\(\int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx\) [839]

Optimal result

Integrand size = 29, antiderivative size = 160 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {2 c \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} \sqrt {g}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {963, 79, 65, 223, 212} \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (c \left (6 d e f-4 d^2 g\right )-e (-2 a e g-b d g+3 b e f)\right )}{3 e^2 \sqrt {d+e x} (e f-d g)^2}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}+\frac {2 c \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} \sqrt {g}} \]

[In]

Int[(a + b*x + c*x^2)/((d + e*x)^(5/2)*Sqrt[f + g*x]),x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 963

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g))), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}-\frac {2 \int \frac {\frac {c d (3 e f-d g)-e (3 b e f-b d g-2 a e g)}{2 e^2}-\frac {3}{2} c \left (f-\frac {d g}{e}\right ) x}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx}{3 (e f-d g)} \\ & = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {c \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{e^2} \\ & = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {(2 c) \text {Subst}\left (\int \frac {1}{\sqrt {f-\frac {d g}{e}+\frac {g x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{e^3} \\ & = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {(2 c) \text {Subst}\left (\int \frac {1}{1-\frac {g x^2}{e}} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e^3} \\ & = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {2 c \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} \sqrt {g}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (c d \left (-3 d^2 g+6 e^2 f x+d e (5 f-4 g x)\right )+e^2 (b (-2 d f-3 e f x+d g x)+a (-e f+3 d g+2 e g x))\right )}{3 e^2 (e f-d g)^2 (d+e x)^{3/2}}+\frac {2 c \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right )}{e^{5/2} \sqrt {g}} \]

[In]

Integrate[(a + b*x + c*x^2)/((d + e*x)^(5/2)*Sqrt[f + g*x]),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(772\) vs. \(2(136)=272\).

Time = 0.48 (sec) , antiderivative size = 773, normalized size of antiderivative = 4.83

method	result	size
default	\(\frac {\sqrt {g x +f}\, \left (3 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e^{2} g^{2} x^{2}-6 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c d \,e^{3} f g \,x^{2}+3 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,e^{4} f^{2} x^{2}+6 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{3} e \,g^{2} x -12 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e^{2} f g x +6 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c d \,e^{3} f^{2} x +3 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{4} g^{2}-6 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{3} e f g +3 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e^{2} f^{2}+4 a \,e^{3} g x \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+2 b d \,e^{2} g x \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}-6 b \,e^{3} f x \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}-8 c \,d^{2} e g x \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+12 c d \,e^{2} f x \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+6 a d \,e^{2} g \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}-2 a \,e^{3} f \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}-4 b d \,e^{2} f \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}-6 c \,d^{3} g \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+10 c \,d^{2} e f \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\right )}{3 \sqrt {e g}\, \left (d g -e f \right )^{2} \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, e^{2} \left (e x +d \right )^{\frac {3}{2}}}\)	\(773\)

[In]

int((c*x^2+b*x+a)/(e*x+d)^(5/2)/(g*x+f)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(g*x+f)^(1/2)*(3*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*g^2
*x^2-6*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d*e^3*f*g*x^2+3*ln(1/2*(2
*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*e^4*f^2*x^2+6*ln(1/2*(2*e*g*x+2*((g*x+f)*
(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^3*e*g^2*x-12*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e
*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*f*g*x+6*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*
f)/(e*g)^(1/2))*c*d*e^3*f^2*x+3*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*
d^4*g^2-6*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^3*e*f*g+3*ln(1/2*(2*
e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*f^2+4*a*e^3*g*x*((g*x+f)*(e*x+d))^
(1/2)*(e*g)^(1/2)+2*b*d*e^2*g*x*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-6*b*e^3*f*x*((g*x+f)*(e*x+d))^(1/2)*(e*g)^
(1/2)-8*c*d^2*e*g*x*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+12*c*d*e^2*f*x*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+6*a
*d*e^2*g*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-2*a*e^3*f*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-4*b*d*e^2*f*((g*x+f
)*(e*x+d))^(1/2)*(e*g)^(1/2)-6*c*d^3*g*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+10*c*d^2*e*f*((g*x+f)*(e*x+d))^(1/2
)*(e*g)^(1/2))/(e*g)^(1/2)/(d*g-e*f)^2/((g*x+f)*(e*x+d))^(1/2)/e^2/(e*x+d)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (137) = 274\).

Time = 3.31 (sec) , antiderivative size = 792, normalized size of antiderivative = 4.95 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\left [\frac {3 \, {\left (c d^{2} e^{2} f^{2} - 2 \, c d^{3} e f g + c d^{4} g^{2} + {\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (c d e^{3} f^{2} - 2 \, c d^{2} e^{2} f g + c d^{3} e g^{2}\right )} x\right )} \sqrt {e g} \log \left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 4 \, {\left (2 \, e g x + e f + d g\right )} \sqrt {e g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (e^{2} f g + d e g^{2}\right )} x\right ) + 4 \, {\left ({\left (5 \, c d^{2} e^{2} - 2 \, b d e^{3} - a e^{4}\right )} f g - 3 \, {\left (c d^{3} e - a d e^{3}\right )} g^{2} + {\left (3 \, {\left (2 \, c d e^{3} - b e^{4}\right )} f g - {\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} g^{2}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{6 \, {\left (d^{2} e^{5} f^{2} g - 2 \, d^{3} e^{4} f g^{2} + d^{4} e^{3} g^{3} + {\left (e^{7} f^{2} g - 2 \, d e^{6} f g^{2} + d^{2} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{6} f^{2} g - 2 \, d^{2} e^{5} f g^{2} + d^{3} e^{4} g^{3}\right )} x\right )}}, -\frac {3 \, {\left (c d^{2} e^{2} f^{2} - 2 \, c d^{3} e f g + c d^{4} g^{2} + {\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (c d e^{3} f^{2} - 2 \, c d^{2} e^{2} f g + c d^{3} e g^{2}\right )} x\right )} \sqrt {-e g} \arctan \left (\frac {{\left (2 \, e g x + e f + d g\right )} \sqrt {-e g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (e^{2} g^{2} x^{2} + d e f g + {\left (e^{2} f g + d e g^{2}\right )} x\right )}}\right ) - 2 \, {\left ({\left (5 \, c d^{2} e^{2} - 2 \, b d e^{3} - a e^{4}\right )} f g - 3 \, {\left (c d^{3} e - a d e^{3}\right )} g^{2} + {\left (3 \, {\left (2 \, c d e^{3} - b e^{4}\right )} f g - {\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} g^{2}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (d^{2} e^{5} f^{2} g - 2 \, d^{3} e^{4} f g^{2} + d^{4} e^{3} g^{3} + {\left (e^{7} f^{2} g - 2 \, d e^{6} f g^{2} + d^{2} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{6} f^{2} g - 2 \, d^{2} e^{5} f g^{2} + d^{3} e^{4} g^{3}\right )} x\right )}}\right ] \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(5/2)/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\int \frac {a + b x + c x^{2}}{\left (d + e x\right )^{\frac {5}{2}} \sqrt {f + g x}}\, dx \]

[In]

integrate((c*x**2+b*x+a)/(e*x+d)**(5/2)/(g*x+f)**(1/2),x)

[Out]

Integral((a + b*x + c*x**2)/((d + e*x)**(5/2)*sqrt(f + g*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(5/2)/(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>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (137) = 274\).

Time = 0.38 (sec) , antiderivative size = 491, normalized size of antiderivative = 3.07 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=-\frac {c \log \left ({\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2}\right )}{\sqrt {e g} e {\left | e \right |}} + \frac {4 \, {\left (6 \, c d e^{4} f^{2} g - 3 \, b e^{5} f^{2} g - 10 \, c d^{2} e^{3} f g^{2} + 4 \, b d e^{4} f g^{2} + 2 \, a e^{5} f g^{2} + 4 \, c d^{3} e^{2} g^{3} - b d^{2} e^{3} g^{3} - 2 \, a d e^{4} g^{3} - 12 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} c d e^{2} f g + 6 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} b e^{3} f g + 6 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} c d^{2} e g^{2} - 6 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} a e^{3} g^{2} + 6 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{4} c d g - 3 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{4} b e g\right )}}{3 \, {\left (e^{2} f - d e g - {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2}\right )}^{3} \sqrt {e g} {\left | e \right |}} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(5/2)/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

-c*log((sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2)/(sqrt(e*g)*e*abs(e)) + 4/3*(6*c*d*e^
4*f^2*g - 3*b*e^5*f^2*g - 10*c*d^2*e^3*f*g^2 + 4*b*d*e^4*f*g^2 + 2*a*e^5*f*g^2 + 4*c*d^3*e^2*g^3 - b*d^2*e^3*g
^3 - 2*a*d*e^4*g^3 - 12*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*c*d*e^2*f*g + 6*(sqr
t(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*b*e^3*f*g + 6*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^
2*f + (e*x + d)*e*g - d*e*g))^2*c*d^2*e*g^2 - 6*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g)
)^2*a*e^3*g^2 + 6*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^4*c*d*g - 3*(sqrt(e*g)*sqrt(
e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^4*b*e*g)/((e^2*f - d*e*g - (sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2
*f + (e*x + d)*e*g - d*e*g))^2)^3*sqrt(e*g)*abs(e))

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\int \frac {c\,x^2+b\,x+a}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]

[In]

int((a + b*x + c*x^2)/((f + g*x)^(1/2)*(d + e*x)^(5/2)),x)

[Out]

int((a + b*x + c*x^2)/((f + g*x)^(1/2)*(d + e*x)^(5/2)), x)