\(\int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx\) [125]

Optimal result

Integrand size = 28, antiderivative size = 596 \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {4 e (e f-5 d g) \sqrt {f+g x} \sqrt {a+c x^2}}{15 c g}+\frac {2 e^2 (f+g x)^{3/2} \sqrt {a+c x^2}}{5 c g}+\frac {2 \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (9 a e^2 g^2+c \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{15 c^{7/4} g^3 \sqrt {a+c x^2}}+\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (15 c^{3/2} d^2 f g+9 \sqrt {-a} a e^2 g^2-a \sqrt {c} e g (7 e f+10 d g)+\sqrt {-a} c \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{15 c^{7/4} g^2 \sqrt {a+c x^2}} \] Output:

-4/15*e*(-5*d*g+e*f)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/g+2/5*e^2*(g*x+f)^(3/ 
2)*(c*x^2+a)^(1/2)/c/g+2/15*(c^(1/2)*f-(-a)^(1/2)*g)*(c^(1/2)*f+(-a)^(1/2) 
*g)^(1/2)*(9*a*e^2*g^2+c*(-15*d^2*g^2-10*d*e*f*g+2*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)*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^(7/4)/g^3 
/(c*x^2+a)^(1/2)+2/15*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(15*c^(3/2)*d^2*f*g+9 
*(-a)^(1/2)*a*e^2*g^2-a*c^(1/2)*e*g*(10*d*g+7*e*f)+(-a)^(1/2)*c*(-15*d^2*g 
^2-10*d*e*f*g+2*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^(7/4)/g^2/(c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.80 (sec) , antiderivative size = 591, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 e \left (a+c x^2\right ) (10 d g+e (f+3 g x))}{c g}+\frac {(f+g x) \left (\frac {2 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (-9 a e^2 g^2+c \left (-2 e^2 f^2+10 d e f g+15 d^2 g^2\right )\right ) \left (a+c x^2\right )}{(f+g x)^2}+\frac {2 \sqrt {c} \left (-i \sqrt {c} f+\sqrt {a} g\right ) \left (-9 a e^2 g^2+c \left (-2 e^2 f^2+10 d e f g+15 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {2 \sqrt {c} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (15 i c d^2 g-9 i a e^2 g+2 \sqrt {a} \sqrt {c} e (e f-5 d g)\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}\right )}{c^2 g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}\right )}{15 \sqrt {a+c x^2}} \] Input:

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

Output:

(Sqrt[f + g*x]*((2*e*(a + c*x^2)*(10*d*g + e*(f + 3*g*x)))/(c*g) + ((f + g 
*x)*((2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-9*a*e^2*g^2 + c*(-2*e^2*f^2 
 + 10*d*e*f*g + 15*d^2*g^2))*(a + c*x^2))/(f + g*x)^2 + (2*Sqrt[c]*((-I)*S 
qrt[c]*f + Sqrt[a]*g)*(-9*a*e^2*g^2 + c*(-2*e^2*f^2 + 10*d*e*f*g + 15*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))]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqr 
t[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] 
)/Sqrt[f + g*x] + (2*Sqrt[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((15*I)*c*d^2*g - 
 (9*I)*a*e^2*g + 2*Sqrt[a]*Sqrt[c]*e*(e*f - 5*d*g))*Sqrt[(g*((I*Sqrt[a])/S 
qrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*E 
llipticF[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[f + g*x]))/(c^2*g^3*S 
qrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])))/(15*Sqrt[a + c*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 2.03 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.28, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {735, 25, 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.

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

\(\Big \downarrow \) 735

\(\displaystyle \frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}-\frac {\int -\frac {5 c f d^2+c e (e f+7 d g) x^2-a e (2 e f+d g)-\left (3 a e^2 g-c d (8 e f+5 d g)\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {5 c f d^2+c e (e f+7 d g) x^2-a e (2 e f+d g)-\left (3 a e^2 g-c d (8 e f+5 d g)\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\)

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 599

\(\displaystyle \frac {\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}-\frac {2 \int -\frac {2 e (e f-5 d g) \left (c f^2+a g^2\right )-\left (9 a e^2 g^2+c \left (2 e^2 f^2-10 d e g f-15 d^2 g^2\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g^3}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \int \frac {2 e (e f-5 d g) \left (c f^2+a g^2\right )-\left (9 a e^2 g^2+c \left (2 e^2 f^2-10 d e g f-15 d^2 g^2\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g^3}+\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}-\frac {2 \left (\frac {\sqrt {a g^2+c f^2} \left (-2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-5 d g)+9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\sqrt {a g^2+c f^2} \left (9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{3 g^3}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}-\frac {2 \left (\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (-2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-5 d g)+9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {a g^2+c f^2} \left (9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{3 g^3}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}-\frac {2 \left (\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (-2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-5 d g)+9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {a g^2+c f^2} \left (9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \left (\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {f+g x} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )}\right )}{\sqrt {c}}\right )}{3 g^3}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\)

Input:

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

Output:

(2*e*(d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(5*c) + ((2*e*(e*f + 7*d*g)* 
Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*g) - (2*(-((Sqrt[c*f^2 + a*g^2]*(9*a*e^2 
*g^2 + c*(2*e^2*f^2 - 10*d*e*f*g - 15*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 + (Sqr 
t[c]*(f + g*x))/Sqrt[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])))/Sqrt[c]) + ((c*f^2 + a*g^2)^(3/4)*(9*a*e^2*g^2 - 2*Sqrt[c]*e*(e*f 
- 5*d*g)*Sqrt[c*f^2 + a*g^2] + c*(2*e^2*f^2 - 10*d*e*f*g - 15*d^2*g^2))*(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))/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)/Sqrt[c*f^2 + a*g^2])/2])/(2*c^ 
(3/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2]) 
))/(3*g^3))/(5*c)
 

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 735

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_) + (c_.)* 
(x_)^2], x_Symbol] :> Simp[2*e*(d + e*x)^(m - 1)*Sqrt[f + g*x]*(Sqrt[a + c* 
x^2]/(c*(2*m + 1))), x] - Simp[1/(c*(2*m + 1))   Int[((d + e*x)^(m - 2)/(Sq 
rt[f + g*x]*Sqrt[a + c*x^2]))*Simp[a*e*(d*g + 2*e*f*(m - 1)) - c*d^2*f*(2*m 
 + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f*m + d*g*(2*m + 1)))*x - c*e*(e*f + 
d*g*(4*m - 1))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && IntegerQ[ 
2*m] && GtQ[m, 1]
 

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 = 3.27 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.17

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{2} x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{5 c}+\frac {2 \left (2 g d e +\frac {1}{5} e^{2} f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{3 c g}+\frac {2 \left (d^{2} f -\frac {2 a f \,e^{2}}{5 c}-\frac {a \left (2 g d e +\frac {1}{5} e^{2} f \right )}{3 c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {2 \left (d^{2} g +2 d e f -\frac {3 a g \,e^{2}}{5 c}-\frac {2 f \left (2 g d e +\frac {1}{5} e^{2} f \right )}{3 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\)	\(700\)
risch	\(\text {Expression too large to display}\)	\(1073\)
default	\(\text {Expression too large to display}\)	\(2470\)

Input:

int((e*x+d)^2*(g*x+f)^(1/2)/(c*x^2+a)^(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)*(2/5*e^2/c*x*(c*g* 
x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(2*g*d*e+1/5*e^2*f)/c/g*(c*g*x^3+c*f*x^2+ 
a*g*x+a*f)^(1/2)+2*(d^2*f-2/5*a*f/c*e^2-1/3*a/c*(2*g*d*e+1/5*e^2*f))*(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*(d^2*g+2* 
d*e*f-3/5*a/c*g*e^2-2/3*f/g*(2*g*d*e+1/5*e^2*f))*(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*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))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.50 \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {2 \, {\left (2 \, {\left (c e^{2} f^{3} - 5 \, c d e f^{2} g - 15 \, a d e g^{3} + 3 \, {\left (5 \, c d^{2} - 2 \, a e^{2}\right )} f g^{2}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 3 \, {\left (2 \, c e^{2} f^{2} g - 10 \, c d e f g^{2} - 3 \, {\left (5 \, c d^{2} - 3 \, a e^{2}\right )} g^{3}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, {\left (3 \, c e^{2} g^{3} x + c e^{2} f g^{2} + 10 \, c d e g^{3}\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{45 \, c^{2} g^{3}} \] Input:

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

Output:

2/45*(2*(c*e^2*f^3 - 5*c*d*e*f^2*g - 15*a*d*e*g^3 + 3*(5*c*d^2 - 2*a*e^2)* 
f*g^2)*sqrt(c*g)*weierstrassPInverse(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*(2*c*e^2*f^2*g - 10*c* 
d*e*f*g^2 - 3*(5*c*d^2 - 3*a*e^2)*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), weierstrassPInver 
se(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*(3*c*e^2*g^3*x + c*e^2*f*g^2 + 10*c*d*e*g^3)*sqrt(c*x^2 
 + a)*sqrt(g*x + f))/(c^2*g^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {-6 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, a \,e^{2} g +10 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, c \,d^{2} g +20 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, c d e f +4 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, c \,e^{2} f x +9 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c \,e^{2} g^{2}-15 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} d^{2} g^{2}-10 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} d e f g +2 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} e^{2} f^{2}+3 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a^{2} e^{2} g^{2}-5 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c \,d^{2} g^{2}-10 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c d e f g -4 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c \,e^{2} f^{2}+10 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} d^{2} f^{2}}{10 c^{2} f} \] Input:

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

Output:

( - 6*sqrt(f + g*x)*sqrt(a + c*x**2)*a*e**2*g + 10*sqrt(f + g*x)*sqrt(a + 
c*x**2)*c*d**2*g + 20*sqrt(f + g*x)*sqrt(a + c*x**2)*c*d*e*f + 4*sqrt(f + 
g*x)*sqrt(a + c*x**2)*c*e**2*f*x + 9*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*g**2 - 15*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*g**2 - 10*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*g + 2*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**2 + 3*int((sqr 
t(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*g**2 - 5*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*g**2 - 10*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*g - 4*int((sqrt(f + g*x)*sqr 
t(a + c*x**2))/(a*f + a*g*x + c*f*x**2 + c*g*x**3),x)*a*c*e**2*f**2 + 10*i 
nt((sqrt(f + g*x)*sqrt(a + c*x**2))/(a*f + a*g*x + c*f*x**2 + c*g*x**3),x) 
*c**2*d**2*f**2)/(10*c**2*f)