\(\int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx\) [124]

Optimal result

Integrand size = 28, antiderivative size = 742 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 e \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}-\frac {6 e^2 (3 e f-7 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}+\frac {2 e^3 (f+g x)^{5/2} \sqrt {a+c x^2}}{7 c g^2}+\frac {2 \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{105 c^{7/4} g^4 \sqrt {a+c x^2}}+\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (105 c^2 d^3 f g^2+25 a^2 e^3 g^3+\sqrt {-a} a \sqrt {c} e^2 g^2 (19 e f+189 d g)-a c e g \left (2 e^2 f^2+147 d e f g+105 d^2 g^2\right )-\sqrt {-a} c^{3/2} \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{105 c^{9/4} g^3 \sqrt {a+c x^2}} \] Output:

-2/105*e*(25*a*e^2*g^2-c*(105*d^2*g^2-42*d*e*f*g+8*e^2*f^2))*(g*x+f)^(1/2) 
*(c*x^2+a)^(1/2)/c^2/g^2-6/35*e^2*(-7*d*g+3*e*f)*(g*x+f)^(3/2)*(c*x^2+a)^( 
1/2)/c/g^2+2/7*e^3*(g*x+f)^(5/2)*(c*x^2+a)^(1/2)/c/g^2+2/105*(c^(1/2)*f-(- 
a)^(1/2)*g)*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(a*e^2*g^2*(189*d*g+19*e*f)-c*( 
105*d^3*g^3+105*d^2*e*f*g^2-42*d*e^2*f^2*g+8*e^3*f^3))*(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^4/(c*x^ 
2+a)^(1/2)+2/105*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(105*c^2*d^3*f*g^2+25*a^2* 
e^3*g^3+(-a)^(1/2)*a*c^(1/2)*e^2*g^2*(189*d*g+19*e*f)-a*c*e*g*(105*d^2*g^2 
+147*d*e*f*g+2*e^2*f^2)-(-a)^(1/2)*c^(3/2)*(105*d^3*g^3+105*d^2*e*f*g^2-42 
*d*e^2*f^2*g+8*e^3*f^3))*(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^(9/4)/g^3/(c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.16 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.05 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (-25 a e^3 g^2+c e \left (105 d^2 g^2+21 d e g (f+3 g x)+e^2 \left (-4 f^2+3 f g x+15 g^2 x^2\right )\right )\right )}{c^2 g^2}-\frac {2 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \left (a+c x^2\right )-\sqrt {c} \left (i a \sqrt {c} e^2 f g^2 (19 e f+189 d g)-a^{3/2} e^2 g^3 (19 e f+189 d g)-i c^{3/2} f \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+\sqrt {a} c g \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (105 i c^{3/2} d^3 g^2+25 a^{3/2} e^3 g^2+3 i a \sqrt {c} e^2 g (2 e f-63 d g)+\sqrt {a} c e \left (-8 e^2 f^2+42 d e f g-105 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c^2 g^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{105 \sqrt {a+c x^2}} \] Input:

Integrate[((d + e*x)^3*Sqrt[f + g*x])/Sqrt[a + c*x^2],x]
 

Output:

(Sqrt[f + g*x]*((2*(a + c*x^2)*(-25*a*e^3*g^2 + c*e*(105*d^2*g^2 + 21*d*e* 
g*(f + 3*g*x) + e^2*(-4*f^2 + 3*f*g*x + 15*g^2*x^2))))/(c^2*g^2) - (2*(g^2 
*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(a*e^2*g^2*(19*e*f + 189*d*g) - c*(8*e^3 
*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3))*(a + c*x^2) - Sqrt 
[c]*(I*a*Sqrt[c]*e^2*f*g^2*(19*e*f + 189*d*g) - a^(3/2)*e^2*g^3*(19*e*f + 
189*d*g) - I*c^(3/2)*f*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105 
*d^3*g^3) + Sqrt[a]*c*g*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 10 
5*d^3*g^3))*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)/(Sqr 
t[c]*f + I*Sqrt[a]*g)] - g*(Sqrt[c]*f + I*Sqrt[a]*g)*((105*I)*c^(3/2)*d^3* 
g^2 + 25*a^(3/2)*e^3*g^2 + (3*I)*a*Sqrt[c]*e^2*g*(2*e*f - 63*d*g) + Sqrt[a 
]*c*e*(-8*e^2*f^2 + 42*d*e*f*g - 105*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^2*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 = 3.13 (sec) , antiderivative size = 916, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {735, 25, 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.

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

\(\Big \downarrow \) 735

\(\displaystyle \frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}-\frac {\int -\frac {(d+e x) \left (7 c f d^2+c e (e f+11 d g) x^2-a e (4 e f+d g)-\left (5 a e^2 g-c d (12 e f+7 d g)\right ) x\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(d+e x) \left (7 c f d^2+c e (e f+11 d g) x^2-a e (4 e f+d g)-\left (5 a e^2 g-c d (12 e f+7 d g)\right ) x\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {2 \int \frac {-c e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e g f-90 d^2 g^2\right )\right ) x^2 g^2+c \left (35 c d^3 f g-a e \left (3 e^2 f^2+53 d e g f+5 d^2 g^2\right )\right ) g^2-c \left (a e^2 (23 e f+63 d g) g^2+c \left (2 e^3 f^3+22 d e^2 g f^2-95 d^2 e g^2 f-35 d^3 g^3\right )\right ) x g}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-c e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e g f-90 d^2 g^2\right )\right ) x^2 g^2+c \left (35 c d^3 f g-a e \left (3 e^2 f^2+53 d e g f+5 d^2 g^2\right )\right ) g^2-c \left (a e^2 (23 e f+63 d g) g^2+c \left (2 e^3 f^3+22 d e^2 g f^2-95 d^2 e g^2 f-35 d^3 g^3\right )\right ) x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {\frac {2 \int \frac {c g^3 \left (g \left (105 c^2 f g d^3+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e g f+105 d^2 g^2\right )\right )-c \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 d^2 e g^2 f+105 d^3 g^3\right )\right ) x\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g^2}-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {1}{3} g \int \frac {g \left (105 c^2 f g d^3+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e g f+105 d^2 g^2\right )\right )-c \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 d^2 e g^2 f+105 d^3 g^3\right )\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 599

\(\displaystyle \frac {\frac {-\frac {2 \int -\frac {e \left (c f^2+a g^2\right ) \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e g f+105 d^2 g^2\right )\right )-c \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 d^2 e g^2 f+105 d^3 g^3\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g}-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {2 \int \frac {e \left (c f^2+a g^2\right ) \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e g f+105 d^2 g^2\right )\right )-c \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 d^2 e g^2 f+105 d^3 g^3\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g}-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\frac {-\frac {2 \left (-\sqrt {a g^2+c f^2} \left (e \sqrt {a g^2+c f^2} \left (25 a e^2 g^2-c \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right )-\sqrt {c} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\sqrt {c} \sqrt {a g^2+c f^2} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{3 g}-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {-\frac {2 \left (-\sqrt {c} \sqrt {a g^2+c f^2} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (e \sqrt {a g^2+c f^2} \left (25 a e^2 g^2-c \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right )-\sqrt {c} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}\right )}{3 g}-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {2 e \sqrt {f+g x} \sqrt {c x^2+a} (d+e x)^2}{7 c}+\frac {\frac {2 (e f+11 d g) (f+g x)^{3/2} \sqrt {c x^2+a} e^2}{5 g^2}+\frac {-\frac {2}{3} e g \sqrt {f+g x} \sqrt {c x^2+a} \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e g f-90 d^2 g^2\right )\right )-\frac {2 \left (-\sqrt {c} \sqrt {c f^2+a g^2} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 d^2 e g^2 f+105 d^3 g^3\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 (e \sqrt {c f^2+a g^2} \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e g f+105 d^2 g^2\right )\right )-\sqrt {c} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 d^2 e g^2 f+105 d^3 g^3\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}}{7 c}\)

Input:

Int[((d + e*x)^3*Sqrt[f + g*x])/Sqrt[a + c*x^2],x]
 

Output:

(2*e*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(7*c) + ((2*e^2*(e*f + 11* 
d*g)*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(5*g^2) + ((-2*e*g*(25*a*e^2*g^2 + c 
*(7*e^2*f^2 + 12*d*e*f*g - 90*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^2*g^2*(19*e*f + 189*d*g) - c*(8*e^ 
3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3))*(-((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)*(e*Sqrt[c*f^2 + a*g^2]*(25*a*e^2 
*g^2 - c*(8*e^2*f^2 - 42*d*e*f*g + 105*d^2*g^2)) - Sqrt[c]*(a*e^2*g^2*(19* 
e*f + 189*d*g) - c*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3 
*g^3)))*(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)*Sq 
rt[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^(1/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + ...
 

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 = 5.06 (sec) , antiderivative size = 882, normalized size of antiderivative = 1.19

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{3} x^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{7 c}+\frac {2 \left (3 d \,e^{2} g +\frac {1}{7} f \,e^{3}\right ) x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{5 c g}+\frac {2 \left (3 g \,d^{2} e +3 d \,e^{2} f -\frac {4 f \left (3 d \,e^{2} g +\frac {1}{7} f \,e^{3}\right )}{5 g}-\frac {5 a g \,e^{3}}{7 c}\right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{3 c g}+\frac {2 \left (d^{3} f -\frac {2 a f \left (3 d \,e^{2} g +\frac {1}{7} f \,e^{3}\right )}{5 c g}-\frac {a \left (3 g \,d^{2} e +3 d \,e^{2} f -\frac {4 f \left (3 d \,e^{2} g +\frac {1}{7} f \,e^{3}\right )}{5 g}-\frac {5 a g \,e^{3}}{7 c}\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^{3} g +3 d^{2} e f -\frac {4 a f \,e^{3}}{7 c}-\frac {3 a \left (3 d \,e^{2} g +\frac {1}{7} f \,e^{3}\right )}{5 c}-\frac {2 f \left (3 g \,d^{2} e +3 d \,e^{2} f -\frac {4 f \left (3 d \,e^{2} g +\frac {1}{7} f \,e^{3}\right )}{5 g}-\frac {5 a g \,e^{3}}{7 c}\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}}\)	\(882\)
risch	\(\text {Expression too large to display}\)	\(1601\)
default	\(\text {Expression too large to display}\)	\(3922\)

Input:

int((e*x+d)^3*(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^3/c*x^2*(c* 
g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/5*(3*d*e^2*g+1/7*f*e^3)/c/g*x*(c*g*x^3+c* 
f*x^2+a*g*x+a*f)^(1/2)+2/3*(3*g*d^2*e+3*d*e^2*f-4/5*f/g*(3*d*e^2*g+1/7*f*e 
^3)-5/7*a/c*g*e^3)/c/g*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2*(d^3*f-2/5*a*f/ 
c/g*(3*d*e^2*g+1/7*f*e^3)-1/3*a/c*(3*g*d^2*e+3*d*e^2*f-4/5*f/g*(3*d*e^2*g+ 
1/7*f*e^3)-5/7*a/c*g*e^3))*(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)*Ellip 
ticF(((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^3*g+3*d^2*e*f-4/7*a*f/c*e^3-3/5*a/c*(3*d*e^2*g+1 
/7*f*e^3)-2/3*f/g*(3*g*d^2*e+3*d*e^2*f-4/5*f/g*(3*d*e^2*g+1/7*f*e^3)-5/7*a 
/c*g*e^3))*(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 = 0.11 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.60 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left ({\left (8 \, c^{2} e^{3} f^{4} - 42 \, c^{2} d e^{2} f^{3} g + {\left (105 \, c^{2} d^{2} e - 13 \, a c e^{3}\right )} f^{2} g^{2} - 42 \, {\left (5 \, c^{2} d^{3} - 6 \, a c d e^{2}\right )} f g^{3} + 15 \, {\left (21 \, a c d^{2} e - 5 \, a^{2} e^{3}\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 3 \, {\left (8 \, c^{2} e^{3} f^{3} g - 42 \, c^{2} d e^{2} f^{2} g^{2} + {\left (105 \, c^{2} d^{2} e - 19 \, a c e^{3}\right )} f g^{3} + 21 \, {\left (5 \, c^{2} d^{3} - 9 \, a c d e^{2}\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) - 3 \, {\left (15 \, c^{2} e^{3} g^{4} x^{2} - 4 \, c^{2} e^{3} f^{2} g^{2} + 21 \, c^{2} d e^{2} f g^{3} + 5 \, {\left (21 \, c^{2} d^{2} e - 5 \, a c e^{3}\right )} g^{4} + 3 \, {\left (c^{2} e^{3} f g^{3} + 21 \, c^{2} d e^{2} g^{4}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{315 \, c^{3} g^{4}} \] Input:

integrate((e*x+d)^3*(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

-2/315*((8*c^2*e^3*f^4 - 42*c^2*d*e^2*f^3*g + (105*c^2*d^2*e - 13*a*c*e^3) 
*f^2*g^2 - 42*(5*c^2*d^3 - 6*a*c*d*e^2)*f*g^3 + 15*(21*a*c*d^2*e - 5*a^2*e 
^3)*g^4)*sqrt(c*g)*weierstrassPInverse(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/2 
7*(c*f^3 + 9*a*f*g^2)/(c*g^3), 1/3*(3*g*x + f)/g) + 3*(8*c^2*e^3*f^3*g - 4 
2*c^2*d*e^2*f^2*g^2 + (105*c^2*d^2*e - 19*a*c*e^3)*f*g^3 + 21*(5*c^2*d^3 - 
 9*a*c*d*e^2)*g^4)*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*( 
15*c^2*e^3*g^4*x^2 - 4*c^2*e^3*f^2*g^2 + 21*c^2*d*e^2*f*g^3 + 5*(21*c^2*d^ 
2*e - 5*a*c*e^3)*g^4 + 3*(c^2*e^3*f*g^3 + 21*c^2*d*e^2*g^4)*x)*sqrt(c*x^2 
+ a)*sqrt(g*x + f))/(c^3*g^4)
 

Sympy [F]

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

integrate((e*x+d)**3*(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)
 

Output:

Integral((d + e*x)**3*sqrt(f + g*x)/sqrt(a + c*x**2), x)
 

Maxima [F]

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

integrate((e*x+d)^3*(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate((e*x + d)^3*sqrt(g*x + f)/sqrt(c*x^2 + a), x)
 

Giac [F]

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

integrate((e*x+d)^3*(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

integrate((e*x + d)^3*sqrt(g*x + f)/sqrt(c*x^2 + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 126*sqrt(f + g*x)*sqrt(a + c*x**2)*a*d*e**2*g**2 - 46*sqrt(f + g*x)*sq 
rt(a + c*x**2)*a*e**3*f*g + 70*sqrt(f + g*x)*sqrt(a + c*x**2)*c*d**3*g**2 
+ 210*sqrt(f + g*x)*sqrt(a + c*x**2)*c*d**2*e*f*g + 84*sqrt(f + g*x)*sqrt( 
a + c*x**2)*c*d*e**2*f*g*x + 4*sqrt(f + g*x)*sqrt(a + c*x**2)*c*e**3*f**2* 
x + 20*sqrt(f + g*x)*sqrt(a + c*x**2)*c*e**3*f*g*x**2 + 189*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 
**2*g**3 + 19*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**3*f*g**2 - 105*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**3*g**3 - 105*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*e*f*g**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)*c**2*d*e**2*f**2*g - 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)*c**2*e**3 
*f**3 + 63*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**2*g**3 + 23*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**3*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**3*g**3 
 - 105*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*e*f*g**2 - 84*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**2*f**2*g - 4*int((sqrt(f + g...