\(\int (d+e x) \sqrt {f+g x} \sqrt {a+c x^2} \, dx\) [112]

Optimal result

Integrand size = 26, antiderivative size = 600 \[ \int (d+e x) \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {4}{105} \left (\frac {5 a e}{c}+\frac {f (4 e f-7 d g)}{g^2}\right ) \sqrt {f+g x} \sqrt {a+c x^2}-\frac {2 (f+g x)^{3/2} (4 e f-7 d g-5 e g x) \sqrt {a+c x^2}}{35 g^2}+\frac {4 \left (\sqrt {-a}-\frac {\sqrt {c} f}{g}\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (c f^2 (4 e f-7 d g)+a g^2 (8 e f+21 d g)\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{105 c^{3/4} g^3 \sqrt {a+c x^2}}-\frac {4 \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (5 a^2 e g^3+a c f g (e f-28 d g)+\sqrt {-a} c^{3/2} f^2 (4 e f-7 d g)+\sqrt {-a} a \sqrt {c} g^2 (8 e f+21 d g)\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{105 c^{5/4} g^3 \sqrt {a+c x^2}} \] Output:

4/105*(5*a*e/c+f*(-7*d*g+4*e*f)/g^2)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)-2/35*(g 
*x+f)^(3/2)*(-5*e*g*x-7*d*g+4*e*f)*(c*x^2+a)^(1/2)/g^2+4/105*((-a)^(1/2)-c 
^(1/2)*f/g)*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(c*f^2*(-7*d*g+4*e*f)+a*g^2*(21 
*d*g+8*e*f))*(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^3/(c*x^2+a)^(1/2)-4/105*(c^(1/2)*f+(-a)^(1/2)*g)^ 
(1/2)*(5*a^2*e*g^3+a*c*f*g*(-28*d*g+e*f)+(-a)^(1/2)*c^(3/2)*f^2*(-7*d*g+4* 
e*f)+(-a)^(1/2)*a*c^(1/2)*g^2*(21*d*g+8*e*f))*(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^3/(c*x^2+a)^(1/2 
)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.60 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.02 \[ \int (d+e x) \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (10 a e g^2+7 c d g (f+3 g x)+c e \left (-4 f^2+3 f g x+15 g^2 x^2\right )\right )}{c g^2}+\frac {4 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (c f^2 (4 e f-7 d g)+a g^2 (8 e f+21 d g)\right ) \left (a+c x^2\right )+i \sqrt {c} \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (c f^2 (-4 e f+7 d g)-a g^2 (8 e f+21 d g)\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\sqrt {a} g \left (i \sqrt {c} f-\sqrt {a} g\right ) \left (5 i a e g^2+i c f (4 e f-7 d g)+3 \sqrt {a} \sqrt {c} g (e f+7 d g)\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c g^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{105 \sqrt {a+c x^2}} \] Input:

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

Output:

(Sqrt[f + g*x]*((2*(a + c*x^2)*(10*a*e*g^2 + 7*c*d*g*(f + 3*g*x) + c*e*(-4 
*f^2 + 3*f*g*x + 15*g^2*x^2)))/(c*g^2) + (4*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/S 
qrt[c]]*(c*f^2*(4*e*f - 7*d*g) + a*g^2*(8*e*f + 21*d*g))*(a + c*x^2) + I*S 
qrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(c*f^2*(-4*e*f + 7*d*g) - a*g^2*(8*e*f + 
21*d*g))*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)*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*(I*Sqrt[c]*f - Sqrt[a]*g)*((5*I)*a*e*g^2 + 
 I*c*f*(4*e*f - 7*d*g) + 3*Sqrt[a]*Sqrt[c]*g*(e*f + 7*d*g))*Sqrt[(g*((I*Sq 
rt[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^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(105*Sqrt[a + c*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 1.06 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.31, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {687, 27, 682, 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.

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

\(\Big \downarrow \) 687

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {4 \int -\frac {c \left (a g \left (5 a e g^2+c f (e f-28 d g)\right )-c \left (c (4 e f-7 d g) f^2+a g^2 (8 e f+21 d g)\right ) x\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{15 c g^2}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{15 g^2}}{7 c}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {2 \int \frac {a g \left (5 a e g^2+c f (e f-28 d g)\right )-c \left (c (4 e f-7 d g) f^2+a g^2 (8 e f+21 d g)\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{15 g^2}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{15 g^2}}{7 c}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 599

\(\displaystyle \frac {\frac {4 \int -\frac {\left (c f^2+a g^2\right ) \left (5 a e g^2+c f (4 e f-7 d g)\right )-c \left (c (4 e f-7 d g) f^2+a g^2 (8 e f+21 d g)\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{15 g^4}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{15 g^2}}{7 c}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {4 \int \frac {\left (c f^2+a g^2\right ) \left (5 a e g^2+c f (4 e f-7 d g)\right )-c \left (c (4 e f-7 d g) f^2+a g^2 (8 e f+21 d g)\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{15 g^4}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{15 g^2}}{7 c}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\frac {4 \left (-\sqrt {a g^2+c f^2} \left (\sqrt {a g^2+c f^2} \left (5 a e g^2+c f (4 e f-7 d g)\right )-\sqrt {c} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\sqrt {c} \sqrt {a g^2+c f^2} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{15 g^4}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{15 g^2}}{7 c}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {4 \left (-\sqrt {c} \sqrt {a g^2+c f^2} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (\sqrt {a g^2+c f^2} \left (5 a e g^2+c f (4 e f-7 d g)\right )-\sqrt {c} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}\right )}{15 g^4}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{15 g^2}}{7 c}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\frac {4 \left (-\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (\sqrt {a g^2+c f^2} \left (5 a e g^2+c f (4 e f-7 d g)\right )-\sqrt {c} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\sqrt {c} \sqrt {a g^2+c f^2} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right ) \left (\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {f+g x} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )}\right )\right )}{15 g^4}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{15 g^2}}{7 c}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {f+g x}}{7 c}\)

Input:

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

Output:

(2*e*Sqrt[f + g*x]*(a + c*x^2)^(3/2))/(7*c) + ((-2*Sqrt[f + g*x]*(5*a*e*g^ 
2 + c*f*(4*e*f - 7*d*g) - 3*c*g*(e*f + 7*d*g)*x)*Sqrt[a + c*x^2])/(15*g^2) 
 + (4*(-(Sqrt[c]*Sqrt[c*f^2 + a*g^2]*(c*f^2*(4*e*f - 7*d*g) + a*g^2*(8*e*f 
 + 21*d*g))*(-((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))/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]))) - ((c*f^2 + a*g^2)^(3/4)*(Sq 
rt[c*f^2 + a*g^2]*(5*a*e*g^2 + c*f*(4*e*f - 7*d*g)) - Sqrt[c]*(c*f^2*(4*e* 
f - 7*d*g) + a*g^2*(8*e*f + 21*d*g)))*(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 + (S 
qrt[c]*f)/Sqrt[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])))/(15*g^4))/(7*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 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 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]
 

Maple [A] (verified)

Time = 2.55 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.32

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e \,x^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{7}+\frac {2 \left (c d g +\frac {1}{7} f c e \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 g}{7}+d f c -\frac {4 f \left (c d g +\frac {1}{7} f c e \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 f -\frac {2 a f \left (c d g +\frac {1}{7} f c e \right )}{5 c g}-\frac {a \left (\frac {2 a e g}{7}+d f c -\frac {4 f \left (c d g +\frac {1}{7} f c e \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 (a d g +\frac {3 a e f}{7}-\frac {3 a \left (c d g +\frac {1}{7} f c e \right )}{5 c}-\frac {2 f \left (\frac {2 a e g}{7}+d f c -\frac {4 f \left (c d g +\frac {1}{7} f c e \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}}\)	\(794\)
risch	\(\text {Expression too large to display}\)	\(1105\)
default	\(\text {Expression too large to display}\)	\(2551\)

Input:

int((e*x+d)*(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)*(2/7*e*x^2*(c*g*x^ 
3+c*f*x^2+a*g*x+a*f)^(1/2)+2/5*(c*d*g+1/7*f*c*e)/c/g*x*(c*g*x^3+c*f*x^2+a* 
g*x+a*f)^(1/2)+2/3*(2/7*a*e*g+d*f*c-4/5*f/g*(c*d*g+1/7*f*c*e))/c/g*(c*g*x^ 
3+c*f*x^2+a*g*x+a*f)^(1/2)+2*(a*d*f-2/5*a/c*f/g*(c*d*g+1/7*f*c*e)-1/3*a/c* 
(2/7*a*e*g+d*f*c-4/5*f/g*(c*d*g+1/7*f*c*e)))*(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*(a*d*g+3/7*a*e*f-3/5*a/c*(c*d*g+1 
/7*f*c*e)-2/3*f/g*(2/7*a*e*g+d*f*c-4/5*f/g*(c*d*g+1/7*f*c*e)))*(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 = 0.10 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.57 \[ \int (d+e x) \sqrt {f+g x} \sqrt {a+c x^2} \, dx=-\frac {2 \, {\left (2 \, {\left (4 \, c^{2} e f^{4} - 7 \, c^{2} d f^{3} g + 11 \, a c e f^{2} g^{2} - 63 \, a c d f g^{3} + 15 \, a^{2} e g^{4}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (4 \, c^{2} e f^{3} g - 7 \, c^{2} d f^{2} g^{2} + 8 \, a c e f g^{3} + 21 \, a c d g^{4}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) - 3 \, {\left (15 \, c^{2} e g^{4} x^{2} - 4 \, c^{2} e f^{2} g^{2} + 7 \, c^{2} d f g^{3} + 10 \, a c e g^{4} + 3 \, {\left (c^{2} e f g^{3} + 7 \, c^{2} d g^{4}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{315 \, c^{2} g^{4}} \] Input:

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

Output:

-2/315*(2*(4*c^2*e*f^4 - 7*c^2*d*f^3*g + 11*a*c*e*f^2*g^2 - 63*a*c*d*f*g^3 
 + 15*a^2*e*g^4)*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) + 6*(4*c^2*e*f^3 
*g - 7*c^2*d*f^2*g^2 + 8*a*c*e*f*g^3 + 21*a*c*d*g^4)*sqrt(c*g)*weierstrass 
Zeta(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), wei 
erstrassPInverse(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*g^4*x^2 - 4*c^2*e*f^2*g^2 + 7*c 
^2*d*f*g^3 + 10*a*c*e*g^4 + 3*(c^2*e*f*g^3 + 7*c^2*d*g^4)*x)*sqrt(c*x^2 + 
a)*sqrt(g*x + f))/(c^2*g^4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x) \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {14 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, a d \,g^{2}+12 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, a e f g +14 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, c d f g x +2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, c e \,f^{2} x +10 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, c e f g \,x^{2}-21 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c d \,g^{3}-8 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c e f \,g^{2}+7 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} d \,f^{2} g -4 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} e \,f^{3}-7 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a^{2} d \,g^{3}-6 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a^{2} e f \,g^{2}+21 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c d \,f^{2} g -2 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c e \,f^{3}}{35 c f g} \] Input:

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

Output:

(14*sqrt(f + g*x)*sqrt(a + c*x**2)*a*d*g**2 + 12*sqrt(f + g*x)*sqrt(a + c* 
x**2)*a*e*f*g + 14*sqrt(f + g*x)*sqrt(a + c*x**2)*c*d*f*g*x + 2*sqrt(f + g 
*x)*sqrt(a + c*x**2)*c*e*f**2*x + 10*sqrt(f + g*x)*sqrt(a + c*x**2)*c*e*f* 
g*x**2 - 21*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*g**3 - 8*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*f*g**2 + 7*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*f* 
*2*g - 4*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*f**3 - 7*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*g**3 - 6*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*f*g**2 + 21*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*f**2*g - 2*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*f**3)/(35*c*f*g)