\(\int \frac {(a+c x^2)^2}{(d+e x)^3 \sqrt {f+g x}} \, dx\) [83]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [B] (verification not implemented)
Sympy [F(-1)]
Maxima [F(-2)]
Giac [A] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 26, antiderivative size = 282 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=-\frac {2 c^2 (e f+3 d g) \sqrt {f+g x}}{e^4 g^2}-\frac {\left (c d^2+a e^2\right )^2 \sqrt {f+g x}}{2 e^4 (e f-d g) (d+e x)^2}+\frac {\left (c d^2+a e^2\right ) \left (3 a e^2 g+c d (16 e f-13 d g)\right ) \sqrt {f+g x}}{4 e^4 (e f-d g)^2 (d+e x)}+\frac {2 c^2 (f+g x)^{3/2}}{3 e^3 g^2}-\frac {\left (3 a^2 e^4 g^2+2 a c e^2 \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )+c^2 d^2 \left (48 e^2 f^2-80 d e f g+35 d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{9/2} (e f-d g)^{5/2}} \] Output:

-2*c^2*(3*d*g+e*f)*(g*x+f)^(1/2)/e^4/g^2-1/2*(a*e^2+c*d^2)^2*(g*x+f)^(1/2) 
/e^4/(-d*g+e*f)/(e*x+d)^2+1/4*(a*e^2+c*d^2)*(3*a*e^2*g+c*d*(-13*d*g+16*e*f 
))*(g*x+f)^(1/2)/e^4/(-d*g+e*f)^2/(e*x+d)+2/3*c^2*(g*x+f)^(3/2)/e^3/g^2-1/ 
4*(3*a^2*e^4*g^2+2*a*c*e^2*(3*d^2*g^2-8*d*e*f*g+8*e^2*f^2)+c^2*d^2*(35*d^2 
*g^2-80*d*e*f*g+48*e^2*f^2))*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2 
))/e^(9/2)/(-d*g+e*f)^(5/2)
 

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=-\frac {\sqrt {f+g x} \left (-3 a^2 e^4 g^2 (-2 e f+5 d g+3 e g x)+6 a c d e^2 g^2 \left (-6 d e f+3 d^2 g-8 e^2 f x+5 d e g x\right )+c^2 \left (105 d^5 g^3+8 e^5 f^2 x^2 (2 f-g x)+5 d^4 e g^2 (-34 f+35 g x)+8 d e^4 f x \left (4 f^2+3 f g x+2 g^2 x^2\right )+8 d^3 e^2 g \left (5 f^2-36 f g x+7 g^2 x^2\right )+8 d^2 e^3 \left (2 f^3+9 f^2 g x-12 f g^2 x^2-g^3 x^3\right )\right )\right )}{12 e^4 g^2 (e f-d g)^2 (d+e x)^2}+\frac {\left (3 a^2 e^4 g^2+2 a c e^2 \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )+c^2 d^2 \left (48 e^2 f^2-80 d e f g+35 d^2 g^2\right )\right ) \arctan \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{4 e^{9/2} (-e f+d g)^{5/2}} \] Input:

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

Output:

-1/12*(Sqrt[f + g*x]*(-3*a^2*e^4*g^2*(-2*e*f + 5*d*g + 3*e*g*x) + 6*a*c*d* 
e^2*g^2*(-6*d*e*f + 3*d^2*g - 8*e^2*f*x + 5*d*e*g*x) + c^2*(105*d^5*g^3 + 
8*e^5*f^2*x^2*(2*f - g*x) + 5*d^4*e*g^2*(-34*f + 35*g*x) + 8*d*e^4*f*x*(4* 
f^2 + 3*f*g*x + 2*g^2*x^2) + 8*d^3*e^2*g*(5*f^2 - 36*f*g*x + 7*g^2*x^2) + 
8*d^2*e^3*(2*f^3 + 9*f^2*g*x - 12*f*g^2*x^2 - g^3*x^3))))/(e^4*g^2*(e*f - 
d*g)^2*(d + e*x)^2) + ((3*a^2*e^4*g^2 + 2*a*c*e^2*(8*e^2*f^2 - 8*d*e*f*g + 
 3*d^2*g^2) + c^2*d^2*(48*e^2*f^2 - 80*d*e*f*g + 35*d^2*g^2))*ArcTan[(Sqrt 
[e]*Sqrt[f + g*x])/Sqrt[-(e*f) + d*g]])/(4*e^(9/2)*(-(e*f) + d*g)^(5/2))
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.27, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {649, 25, 1471, 25, 2345, 25, 1467, 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.

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

\(\Big \downarrow \) 649

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1471

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2345

\(\displaystyle \frac {2 \left (-\frac {\frac {g^3 \sqrt {f+g x} \left (a e^2+c d^2\right ) \left (3 a e^2 g+c d (16 e f-13 d g)\right )}{2 e^4 (e f-d g) (-d g-e (f+g x)+e f)}-\frac {\int -\frac {3 a^2 g^4+\frac {2 a c \left (8 e^2 f^2-8 d e g f+3 d^2 g^2\right ) g^2}{e^2}+\frac {8 c^2 (e f-d g)^2 (f+g x)^2}{e^2}+c^2 \left (8 f^4-\frac {16 d^3 g^3 f}{e^3}+\frac {11 d^4 g^4}{e^4}\right )-\frac {16 c^2 (e f-d g)^2 (e f+d g) (f+g x)}{e^3}}{e f-d g-e (f+g x)}d\sqrt {f+g x}}{2 (e f-d g)}}{4 (e f-d g)}-\frac {g^4 \sqrt {f+g x} \left (a e^2+c d^2\right )^2}{4 e^4 (e f-d g) (-d g-e (f+g x)+e f)^2}\right )}{g^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (-\frac {\frac {\int \frac {3 a^2 g^4+\frac {2 a c \left (8 e^2 f^2-8 d e g f+3 d^2 g^2\right ) g^2}{e^2}+\frac {8 c^2 (e f-d g)^2 (f+g x)^2}{e^2}+\frac {c^2 \left (8 e^4 f^4-16 d^3 e g^3 f+11 d^4 g^4\right )}{e^4}-\frac {16 c^2 (e f-d g)^2 (e f+d g) (f+g x)}{e^3}}{e f-d g-e (f+g x)}d\sqrt {f+g x}}{2 (e f-d g)}+\frac {g^3 \sqrt {f+g x} \left (a e^2+c d^2\right ) \left (3 a e^2 g+c d (16 e f-13 d g)\right )}{2 e^4 (e f-d g) (-d g-e (f+g x)+e f)}}{4 (e f-d g)}-\frac {g^4 \sqrt {f+g x} \left (a e^2+c d^2\right )^2}{4 e^4 (e f-d g) (-d g-e (f+g x)+e f)^2}\right )}{g^2}\)

\(\Big \downarrow \) 1467

\(\displaystyle \frac {2 \left (-\frac {\frac {\int \left (\frac {8 c^2 (e f+3 d g) (e f-d g)^2}{e^4}-\frac {8 c^2 (f+g x) (e f-d g)^2}{e^3}+\frac {3 a^2 g^4 e^4+16 a c f^2 g^2 e^4-16 a c d f g^3 e^3+6 a c d^2 g^4 e^2+48 c^2 d^2 f^2 g^2 e^2-80 c^2 d^3 f g^3 e+35 c^2 d^4 g^4}{e^4 (e f-d g-e (f+g x))}\right )d\sqrt {f+g x}}{2 (e f-d g)}+\frac {g^3 \sqrt {f+g x} \left (a e^2+c d^2\right ) \left (3 a e^2 g+c d (16 e f-13 d g)\right )}{2 e^4 (e f-d g) (-d g-e (f+g x)+e f)}}{4 (e f-d g)}-\frac {g^4 \sqrt {f+g x} \left (a e^2+c d^2\right )^2}{4 e^4 (e f-d g) (-d g-e (f+g x)+e f)^2}\right )}{g^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {\frac {\frac {g^2 \left (3 a^2 e^4 g^2+2 a c e^2 \left (3 d^2 g^2-8 d e f g+8 e^2 f^2\right )+c^2 d^2 \left (35 d^2 g^2-80 d e f g+48 e^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{9/2} \sqrt {e f-d g}}+\frac {8 c^2 \sqrt {f+g x} (e f-d g)^2 (3 d g+e f)}{e^4}-\frac {8 c^2 (f+g x)^{3/2} (e f-d g)^2}{3 e^3}}{2 (e f-d g)}+\frac {g^3 \sqrt {f+g x} \left (a e^2+c d^2\right ) \left (3 a e^2 g+c d (16 e f-13 d g)\right )}{2 e^4 (e f-d g) (-d g-e (f+g x)+e f)}}{4 (e f-d g)}-\frac {g^4 \sqrt {f+g x} \left (a e^2+c d^2\right )^2}{4 e^4 (e f-d g) (-d g-e (f+g x)+e f)^2}\right )}{g^2}\)

Input:

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

Output:

(2*(-1/4*((c*d^2 + a*e^2)^2*g^4*Sqrt[f + g*x])/(e^4*(e*f - d*g)*(e*f - d*g 
 - e*(f + g*x))^2) - (((c*d^2 + a*e^2)*g^3*(3*a*e^2*g + c*d*(16*e*f - 13*d 
*g))*Sqrt[f + g*x])/(2*e^4*(e*f - d*g)*(e*f - d*g - e*(f + g*x))) + ((8*c^ 
2*(e*f - d*g)^2*(e*f + 3*d*g)*Sqrt[f + g*x])/e^4 - (8*c^2*(e*f - d*g)^2*(f 
 + g*x)^(3/2))/(3*e^3) + (g^2*(3*a^2*e^4*g^2 + 2*a*c*e^2*(8*e^2*f^2 - 8*d* 
e*f*g + 3*d^2*g^2) + c^2*d^2*(48*e^2*f^2 - 80*d*e*f*g + 35*d^2*g^2))*ArcTa 
nh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(9/2)*Sqrt[e*f - d*g]))/(2 
*(e*f - d*g)))/(4*(e*f - d*g))))/g^2
 

Defintions of rubi rules used

rule 25
Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1467
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
 x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], 
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e 
 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 1471
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2345
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 
Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.24

method result size
risch \(-\frac {2 c^{2} \left (-e g x +9 d g +2 e f \right ) \sqrt {g x +f}}{3 g^{2} e^{4}}+\frac {\frac {\frac {2 e g \left (3 a^{2} e^{4} g -10 a c \,d^{2} e^{2} g +16 a c d \,e^{3} f -13 c^{2} d^{4} g +16 c^{2} d^{3} e f \right ) \left (g x +f \right )^{\frac {3}{2}}}{8 d^{2} g^{2}-16 d e f g +8 e^{2} f^{2}}+\frac {2 \left (5 a^{2} e^{4} g -6 a c \,d^{2} e^{2} g +16 a c d \,e^{3} f -11 c^{2} d^{4} g +16 c^{2} d^{3} e f \right ) g \sqrt {g x +f}}{8 d g -8 e f}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (3 a^{2} e^{4} g^{2}+6 a c \,d^{2} e^{2} g^{2}-16 a c d \,e^{3} f g +16 a c \,e^{4} f^{2}+35 c^{2} d^{4} g^{2}-80 c^{2} d^{3} e f g +48 c^{2} d^{2} e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}}}{e^{4}}\) \(351\)
pseudoelliptic \(\frac {5 \left (\left (-7 \left (-\frac {8}{105} e^{3} x^{3}+\frac {8}{15} d \,e^{2} x^{2}+\frac {5}{3} d^{2} e x +d^{3}\right ) d^{2} c^{2}-\frac {6 \left (\frac {5 e x}{3}+d \right ) e^{2} a \,d^{2} c}{5}+a^{2} e^{4} \left (\frac {3 e x}{5}+d \right )\right ) g^{3}-\frac {2 e f \left (\left (\frac {8}{3} d \,e^{3} x^{3}-16 d^{2} e^{2} x^{2}-48 d^{3} e x -\frac {85}{3} d^{4}\right ) c^{2}-6 \left (\frac {4 e x}{3}+d \right ) e^{2} a d c +a^{2} e^{4}\right ) g^{2}}{5}-\frac {8 e^{2} f^{2} c^{2} \left (e x +d \right )^{2} \left (-\frac {e x}{5}+d \right ) g}{3}-\frac {16 c^{2} e^{3} f^{3} \left (e x +d \right )^{2}}{15}\right ) \sqrt {\left (d g -e f \right ) e}\, \sqrt {g x +f}+g^{2} \left (e x +d \right )^{2} \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) \left (3 a^{2} e^{4} g^{2}+6 a c \,d^{2} e^{2} g^{2}-16 a c d \,e^{3} f g +16 a c \,e^{4} f^{2}+35 c^{2} d^{4} g^{2}-80 c^{2} d^{3} e f g +48 c^{2} d^{2} e^{2} f^{2}\right )}{4 \left (d g -e f \right )^{2} \sqrt {\left (d g -e f \right ) e}\, e^{4} \left (e x +d \right )^{2} g^{2}}\) \(354\)
derivativedivides \(\frac {-\frac {2 c^{2} \left (-\frac {e \left (g x +f \right )^{\frac {3}{2}}}{3}+3 d g \sqrt {g x +f}+e f \sqrt {g x +f}\right )}{e^{4}}+\frac {2 g^{2} \left (\frac {\frac {e g \left (3 a^{2} e^{4} g -10 a c \,d^{2} e^{2} g +16 a c d \,e^{3} f -13 c^{2} d^{4} g +16 c^{2} d^{3} e f \right ) \left (g x +f \right )^{\frac {3}{2}}}{8 d^{2} g^{2}-16 d e f g +8 e^{2} f^{2}}+\frac {\left (5 a^{2} e^{4} g -6 a c \,d^{2} e^{2} g +16 a c d \,e^{3} f -11 c^{2} d^{4} g +16 c^{2} d^{3} e f \right ) g \sqrt {g x +f}}{8 d g -8 e f}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (3 a^{2} e^{4} g^{2}+6 a c \,d^{2} e^{2} g^{2}-16 a c d \,e^{3} f g +16 a c \,e^{4} f^{2}+35 c^{2} d^{4} g^{2}-80 c^{2} d^{3} e f g +48 c^{2} d^{2} e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{8 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}}\right )}{e^{4}}}{g^{2}}\) \(366\)
default \(\frac {-\frac {2 c^{2} \left (-\frac {e \left (g x +f \right )^{\frac {3}{2}}}{3}+3 d g \sqrt {g x +f}+e f \sqrt {g x +f}\right )}{e^{4}}+\frac {2 g^{2} \left (\frac {\frac {e g \left (3 a^{2} e^{4} g -10 a c \,d^{2} e^{2} g +16 a c d \,e^{3} f -13 c^{2} d^{4} g +16 c^{2} d^{3} e f \right ) \left (g x +f \right )^{\frac {3}{2}}}{8 d^{2} g^{2}-16 d e f g +8 e^{2} f^{2}}+\frac {\left (5 a^{2} e^{4} g -6 a c \,d^{2} e^{2} g +16 a c d \,e^{3} f -11 c^{2} d^{4} g +16 c^{2} d^{3} e f \right ) g \sqrt {g x +f}}{8 d g -8 e f}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (3 a^{2} e^{4} g^{2}+6 a c \,d^{2} e^{2} g^{2}-16 a c d \,e^{3} f g +16 a c \,e^{4} f^{2}+35 c^{2} d^{4} g^{2}-80 c^{2} d^{3} e f g +48 c^{2} d^{2} e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{8 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}}\right )}{e^{4}}}{g^{2}}\) \(366\)

Input:

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

Output:

-2/3*c^2*(-e*g*x+9*d*g+2*e*f)*(g*x+f)^(1/2)/g^2/e^4+1/e^4*(2*(1/8*e*g*(3*a 
^2*e^4*g-10*a*c*d^2*e^2*g+16*a*c*d*e^3*f-13*c^2*d^4*g+16*c^2*d^3*e*f)/(d^2 
*g^2-2*d*e*f*g+e^2*f^2)*(g*x+f)^(3/2)+1/8*(5*a^2*e^4*g-6*a*c*d^2*e^2*g+16* 
a*c*d*e^3*f-11*c^2*d^4*g+16*c^2*d^3*e*f)*g/(d*g-e*f)*(g*x+f)^(1/2))/(e*(g* 
x+f)+d*g-e*f)^2+1/4*(3*a^2*e^4*g^2+6*a*c*d^2*e^2*g^2-16*a*c*d*e^3*f*g+16*a 
*c*e^4*f^2+35*c^2*d^4*g^2-80*c^2*d^3*e*f*g+48*c^2*d^2*e^2*f^2)/(d^2*g^2-2* 
d*e*f*g+e^2*f^2)/((d*g-e*f)*e)^(1/2)*arctan(e*(g*x+f)^(1/2)/((d*g-e*f)*e)^ 
(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (256) = 512\).

Time = 0.15 (sec) , antiderivative size = 1791, normalized size of antiderivative = 6.35 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/24*(3*(16*(3*c^2*d^4*e^2 + a*c*d^2*e^4)*f^2*g^2 - 16*(5*c^2*d^5*e + a*c 
*d^3*e^3)*f*g^3 + (35*c^2*d^6 + 6*a*c*d^4*e^2 + 3*a^2*d^2*e^4)*g^4 + (16*( 
3*c^2*d^2*e^4 + a*c*e^6)*f^2*g^2 - 16*(5*c^2*d^3*e^3 + a*c*d*e^5)*f*g^3 + 
(35*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + 3*a^2*e^6)*g^4)*x^2 + 2*(16*(3*c^2*d^3*e 
^3 + a*c*d*e^5)*f^2*g^2 - 16*(5*c^2*d^4*e^2 + a*c*d^2*e^4)*f*g^3 + (35*c^2 
*d^5*e + 6*a*c*d^3*e^3 + 3*a^2*d*e^5)*g^4)*x)*sqrt(e^2*f - d*e*g)*log((e*g 
*x + 2*e*f - d*g - 2*sqrt(e^2*f - d*e*g)*sqrt(g*x + f))/(e*x + d)) - 2*(16 
*c^2*d^2*e^5*f^4 + 24*c^2*d^3*e^4*f^3*g - 6*(35*c^2*d^4*e^3 + 6*a*c*d^2*e^ 
5 - a^2*e^7)*f^2*g^2 + (275*c^2*d^5*e^2 + 54*a*c*d^3*e^4 - 21*a^2*d*e^6)*f 
*g^3 - 3*(35*c^2*d^6*e + 6*a*c*d^4*e^3 - 5*a^2*d^2*e^5)*g^4 - 8*(c^2*e^7*f 
^3*g - 3*c^2*d*e^6*f^2*g^2 + 3*c^2*d^2*e^5*f*g^3 - c^2*d^3*e^4*g^4)*x^3 + 
8*(2*c^2*e^7*f^4 + c^2*d*e^6*f^3*g - 15*c^2*d^2*e^5*f^2*g^2 + 19*c^2*d^3*e 
^4*f*g^3 - 7*c^2*d^4*e^3*g^4)*x^2 + (32*c^2*d*e^6*f^4 + 40*c^2*d^2*e^5*f^3 
*g - 24*(15*c^2*d^3*e^4 + 2*a*c*d*e^6)*f^2*g^2 + (463*c^2*d^4*e^3 + 78*a*c 
*d^2*e^5 - 9*a^2*e^7)*f*g^3 - (175*c^2*d^5*e^2 + 30*a*c*d^3*e^4 - 9*a^2*d* 
e^6)*g^4)*x)*sqrt(g*x + f))/(d^2*e^8*f^3*g^2 - 3*d^3*e^7*f^2*g^3 + 3*d^4*e 
^6*f*g^4 - d^5*e^5*g^5 + (e^10*f^3*g^2 - 3*d*e^9*f^2*g^3 + 3*d^2*e^8*f*g^4 
 - d^3*e^7*g^5)*x^2 + 2*(d*e^9*f^3*g^2 - 3*d^2*e^8*f^2*g^3 + 3*d^3*e^7*f*g 
^4 - d^4*e^6*g^5)*x), 1/12*(3*(16*(3*c^2*d^4*e^2 + a*c*d^2*e^4)*f^2*g^2 - 
16*(5*c^2*d^5*e + a*c*d^3*e^3)*f*g^3 + (35*c^2*d^6 + 6*a*c*d^4*e^2 + 3*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((c*x^2+a)^2/(e*x+d)^3/(g*x+f)^(1/2),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 [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.82 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=\frac {{\left (48 \, c^{2} d^{2} e^{2} f^{2} + 16 \, a c e^{4} f^{2} - 80 \, c^{2} d^{3} e f g - 16 \, a c d e^{3} f g + 35 \, c^{2} d^{4} g^{2} + 6 \, a c d^{2} e^{2} g^{2} + 3 \, a^{2} e^{4} g^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{4 \, {\left (e^{6} f^{2} - 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} \sqrt {-e^{2} f + d e g}} + \frac {16 \, {\left (g x + f\right )}^{\frac {3}{2}} c^{2} d^{3} e^{2} f g + 16 \, {\left (g x + f\right )}^{\frac {3}{2}} a c d e^{4} f g - 16 \, \sqrt {g x + f} c^{2} d^{3} e^{2} f^{2} g - 16 \, \sqrt {g x + f} a c d e^{4} f^{2} g - 13 \, {\left (g x + f\right )}^{\frac {3}{2}} c^{2} d^{4} e g^{2} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} a c d^{2} e^{3} g^{2} + 3 \, {\left (g x + f\right )}^{\frac {3}{2}} a^{2} e^{5} g^{2} + 27 \, \sqrt {g x + f} c^{2} d^{4} e f g^{2} + 22 \, \sqrt {g x + f} a c d^{2} e^{3} f g^{2} - 5 \, \sqrt {g x + f} a^{2} e^{5} f g^{2} - 11 \, \sqrt {g x + f} c^{2} d^{5} g^{3} - 6 \, \sqrt {g x + f} a c d^{3} e^{2} g^{3} + 5 \, \sqrt {g x + f} a^{2} d e^{4} g^{3}}{4 \, {\left (e^{6} f^{2} - 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} {\left ({\left (g x + f\right )} e - e f + d g\right )}^{2}} + \frac {2 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} c^{2} e^{6} g^{4} - 3 \, \sqrt {g x + f} c^{2} e^{6} f g^{4} - 9 \, \sqrt {g x + f} c^{2} d e^{5} g^{5}\right )}}{3 \, e^{9} g^{6}} \] Input:

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

Output:

1/4*(48*c^2*d^2*e^2*f^2 + 16*a*c*e^4*f^2 - 80*c^2*d^3*e*f*g - 16*a*c*d*e^3 
*f*g + 35*c^2*d^4*g^2 + 6*a*c*d^2*e^2*g^2 + 3*a^2*e^4*g^2)*arctan(sqrt(g*x 
 + f)*e/sqrt(-e^2*f + d*e*g))/((e^6*f^2 - 2*d*e^5*f*g + d^2*e^4*g^2)*sqrt( 
-e^2*f + d*e*g)) + 1/4*(16*(g*x + f)^(3/2)*c^2*d^3*e^2*f*g + 16*(g*x + f)^ 
(3/2)*a*c*d*e^4*f*g - 16*sqrt(g*x + f)*c^2*d^3*e^2*f^2*g - 16*sqrt(g*x + f 
)*a*c*d*e^4*f^2*g - 13*(g*x + f)^(3/2)*c^2*d^4*e*g^2 - 10*(g*x + f)^(3/2)* 
a*c*d^2*e^3*g^2 + 3*(g*x + f)^(3/2)*a^2*e^5*g^2 + 27*sqrt(g*x + f)*c^2*d^4 
*e*f*g^2 + 22*sqrt(g*x + f)*a*c*d^2*e^3*f*g^2 - 5*sqrt(g*x + f)*a^2*e^5*f* 
g^2 - 11*sqrt(g*x + f)*c^2*d^5*g^3 - 6*sqrt(g*x + f)*a*c*d^3*e^2*g^3 + 5*s 
qrt(g*x + f)*a^2*d*e^4*g^3)/((e^6*f^2 - 2*d*e^5*f*g + d^2*e^4*g^2)*((g*x + 
 f)*e - e*f + d*g)^2) + 2/3*((g*x + f)^(3/2)*c^2*e^6*g^4 - 3*sqrt(g*x + f) 
*c^2*e^6*f*g^4 - 9*sqrt(g*x + f)*c^2*d*e^5*g^5)/(e^9*g^6)
 

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=\frac {\frac {\sqrt {f+g\,x}\,\left (5\,a^2\,e^4\,g^2-6\,a\,c\,d^2\,e^2\,g^2+16\,f\,a\,c\,d\,e^3\,g-11\,c^2\,d^4\,g^2+16\,f\,c^2\,d^3\,e\,g\right )}{4\,\left (d\,g-e\,f\right )}+\frac {{\left (f+g\,x\right )}^{3/2}\,\left (3\,a^2\,e^5\,g^2-10\,a\,c\,d^2\,e^3\,g^2+16\,f\,a\,c\,d\,e^4\,g-13\,c^2\,d^4\,e\,g^2+16\,f\,c^2\,d^3\,e^2\,g\right )}{4\,{\left (d\,g-e\,f\right )}^2}}{e^6\,{\left (f+g\,x\right )}^2-\left (f+g\,x\right )\,\left (2\,e^6\,f-2\,d\,e^5\,g\right )+e^6\,f^2+d^2\,e^4\,g^2-2\,d\,e^5\,f\,g}-\sqrt {f+g\,x}\,\left (\frac {8\,c^2\,f}{e^3\,g^2}+\frac {6\,c^2\,\left (d\,g-e\,f\right )}{e^4\,g^2}\right )+\frac {2\,c^2\,{\left (f+g\,x\right )}^{3/2}}{3\,e^3\,g^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (3\,a^2\,e^4\,g^2+6\,a\,c\,d^2\,e^2\,g^2-16\,a\,c\,d\,e^3\,f\,g+16\,a\,c\,e^4\,f^2+35\,c^2\,d^4\,g^2-80\,c^2\,d^3\,e\,f\,g+48\,c^2\,d^2\,e^2\,f^2\right )}{4\,e^{9/2}\,{\left (d\,g-e\,f\right )}^{5/2}} \] Input:

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

Output:

(((f + g*x)^(1/2)*(5*a^2*e^4*g^2 - 11*c^2*d^4*g^2 + 16*c^2*d^3*e*f*g - 6*a 
*c*d^2*e^2*g^2 + 16*a*c*d*e^3*f*g))/(4*(d*g - e*f)) + ((f + g*x)^(3/2)*(3* 
a^2*e^5*g^2 - 13*c^2*d^4*e*g^2 - 10*a*c*d^2*e^3*g^2 + 16*c^2*d^3*e^2*f*g + 
 16*a*c*d*e^4*f*g))/(4*(d*g - e*f)^2))/(e^6*(f + g*x)^2 - (f + g*x)*(2*e^6 
*f - 2*d*e^5*g) + e^6*f^2 + d^2*e^4*g^2 - 2*d*e^5*f*g) - (f + g*x)^(1/2)*( 
(8*c^2*f)/(e^3*g^2) + (6*c^2*(d*g - e*f))/(e^4*g^2)) + (2*c^2*(f + g*x)^(3 
/2))/(3*e^3*g^2) + (atan((e^(1/2)*(f + g*x)^(1/2))/(d*g - e*f)^(1/2))*(3*a 
^2*e^4*g^2 + 35*c^2*d^4*g^2 + 48*c^2*d^2*e^2*f^2 + 16*a*c*e^4*f^2 - 80*c^2 
*d^3*e*f*g + 6*a*c*d^2*e^2*g^2 - 16*a*c*d*e^3*f*g))/(4*e^(9/2)*(d*g - e*f) 
^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1841, normalized size of antiderivative = 6.53 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3 \sqrt {f+g x}} \, dx =\text {Too large to display} \] Input:

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

Output:

(9*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f) 
))*a**2*d**2*e**4*g**4 + 18*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e) 
/(sqrt(e)*sqrt(d*g - e*f)))*a**2*d*e**5*g**4*x + 9*sqrt(e)*sqrt(d*g - e*f) 
*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*a**2*e**6*g**4*x**2 + 1 
8*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)) 
)*a*c*d**4*e**2*g**4 - 48*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**3*f*g**3 + 36*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**3*g**4*x + 
 48*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**4*f**2*g**2 - 96*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**4*f*g**3*x + 18*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**4 
*g**4*x**2 + 96*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sq 
rt(d*g - e*f)))*a*c*d*e**5*f**2*g**2*x - 48*sqrt(e)*sqrt(d*g - e*f)*atan(( 
sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*a*c*d*e**5*f*g**3*x**2 + 48*sq 
rt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*a* 
c*e**6*f**2*g**2*x**2 + 105*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e) 
/(sqrt(e)*sqrt(d*g - e*f)))*c**2*d**6*g**4 - 240*sqrt(e)*sqrt(d*g - e*f)*a 
tan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*c**2*d**5*e*f*g**3 + 210* 
sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)...