\(\int \frac {a+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, dx\) [595]

Optimal result

Integrand size = 24, antiderivative size = 178 \[ \int \frac {a+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=-\frac {\left (a+\frac {c d^2}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {\left (3 a e^2 g+c d (8 e f-5 d g)\right ) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}-\frac {\left (3 a e^2 g^2+c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {912, 1171, 393, 214} \[ \int \frac {a+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=-\frac {\left (3 a e^2 g^2+c \left (3 d^2 g^2-8 d e f g+8 e^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}}-\frac {\sqrt {f+g x} \left (a e^2+c d^2\right )}{2 e^2 (d+e x)^2 (e f-d g)}+\frac {\sqrt {f+g x} \left (3 a e^2 g+c d (8 e f-5 d g)\right )}{4 e^2 (d+e x) (e f-d g)^2} \]

[In]

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

[Out]

-1/2*((c*d^2 + a*e^2)*Sqrt[f + g*x])/(e^2*(e*f - d*g)*(d + e*x)^2) + ((3*a*e^2*g + c*d*(8*e*f - 5*d*g))*Sqrt[f
+ g*x])/(4*e^2*(e*f - d*g)^2*(d + e*x)) - ((3*a*e^2*g^2 + c*(8*e^2*f^2 - 8*d*e*f*g + 3*d^2*g^2))*ArcTanh[(Sqr
t[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(4*e^(5/2)*(e*f - d*g)^(5/2))

Rule 214

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

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 912

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 + a*e^2)/e^2 - 2*c*
d*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\frac {c f^2+a g^2}{g^2}-\frac {2 c f x^2}{g^2}+\frac {c x^4}{g^2}}{\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^3} \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = -\frac {\left (c d^2+a e^2\right ) \sqrt {f+g x}}{2 e^2 (e f-d g) (d+e x)^2}+\frac {\text {Subst}\left (\int \frac {-3 a+\frac {c d^2}{e^2}-\frac {4 c f^2}{g^2}+\frac {4 c (e f-d g) x^2}{e g^2}}{\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2} \, dx,x,\sqrt {f+g x}\right )}{2 (e f-d g)} \\ & = -\frac {\left (c d^2+a e^2\right ) \sqrt {f+g x}}{2 e^2 (e f-d g) (d+e x)^2}+\frac {\left (3 a e^2 g+c d (8 e f-5 d g)\right ) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}+\frac {\left (3 a e^2 g^2+c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{4 e^2 g (e f-d g)^2} \\ & = -\frac {\left (c d^2+a e^2\right ) \sqrt {f+g x}}{2 e^2 (e f-d g) (d+e x)^2}+\frac {\left (3 a e^2 g+c d (8 e f-5 d g)\right ) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}-\frac {\left (3 a e^2 g^2+c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.93 \[ \int \frac {a+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=\frac {\frac {\sqrt {e} \sqrt {f+g x} \left (a e^2 (-2 e f+5 d g+3 e g x)+c d \left (-3 d^2 g+8 e^2 f x+d e (6 f-5 g x)\right )\right )}{(e f-d g)^2 (d+e x)^2}+\frac {\left (3 a e^2 g^2+c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \arctan \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{(-e f+d g)^{5/2}}}{4 e^{5/2}} \]

[In]

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

[Out]

((Sqrt[e]*Sqrt[f + g*x]*(a*e^2*(-2*e*f + 5*d*g + 3*e*g*x) + c*d*(-3*d^2*g + 8*e^2*f*x + d*e*(6*f - 5*g*x))))/(
(e*f - d*g)^2*(d + e*x)^2) + ((3*a*e^2*g^2 + c*(8*e^2*f^2 - 8*d*e*f*g + 3*d^2*g^2))*ArcTan[(Sqrt[e]*Sqrt[f + g
*x])/Sqrt[-(e*f) + d*g]])/(-(e*f) + d*g)^(5/2))/(4*e^(5/2))

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {\frac {3 \left (\left (a \,g^{2}+\frac {8 c \,f^{2}}{3}\right ) e^{2}-\frac {8 c d e f g}{3}+c \,d^{2} g^{2}\right ) \left (e x +d \right )^{2} \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{4}+\frac {5 \left (-\frac {2 a \left (-\frac {3 g x}{2}+f \right ) e^{3}}{5}+d \left (\frac {8 c f x}{5}+a g \right ) e^{2}+\frac {6 c \left (-\frac {5 g x}{6}+f \right ) d^{2} e}{5}-\frac {3 c \,d^{3} g}{5}\right ) \sqrt {\left (d g -e f \right ) e}\, \sqrt {g x +f}}{4}}{\sqrt {\left (d g -e f \right ) e}\, \left (d g -e f \right )^{2} \left (e x +d \right )^{2} e^{2}}\)	\(166\)
derivativedivides	\(\frac {\frac {g \left (3 a \,e^{2} g -5 c \,d^{2} g +8 c d e f \right ) \left (g x +f \right )^{\frac {3}{2}}}{4 e \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )}+\frac {\left (5 a \,e^{2} g -3 c \,d^{2} g +8 c d e f \right ) g \sqrt {g x +f}}{4 e^{2} \left (d g -e f \right )}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (3 a \,e^{2} g^{2}+3 c \,d^{2} g^{2}-8 c d e f g +8 c \,e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) e^{2} \sqrt {\left (d g -e f \right ) e}}\)	\(220\)
default	\(\frac {\frac {g \left (3 a \,e^{2} g -5 c \,d^{2} g +8 c d e f \right ) \left (g x +f \right )^{\frac {3}{2}}}{4 e \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )}+\frac {\left (5 a \,e^{2} g -3 c \,d^{2} g +8 c d e f \right ) g \sqrt {g x +f}}{4 e^{2} \left (d g -e f \right )}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (3 a \,e^{2} g^{2}+3 c \,d^{2} g^{2}-8 c d e f g +8 c \,e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) e^{2} \sqrt {\left (d g -e f \right ) e}}\)	\(220\)

[In]

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

[Out]

5/4/((d*g-e*f)*e)^(1/2)/(d*g-e*f)^2*(3/5*((a*g^2+8/3*c*f^2)*e^2-8/3*c*d*e*f*g+c*d^2*g^2)*(e*x+d)^2*arctan(e*(g
*x+f)^(1/2)/((d*g-e*f)*e)^(1/2))+(-2/5*a*(-3/2*g*x+f)*e^3+d*(8/5*c*f*x+a*g)*e^2+6/5*c*(-5/6*g*x+f)*d^2*e-3/5*c
*d^3*g)*((d*g-e*f)*e)^(1/2)*(g*x+f)^(1/2))/(e*x+d)^2/e^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (158) = 316\).

Time = 0.30 (sec) , antiderivative size = 896, normalized size of antiderivative = 5.03 \[ \int \frac {a+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=\left [\frac {{\left (8 \, c d^{2} e^{2} f^{2} - 8 \, c d^{3} e f g + 3 \, {\left (c d^{4} + a d^{2} e^{2}\right )} g^{2} + {\left (8 \, c e^{4} f^{2} - 8 \, c d e^{3} f g + 3 \, {\left (c d^{2} e^{2} + a e^{4}\right )} g^{2}\right )} x^{2} + 2 \, {\left (8 \, c d e^{3} f^{2} - 8 \, c d^{2} e^{2} f g + 3 \, {\left (c d^{3} e + a d e^{3}\right )} g^{2}\right )} x\right )} \sqrt {e^{2} f - d e g} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) + 2 \, {\left (2 \, {\left (3 \, c d^{2} e^{3} - a e^{5}\right )} f^{2} - {\left (9 \, c d^{3} e^{2} - 7 \, a d e^{4}\right )} f g + {\left (3 \, c d^{4} e - 5 \, a d^{2} e^{3}\right )} g^{2} + {\left (8 \, c d e^{4} f^{2} - {\left (13 \, c d^{2} e^{3} - 3 \, a e^{5}\right )} f g + {\left (5 \, c d^{3} e^{2} - 3 \, a d e^{4}\right )} g^{2}\right )} x\right )} \sqrt {g x + f}}{8 \, {\left (d^{2} e^{6} f^{3} - 3 \, d^{3} e^{5} f^{2} g + 3 \, d^{4} e^{4} f g^{2} - d^{5} e^{3} g^{3} + {\left (e^{8} f^{3} - 3 \, d e^{7} f^{2} g + 3 \, d^{2} e^{6} f g^{2} - d^{3} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{7} f^{3} - 3 \, d^{2} e^{6} f^{2} g + 3 \, d^{3} e^{5} f g^{2} - d^{4} e^{4} g^{3}\right )} x\right )}}, \frac {{\left (8 \, c d^{2} e^{2} f^{2} - 8 \, c d^{3} e f g + 3 \, {\left (c d^{4} + a d^{2} e^{2}\right )} g^{2} + {\left (8 \, c e^{4} f^{2} - 8 \, c d e^{3} f g + 3 \, {\left (c d^{2} e^{2} + a e^{4}\right )} g^{2}\right )} x^{2} + 2 \, {\left (8 \, c d e^{3} f^{2} - 8 \, c d^{2} e^{2} f g + 3 \, {\left (c d^{3} e + a d e^{3}\right )} g^{2}\right )} x\right )} \sqrt {-e^{2} f + d e g} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) + {\left (2 \, {\left (3 \, c d^{2} e^{3} - a e^{5}\right )} f^{2} - {\left (9 \, c d^{3} e^{2} - 7 \, a d e^{4}\right )} f g + {\left (3 \, c d^{4} e - 5 \, a d^{2} e^{3}\right )} g^{2} + {\left (8 \, c d e^{4} f^{2} - {\left (13 \, c d^{2} e^{3} - 3 \, a e^{5}\right )} f g + {\left (5 \, c d^{3} e^{2} - 3 \, a d e^{4}\right )} g^{2}\right )} x\right )} \sqrt {g x + f}}{4 \, {\left (d^{2} e^{6} f^{3} - 3 \, d^{3} e^{5} f^{2} g + 3 \, d^{4} e^{4} f g^{2} - d^{5} e^{3} g^{3} + {\left (e^{8} f^{3} - 3 \, d e^{7} f^{2} g + 3 \, d^{2} e^{6} f g^{2} - d^{3} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{7} f^{3} - 3 \, d^{2} e^{6} f^{2} g + 3 \, d^{3} e^{5} f g^{2} - d^{4} e^{4} g^{3}\right )} x\right )}}\right ] \]

[In]

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

[Out]

[1/8*((8*c*d^2*e^2*f^2 - 8*c*d^3*e*f*g + 3*(c*d^4 + a*d^2*e^2)*g^2 + (8*c*e^4*f^2 - 8*c*d*e^3*f*g + 3*(c*d^2*e
^2 + a*e^4)*g^2)*x^2 + 2*(8*c*d*e^3*f^2 - 8*c*d^2*e^2*f*g + 3*(c*d^3*e + a*d*e^3)*g^2)*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*(2*(3*c*d^2*e^3 - a*e^5)*f^2 -
(9*c*d^3*e^2 - 7*a*d*e^4)*f*g + (3*c*d^4*e - 5*a*d^2*e^3)*g^2 + (8*c*d*e^4*f^2 - (13*c*d^2*e^3 - 3*a*e^5)*f*g
+ (5*c*d^3*e^2 - 3*a*d*e^4)*g^2)*x)*sqrt(g*x + f))/(d^2*e^6*f^3 - 3*d^3*e^5*f^2*g + 3*d^4*e^4*f*g^2 - d^5*e^3*
g^3 + (e^8*f^3 - 3*d*e^7*f^2*g + 3*d^2*e^6*f*g^2 - d^3*e^5*g^3)*x^2 + 2*(d*e^7*f^3 - 3*d^2*e^6*f^2*g + 3*d^3*e
^5*f*g^2 - d^4*e^4*g^3)*x), 1/4*((8*c*d^2*e^2*f^2 - 8*c*d^3*e*f*g + 3*(c*d^4 + a*d^2*e^2)*g^2 + (8*c*e^4*f^2 -
8*c*d*e^3*f*g + 3*(c*d^2*e^2 + a*e^4)*g^2)*x^2 + 2*(8*c*d*e^3*f^2 - 8*c*d^2*e^2*f*g + 3*(c*d^3*e + a*d*e^3)*g
^2)*x)*sqrt(-e^2*f + d*e*g)*arctan(sqrt(-e^2*f + d*e*g)*sqrt(g*x + f)/(e*g*x + e*f)) + (2*(3*c*d^2*e^3 - a*e^5
)*f^2 - (9*c*d^3*e^2 - 7*a*d*e^4)*f*g + (3*c*d^4*e - 5*a*d^2*e^3)*g^2 + (8*c*d*e^4*f^2 - (13*c*d^2*e^3 - 3*a*e
^5)*f*g + (5*c*d^3*e^2 - 3*a*d*e^4)*g^2)*x)*sqrt(g*x + f))/(d^2*e^6*f^3 - 3*d^3*e^5*f^2*g + 3*d^4*e^4*f*g^2 -
d^5*e^3*g^3 + (e^8*f^3 - 3*d*e^7*f^2*g + 3*d^2*e^6*f*g^2 - d^3*e^5*g^3)*x^2 + 2*(d*e^7*f^3 - 3*d^2*e^6*f^2*g +
3*d^3*e^5*f*g^2 - d^4*e^4*g^3)*x)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.60 \[ \int \frac {a+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=\frac {{\left (8 \, c e^{2} f^{2} - 8 \, c d e f g + 3 \, c d^{2} g^{2} + 3 \, a e^{2} g^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{4 \, {\left (e^{4} f^{2} - 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} \sqrt {-e^{2} f + d e g}} + \frac {8 \, {\left (g x + f\right )}^{\frac {3}{2}} c d e^{2} f g - 8 \, \sqrt {g x + f} c d e^{2} f^{2} g - 5 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} e g^{2} + 3 \, {\left (g x + f\right )}^{\frac {3}{2}} a e^{3} g^{2} + 11 \, \sqrt {g x + f} c d^{2} e f g^{2} - 5 \, \sqrt {g x + f} a e^{3} f g^{2} - 3 \, \sqrt {g x + f} c d^{3} g^{3} + 5 \, \sqrt {g x + f} a d e^{2} g^{3}}{4 \, {\left (e^{4} f^{2} - 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} {\left ({\left (g x + f\right )} e - e f + d g\right )}^{2}} \]

[In]

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

[Out]

1/4*(8*c*e^2*f^2 - 8*c*d*e*f*g + 3*c*d^2*g^2 + 3*a*e^2*g^2)*arctan(sqrt(g*x + f)*e/sqrt(-e^2*f + d*e*g))/((e^4
*f^2 - 2*d*e^3*f*g + d^2*e^2*g^2)*sqrt(-e^2*f + d*e*g)) + 1/4*(8*(g*x + f)^(3/2)*c*d*e^2*f*g - 8*sqrt(g*x + f)
*c*d*e^2*f^2*g - 5*(g*x + f)^(3/2)*c*d^2*e*g^2 + 3*(g*x + f)^(3/2)*a*e^3*g^2 + 11*sqrt(g*x + f)*c*d^2*e*f*g^2
- 5*sqrt(g*x + f)*a*e^3*f*g^2 - 3*sqrt(g*x + f)*c*d^3*g^3 + 5*sqrt(g*x + f)*a*d*e^2*g^3)/((e^4*f^2 - 2*d*e^3*f
*g + d^2*e^2*g^2)*((g*x + f)*e - e*f + d*g)^2)

Mupad [B] (verification not implemented)

Time = 12.01 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.26 \[ \int \frac {a+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, dx=\frac {\frac {\sqrt {f+g\,x}\,\left (-3\,c\,d^2\,g^2+8\,c\,f\,d\,e\,g+5\,a\,e^2\,g^2\right )}{4\,e^2\,\left (d\,g-e\,f\right )}+\frac {{\left (f+g\,x\right )}^{3/2}\,\left (-5\,c\,d^2\,g^2+8\,c\,f\,d\,e\,g+3\,a\,e^2\,g^2\right )}{4\,e\,{\left (d\,g-e\,f\right )}^2}}{e^2\,{\left (f+g\,x\right )}^2-\left (f+g\,x\right )\,\left (2\,e^2\,f-2\,d\,e\,g\right )+d^2\,g^2+e^2\,f^2-2\,d\,e\,f\,g}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (3\,c\,d^2\,g^2-8\,c\,d\,e\,f\,g+8\,c\,e^2\,f^2+3\,a\,e^2\,g^2\right )}{4\,e^{5/2}\,{\left (d\,g-e\,f\right )}^{5/2}} \]

[In]

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

[Out]

(((f + g*x)^(1/2)*(5*a*e^2*g^2 - 3*c*d^2*g^2 + 8*c*d*e*f*g))/(4*e^2*(d*g - e*f)) + ((f + g*x)^(3/2)*(3*a*e^2*g
^2 - 5*c*d^2*g^2 + 8*c*d*e*f*g))/(4*e*(d*g - e*f)^2))/(e^2*(f + g*x)^2 - (f + g*x)*(2*e^2*f - 2*d*e*g) + d^2*g
^2 + e^2*f^2 - 2*d*e*f*g) + (atan((e^(1/2)*(f + g*x)^(1/2))/(d*g - e*f)^(1/2))*(3*a*e^2*g^2 + 3*c*d^2*g^2 + 8*
c*e^2*f^2 - 8*c*d*e*f*g))/(4*e^(5/2)*(d*g - e*f)^(5/2))