\(\int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx\) [118]

Optimal result

Integrand size = 28, antiderivative size = 672 \[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (5 \left (7 c d^2-a e^2\right ) g^2+8 c e f (3 e f-7 d g)-6 c e g (3 e f-7 d g) x\right ) \sqrt {a+c x^2}}{105 c g^3}+\frac {2 e^2 \sqrt {f+g x} \left (a+c x^2\right )^{3/2}}{7 c g}-\frac {4 \left (\sqrt {-a}-\frac {\sqrt {c} f}{g}\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (5 \left (7 c d^2-a e^2\right ) f g^2+2 e (3 e f-7 d g) \left (4 c f^2+3 a g^2\right )\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{105 c^{3/4} g^4 \sqrt {a+c x^2}}-\frac {4 \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (5 a^2 e^2 g^3-\sqrt {-a} a \sqrt {c} e g^2 (13 e f-42 d g)-\sqrt {-a} c^{3/2} f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )-a c g \left (6 e^2 f^2-14 d e f g+35 d^2 g^2\right )\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{105 c^{5/4} g^4 \sqrt {a+c x^2}} \] Output:

2/105*(g*x+f)^(1/2)*(5*(-a*e^2+7*c*d^2)*g^2+8*c*e*f*(-7*d*g+3*e*f)-6*c*e*g 
*(-7*d*g+3*e*f)*x)*(c*x^2+a)^(1/2)/c/g^3+2/7*e^2*(g*x+f)^(1/2)*(c*x^2+a)^( 
3/2)/c/g-4/105*((-a)^(1/2)-c^(1/2)*f/g)*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(5* 
(-a*e^2+7*c*d^2)*f*g^2+2*e*(-7*d*g+3*e*f)*(3*a*g^2+4*c*f^2))*(1-c^(1/2)*(g 
*x+f)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^( 
1/2)*g))^(1/2)*EllipticE(c^(1/4)*(g*x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1 
/2),((c^(1/2)*f+(-a)^(1/2)*g)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2))/c^(3/4)/g^4 
/(c*x^2+a)^(1/2)-4/105*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(5*a^2*e^2*g^3-(-a)^ 
(1/2)*a*c^(1/2)*e*g^2*(-42*d*g+13*e*f)-(-a)^(1/2)*c^(3/2)*f*(35*d^2*g^2-56 
*d*e*f*g+24*e^2*f^2)-a*c*g*(35*d^2*g^2-14*d*e*f*g+6*e^2*f^2))*(1-c^(1/2)*( 
g*x+f)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^ 
(1/2)*g))^(1/2)*EllipticF(c^(1/4)*(g*x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^( 
1/2),((c^(1/2)*f+(-a)^(1/2)*g)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2))/c^(5/4)/g^ 
4/(c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 27.42 (sec) , antiderivative size = 665, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (10 a e^2 g^2+c \left (35 d^2 g^2+14 d e g (-4 f+3 g x)+3 e^2 \left (8 f^2-6 f g x+5 g^2 x^2\right )\right )\right )}{c g^3}-\frac {4 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (a e g^2 (13 e f-42 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) \left (a+c x^2\right )-i \sqrt {c} \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (a e g^2 (13 e f-42 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\sqrt {a} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (5 a e^2 g^2+6 i \sqrt {a} \sqrt {c} e g (3 e f-7 d g)+c \left (-24 e^2 f^2+56 d e f g-35 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c g^5 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{105 \sqrt {a+c x^2}} \] Input:

Integrate[((d + e*x)^2*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]
 

Output:

(Sqrt[f + g*x]*((2*(a + c*x^2)*(10*a*e^2*g^2 + c*(35*d^2*g^2 + 14*d*e*g*(- 
4*f + 3*g*x) + 3*e^2*(8*f^2 - 6*f*g*x + 5*g^2*x^2))))/(c*g^3) - (4*(g^2*Sq 
rt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(a*e*g^2*(13*e*f - 42*d*g) + c*f*(24*e^2*f^ 
2 - 56*d*e*f*g + 35*d^2*g^2))*(a + c*x^2) - I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[ 
a]*g)*(a*e*g^2*(13*e*f - 42*d*g) + c*f*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g 
^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/S 
qrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I 
*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f 
+ I*Sqrt[a]*g)] + Sqrt[a]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*(5*a*e^2*g^2 + (6*I) 
*Sqrt[a]*Sqrt[c]*e*g*(3*e*f - 7*d*g) + c*(-24*e^2*f^2 + 56*d*e*f*g - 35*d^ 
2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g 
)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - 
 (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c] 
*f + I*Sqrt[a]*g)]))/(c*g^5*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/ 
(105*Sqrt[a + c*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 1.66 (sec) , antiderivative size = 878, normalized size of antiderivative = 1.31, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {723, 27, 2185, 27, 2185, 27, 599, 25, 1511, 1416, 1509}

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.

\(\displaystyle \int \frac {\sqrt {a+c x^2} (d+e x)^2}{\sqrt {f+g x}} \, dx\)

\(\Big \downarrow \) 723

\(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g}-\frac {\int \frac {2 (d+e x) \left (c (3 e f-2 d g) x^2+(c d f-a e g) x+a (2 e f-3 d g)\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 g}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g}-\frac {2 \int \frac {(d+e x) \left (c (3 e f-2 d g) x^2+(c d f-a e g) x+a (2 e f-3 d g)\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 g}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g}-\frac {2 \left (\frac {2 \int -\frac {c \left (5 a e^2 g^2+c \left (21 e^2 f^2-34 d e g f+10 d^2 g^2\right )\right ) x^2 g^2+a c \left (9 e^2 f^2-16 d e g f+15 d^2 g^2\right ) g^2-c \left (a e g^2 (e f-14 d g)-c f \left (6 e^2 f^2-4 d e g f-5 d^2 g^2\right )\right ) x g}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}+\frac {2 e \sqrt {a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{5 g^2}\right )}{7 g}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g}-\frac {2 \left (\frac {2 e \sqrt {a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{5 g^2}-\frac {\int \frac {c \left (5 a e^2 g^2+c \left (21 e^2 f^2-34 d e g f+10 d^2 g^2\right )\right ) x^2 g^2+a c \left (9 e^2 f^2-16 d e g f+15 d^2 g^2\right ) g^2-c \left (a e g^2 (e f-14 d g)-c f \left (6 e^2 f^2-4 d e g f-5 d^2 g^2\right )\right ) x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}\right )}{7 g}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g}-\frac {2 \left (\frac {2 e \sqrt {a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{5 g^2}-\frac {\frac {2 \int -\frac {c g^3 \left (a g \left (5 a e^2 g^2-c \left (6 e^2 f^2-14 d e g f+35 d^2 g^2\right )\right )+c \left (a e (13 e f-42 d g) g^2+c f \left (24 e^2 f^2-56 d e g f+35 d^2 g^2\right )\right ) x\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g^2}+\frac {2}{3} g \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e^2 g^2+c \left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )}{5 c g^3}\right )}{7 g}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g}-\frac {2 \left (\frac {2 e \sqrt {a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{5 g^2}-\frac {\frac {2}{3} g \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e^2 g^2+c \left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )-\frac {1}{3} g \int \frac {a g \left (5 a e^2 g^2-c \left (6 e^2 f^2-14 d e g f+35 d^2 g^2\right )\right )+c \left (a e (13 e f-42 d g) g^2+c f \left (24 e^2 f^2-56 d e g f+35 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}\right )}{7 g}\)

\(\Big \downarrow \) 599

\(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g}-\frac {2 \left (\frac {2 e \sqrt {a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{5 g^2}-\frac {\frac {2 \int -\frac {\left (c f^2+a g^2\right ) \left (5 a e^2 g^2-c \left (24 e^2 f^2-56 d e g f+35 d^2 g^2\right )\right )+c \left (a e (13 e f-42 d g) g^2+c f \left (24 e^2 f^2-56 d e g f+35 d^2 g^2\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g}+\frac {2}{3} g \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e^2 g^2+c \left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )}{5 c g^3}\right )}{7 g}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g}-\frac {2 \left (\frac {2 e \sqrt {a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{5 g^2}-\frac {\frac {2}{3} g \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e^2 g^2+c \left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )-\frac {2 \int \frac {\left (c f^2+a g^2\right ) \left (5 a e^2 g^2-c \left (24 e^2 f^2-56 d e g f+35 d^2 g^2\right )\right )+c \left (a e (13 e f-42 d g) g^2+c f \left (24 e^2 f^2-56 d e g f+35 d^2 g^2\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g}}{5 c g^3}\right )}{7 g}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g}-\frac {2 \left (\frac {2 e \sqrt {a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{5 g^2}-\frac {\frac {2 \left (\sqrt {c} \sqrt {a g^2+c f^2} \left (a e g^2 (13 e f-42 d g)+c f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\sqrt {a g^2+c f^2} \left (\sqrt {a g^2+c f^2} \left (5 a e^2 g^2-c \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right )+\sqrt {c} \left (a e g^2 (13 e f-42 d g)+c f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{3 g}+\frac {2}{3} g \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e^2 g^2+c \left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )}{5 c g^3}\right )}{7 g}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g}-\frac {2 \left (\frac {2 e \sqrt {a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{5 g^2}-\frac {\frac {2 \left (\sqrt {c} \sqrt {a g^2+c f^2} \left (a e g^2 (13 e f-42 d g)+c f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (\sqrt {a g^2+c f^2} \left (5 a e^2 g^2-c \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right )+\sqrt {c} \left (a e g^2 (13 e f-42 d g)+c f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}\right )}{3 g}+\frac {2}{3} g \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e^2 g^2+c \left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )}{5 c g^3}\right )}{7 g}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {c x^2+a}}{7 g}-\frac {2 \left (\frac {2 e (3 e f-2 d g) (f+g x)^{3/2} \sqrt {c x^2+a}}{5 g^2}-\frac {\frac {2}{3} g \sqrt {f+g x} \sqrt {c x^2+a} \left (5 a e^2 g^2+c \left (21 e^2 f^2-34 d e g f+10 d^2 g^2\right )\right )+\frac {2 \left (\sqrt {c} \sqrt {c f^2+a g^2} \left (a e (13 e f-42 d g) g^2+c f \left (24 e^2 f^2-56 d e g f+35 d^2 g^2\right )\right ) \left (\frac {\sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}-\frac {\sqrt {f+g x} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\right )-\frac {\left (c f^2+a g^2\right )^{3/4} \left (\sqrt {c f^2+a g^2} \left (5 a e^2 g^2-c \left (24 e^2 f^2-56 d e g f+35 d^2 g^2\right )\right )+\sqrt {c} \left (a e (13 e f-42 d g) g^2+c f \left (24 e^2 f^2-56 d e g f+35 d^2 g^2\right )\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{3 g}}{5 c g^3}\right )}{7 g}\)

Input:

Int[((d + e*x)^2*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]
 

Output:

(2*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(7*g) - (2*((2*e*(3*e*f - 2* 
d*g)*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(5*g^2) - ((2*g*(5*a*e^2*g^2 + c*(21 
*e^2*f^2 - 34*d*e*f*g + 10*d^2*g^2))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/3 + (2 
*(Sqrt[c]*Sqrt[c*f^2 + a*g^2]*(a*e*g^2*(13*e*f - 42*d*g) + c*f*(24*e^2*f^2 
 - 56*d*e*f*g + 35*d^2*g^2))*(-((Sqrt[f + g*x]*Sqrt[a + (c*f^2)/g^2 - (2*c 
*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])/((a + (c*f^2)/g^2)*(1 + (Sqrt[c] 
*(f + g*x))/Sqrt[c*f^2 + a*g^2]))) + ((c*f^2 + a*g^2)^(1/4)*(1 + (Sqrt[c]* 
(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/ 
g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqr 
t[c*f^2 + a*g^2])^2)]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + 
a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2])/2])/(c^(1/4)*Sqrt[a + 
 (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])) - ((c*f^2 + 
a*g^2)^(3/4)*(Sqrt[c*f^2 + a*g^2]*(5*a*e^2*g^2 - c*(24*e^2*f^2 - 56*d*e*f* 
g + 35*d^2*g^2)) + Sqrt[c]*(a*e*g^2*(13*e*f - 42*d*g) + c*f*(24*e^2*f^2 - 
56*d*e*f*g + 35*d^2*g^2)))*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*S 
qrt[(a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((a + 
(c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*EllipticF[2 
*ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/S 
qrt[c*f^2 + a*g^2])/2])/(2*c^(1/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x) 
)/g^2 + (c*(f + g*x)^2)/g^2])))/(3*g))/(5*c*g^3)))/(7*g)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 723

Int[(((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(a_) + (c_.)*(x_)^2])/Sqrt[(f_.) + (g_ 
.)*(x_)], x_Symbol] :> Simp[2*(d + e*x)^m*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/(g 
*(2*m + 3))), x] - Simp[1/(g*(2*m + 3))   Int[((d + e*x)^(m - 1)/(Sqrt[f + 
g*x]*Sqrt[a + c*x^2]))*Simp[2*a*(e*f*m - d*g*(m + 1)) + (2*c*d*f - 2*a*e*g) 
*x - (2*c*(d*g*m - e*f*(m + 1)))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g 
}, x] && IntegerQ[2*m] && GtQ[m, 0]
 

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 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 = 3.95 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.26

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{2} x^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{7 g}+\frac {2 \left (2 d e c -\frac {6 c f \,e^{2}}{7 g}\right ) x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{5 c g}+\frac {2 \left (\frac {2 a \,e^{2}}{7}+c \,d^{2}-\frac {4 f \left (2 d e c -\frac {6 c f \,e^{2}}{7 g}\right )}{5 g}\right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{3 c g}+\frac {2 \left (a \,d^{2}-\frac {2 a f \left (2 d e c -\frac {6 c f \,e^{2}}{7 g}\right )}{5 c g}-\frac {a \left (\frac {2 a \,e^{2}}{7}+c \,d^{2}-\frac {4 f \left (2 d e c -\frac {6 c f \,e^{2}}{7 g}\right )}{5 g}\right )}{3 c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {2 \left (2 a d e -\frac {4 a f \,e^{2}}{7 g}-\frac {3 a \left (2 d e c -\frac {6 c f \,e^{2}}{7 g}\right )}{5 c}-\frac {2 f \left (\frac {2 a \,e^{2}}{7}+c \,d^{2}-\frac {4 f \left (2 d e c -\frac {6 c f \,e^{2}}{7 g}\right )}{5 g}\right )}{3 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\)	\(848\)
risch	\(\text {Expression too large to display}\)	\(1363\)
default	\(\text {Expression too large to display}\)	\(3278\)

Input:

int((e*x+d)^2*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/7*e^2/g*x^2*(c* 
g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/5*(2*d*e*c-6/7*c*f/g*e^2)/c/g*x*(c*g*x^3+ 
c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(2/7*a*e^2+c*d^2-4/5*f/g*(2*d*e*c-6/7*c*f/g*e 
^2))/c/g*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2*(a*d^2-2/5*a*f/c/g*(2*d*e*c-6 
/7*c*f/g*e^2)-1/3*a/c*(2/7*a*e^2+c*d^2-4/5*f/g*(2*d*e*c-6/7*c*f/g*e^2)))*( 
f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/ 
c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c)) 
^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)*EllipticF(((x+f/g)/(f/g-(-a*c)^(1 
/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+2*(2*a* 
d*e-4/7*a*f/g*e^2-3/5*a/c*(2*d*e*c-6/7*c*f/g*e^2)-2/3*f/g*(2/7*a*e^2+c*d^2 
-4/5*f/g*(2*d*e*c-6/7*c*f/g*e^2)))*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a* 
c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(- 
a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/ 
2)*((-f/g-(-a*c)^(1/2)/c)*EllipticE(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),( 
(-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c*Ellipti 
cF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c 
)^(1/2)/c))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.61 \[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (2 \, {\left (24 \, c^{2} e^{2} f^{4} - 56 \, c^{2} d e f^{3} g - 84 \, a c d e f g^{3} + {\left (35 \, c^{2} d^{2} + 31 \, a c e^{2}\right )} f^{2} g^{2} + 15 \, {\left (7 \, a c d^{2} - a^{2} e^{2}\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (24 \, c^{2} e^{2} f^{3} g - 56 \, c^{2} d e f^{2} g^{2} - 42 \, a c d e g^{4} + {\left (35 \, c^{2} d^{2} + 13 \, a c e^{2}\right )} f g^{3}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, {\left (15 \, c^{2} e^{2} g^{4} x^{2} + 24 \, c^{2} e^{2} f^{2} g^{2} - 56 \, c^{2} d e f g^{3} + 5 \, {\left (7 \, c^{2} d^{2} + 2 \, a c e^{2}\right )} g^{4} - 6 \, {\left (3 \, c^{2} e^{2} f g^{3} - 7 \, c^{2} d e g^{4}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{315 \, c^{2} g^{5}} \] Input:

integrate((e*x+d)^2*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="fricas")
 

Output:

2/315*(2*(24*c^2*e^2*f^4 - 56*c^2*d*e*f^3*g - 84*a*c*d*e*f*g^3 + (35*c^2*d 
^2 + 31*a*c*e^2)*f^2*g^2 + 15*(7*a*c*d^2 - a^2*e^2)*g^4)*sqrt(c*g)*weierst 
rassPInverse(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^2)/(c*g 
^3), 1/3*(3*g*x + f)/g) + 6*(24*c^2*e^2*f^3*g - 56*c^2*d*e*f^2*g^2 - 42*a* 
c*d*e*g^4 + (35*c^2*d^2 + 13*a*c*e^2)*f*g^3)*sqrt(c*g)*weierstrassZeta(4/3 
*(c*f^2 - 3*a*g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^2)/(c*g^3), weierstrass 
PInverse(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^2)/(c*g^3), 
 1/3*(3*g*x + f)/g)) + 3*(15*c^2*e^2*g^4*x^2 + 24*c^2*e^2*f^2*g^2 - 56*c^2 
*d*e*f*g^3 + 5*(7*c^2*d^2 + 2*a*c*e^2)*g^4 - 6*(3*c^2*e^2*f*g^3 - 7*c^2*d* 
e*g^4)*x)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c^2*g^5)
 

Sympy [F]

\[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {a + c x^{2}} \left (d + e x\right )^{2}}{\sqrt {f + g x}}\, dx \] Input:

integrate((e*x+d)**2*(c*x**2+a)**(1/2)/(g*x+f)**(1/2),x)
 

Output:

Integral(sqrt(a + c*x**2)*(d + e*x)**2/sqrt(f + g*x), x)
 

Maxima [F]

\[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2}}{\sqrt {g x + f}} \,d x } \] Input:

integrate((e*x+d)^2*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^2 + a)*(e*x + d)^2/sqrt(g*x + f), x)
 

Giac [F]

\[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2}}{\sqrt {g x + f}} \,d x } \] Input:

integrate((e*x+d)^2*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^2 + a)*(e*x + d)^2/sqrt(g*x + f), x)
 

Mupad [F(-1)]

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

int(((a + c*x^2)^(1/2)*(d + e*x)^2)/(f + g*x)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int(((a + c*x^2)^(1/2)*(d + e*x)^2)/(f + g*x)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {28 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, a d e \,g^{2}-2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, a \,e^{2} f g +28 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, c d e f g x -12 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, c \,e^{2} f^{2} x +10 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, c \,e^{2} f g \,x^{2}-42 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c d e \,g^{3}+13 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c \,e^{2} f \,g^{2}+35 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} d^{2} f \,g^{2}-56 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} d e \,f^{2} g +24 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} e^{2} f^{3}-14 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a^{2} d e \,g^{3}+\left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a^{2} e^{2} f \,g^{2}+35 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c \,d^{2} f \,g^{2}-28 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c d e \,f^{2} g +12 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c \,e^{2} f^{3}}{35 c f \,g^{2}} \] Input:

int((e*x+d)^2*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x)
 

Output:

(28*sqrt(f + g*x)*sqrt(a + c*x**2)*a*d*e*g**2 - 2*sqrt(f + g*x)*sqrt(a + c 
*x**2)*a*e**2*f*g + 28*sqrt(f + g*x)*sqrt(a + c*x**2)*c*d*e*f*g*x - 12*sqr 
t(f + g*x)*sqrt(a + c*x**2)*c*e**2*f**2*x + 10*sqrt(f + g*x)*sqrt(a + c*x* 
*2)*c*e**2*f*g*x**2 - 42*int((sqrt(f + g*x)*sqrt(a + c*x**2)*x**2)/(a*f + 
a*g*x + c*f*x**2 + c*g*x**3),x)*a*c*d*e*g**3 + 13*int((sqrt(f + g*x)*sqrt( 
a + c*x**2)*x**2)/(a*f + a*g*x + c*f*x**2 + c*g*x**3),x)*a*c*e**2*f*g**2 + 
 35*int((sqrt(f + g*x)*sqrt(a + c*x**2)*x**2)/(a*f + a*g*x + c*f*x**2 + c* 
g*x**3),x)*c**2*d**2*f*g**2 - 56*int((sqrt(f + g*x)*sqrt(a + c*x**2)*x**2) 
/(a*f + a*g*x + c*f*x**2 + c*g*x**3),x)*c**2*d*e*f**2*g + 24*int((sqrt(f + 
 g*x)*sqrt(a + c*x**2)*x**2)/(a*f + a*g*x + c*f*x**2 + c*g*x**3),x)*c**2*e 
**2*f**3 - 14*int((sqrt(f + g*x)*sqrt(a + c*x**2))/(a*f + a*g*x + c*f*x**2 
 + c*g*x**3),x)*a**2*d*e*g**3 + int((sqrt(f + g*x)*sqrt(a + c*x**2))/(a*f 
+ a*g*x + c*f*x**2 + c*g*x**3),x)*a**2*e**2*f*g**2 + 35*int((sqrt(f + g*x) 
*sqrt(a + c*x**2))/(a*f + a*g*x + c*f*x**2 + c*g*x**3),x)*a*c*d**2*f*g**2 
- 28*int((sqrt(f + g*x)*sqrt(a + c*x**2))/(a*f + a*g*x + c*f*x**2 + c*g*x* 
*3),x)*a*c*d*e*f**2*g + 12*int((sqrt(f + g*x)*sqrt(a + c*x**2))/(a*f + a*g 
*x + c*f*x**2 + c*g*x**3),x)*a*c*e**2*f**3)/(35*c*f*g**2)