\(\int \frac {(f+g x)^{3/2} (a+c x^2)^2}{(d+e x)^2} \, dx\) [75]

Optimal result

Integrand size = 26, antiderivative size = 293 \[ \int \frac {(f+g x)^{3/2} \left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {\left (c d^2+a e^2\right ) \left (3 a e^2 g-c d (8 e f-11 d g)\right ) \sqrt {f+g x}}{e^6}-\frac {8 c d \left (c d^2+a e^2\right ) (f+g x)^{3/2}}{3 e^5}-\frac {\left (c d^2+a e^2\right )^2 (f+g x)^{3/2}}{e^5 (d+e x)}+\frac {2 c \left (2 a e^2 g^2+c \left (e^2 f^2+2 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 e^4 g^3}-\frac {4 c^2 (e f+d g) (f+g x)^{7/2}}{7 e^3 g^3}+\frac {2 c^2 (f+g x)^{9/2}}{9 e^2 g^3}-\frac {\left (c d^2+a e^2\right ) \sqrt {e f-d g} \left (3 a e^2 g-c d (8 e f-11 d g)\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{13/2}} \] Output:

(a*e^2+c*d^2)*(3*a*e^2*g-c*d*(-11*d*g+8*e*f))*(g*x+f)^(1/2)/e^6-8/3*c*d*(a 
*e^2+c*d^2)*(g*x+f)^(3/2)/e^5-(a*e^2+c*d^2)^2*(g*x+f)^(3/2)/e^5/(e*x+d)+2/ 
5*c*(2*a*e^2*g^2+c*(3*d^2*g^2+2*d*e*f*g+e^2*f^2))*(g*x+f)^(5/2)/e^4/g^3-4/ 
7*c^2*(d*g+e*f)*(g*x+f)^(7/2)/e^3/g^3+2/9*c^2*(g*x+f)^(9/2)/e^2/g^3-(a*e^2 
+c*d^2)*(-d*g+e*f)^(1/2)*(3*a*e^2*g-c*d*(-11*d*g+8*e*f))*arctanh(e^(1/2)*( 
g*x+f)^(1/2)/(-d*g+e*f)^(1/2))/e^(13/2)
 

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.20 \[ \int \frac {(f+g x)^{3/2} \left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {\sqrt {f+g x} \left (315 a^2 e^4 g^3 (-e f+3 d g+2 e g x)+42 a c e^2 g^2 \left (105 d^3 g^2+6 e^3 x (f+g x)^2+5 d^2 e g (-19 f+14 g x)+2 d e^2 \left (3 f^2-34 f g x-7 g^2 x^2\right )\right )+c^2 \left (3465 d^5 g^4+18 d^2 e^3 g (f+g x)^2 (4 f+11 g x)+105 d^4 e g^3 (-35 f+22 g x)+2 d e^4 (f+g x)^2 \left (8 f^2+16 f g x-55 g^2 x^2\right )-42 d^3 e^2 g^2 \left (-9 f^2+62 f g x+11 g^2 x^2\right )+2 e^5 x (f+g x)^2 \left (8 f^2-20 f g x+35 g^2 x^2\right )\right )\right )}{315 e^6 g^3 (d+e x)}-\frac {\left (c d^2+a e^2\right ) \sqrt {-e f+d g} \left (3 a e^2 g+c d (-8 e f+11 d g)\right ) \arctan \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{e^{13/2}} \] Input:

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

Output:

(Sqrt[f + g*x]*(315*a^2*e^4*g^3*(-(e*f) + 3*d*g + 2*e*g*x) + 42*a*c*e^2*g^ 
2*(105*d^3*g^2 + 6*e^3*x*(f + g*x)^2 + 5*d^2*e*g*(-19*f + 14*g*x) + 2*d*e^ 
2*(3*f^2 - 34*f*g*x - 7*g^2*x^2)) + c^2*(3465*d^5*g^4 + 18*d^2*e^3*g*(f + 
g*x)^2*(4*f + 11*g*x) + 105*d^4*e*g^3*(-35*f + 22*g*x) + 2*d*e^4*(f + g*x) 
^2*(8*f^2 + 16*f*g*x - 55*g^2*x^2) - 42*d^3*e^2*g^2*(-9*f^2 + 62*f*g*x + 1 
1*g^2*x^2) + 2*e^5*x*(f + g*x)^2*(8*f^2 - 20*f*g*x + 35*g^2*x^2))))/(315*e 
^6*g^3*(d + e*x)) - ((c*d^2 + a*e^2)*Sqrt[-(e*f) + d*g]*(3*a*e^2*g + c*d*( 
-8*e*f + 11*d*g))*ArcTan[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[-(e*f) + d*g]])/e^(1 
3/2)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {649, 1580, 2341, 2009}

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.

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

\(\Big \downarrow \) 649

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

\(\Big \downarrow \) 1580

\(\displaystyle \frac {2 \left (\frac {g^4 \sqrt {f+g x} \left (a e^2+c d^2\right )^2 (e f-d g)}{2 e^6 (-d g-e (f+g x)+e f)}-\frac {\int \frac {2 c^2 e^5 (f+g x)^5-2 c^2 e^4 (3 e f+d g) (f+g x)^4+2 c e^3 \left (2 a e^2 g^2+c \left (3 e^2 f^2+2 d e g f+d^2 g^2\right )\right ) (f+g x)^3-2 c e^2 (e f+d g) \left (c e^2 f^2+c d^2 g^2+2 a e^2 g^2\right ) (f+g x)^2+2 e \left (c d^2+a e^2\right )^2 g^4 (f+g x)+\left (c d^2+a e^2\right )^2 g^4 (e f-d g)}{e f-d g-e (f+g x)}d\sqrt {f+g x}}{2 e^6}\right )}{g^3}\)

\(\Big \downarrow \) 2341

\(\displaystyle \frac {2 \left (\frac {g^4 \sqrt {f+g x} \left (a e^2+c d^2\right )^2 (e f-d g)}{2 e^6 (-d g-e (f+g x)+e f)}-\frac {\int \left (-2 c^2 e^4 (f+g x)^4+4 c^2 e^3 (e f+d g) (f+g x)^3-2 c e^2 \left (2 a e^2 g^2+c \left (e^2 f^2+2 d e g f+3 d^2 g^2\right )\right ) (f+g x)^2+8 c d e \left (c d^2+a e^2\right ) g^3 (f+g x)-2 \left (c d^2+a e^2\right ) g^3 \left (a e^2 g-c d (4 e f-5 d g)\right )+\frac {3 a^2 f g^4 e^5-3 a^2 d g^5 e^4-8 a c d f^2 g^3 e^4+22 a c d^2 f g^4 e^3-14 a c d^3 g^5 e^2-8 c^2 d^3 f^2 g^3 e^2+19 c^2 d^4 f g^4 e-11 c^2 d^5 g^5}{e f-d g-e (f+g x)}\right )d\sqrt {f+g x}}{2 e^6}\right )}{g^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {g^4 \sqrt {f+g x} \left (a e^2+c d^2\right )^2 (e f-d g)}{2 e^6 (-d g-e (f+g x)+e f)}-\frac {\frac {g^3 \left (a e^2+c d^2\right ) \sqrt {e f-d g} \left (3 a e^2 g-c d (8 e f-11 d g)\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e}}-\frac {2}{5} c e^2 (f+g x)^{5/2} \left (2 a e^2 g^2+c \left (3 d^2 g^2+2 d e f g+e^2 f^2\right )\right )+\frac {8}{3} c d e g^3 (f+g x)^{3/2} \left (a e^2+c d^2\right )-2 g^3 \sqrt {f+g x} \left (a e^2+c d^2\right ) \left (a e^2 g-c d (4 e f-5 d g)\right )+\frac {4}{7} c^2 e^3 (f+g x)^{7/2} (d g+e f)-\frac {2}{9} c^2 e^4 (f+g x)^{9/2}}{2 e^6}\right )}{g^3}\)

Input:

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

Output:

(2*(((c*d^2 + a*e^2)^2*g^4*(e*f - d*g)*Sqrt[f + g*x])/(2*e^6*(e*f - d*g - 
e*(f + g*x))) - (-2*(c*d^2 + a*e^2)*g^3*(a*e^2*g - c*d*(4*e*f - 5*d*g))*Sq 
rt[f + g*x] + (8*c*d*e*(c*d^2 + a*e^2)*g^3*(f + g*x)^(3/2))/3 - (2*c*e^2*( 
2*a*e^2*g^2 + c*(e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(5/2))/5 + (4 
*c^2*e^3*(e*f + d*g)*(f + g*x)^(7/2))/7 - (2*c^2*e^4*(f + g*x)^(9/2))/9 + 
((c*d^2 + a*e^2)*g^3*Sqrt[e*f - d*g]*(3*a*e^2*g - c*d*(8*e*f - 11*d*g))*Ar 
cTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/Sqrt[e])/(2*e^6)))/g^3
 

Defintions of rubi rules used

rule 649

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^(2*m + 1)*(e*f 
- d*g + g*x^2)^n*(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x 
]], x] /; FreeQ[{a, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && Integ 
erQ[m + 1/2]
 

rule 1580

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_) 
^4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d 
 + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[1/(2*e^(2*p + m/2)* 
(q + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2* 
e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b 
*d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, b, c, d, e 
}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2341

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.35

method	result	size
pseudoelliptic	\(\frac {3 \sqrt {\left (d g -e f \right ) e}\, \left (\left (\left (\frac {2}{27} e^{5} x^{5}-\frac {22}{189} d \,e^{4} x^{4}+\frac {22}{105} d^{2} e^{3} x^{3}-\frac {22}{45} d^{3} e^{2} x^{2}+\frac {22}{9} d^{4} e x +\frac {11}{3} d^{5}\right ) g^{4}-\frac {35 e f \left (-\frac {4}{147} e^{4} x^{4}+\frac {188}{3675} d \,e^{3} x^{3}-\frac {156}{1225} d^{2} e^{2} x^{2}+\frac {124}{175} d^{3} e x +d^{4}\right ) g^{3}}{9}+\frac {2 e^{2} f^{2} \left (\frac {1}{63} e^{2} x^{2}-\frac {2}{21} d e x +d^{2}\right ) \left (e x +d \right ) g^{2}}{5}+\frac {8 \left (-\frac {e x}{9}+d \right ) e^{3} f^{3} \left (e x +d \right ) g}{105}+\frac {16 e^{4} f^{4} \left (e x +d \right )}{945}\right ) c^{2}+\frac {14 e^{2} g^{2} \left (\left (\frac {2}{35} e^{3} x^{3}-\frac {2}{15} d \,e^{2} x^{2}+\frac {2}{3} d^{2} e x +d^{3}\right ) g^{2}-\frac {19 e f \left (-\frac {12}{95} e^{2} x^{2}+\frac {68}{95} d e x +d^{2}\right ) g}{21}+\frac {2 e^{2} f^{2} \left (e x +d \right )}{35}\right ) a c}{3}+e^{4} g^{3} \left (\left (\frac {2 e x}{3}+d \right ) g -\frac {e f}{3}\right ) a^{2}\right ) \sqrt {g x +f}-3 \left (a \,e^{2}+c \,d^{2}\right ) g^{3} \left (\left (\frac {11}{3} d^{2} g -\frac {8}{3} d e f \right ) c +a \,e^{2} g \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) \left (e x +d \right ) \left (d g -e f \right )}{g^{3} e^{6} \left (e x +d \right ) \sqrt {\left (d g -e f \right ) e}}\)	\(396\)
risch	\(\frac {2 \left (35 c^{2} e^{4} g^{4} x^{4}-90 c^{2} d \,e^{3} g^{4} x^{3}+50 c^{2} e^{4} f \,g^{3} x^{3}+126 a c \,e^{4} g^{4} x^{2}+189 c^{2} d^{2} e^{2} g^{4} x^{2}-144 c^{2} d \,e^{3} f \,g^{3} x^{2}+3 c^{2} e^{4} f^{2} g^{2} x^{2}-420 a c d \,e^{3} g^{4} x +252 a c \,e^{4} f \,g^{3} x -420 c^{2} d^{3} e \,g^{4} x +378 c^{2} d^{2} e^{2} f \,g^{3} x -18 c^{2} d \,e^{3} f^{2} g^{2} x -4 c^{2} e^{4} f^{3} g x +315 a^{2} e^{4} g^{4}+1890 a c \,d^{2} e^{2} g^{4}-1680 a c d \,e^{3} f \,g^{3}+126 a c \,e^{4} f^{2} g^{2}+1575 c^{2} d^{4} g^{4}-1680 c^{2} d^{3} e f \,g^{3}+189 c^{2} d^{2} e^{2} f^{2} g^{2}+36 c^{2} d \,e^{3} f^{3} g +8 c^{2} e^{4} f^{4}\right ) \sqrt {g x +f}}{315 g^{3} e^{6}}-\frac {\left (2 a d \,e^{2} g -2 a \,e^{3} f +2 c \,d^{3} g -2 c \,d^{2} e f \right ) \left (\frac {\left (-\frac {1}{2} a \,e^{2} g -\frac {1}{2} c \,d^{2} g \right ) \sqrt {g x +f}}{e \left (g x +f \right )+d g -e f}+\frac {\left (3 a \,e^{2} g +11 c \,d^{2} g -8 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 \sqrt {\left (d g -e f \right ) e}}\right )}{e^{6}}\)	\(455\)
derivativedivides	\(\frac {\frac {2 \left (\frac {c^{2} \left (g x +f \right )^{\frac {9}{2}} e^{4}}{9}-\frac {2 c^{2} d \,e^{3} g \left (g x +f \right )^{\frac {7}{2}}}{7}-\frac {2 c^{2} e^{4} f \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 a c \,e^{4} g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {3 c^{2} d^{2} e^{2} g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 c^{2} d \,e^{3} f g \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {c^{2} e^{4} f^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {4 a c d \,e^{3} g^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}-\frac {4 c^{2} d^{3} e \,g^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}+a^{2} e^{4} g^{4} \sqrt {g x +f}+6 a c \,d^{2} e^{2} g^{4} \sqrt {g x +f}-4 a c d f \,g^{3} e^{3} \sqrt {g x +f}+5 c^{2} d^{4} g^{4} \sqrt {g x +f}-4 c^{2} d^{3} f \,g^{3} e \sqrt {g x +f}\right )}{e^{6}}-\frac {2 g^{3} \left (\frac {\left (-\frac {1}{2} a^{2} d \,e^{4} g^{2}+\frac {1}{2} a^{2} e^{5} f g -a c \,d^{3} e^{2} g^{2}+a c \,d^{2} e^{3} f g -\frac {1}{2} c^{2} d^{5} g^{2}+\frac {1}{2} c^{2} d^{4} e f g \right ) \sqrt {g x +f}}{e \left (g x +f \right )+d g -e f}+\frac {\left (3 a^{2} d \,e^{4} g^{2}-3 a^{2} e^{5} f g +14 a c \,d^{3} e^{2} g^{2}-22 a c \,d^{2} e^{3} f g +8 a c d \,e^{4} f^{2}+11 c^{2} d^{5} g^{2}-19 c^{2} d^{4} e f g +8 c^{2} d^{3} e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 \sqrt {\left (d g -e f \right ) e}}\right )}{e^{6}}}{g^{3}}\)	\(499\)
default	\(\frac {\frac {2 \left (\frac {c^{2} \left (g x +f \right )^{\frac {9}{2}} e^{4}}{9}-\frac {2 c^{2} d \,e^{3} g \left (g x +f \right )^{\frac {7}{2}}}{7}-\frac {2 c^{2} e^{4} f \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 a c \,e^{4} g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {3 c^{2} d^{2} e^{2} g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 c^{2} d \,e^{3} f g \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {c^{2} e^{4} f^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {4 a c d \,e^{3} g^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}-\frac {4 c^{2} d^{3} e \,g^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}+a^{2} e^{4} g^{4} \sqrt {g x +f}+6 a c \,d^{2} e^{2} g^{4} \sqrt {g x +f}-4 a c d f \,g^{3} e^{3} \sqrt {g x +f}+5 c^{2} d^{4} g^{4} \sqrt {g x +f}-4 c^{2} d^{3} f \,g^{3} e \sqrt {g x +f}\right )}{e^{6}}-\frac {2 g^{3} \left (\frac {\left (-\frac {1}{2} a^{2} d \,e^{4} g^{2}+\frac {1}{2} a^{2} e^{5} f g -a c \,d^{3} e^{2} g^{2}+a c \,d^{2} e^{3} f g -\frac {1}{2} c^{2} d^{5} g^{2}+\frac {1}{2} c^{2} d^{4} e f g \right ) \sqrt {g x +f}}{e \left (g x +f \right )+d g -e f}+\frac {\left (3 a^{2} d \,e^{4} g^{2}-3 a^{2} e^{5} f g +14 a c \,d^{3} e^{2} g^{2}-22 a c \,d^{2} e^{3} f g +8 a c d \,e^{4} f^{2}+11 c^{2} d^{5} g^{2}-19 c^{2} d^{4} e f g +8 c^{2} d^{3} e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 \sqrt {\left (d g -e f \right ) e}}\right )}{e^{6}}}{g^{3}}\)	\(499\)

Input:

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

Output:

3*(((d*g-e*f)*e)^(1/2)*(((2/27*e^5*x^5-22/189*d*e^4*x^4+22/105*d^2*e^3*x^3 
-22/45*d^3*e^2*x^2+22/9*d^4*e*x+11/3*d^5)*g^4-35/9*e*f*(-4/147*e^4*x^4+188 
/3675*d*e^3*x^3-156/1225*d^2*e^2*x^2+124/175*d^3*e*x+d^4)*g^3+2/5*e^2*f^2* 
(1/63*e^2*x^2-2/21*d*e*x+d^2)*(e*x+d)*g^2+8/105*(-1/9*e*x+d)*e^3*f^3*(e*x+ 
d)*g+16/945*e^4*f^4*(e*x+d))*c^2+14/3*e^2*g^2*((2/35*e^3*x^3-2/15*d*e^2*x^ 
2+2/3*d^2*e*x+d^3)*g^2-19/21*e*f*(-12/95*e^2*x^2+68/95*d*e*x+d^2)*g+2/35*e 
^2*f^2*(e*x+d))*a*c+e^4*g^3*((2/3*e*x+d)*g-1/3*e*f)*a^2)*(g*x+f)^(1/2)-(a* 
e^2+c*d^2)*g^3*((11/3*d^2*g-8/3*d*e*f)*c+a*e^2*g)*arctan(e*(g*x+f)^(1/2)/( 
(d*g-e*f)*e)^(1/2))*(e*x+d)*(d*g-e*f))/((d*g-e*f)*e)^(1/2)/g^3/e^6/(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (263) = 526\).

Time = 0.14 (sec) , antiderivative size = 1255, normalized size of antiderivative = 4.28 \[ \int \frac {(f+g x)^{3/2} \left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/630*(315*(8*(c^2*d^4*e + a*c*d^2*e^3)*f*g^3 - (11*c^2*d^5 + 14*a*c*d^3 
*e^2 + 3*a^2*d*e^4)*g^4 + (8*(c^2*d^3*e^2 + a*c*d*e^4)*f*g^3 - (11*c^2*d^4 
*e + 14*a*c*d^2*e^3 + 3*a^2*e^5)*g^4)*x)*sqrt((e*f - d*g)/e)*log((e*g*x + 
2*e*f - d*g - 2*sqrt(g*x + f)*e*sqrt((e*f - d*g)/e))/(e*x + d)) - 2*(70*c^ 
2*e^5*g^4*x^5 + 16*c^2*d*e^4*f^4 + 72*c^2*d^2*e^3*f^3*g + 126*(3*c^2*d^3*e 
^2 + 2*a*c*d*e^4)*f^2*g^2 - 105*(35*c^2*d^4*e + 38*a*c*d^2*e^3 + 3*a^2*e^5 
)*f*g^3 + 315*(11*c^2*d^5 + 14*a*c*d^3*e^2 + 3*a^2*d*e^4)*g^4 + 10*(10*c^2 
*e^5*f*g^3 - 11*c^2*d*e^4*g^4)*x^4 + 2*(3*c^2*e^5*f^2*g^2 - 94*c^2*d*e^4*f 
*g^3 + 9*(11*c^2*d^2*e^3 + 14*a*c*e^5)*g^4)*x^3 - 2*(4*c^2*e^5*f^3*g + 15* 
c^2*d*e^4*f^2*g^2 - 18*(13*c^2*d^2*e^3 + 14*a*c*e^5)*f*g^3 + 21*(11*c^2*d^ 
3*e^2 + 14*a*c*d*e^4)*g^4)*x^2 + 2*(8*c^2*e^5*f^4 + 32*c^2*d*e^4*f^3*g + 9 
*(19*c^2*d^2*e^3 + 14*a*c*e^5)*f^2*g^2 - 42*(31*c^2*d^3*e^2 + 34*a*c*d*e^4 
)*f*g^3 + 105*(11*c^2*d^4*e + 14*a*c*d^2*e^3 + 3*a^2*e^5)*g^4)*x)*sqrt(g*x 
 + f))/(e^7*g^3*x + d*e^6*g^3), 1/315*(315*(8*(c^2*d^4*e + a*c*d^2*e^3)*f* 
g^3 - (11*c^2*d^5 + 14*a*c*d^3*e^2 + 3*a^2*d*e^4)*g^4 + (8*(c^2*d^3*e^2 + 
a*c*d*e^4)*f*g^3 - (11*c^2*d^4*e + 14*a*c*d^2*e^3 + 3*a^2*e^5)*g^4)*x)*sqr 
t(-(e*f - d*g)/e)*arctan(-sqrt(g*x + f)*e*sqrt(-(e*f - d*g)/e)/(e*f - d*g) 
) + (70*c^2*e^5*g^4*x^5 + 16*c^2*d*e^4*f^4 + 72*c^2*d^2*e^3*f^3*g + 126*(3 
*c^2*d^3*e^2 + 2*a*c*d*e^4)*f^2*g^2 - 105*(35*c^2*d^4*e + 38*a*c*d^2*e^3 + 
 3*a^2*e^5)*f*g^3 + 315*(11*c^2*d^5 + 14*a*c*d^3*e^2 + 3*a^2*d*e^4)*g^4...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((g*x+f)^(3/2)*(c*x^2+a)^2/(e*x+d)^2,x, algorithm="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*(d*g-e*f)>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (263) = 526\).

Time = 0.13 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.88 \[ \int \frac {(f+g x)^{3/2} \left (a+c x^2\right )^2}{(d+e x)^2} \, dx=-\frac {{\left (8 \, c^{2} d^{3} e^{2} f^{2} + 8 \, a c d e^{4} f^{2} - 19 \, c^{2} d^{4} e f g - 22 \, a c d^{2} e^{3} f g - 3 \, a^{2} e^{5} f g + 11 \, c^{2} d^{5} g^{2} + 14 \, a c d^{3} e^{2} g^{2} + 3 \, a^{2} d e^{4} g^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{\sqrt {-e^{2} f + d e g} e^{6}} - \frac {\sqrt {g x + f} c^{2} d^{4} e f g + 2 \, \sqrt {g x + f} a c d^{2} e^{3} f g + \sqrt {g x + f} a^{2} e^{5} f g - \sqrt {g x + f} c^{2} d^{5} g^{2} - 2 \, \sqrt {g x + f} a c d^{3} e^{2} g^{2} - \sqrt {g x + f} a^{2} d e^{4} g^{2}}{{\left ({\left (g x + f\right )} e - e f + d g\right )} e^{6}} + \frac {2 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} c^{2} e^{16} g^{24} - 90 \, {\left (g x + f\right )}^{\frac {7}{2}} c^{2} e^{16} f g^{24} + 63 \, {\left (g x + f\right )}^{\frac {5}{2}} c^{2} e^{16} f^{2} g^{24} - 90 \, {\left (g x + f\right )}^{\frac {7}{2}} c^{2} d e^{15} g^{25} + 126 \, {\left (g x + f\right )}^{\frac {5}{2}} c^{2} d e^{15} f g^{25} + 189 \, {\left (g x + f\right )}^{\frac {5}{2}} c^{2} d^{2} e^{14} g^{26} + 126 \, {\left (g x + f\right )}^{\frac {5}{2}} a c e^{16} g^{26} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} c^{2} d^{3} e^{13} g^{27} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} a c d e^{15} g^{27} - 1260 \, \sqrt {g x + f} c^{2} d^{3} e^{13} f g^{27} - 1260 \, \sqrt {g x + f} a c d e^{15} f g^{27} + 1575 \, \sqrt {g x + f} c^{2} d^{4} e^{12} g^{28} + 1890 \, \sqrt {g x + f} a c d^{2} e^{14} g^{28} + 315 \, \sqrt {g x + f} a^{2} e^{16} g^{28}\right )}}{315 \, e^{18} g^{27}} \] Input:

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

Output:

-(8*c^2*d^3*e^2*f^2 + 8*a*c*d*e^4*f^2 - 19*c^2*d^4*e*f*g - 22*a*c*d^2*e^3* 
f*g - 3*a^2*e^5*f*g + 11*c^2*d^5*g^2 + 14*a*c*d^3*e^2*g^2 + 3*a^2*d*e^4*g^ 
2)*arctan(sqrt(g*x + f)*e/sqrt(-e^2*f + d*e*g))/(sqrt(-e^2*f + d*e*g)*e^6) 
 - (sqrt(g*x + f)*c^2*d^4*e*f*g + 2*sqrt(g*x + f)*a*c*d^2*e^3*f*g + sqrt(g 
*x + f)*a^2*e^5*f*g - sqrt(g*x + f)*c^2*d^5*g^2 - 2*sqrt(g*x + f)*a*c*d^3* 
e^2*g^2 - sqrt(g*x + f)*a^2*d*e^4*g^2)/(((g*x + f)*e - e*f + d*g)*e^6) + 2 
/315*(35*(g*x + f)^(9/2)*c^2*e^16*g^24 - 90*(g*x + f)^(7/2)*c^2*e^16*f*g^2 
4 + 63*(g*x + f)^(5/2)*c^2*e^16*f^2*g^24 - 90*(g*x + f)^(7/2)*c^2*d*e^15*g 
^25 + 126*(g*x + f)^(5/2)*c^2*d*e^15*f*g^25 + 189*(g*x + f)^(5/2)*c^2*d^2* 
e^14*g^26 + 126*(g*x + f)^(5/2)*a*c*e^16*g^26 - 420*(g*x + f)^(3/2)*c^2*d^ 
3*e^13*g^27 - 420*(g*x + f)^(3/2)*a*c*d*e^15*g^27 - 1260*sqrt(g*x + f)*c^2 
*d^3*e^13*f*g^27 - 1260*sqrt(g*x + f)*a*c*d*e^15*f*g^27 + 1575*sqrt(g*x + 
f)*c^2*d^4*e^12*g^28 + 1890*sqrt(g*x + f)*a*c*d^2*e^14*g^28 + 315*sqrt(g*x 
 + f)*a^2*e^16*g^28)/(e^18*g^27)
 

Mupad [B] (verification not implemented)

Time = 5.80 (sec) , antiderivative size = 965, normalized size of antiderivative = 3.29 \[ \int \frac {(f+g x)^{3/2} \left (a+c x^2\right )^2}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(f + g*x)^(1/2)*((2*(d*g - e*f)*((8*c^2*f^3 + 8*a*c*f*g^2)/(e^2*g^3) - ((d 
*g - e*f)^2*((8*c^2*f)/(e^2*g^3) + (4*c^2*(d*g - e*f))/(e^3*g^3)))/e^2 + ( 
2*(d*g - e*f)*((12*c^2*f^2 + 4*a*c*g^2)/(e^2*g^3) + (2*(d*g - e*f)*((8*c^2 
*f)/(e^2*g^3) + (4*c^2*(d*g - e*f))/(e^3*g^3)))/e - (2*c^2*(d*g - e*f)^2)/ 
(e^4*g^3)))/e))/e - ((d*g - e*f)^2*((12*c^2*f^2 + 4*a*c*g^2)/(e^2*g^3) + ( 
2*(d*g - e*f)*((8*c^2*f)/(e^2*g^3) + (4*c^2*(d*g - e*f))/(e^3*g^3)))/e - ( 
2*c^2*(d*g - e*f)^2)/(e^4*g^3)))/e^2 + (2*(a*g^2 + c*f^2)^2)/(e^2*g^3)) - 
(f + g*x)^(3/2)*((8*c^2*f^3 + 8*a*c*f*g^2)/(3*e^2*g^3) - ((d*g - e*f)^2*(( 
8*c^2*f)/(e^2*g^3) + (4*c^2*(d*g - e*f))/(e^3*g^3)))/(3*e^2) + (2*(d*g - e 
*f)*((12*c^2*f^2 + 4*a*c*g^2)/(e^2*g^3) + (2*(d*g - e*f)*((8*c^2*f)/(e^2*g 
^3) + (4*c^2*(d*g - e*f))/(e^3*g^3)))/e - (2*c^2*(d*g - e*f)^2)/(e^4*g^3)) 
)/(3*e)) - (f + g*x)^(7/2)*((8*c^2*f)/(7*e^2*g^3) + (4*c^2*(d*g - e*f))/(7 
*e^3*g^3)) + (f + g*x)^(5/2)*((12*c^2*f^2 + 4*a*c*g^2)/(5*e^2*g^3) + (2*(d 
*g - e*f)*((8*c^2*f)/(e^2*g^3) + (4*c^2*(d*g - e*f))/(e^3*g^3)))/(5*e) - ( 
2*c^2*(d*g - e*f)^2)/(5*e^4*g^3)) + ((f + g*x)^(1/2)*(c^2*d^5*g^2 + a^2*d* 
e^4*g^2 - a^2*e^5*f*g - c^2*d^4*e*f*g + 2*a*c*d^3*e^2*g^2 - 2*a*c*d^2*e^3* 
f*g))/(e^7*(f + g*x) - e^7*f + d*e^6*g) + (2*c^2*(f + g*x)^(9/2))/(9*e^2*g 
^3) - (atan((e^(1/2)*(f + g*x)^(1/2)*(a*e^2 + c*d^2)*(d*g - e*f)^(1/2)*(3* 
a*e^2*g + 11*c*d^2*g - 8*c*d*e*f))/(11*c^2*d^5*g^2 + 3*a^2*d*e^4*g^2 + 8*c 
^2*d^3*e^2*f^2 - 3*a^2*e^5*f*g + 8*a*c*d*e^4*f^2 - 19*c^2*d^4*e*f*g + 1...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1156, normalized size of antiderivative = 3.95 \[ \int \frac {(f+g x)^{3/2} \left (a+c x^2\right )^2}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 945*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - 
 e*f)))*a**2*d*e**4*g**4 - 945*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x) 
*e)/(sqrt(e)*sqrt(d*g - e*f)))*a**2*e**5*g**4*x - 4410*sqrt(e)*sqrt(d*g - 
e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*a*c*d**3*e**2*g**4 
+ 2520*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - 
e*f)))*a*c*d**2*e**3*f*g**3 - 4410*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + 
g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*a*c*d**2*e**3*g**4*x + 2520*sqrt(e)*sqr 
t(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*a*c*d*e**4* 
f*g**3*x - 3465*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sq 
rt(d*g - e*f)))*c**2*d**5*g**4 + 2520*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f 
 + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*c**2*d**4*e*f*g**3 - 3465*sqrt(e)*sq 
rt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*c**2*d**4* 
e*g**4*x + 2520*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sq 
rt(d*g - e*f)))*c**2*d**3*e**2*f*g**3*x + 945*sqrt(f + g*x)*a**2*d*e**5*g* 
*4 - 315*sqrt(f + g*x)*a**2*e**6*f*g**3 + 630*sqrt(f + g*x)*a**2*e**6*g**4 
*x + 4410*sqrt(f + g*x)*a*c*d**3*e**3*g**4 - 3990*sqrt(f + g*x)*a*c*d**2*e 
**4*f*g**3 + 2940*sqrt(f + g*x)*a*c*d**2*e**4*g**4*x + 252*sqrt(f + g*x)*a 
*c*d*e**5*f**2*g**2 - 2856*sqrt(f + g*x)*a*c*d*e**5*f*g**3*x - 588*sqrt(f 
+ g*x)*a*c*d*e**5*g**4*x**2 + 252*sqrt(f + g*x)*a*c*e**6*f**2*g**2*x + 504 
*sqrt(f + g*x)*a*c*e**6*f*g**3*x**2 + 252*sqrt(f + g*x)*a*c*e**6*g**4*x...