3.7.23 \(\int (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2} \, dx\) [623]

3.7.23.1 Optimal result

Integrand size = 28, antiderivative size = 635 \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=-\frac {2 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 e}+\frac {4 \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^3}+\frac {2 e (e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^3}+\frac {4 \sqrt {-a} \left (21 a^2 e^2 g^4+3 a c g^2 \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )+c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (3 a e g^2 (e f-10 d g)+c f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt {f+g x} \sqrt {a+c x^2}} \]

output

4/315*(7*a*e^2*g^2-c*(21*d^2*g^2-24*d*e*f*g+8*e^2*f^2))*(g*x+f)^(3/2)*(c*x 
^2+a)^(1/2)/c/g^3+2/63*e*(-3*d*g+e*f)*(g*x+f)^(5/2)*(c*x^2+a)^(1/2)/g^3-2/ 
315*(6*a*e^2*g^2*(-10*d*g+e*f)-c*(-35*d^3*g^3+63*d^2*e*f*g^2-57*d*e^2*f^2* 
g+19*e^3*f^3))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/e/g^3+2/9*(e*x+d)^3*(g*x+f) 
^(1/2)*(c*x^2+a)^(1/2)/e+4/315*(21*a^2*e^2*g^4+3*a*c*g^2*(-21*d^2*g^2-16*d 
*e*f*g+3*e^2*f^2)+c^2*f^2*(21*d^2*g^2-24*d*e*f*g+8*e^2*f^2))*EllipticE(1/2 
*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2) 
))^(1/2))*(-a)^(1/2)*(g*x+f)^(1/2)*(1+c*x^2/a)^(1/2)/c^(3/2)/g^4/(c*x^2+a) 
^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)-4/315*(a*g^2+c*f^2 
)*(3*a*e*g^2*(-10*d*g+e*f)+c*f*(21*d^2*g^2-24*d*e*f*g+8*e^2*f^2))*Elliptic 
F(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^ 
(1/2)))^(1/2))*(-a)^(1/2)*(1+c*x^2/a)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2) 
+f*c^(1/2)))^(1/2)/c^(3/2)/g^4/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)
 

3.7.23.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.89 (sec) , antiderivative size = 809, normalized size of antiderivative = 1.27 \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (2 a e g^2 (4 e f+30 d g+7 e g x)+c \left (21 d^2 g^2 (f+3 g x)+6 d e g \left (-4 f^2+3 f g x+15 g^2 x^2\right )+e^2 \left (8 f^3-6 f^2 g x+5 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{c g^3}-\frac {4 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (21 a^2 e^2 g^4+c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 a c g^2 \left (-3 e^2 f^2+16 d e f g+21 d^2 g^2\right )\right ) \left (a+c x^2\right )-i \sqrt {c} \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (21 a^2 e^2 g^4+c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 a c g^2 \left (-3 e^2 f^2+16 d e f g+21 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\sqrt {a} \sqrt {c} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (21 i a^{3/2} e^2 g^3-3 a \sqrt {c} e g^2 (e f-10 d g)+c^{3/2} f \left (-8 e^2 f^2+24 d e f g-21 d^2 g^2\right )-3 i \sqrt {a} c g \left (-2 e^2 f^2+6 d e f g+21 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c^2 g^5 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{315 \sqrt {a+c x^2}} \]

input

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

output

(Sqrt[f + g*x]*((2*(a + c*x^2)*(2*a*e*g^2*(4*e*f + 30*d*g + 7*e*g*x) + c*( 
21*d^2*g^2*(f + 3*g*x) + 6*d*e*g*(-4*f^2 + 3*f*g*x + 15*g^2*x^2) + e^2*(8* 
f^3 - 6*f^2*g*x + 5*f*g^2*x^2 + 35*g^3*x^3))))/(c*g^3) - (4*(g^2*Sqrt[-f - 
 (I*Sqrt[a]*g)/Sqrt[c]]*(21*a^2*e^2*g^4 + c^2*f^2*(8*e^2*f^2 - 24*d*e*f*g 
+ 21*d^2*g^2) - 3*a*c*g^2*(-3*e^2*f^2 + 16*d*e*f*g + 21*d^2*g^2))*(a + c*x 
^2) - I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(21*a^2*e^2*g^4 + c^2*f^2*(8*e^2 
*f^2 - 24*d*e*f*g + 21*d^2*g^2) - 3*a*c*g^2*(-3*e^2*f^2 + 16*d*e*f*g + 21* 
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)*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]*Sqrt[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((21*I) 
*a^(3/2)*e^2*g^3 - 3*a*Sqrt[c]*e*g^2*(e*f - 10*d*g) + c^(3/2)*f*(-8*e^2*f^ 
2 + 24*d*e*f*g - 21*d^2*g^2) - (3*I)*Sqrt[a]*c*g*(-2*e^2*f^2 + 6*d*e*f*g + 
 21*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqr 
t[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqr 
t[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(S 
qrt[c]*f + I*Sqrt[a]*g)]))/(c^2*g^5*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + 
g*x))))/(315*Sqrt[a + c*x^2])
 

3.7.23.3 Rubi [A] (warning: unable to verify)

Time = 2.63 (sec) , antiderivative size = 1041, normalized size of antiderivative = 1.64, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {722, 2185, 27, 2185, 27, 2185, 27, 599, 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.

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

\(\Big \downarrow \) 722

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {2 \int -\frac {-2 c e g^3 \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) x^3-c g^2 \left (4 a e^2 g^2 (4 e f+9 d g)-c \left (11 e^3 f^3-33 d e^2 g f^2+21 d^2 e g^2 f+21 d^3 g^3\right )\right ) x^2+2 c f g \left (a e^2 (5 e f-36 d g) g^2+c \left (e^3 f^3-3 d e^2 g f^2+7 d^3 g^3\right )\right ) x+a c g^2 \left (5 e^3 f^3-15 d e^2 g f^2-21 d^2 e g^2 f+7 d^3 g^3\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 c g^4}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{7 g^3}}{9 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{7 g^3}-\frac {\int \frac {-2 c e g^3 \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) x^3-c g^2 \left (4 a e^2 g^2 (4 e f+9 d g)-c \left (11 e^3 f^3-33 d e^2 g f^2+21 d^2 e g^2 f+21 d^3 g^3\right )\right ) x^2+2 c f g \left (a e^2 (5 e f-36 d g) g^2+c \left (e^3 f^3-3 d e^2 g f^2+7 d^3 g^3\right )\right ) x+a c g^2 \left (5 e^3 f^3-15 d e^2 g f^2-21 d^2 e g^2 f+7 d^3 g^3\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 c g^4}}{9 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 e}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{7 g^3}-\frac {\frac {2 \int \frac {3 c^2 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 g f^2+63 d^2 e g^2 f-35 d^3 g^3\right )\right ) x^2 g^5+a c \left (42 a e^3 f g^2-c \left (23 e^3 f^3-69 d e^2 g f^2+231 d^2 e g^2 f-35 d^3 g^3\right )\right ) g^5+2 c \left (21 a^2 e^3 g^4+3 a c e \left (5 e^2 f^2-36 d e g f-21 d^2 g^2\right ) g^2-c^2 f \left (11 e^3 f^3-33 d e^2 g f^2+42 d^2 e g^2 f-35 d^3 g^3\right )\right ) x g^4}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}-\frac {4}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{7 c g^4}}{9 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{7 g^3}-\frac {\frac {\int \frac {3 c^2 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 g f^2+63 d^2 e g^2 f-35 d^3 g^3\right )\right ) x^2 g^5+a c \left (42 a e^3 f g^2-c \left (23 e^3 f^3-69 d e^2 g f^2+231 d^2 e g^2 f-35 d^3 g^3\right )\right ) g^5+2 c \left (21 a^2 e^3 g^4+3 a c e \left (5 e^2 f^2-36 d e g f-21 d^2 g^2\right ) g^2-c^2 f \left (11 e^3 f^3-33 d e^2 g f^2+42 d^2 e g^2 f-35 d^3 g^3\right )\right ) x g^4}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}-\frac {4}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{7 c g^4}}{9 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 e}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{7 g^3}-\frac {\frac {\frac {2 \int \frac {3 c^2 e g^6 \left (2 a g \left (3 a e g^2 (3 e f+5 d g)-c f \left (e^2 f^2-3 d e g f+42 d^2 g^2\right )\right )+\left (21 a^2 e^2 g^4+3 a c \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) x\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g^2}+2 c g^4 \sqrt {a+c x^2} \sqrt {f+g x} \left (6 a e^2 g^2 (e f-10 d g)-c \left (-35 d^3 g^3+63 d^2 e f g^2-57 d e^2 f^2 g+19 e^3 f^3\right )\right )}{5 c g^3}-\frac {4}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{7 c g^4}}{9 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{7 g^3}-\frac {\frac {2 c e g^4 \int \frac {2 a g \left (3 a e g^2 (3 e f+5 d g)-c f \left (e^2 f^2-3 d e g f+42 d^2 g^2\right )\right )+\left (21 a^2 e^2 g^4+3 a c \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx+2 c g^4 \sqrt {a+c x^2} \sqrt {f+g x} \left (6 a e^2 g^2 (e f-10 d g)-c \left (-35 d^3 g^3+63 d^2 e f g^2-57 d e^2 f^2 g+19 e^3 f^3\right )\right )}{5 c g^3}-\frac {4}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{7 c g^4}}{9 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 e}\)

\(\Big \downarrow \) 599

\(\displaystyle \frac {\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{7 g^3}-\frac {\frac {2 c g^4 \sqrt {a+c x^2} \sqrt {f+g x} \left (6 a e^2 g^2 (e f-10 d g)-c \left (-35 d^3 g^3+63 d^2 e f g^2-57 d e^2 f^2 g+19 e^3 f^3\right )\right )-4 c e g^2 \int \frac {\left (c f^2+a g^2\right ) \left (3 a e (e f-10 d g) g^2+c f \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right )-\left (21 a^2 e^2 g^4+3 a c \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{5 c g^3}-\frac {4}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{7 c g^4}}{9 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 e}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{7 g^3}-\frac {\frac {2 c g^4 \sqrt {a+c x^2} \sqrt {f+g x} \left (6 a e^2 g^2 (e f-10 d g)-c \left (-35 d^3 g^3+63 d^2 e f g^2-57 d e^2 f^2 g+19 e^3 f^3\right )\right )-4 c e g^2 \left (\frac {\sqrt {a g^2+c f^2} \left (21 a^2 e^2 g^4+3 a c g^2 \left (-21 d^2 g^2-16 d e f g+3 e^2 f^2\right )+c^2 f^2 \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\sqrt {a g^2+c f^2} \left (21 a^2 e^2 g^4+3 a c g^2 \left (-21 d^2 g^2-16 d e f g+3 e^2 f^2\right )-\sqrt {c} \sqrt {a g^2+c f^2} \left (3 a e g^2 (e f-10 d g)+c f \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )+c^2 f^2 \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{5 c g^3}-\frac {4}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{7 c g^4}}{9 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 e}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {2 \sqrt {f+g x} \sqrt {c x^2+a} (d+e x)^3}{9 e}+\frac {\frac {2 e^2 (e f-3 d g) (f+g x)^{5/2} \sqrt {c x^2+a}}{7 g^3}-\frac {\frac {2 c g^4 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 g f^2+63 d^2 e g^2 f-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {c x^2+a}-4 c e g^2 \left (\frac {\sqrt {c f^2+a g^2} \left (21 a^2 e^2 g^4+3 a c \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\left (c f^2+a g^2\right )^{3/4} \left (21 a^2 e^2 g^4+3 a c \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )-\sqrt {c} \sqrt {c f^2+a g^2} \left (3 a e (e f-10 d g) g^2+c f \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{5 c g^3}-\frac {4}{5} e g \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {c x^2+a}}{7 c g^4}}{9 e}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {2 \sqrt {f+g x} \sqrt {c x^2+a} (d+e x)^3}{9 e}+\frac {\frac {2 e^2 (e f-3 d g) (f+g x)^{5/2} \sqrt {c x^2+a}}{7 g^3}-\frac {\frac {2 c g^4 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 g f^2+63 d^2 e g^2 f-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {c x^2+a}-4 c e g^2 \left (\frac {\sqrt {c f^2+a g^2} \left (21 a^2 e^2 g^4+3 a c \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 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 )}{\sqrt {c}}-\frac {\left (c f^2+a g^2\right )^{3/4} \left (21 a^2 e^2 g^4+3 a c \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )-\sqrt {c} \sqrt {c f^2+a g^2} \left (3 a e (e f-10 d g) g^2+c f \left (8 e^2 f^2-24 d e g f+21 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 c^{3/4} \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 )}{5 c g^3}-\frac {4}{5} e g \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {c x^2+a}}{7 c g^4}}{9 e}\)

input

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

output

(2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(9*e) + ((2*e^2*(e*f - 3*d*g 
)*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(7*g^3) - ((-4*e*g*(7*a*e^2*g^2 - c*(8* 
e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/5 + ( 
2*c*g^4*(6*a*e^2*g^2*(e*f - 10*d*g) - c*(19*e^3*f^3 - 57*d*e^2*f^2*g + 63* 
d^2*e*f*g^2 - 35*d^3*g^3))*Sqrt[f + g*x]*Sqrt[a + c*x^2] - 4*c*e*g^2*((Sqr 
t[c*f^2 + a*g^2]*(21*a^2*e^2*g^4 + 3*a*c*g^2*(3*e^2*f^2 - 16*d*e*f*g - 21* 
d^2*g^2) + c^2*f^2*(8*e^2*f^2 - 24*d*e*f*g + 21*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))/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)*(21*a^2*e^2*g^4 + 3*a*c 
*g^2*(3*e^2*f^2 - 16*d*e*f*g - 21*d^2*g^2) + c^2*f^2*(8*e^2*f^2 - 24*d*e*f 
*g + 21*d^2*g^2) - Sqrt[c]*Sqrt[c*f^2 + a*g^2]*(3*a*e*g^2*(e*f - 10*d*g) + 
 c*f*(8*e^2*f^2 - 24*d*e*f*g + 21*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*...
 

3.7.23.3.1 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 722

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*( 
x_)^2], x_Symbol] :> Simp[2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^2 
]/(e*(2*m + 5))), x] + Simp[1/(e*(2*m + 5))   Int[((d + e*x)^m/(Sqrt[f + g* 
x]*Sqrt[a + c*x^2]))*Simp[3*a*e*f - a*d*g - 2*(c*d*f - a*e*g)*x + (c*e*f - 
3*c*d*g)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegerQ[2*m 
] &&  !LtQ[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]))
 

3.7.23.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1141\) vs. \(2(551)=1102\).

Time = 1.40 (sec) , antiderivative size = 1142, normalized size of antiderivative = 1.80

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1142\)
risch	\(\text {Expression too large to display}\)	\(1677\)
default	\(\text {Expression too large to display}\)	\(4352\)

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)*(2/9*e^2*x^3*(c*g* 
x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/7*(2*c*d*e*g+1/9*c*e^2*f)/c/g*x^2*(c*g*x^3+ 
c*f*x^2+a*g*x+a*f)^(1/2)+2/5*(2/9*a*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f/g*(2*c*d 
*e*g+1/9*c*e^2*f))/c/g*x*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(2*a*d*e*g+ 
1/3*a*e^2*f+c*d^2*f-4/5*f/g*(2/9*a*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f/g*(2*c*d* 
e*g+1/9*c*e^2*f))-5/7*a/c*(2*c*d*e*g+1/9*c*e^2*f))/c/g*(c*g*x^3+c*f*x^2+a* 
g*x+a*f)^(1/2)+2*(a*d^2*f-2/5*f*a/c/g*(2/9*a*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f 
/g*(2*c*d*e*g+1/9*c*e^2*f))-1/3*a/c*(2*a*d*e*g+1/3*a*e^2*f+c*d^2*f-4/5*f/g 
*(2/9*a*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f/g*(2*c*d*e*g+1/9*c*e^2*f))-5/7*a/c*( 
2*c*d*e*g+1/9*c*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)*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))+2*(a*d^2*g+2*a*d*e*f-4/7*f*a/c/g*(2*c*d*e*g+1/9*c*e^2*f 
)-3/5*a/c*(2/9*a*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f/g*(2*c*d*e*g+1/9*c*e^2*f))- 
2/3*f/g*(2*a*d*e*g+1/3*a*e^2*f+c*d^2*f-4/5*f/g*(2/9*a*e^2*g+c*d^2*g+2*c*d* 
e*f-6/7*f/g*(2*c*d*e*g+1/9*c*e^2*f))-5/7*a/c*(2*c*d*e*g+1/9*c*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...
 

3.7.23.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 510, normalized size of antiderivative = 0.80 \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {2 \, {\left (2 \, {\left (8 \, c^{2} e^{2} f^{5} - 24 \, c^{2} d e f^{4} g - 66 \, a c d e f^{2} g^{3} - 90 \, a^{2} d e g^{5} + 3 \, {\left (7 \, c^{2} d^{2} + 5 \, a c e^{2}\right )} f^{3} g^{2} + 3 \, {\left (63 \, a c d^{2} - 11 \, a^{2} e^{2}\right )} f g^{4}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (8 \, c^{2} e^{2} f^{4} g - 24 \, c^{2} d e f^{3} g^{2} - 48 \, a c d e f g^{4} + 3 \, {\left (7 \, c^{2} d^{2} + 3 \, a c e^{2}\right )} f^{2} g^{3} - 21 \, {\left (3 \, a c d^{2} - a^{2} e^{2}\right )} g^{5}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, {\left (35 \, c^{2} e^{2} g^{5} x^{3} + 8 \, c^{2} e^{2} f^{3} g^{2} - 24 \, c^{2} d e f^{2} g^{3} + 60 \, a c d e g^{5} + {\left (21 \, c^{2} d^{2} + 8 \, a c e^{2}\right )} f g^{4} + 5 \, {\left (c^{2} e^{2} f g^{4} + 18 \, c^{2} d e g^{5}\right )} x^{2} - {\left (6 \, c^{2} e^{2} f^{2} g^{3} - 18 \, c^{2} d e f g^{4} - 7 \, {\left (9 \, c^{2} d^{2} + 2 \, a c e^{2}\right )} g^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{945 \, c^{2} g^{5}} \]

input

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

output

2/945*(2*(8*c^2*e^2*f^5 - 24*c^2*d*e*f^4*g - 66*a*c*d*e*f^2*g^3 - 90*a^2*d 
*e*g^5 + 3*(7*c^2*d^2 + 5*a*c*e^2)*f^3*g^2 + 3*(63*a*c*d^2 - 11*a^2*e^2)*f 
*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*(8*c^2*e^2*f^4*g - 24*c 
^2*d*e*f^3*g^2 - 48*a*c*d*e*f*g^4 + 3*(7*c^2*d^2 + 3*a*c*e^2)*f^2*g^3 - 21 
*(3*a*c*d^2 - a^2*e^2)*g^5)*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), 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*(35*c^2*e^2*g^5*x^3 + 8*c^2*e^2*f^3*g^2 - 24*c^2*d*e*f^2*g^3 + 60* 
a*c*d*e*g^5 + (21*c^2*d^2 + 8*a*c*e^2)*f*g^4 + 5*(c^2*e^2*f*g^4 + 18*c^2*d 
*e*g^5)*x^2 - (6*c^2*e^2*f^2*g^3 - 18*c^2*d*e*f*g^4 - 7*(9*c^2*d^2 + 2*a*c 
*e^2)*g^5)*x)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c^2*g^5)
 

3.7.23.6 Sympy [F]

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

input

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

output

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

3.7.23.7 Maxima [F]

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

input

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

output

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

3.7.23.8 Giac [F]

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

input

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

output

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

3.7.23.9 Mupad [F(-1)]

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

input

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

output

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