\(\int \frac {(f+g x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{d+e x} \, dx\) [267]

Optimal result

Integrand size = 44, antiderivative size = 716 \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right ) \left (7 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (12 e f-5 d g)+3 a c^2 d^2 e^2 g \left (24 e^2 f^2-24 d e f g+7 d^2 g^2\right )-c^3 d^3 \left (64 e^3 f^3-120 d e^2 f^2 g+84 d^2 e f g^2-21 d^3 g^3\right )\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^5}+\frac {1}{20} \left (\frac {a g}{c d}+\frac {2 e f-3 d g}{e^2}\right ) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}+\frac {\left (35 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (60 e f-11 d g)+3 a c^2 d^2 e^2 g \left (104 e^2 f^2-48 d e f g+7 d^2 g^2\right )+c^3 d^3 \left (64 e^3 f^3-456 d e^2 f^2 g+420 d^2 e f g^2-105 d^3 g^3\right )-6 c d e g \left (7 a^2 e^4 g^2-2 a c d e^2 g (10 e f-3 d g)-c^2 d^2 \left (8 e^2 f^2-36 d e f g+21 d^2 g^2\right )\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}-\frac {\left (c d^2-a e^2\right )^3 \left (7 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (12 e f-5 d g)+3 a c^2 d^2 e^2 g \left (24 e^2 f^2-24 d e f g+7 d^2 g^2\right )-c^3 d^3 \left (64 e^3 f^3-120 d e^2 f^2 g+84 d^2 e f g^2-21 d^3 g^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 c^{9/2} d^{9/2} e^{11/2}} \] Output:

1/512*(-a*e^2+c*d^2)*(7*a^3*e^6*g^3-3*a^2*c*d*e^4*g^2*(-5*d*g+12*e*f)+3*a* 
c^2*d^2*e^2*g*(7*d^2*g^2-24*d*e*f*g+24*e^2*f^2)-c^3*d^3*(-21*d^3*g^3+84*d^ 
2*e*f*g^2-120*d*e^2*f^2*g+64*e^3*f^3))*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e 
^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^5+1/20*(a*g/c/d+(-3*d*g+2*e*f)/e^2) 
*(g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/6*(g*x+f)^3*(a*d*e+(a 
*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e+1/960*(35*a^3*e^6*g^3-3*a^2*c*d*e^4*g^2*( 
-11*d*g+60*e*f)+3*a*c^2*d^2*e^2*g*(7*d^2*g^2-48*d*e*f*g+104*e^2*f^2)+c^3*d 
^3*(-105*d^3*g^3+420*d^2*e*f*g^2-456*d*e^2*f^2*g+64*e^3*f^3)-6*c*d*e*g*(7* 
a^2*e^4*g^2-2*a*c*d*e^2*g*(-3*d*g+10*e*f)-c^2*d^2*(21*d^2*g^2-36*d*e*f*g+8 
*e^2*f^2))*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3/e^4-1/512*(- 
a*e^2+c*d^2)^3*(7*a^3*e^6*g^3-3*a^2*c*d*e^4*g^2*(-5*d*g+12*e*f)+3*a*c^2*d^ 
2*e^2*g*(7*d^2*g^2-24*d*e*f*g+24*e^2*f^2)-c^3*d^3*(-21*d^3*g^3+84*d^2*e*f* 
g^2-120*d*e^2*f^2*g+64*e^3*f^3))*arctanh(c^(1/2)*d^(1/2)*(e*x+d)/e^(1/2)/( 
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(9/2)/d^(9/2)/e^(11/2)
 

Mathematica [A] (verified)

Time = 3.65 (sec) , antiderivative size = 753, normalized size of antiderivative = 1.05 \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (-105 a^5 e^{10} g^3+5 a^4 c d e^8 g^2 (11 d g+2 e (54 f+7 g x))+2 a^3 c^2 d^2 e^6 g \left (27 d^2 g^2-4 d e g (45 f+4 g x)-4 e^2 \left (135 f^2+45 f g x+7 g^2 x^2\right )\right )+6 a^2 c^3 d^3 e^4 \left (13 d^3 g^3-6 d^2 e g^2 (12 f+g x)+4 d e^2 g \left (45 f^2+9 f g x+g^2 x^2\right )+8 e^3 \left (20 f^3+15 f^2 g x+6 f g^2 x^2+g^3 x^3\right )\right )+c^5 d^5 \left (315 d^5 g^3-210 d^4 e g^2 (6 f+g x)+24 d^3 e^2 g \left (75 f^2+35 f g x+7 g^2 x^2\right )+64 d e^4 x \left (10 f^3+15 f^2 g x+9 f g^2 x^2+2 g^3 x^3\right )-48 d^2 e^3 \left (20 f^3+25 f^2 g x+14 f g^2 x^2+3 g^3 x^3\right )+128 e^5 x^2 \left (20 f^3+45 f^2 g x+36 f g^2 x^2+10 g^3 x^3\right )\right )+a c^4 d^4 e^2 \left (-525 d^4 g^3+24 d^3 e g^2 (95 f+14 g x)-24 d^2 e^2 g \left (155 f^2+61 f g x+11 g^2 x^2\right )+32 d e^3 \left (80 f^3+75 f^2 g x+36 f g^2 x^2+7 g^3 x^3\right )+64 e^4 x \left (70 f^3+135 f^2 g x+99 f g^2 x^2+26 g^3 x^3\right )\right )\right )-\frac {15 \left (c d^2-a e^2\right )^3 \left (7 a^3 e^6 g^3+3 a^2 c d e^4 g^2 (-12 e f+5 d g)+3 a c^2 d^2 e^2 g \left (24 e^2 f^2-24 d e f g+7 d^2 g^2\right )+c^3 d^3 \left (-64 e^3 f^3+120 d e^2 f^2 g-84 d^2 e f g^2+21 d^3 g^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{7680 c^{9/2} d^{9/2} e^{11/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-105*a^5*e^10*g^3 
 + 5*a^4*c*d*e^8*g^2*(11*d*g + 2*e*(54*f + 7*g*x)) + 2*a^3*c^2*d^2*e^6*g*( 
27*d^2*g^2 - 4*d*e*g*(45*f + 4*g*x) - 4*e^2*(135*f^2 + 45*f*g*x + 7*g^2*x^ 
2)) + 6*a^2*c^3*d^3*e^4*(13*d^3*g^3 - 6*d^2*e*g^2*(12*f + g*x) + 4*d*e^2*g 
*(45*f^2 + 9*f*g*x + g^2*x^2) + 8*e^3*(20*f^3 + 15*f^2*g*x + 6*f*g^2*x^2 + 
 g^3*x^3)) + c^5*d^5*(315*d^5*g^3 - 210*d^4*e*g^2*(6*f + g*x) + 24*d^3*e^2 
*g*(75*f^2 + 35*f*g*x + 7*g^2*x^2) + 64*d*e^4*x*(10*f^3 + 15*f^2*g*x + 9*f 
*g^2*x^2 + 2*g^3*x^3) - 48*d^2*e^3*(20*f^3 + 25*f^2*g*x + 14*f*g^2*x^2 + 3 
*g^3*x^3) + 128*e^5*x^2*(20*f^3 + 45*f^2*g*x + 36*f*g^2*x^2 + 10*g^3*x^3)) 
 + a*c^4*d^4*e^2*(-525*d^4*g^3 + 24*d^3*e*g^2*(95*f + 14*g*x) - 24*d^2*e^2 
*g*(155*f^2 + 61*f*g*x + 11*g^2*x^2) + 32*d*e^3*(80*f^3 + 75*f^2*g*x + 36* 
f*g^2*x^2 + 7*g^3*x^3) + 64*e^4*x*(70*f^3 + 135*f^2*g*x + 99*f*g^2*x^2 + 2 
6*g^3*x^3))) - (15*(c*d^2 - a*e^2)^3*(7*a^3*e^6*g^3 + 3*a^2*c*d*e^4*g^2*(- 
12*e*f + 5*d*g) + 3*a*c^2*d^2*e^2*g*(24*e^2*f^2 - 24*d*e*f*g + 7*d^2*g^2) 
+ c^3*d^3*(-64*e^3*f^3 + 120*d*e^2*f^2*g - 84*d^2*e*f*g^2 + 21*d^3*g^3))*A 
rcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqrt 
[a*e + c*d*x]*Sqrt[d + e*x])))/(7680*c^(9/2)*d^(9/2)*e^(11/2))
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 650, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {1215, 1236, 27, 1236, 27, 1225, 1087, 1092, 219}

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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int -\frac {3}{2} c d (f+g x)^2 \left (c f d^2-a e (3 e f-2 d g)-\left (a g e^2+c d (2 e f-3 d g)\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{6 c d e}+\frac {(f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\int (f+g x)^2 \left (c f d^2-a e (3 e f-2 d g)-\left (a g e^2+c d (2 e f-3 d g)\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{4 e}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {(f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {\int \frac {1}{2} (f+g x) \left (c^2 f (16 e f-9 d g) d^3-2 a c e \left (12 e^2 f^2-11 d e g f+6 d^2 g^2\right ) d+a^2 e^3 g (3 e f+4 d g)+\left (7 a^2 g^2 e^4-2 a c d g (10 e f-3 d g) e^2-c^2 d^2 \left (8 e^2 f^2-36 d e g f+21 d^2 g^2\right )\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{5 c d e}-\frac {1}{5} (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (\frac {a e g}{c d}-\frac {3 d g}{e}+2 f\right )}{4 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {\int (f+g x) \left (c^2 f (16 e f-9 d g) d^3-2 a c e \left (12 e^2 f^2-11 d e g f+6 d^2 g^2\right ) d+a^2 e^3 g (3 e f+4 d g)+\left (7 a^2 g^2 e^4-2 a c d g (10 e f-3 d g) e^2-c^2 d^2 \left (8 e^2 f^2-36 d e g f+21 d^2 g^2\right )\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{10 c d e}-\frac {1}{5} (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (\frac {a e g}{c d}-\frac {3 d g}{e}+2 f\right )}{4 e}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {(f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {-\frac {5 \left (c d^2-a e^2\right ) \left (7 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (12 e f-5 d g)+3 a c^2 d^2 e^2 g \left (7 d^2 g^2-24 d e f g+24 e^2 f^2\right )-c^3 d^3 \left (-21 d^3 g^3+84 d^2 e f g^2-120 d e^2 f^2 g+64 e^3 f^3\right )\right ) \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c^2 d^2 e^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (35 a^3 e^6 g^3-6 c d e g x \left (7 a^2 e^4 g^2-2 a c d e^2 g (10 e f-3 d g)-c^2 d^2 \left (21 d^2 g^2-36 d e f g+8 e^2 f^2\right )\right )-3 a^2 c d e^4 g^2 (60 e f-11 d g)+3 a c^2 d^2 e^2 g \left (7 d^2 g^2-48 d e f g+104 e^2 f^2\right )+c^3 d^3 \left (-105 d^3 g^3+420 d^2 e f g^2-456 d e^2 f^2 g+64 e^3 f^3\right )\right )}{24 c^2 d^2 e^2}}{10 c d e}-\frac {1}{5} (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (\frac {a e g}{c d}-\frac {3 d g}{e}+2 f\right )}{4 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {-\frac {5 \left (c d^2-a e^2\right ) \left (7 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (12 e f-5 d g)+3 a c^2 d^2 e^2 g \left (7 d^2 g^2-24 d e f g+24 e^2 f^2\right )-c^3 d^3 \left (-21 d^3 g^3+84 d^2 e f g^2-120 d e^2 f^2 g+64 e^3 f^3\right )\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c^2 d^2 e^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (35 a^3 e^6 g^3-6 c d e g x \left (7 a^2 e^4 g^2-2 a c d e^2 g (10 e f-3 d g)-c^2 d^2 \left (21 d^2 g^2-36 d e f g+8 e^2 f^2\right )\right )-3 a^2 c d e^4 g^2 (60 e f-11 d g)+3 a c^2 d^2 e^2 g \left (7 d^2 g^2-48 d e f g+104 e^2 f^2\right )+c^3 d^3 \left (-105 d^3 g^3+420 d^2 e f g^2-456 d e^2 f^2 g+64 e^3 f^3\right )\right )}{24 c^2 d^2 e^2}}{10 c d e}-\frac {1}{5} (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (\frac {a e g}{c d}-\frac {3 d g}{e}+2 f\right )}{4 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {-\frac {5 \left (c d^2-a e^2\right ) \left (7 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (12 e f-5 d g)+3 a c^2 d^2 e^2 g \left (7 d^2 g^2-24 d e f g+24 e^2 f^2\right )-c^3 d^3 \left (-21 d^3 g^3+84 d^2 e f g^2-120 d e^2 f^2 g+64 e^3 f^3\right )\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c^2 d^2 e^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (35 a^3 e^6 g^3-6 c d e g x \left (7 a^2 e^4 g^2-2 a c d e^2 g (10 e f-3 d g)-c^2 d^2 \left (21 d^2 g^2-36 d e f g+8 e^2 f^2\right )\right )-3 a^2 c d e^4 g^2 (60 e f-11 d g)+3 a c^2 d^2 e^2 g \left (7 d^2 g^2-48 d e f g+104 e^2 f^2\right )+c^3 d^3 \left (-105 d^3 g^3+420 d^2 e f g^2-456 d e^2 f^2 g+64 e^3 f^3\right )\right )}{24 c^2 d^2 e^2}}{10 c d e}-\frac {1}{5} (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (\frac {a e g}{c d}-\frac {3 d g}{e}+2 f\right )}{4 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {-\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right ) \left (7 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (12 e f-5 d g)+3 a c^2 d^2 e^2 g \left (7 d^2 g^2-24 d e f g+24 e^2 f^2\right )-c^3 d^3 \left (-21 d^3 g^3+84 d^2 e f g^2-120 d e^2 f^2 g+64 e^3 f^3\right )\right )}{16 c^2 d^2 e^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (35 a^3 e^6 g^3-6 c d e g x \left (7 a^2 e^4 g^2-2 a c d e^2 g (10 e f-3 d g)-c^2 d^2 \left (21 d^2 g^2-36 d e f g+8 e^2 f^2\right )\right )-3 a^2 c d e^4 g^2 (60 e f-11 d g)+3 a c^2 d^2 e^2 g \left (7 d^2 g^2-48 d e f g+104 e^2 f^2\right )+c^3 d^3 \left (-105 d^3 g^3+420 d^2 e f g^2-456 d e^2 f^2 g+64 e^3 f^3\right )\right )}{24 c^2 d^2 e^2}}{10 c d e}-\frac {1}{5} (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (\frac {a e g}{c d}-\frac {3 d g}{e}+2 f\right )}{4 e}\)

Input:

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

Output:

((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(6*e) - (-1/5* 
((2*f - (3*d*g)/e + (a*e*g)/(c*d))*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x 
+ c*d*e*x^2)^(3/2)) + (-1/24*((35*a^3*e^6*g^3 - 3*a^2*c*d*e^4*g^2*(60*e*f 
- 11*d*g) + 3*a*c^2*d^2*e^2*g*(104*e^2*f^2 - 48*d*e*f*g + 7*d^2*g^2) + c^3 
*d^3*(64*e^3*f^3 - 456*d*e^2*f^2*g + 420*d^2*e*f*g^2 - 105*d^3*g^3) - 6*c* 
d*e*g*(7*a^2*e^4*g^2 - 2*a*c*d*e^2*g*(10*e*f - 3*d*g) - c^2*d^2*(8*e^2*f^2 
 - 36*d*e*f*g + 21*d^2*g^2))*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3 
/2))/(c^2*d^2*e^2) - (5*(c*d^2 - a*e^2)*(7*a^3*e^6*g^3 - 3*a^2*c*d*e^4*g^2 
*(12*e*f - 5*d*g) + 3*a*c^2*d^2*e^2*g*(24*e^2*f^2 - 24*d*e*f*g + 7*d^2*g^2 
) - c^3*d^3*(64*e^3*f^3 - 120*d*e^2*f^2*g + 84*d^2*e*f*g^2 - 21*d^3*g^3))* 
(((c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) 
/(4*c*d*e) - ((c*d^2 - a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqr 
t[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*c^( 
3/2)*d^(3/2)*e^(3/2))))/(16*c^2*d^2*e^2))/(10*c*d*e))/(4*e)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1215

Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( 
(d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + 
 c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - 
 b*d*e + a*e^2, 0] && GtQ[p, 0]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1979\) vs. \(2(684)=1368\).

Time = 2.48 (sec) , antiderivative size = 1980, normalized size of antiderivative = 2.77

method	result	size
default	\(\text {Expression too large to display}\)	\(1980\)

Input:

int((g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d),x,method=_RE 
TURNVERBOSE)
 

Output:

g/e^3*(d^2*g^2*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2 
*e)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x 
+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2 
*e^2-(a*e^2+c*d^2)^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2) 
+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2)))-e*g*(d*g-3*e*f)* 
(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/d/e/c-1/2*(a*e^2+c*d^2)/d/e/c 
*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/c/d/ 
e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2)* 
(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c* 
d^2)^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2 
+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2))))+e^2*g^2*(1/6*x*(a*d*e+(a*e^2+ 
c*d^2)*x+c*d*x^2*e)^(5/2)/d/e/c-7/12*(a*e^2+c*d^2)/d/e/c*(1/5*(a*d*e+(a*e^ 
2+c*d^2)*x+c*d*x^2*e)^(5/2)/d/e/c-1/2*(a*e^2+c*d^2)/d/e/c*(1/8*(2*c*d*e*x+ 
a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/c/d/e+3/16*(4*a*c*d^2 
*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d 
^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*ln( 
(1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2 
*e)^(1/2))/(d*e*c)^(1/2))))-1/6*a/c*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a 
*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/ 
d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1...
 

Fricas [A] (verification not implemented)

Time = 1.28 (sec) , antiderivative size = 2430, normalized size of antiderivative = 3.39 \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/30720*(15*(64*(c^6*d^9*e^3 - 3*a*c^5*d^7*e^5 + 3*a^2*c^4*d^5*e^7 - a^3* 
c^3*d^3*e^9)*f^3 - 24*(5*c^6*d^10*e^2 - 12*a*c^5*d^8*e^4 + 6*a^2*c^4*d^6*e 
^6 + 4*a^3*c^3*d^4*e^8 - 3*a^4*c^2*d^2*e^10)*f^2*g + 12*(7*c^6*d^11*e - 15 
*a*c^5*d^9*e^3 + 6*a^2*c^4*d^7*e^5 + 2*a^3*c^3*d^5*e^7 + 3*a^4*c^2*d^3*e^9 
 - 3*a^5*c*d*e^11)*f*g^2 - (21*c^6*d^12 - 42*a*c^5*d^10*e^2 + 15*a^2*c^4*d 
^8*e^4 + 4*a^3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 + 6*a^5*c*d^2*e^10 - 7*a^6* 
e^12)*g^3)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a 
^2*e^4 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 
+ a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(1280*c^6*d^6*e^6* 
g^3*x^5 + 128*(36*c^6*d^6*e^6*f*g^2 + (c^6*d^7*e^5 + 13*a*c^5*d^5*e^7)*g^3 
)*x^4 - 320*(3*c^6*d^8*e^4 - 8*a*c^5*d^6*e^6 - 3*a^2*c^4*d^4*e^8)*f^3 + 12 
0*(15*c^6*d^9*e^3 - 31*a*c^5*d^7*e^5 + 9*a^2*c^4*d^5*e^7 - 9*a^3*c^3*d^3*e 
^9)*f^2*g - 12*(105*c^6*d^10*e^2 - 190*a*c^5*d^8*e^4 + 36*a^2*c^4*d^6*e^6 
+ 30*a^3*c^3*d^4*e^8 - 45*a^4*c^2*d^2*e^10)*f*g^2 + (315*c^6*d^11*e - 525* 
a*c^5*d^9*e^3 + 78*a^2*c^4*d^7*e^5 + 54*a^3*c^3*d^5*e^7 + 55*a^4*c^2*d^3*e 
^9 - 105*a^5*c*d*e^11)*g^3 + 16*(360*c^6*d^6*e^6*f^2*g + 36*(c^6*d^7*e^5 + 
 11*a*c^5*d^5*e^7)*f*g^2 - (9*c^6*d^8*e^4 - 14*a*c^5*d^6*e^6 - 3*a^2*c^4*d 
^4*e^8)*g^3)*x^3 + 8*(320*c^6*d^6*e^6*f^3 + 120*(c^6*d^7*e^5 + 9*a*c^5*d^5 
*e^7)*f^2*g - 12*(7*c^6*d^8*e^4 - 12*a*c^5*d^6*e^6 - 3*a^2*c^4*d^4*e^8)*f* 
g^2 + (21*c^6*d^9*e^3 - 33*a*c^5*d^7*e^5 + 3*a^2*c^4*d^5*e^7 - 7*a^3*c^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 1337, normalized size of antiderivative = 1.87 \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

1/7680*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*c*d*g^3 
*x + (36*c^6*d^6*e^5*f*g^2 + c^6*d^7*e^4*g^3 + 13*a*c^5*d^5*e^6*g^3)/(c^5* 
d^5*e^5))*x + (360*c^6*d^6*e^5*f^2*g + 36*c^6*d^7*e^4*f*g^2 + 396*a*c^5*d^ 
5*e^6*f*g^2 - 9*c^6*d^8*e^3*g^3 + 14*a*c^5*d^6*e^5*g^3 + 3*a^2*c^4*d^4*e^7 
*g^3)/(c^5*d^5*e^5))*x + (320*c^6*d^6*e^5*f^3 + 120*c^6*d^7*e^4*f^2*g + 10 
80*a*c^5*d^5*e^6*f^2*g - 84*c^6*d^8*e^3*f*g^2 + 144*a*c^5*d^6*e^5*f*g^2 + 
36*a^2*c^4*d^4*e^7*f*g^2 + 21*c^6*d^9*e^2*g^3 - 33*a*c^5*d^7*e^4*g^3 + 3*a 
^2*c^4*d^5*e^6*g^3 - 7*a^3*c^3*d^3*e^8*g^3)/(c^5*d^5*e^5))*x + (320*c^6*d^ 
7*e^4*f^3 + 2240*a*c^5*d^5*e^6*f^3 - 600*c^6*d^8*e^3*f^2*g + 1200*a*c^5*d^ 
6*e^5*f^2*g + 360*a^2*c^4*d^4*e^7*f^2*g + 420*c^6*d^9*e^2*f*g^2 - 732*a*c^ 
5*d^7*e^4*f*g^2 + 108*a^2*c^4*d^5*e^6*f*g^2 - 180*a^3*c^3*d^3*e^8*f*g^2 - 
105*c^6*d^10*e*g^3 + 168*a*c^5*d^8*e^3*g^3 - 18*a^2*c^4*d^6*e^5*g^3 - 16*a 
^3*c^3*d^4*e^7*g^3 + 35*a^4*c^2*d^2*e^9*g^3)/(c^5*d^5*e^5))*x - (960*c^6*d 
^8*e^3*f^3 - 2560*a*c^5*d^6*e^5*f^3 - 960*a^2*c^4*d^4*e^7*f^3 - 1800*c^6*d 
^9*e^2*f^2*g + 3720*a*c^5*d^7*e^4*f^2*g - 1080*a^2*c^4*d^5*e^6*f^2*g + 108 
0*a^3*c^3*d^3*e^8*f^2*g + 1260*c^6*d^10*e*f*g^2 - 2280*a*c^5*d^8*e^3*f*g^2 
 + 432*a^2*c^4*d^6*e^5*f*g^2 + 360*a^3*c^3*d^4*e^7*f*g^2 - 540*a^4*c^2*d^2 
*e^9*f*g^2 - 315*c^6*d^11*g^3 + 525*a*c^5*d^9*e^2*g^3 - 78*a^2*c^4*d^7*e^4 
*g^3 - 54*a^3*c^3*d^5*e^6*g^3 - 55*a^4*c^2*d^3*e^8*g^3 + 105*a^5*c*d*e^10* 
g^3)/(c^5*d^5*e^5)) - 1/1024*(64*c^6*d^9*e^3*f^3 - 192*a*c^5*d^7*e^5*f^...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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