-C*e*f+D*e^2)/f^3-1/3*D/f/b*a)*(-(-a 
*b)^(1/2)/b+c/d)*((x+c/d)/(-(-a*b)^(1/2)/b+c/d))^(1/2)*((x-(-a*b)^(1/2)/b) 
/(-c/d-(-a*b)^(1/2)/b))^(1/2)*((x+(-a*b)^(1/2)/b)/(-c/d+(-a*b)^(1/2)/b))^( 
1/2)/(b*d*x^3+b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(-(-a*b)^(1/2)/b 
+c/d))^(1/2),((-c/d+(-a*b)^(1/2)/b)/(-c/d-(-a*b)^(1/2)/b))^(1/2))+2*(1/f^2 
*(C*f-D*e)-2/3*D/f/d*c)*(-(-a*b)^(1/2)/b+c/d)*((x+c/d)/(-(-a*b)^(1/2)/b+c/ 
d))^(1/2)*((x-(-a*b)^(1/2)/b)/(-c/d-(-a*b)^(1/2)/b))^(1/2)*((x+(-a*b)^(1/2 
)/b)/(-c/d+(-a*b)^(1/2)/b))^(1/2)/(b*d*x^3+b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d 
-(-a*b)^(1/2)/b)*EllipticE(((x+c/d)/(-(-a*b)^(1/2)/b+c/d))^(1/2),((-c/d+(- 
a*b)^(1/2)/b)/(-c/d-(-a*b)^(1/2)/b))^(1/2))+(-a*b)^(1/2)/b*EllipticF(((x+c 
/d)/(-(-a*b)^(1/2)/b+c/d))^(1/2),((-c/d+(-a*b)^(1/2)/b)/(-c/d-(-a*b)^(1/2) 
/b))^(1/2)))+2*(A*f^3-B*e*f^2+C*e^2*f-D*e^3)/f^4*(-(-a*b)^(1/2)/b+c/d)*((x 
+c/d)/(-(-a*b)^(1/2)/b+c/d))^(1/2)*((x-(-a*b)^(1/2)/b)/(-c/d-(-a*b)^(1/2)/ 
b))^(1/2)*((x+(-a*b)^(1/2)/b)/(-c/d+(-a*b)^(1/2)/b))^(1/2)/(b*d*x^3+b*c*x^ 
2+a*d*x+a*c)^(1/2)/(-c/d+e/f)*EllipticPi(((x+c/d)/(-(-a*b)^(1/2)/b+c/d))^( 
1/2),(-c/d+(-a*b)^(1/2)/b)/(-c/d+e/f),((-c/d+(-a*b)^(1/2)/b)/(-c/d-(-a*b)^ 
(1/2)/b))^(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)/(b*x^2+a)^(1/2),x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx=\int \frac {A + B x + C x^{2} + D x^{3}}{\sqrt {a + b x^{2}} \sqrt {c + d x} \left (e + f x\right )}\, dx \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2)/(f*x+e)/(b*x**2+a)**(1/2),x 
)
 

Output:

Integral((A + B*x + C*x**2 + D*x**3)/(sqrt(a + b*x**2)*sqrt(c + d*x)*(e + 
f*x)), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{\sqrt {b x^{2} + a} \sqrt {d x + c} {\left (f x + e\right )}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)/(b*x^2+a)^(1/2),x, alg 
orithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{\sqrt {b x^{2} + a} \sqrt {d x + c} {\left (f x + e\right )}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)/(b*x^2+a)^(1/2),x, alg 
orithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{\left (e+f\,x\right )\,\sqrt {b\,x^2+a}\,\sqrt {c+d\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx=\int \frac {D x^{3}+C \,x^{2}+B x +A}{\sqrt {d x +c}\, \left (f x +e \right ) \sqrt {b \,x^{2}+a}}d x \] Input:

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

Output:

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