\(\int \frac {\sqrt {d+e x} (a+b x+c x^2)}{(e+f x)^{3/2}} \, dx\) [842]

Optimal result

Integrand size = 29, antiderivative size = 249 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{(e+f x)^{3/2}} \, dx=\frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \sqrt {d+e x} \sqrt {e+f x}}{4 e f^3 \left (e^2-d f\right )}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}-\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {e} \sqrt {e+f x}}\right )}{4 e^{3/2} f^{7/2}} \]

[Out]

-1/4*(4*e*f*(-2*a*e*f-b*d*f+3*b*e^2)-c*(-d^2*f^2-6*d*e^2*f+15*e^4))*arctanh(f^(1/2)*(e*x+d)^(1/2)/e^(1/2)/(f*x
+e)^(1/2))/e^(3/2)/f^(7/2)+2*(a+e*(-b*f+c*e)/f^2)*(e*x+d)^(3/2)/(-d*f+e^2)/(f*x+e)^(1/2)+1/2*c*(e*x+d)^(3/2)*(
f*x+e)^(1/2)/e/f^2+1/4*(4*e*f*(-2*a*e*f-b*d*f+3*b*e^2)-c*(-d^2*f^2-6*d*e^2*f+15*e^4))*(e*x+d)^(1/2)*(f*x+e)^(1
/2)/e/f^3/(-d*f+e^2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {963, 81, 52, 65, 223, 212} \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{(e+f x)^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {e} \sqrt {e+f x}}\right ) \left (4 e f \left (-2 a e f-b d f+3 b e^2\right )-c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{4 e^{3/2} f^{7/2}}+\frac {\sqrt {d+e x} \sqrt {e+f x} \left (4 e f \left (-2 a e f-b d f+3 b e^2\right )-c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{4 e f^3 \left (e^2-d f\right )}+\frac {2 (d+e x)^{3/2} \left (a+\frac {e (c e-b f)}{f^2}\right )}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2} \]

[In]

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

[Out]

(2*(a + (e*(c*e - b*f))/f^2)*(d + e*x)^(3/2))/((e^2 - d*f)*Sqrt[e + f*x]) + ((4*e*f*(3*b*e^2 - b*d*f - 2*a*e*f
) - c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*Sqrt[d + e*x]*Sqrt[e + f*x])/(4*e*f^3*(e^2 - d*f)) + (c*(d + e*x)^(3/2)*
Sqrt[e + f*x])/(2*e*f^2) - ((4*e*f*(3*b*e^2 - b*d*f - 2*a*e*f) - c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*ArcTanh[(Sq
rt[f]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[e + f*x])])/(4*e^(3/2)*f^(7/2))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 81

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

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 {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {2 \int \frac {\sqrt {d+e x} \left (\frac {f \left (3 b e^2-b d f-2 a e f\right )-c \left (3 e^3-d e f\right )}{2 f^2}-\frac {1}{2} c \left (d-\frac {e^2}{f}\right ) x\right )}{\sqrt {e+f x}} \, dx}{e^2-d f} \\ & = \frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {e+f x}} \, dx}{4 e f^2 \left (e^2-d f\right )} \\ & = \frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \sqrt {d+e x} \sqrt {e+f x}}{4 e f^3 \left (e^2-d f\right )}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}-\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {e+f x}} \, dx}{8 e f^3} \\ & = \frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \sqrt {d+e x} \sqrt {e+f x}}{4 e f^3 \left (e^2-d f\right )}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}-\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {e-\frac {d f}{e}+\frac {f x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{4 e^2 f^3} \\ & = \frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \sqrt {d+e x} \sqrt {e+f x}}{4 e f^3 \left (e^2-d f\right )}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}-\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {f x^2}{e}} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {e+f x}}\right )}{4 e^2 f^3} \\ & = \frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \sqrt {d+e x} \sqrt {e+f x}}{4 e f^3 \left (e^2-d f\right )}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}-\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {e} \sqrt {e+f x}}\right )}{4 e^{3/2} f^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{(e+f x)^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (4 e f (3 b e-2 a f+b f x)+c \left (-15 e^3-5 e^2 f x+d f^2 x+e f \left (d+2 f x^2\right )\right )\right )}{4 e f^3 \sqrt {e+f x}}+\frac {\left (4 e f \left (-3 b e^2+b d f+2 a e f\right )+c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {e} \sqrt {e+f x}}\right )}{4 e^{3/2} f^{7/2}} \]

[In]

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

[Out]

(Sqrt[d + e*x]*(4*e*f*(3*b*e - 2*a*f + b*f*x) + c*(-15*e^3 - 5*e^2*f*x + d*f^2*x + e*f*(d + 2*f*x^2))))/(4*e*f
^3*Sqrt[e + f*x]) + ((4*e*f*(-3*b*e^2 + b*d*f + 2*a*e*f) + c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*ArcTanh[(Sqrt[f]*
Sqrt[d + e*x])/(Sqrt[e]*Sqrt[e + f*x])])/(4*e^(3/2)*f^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(833\) vs. \(2(219)=438\).

Time = 0.47 (sec) , antiderivative size = 834, normalized size of antiderivative = 3.35

method	result	size
default	\(\frac {\sqrt {e x +d}\, \left (8 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) a \,e^{2} f^{3} x +4 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) b d e \,f^{3} x -12 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) b \,e^{3} f^{2} x -\ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c \,d^{2} f^{3} x -6 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c d \,e^{2} f^{2} x +15 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c \,e^{4} f x +4 c e \,f^{2} x^{2} \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+8 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) a \,e^{3} f^{2}+4 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) b d \,e^{2} f^{2}-12 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) b \,e^{4} f -\ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c \,d^{2} e \,f^{2}-6 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c d \,e^{3} f +15 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c \,e^{5}+8 b e \,f^{2} x \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+2 c d \,f^{2} x \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}-10 c \,e^{2} f x \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}-16 a e \,f^{2} \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+24 b \,e^{2} f \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+2 c d e f \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}-30 c \,e^{3} \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}\right )}{8 \sqrt {e f}\, e \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, f^{3} \sqrt {f x +e}}\)	\(834\)

[In]

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

[Out]

1/8*(e*x+d)^(1/2)*(8*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*a*e^2*f^3*x+4
*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*b*d*e*f^3*x-12*ln(1/2*(2*e*f*x+2*
((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*b*e^3*f^2*x-ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2
)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*c*d^2*f^3*x-6*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^
2)/(e*f)^(1/2))*c*d*e^2*f^2*x+15*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*c
*e^4*f*x+4*c*e*f^2*x^2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+8*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(
1/2)+d*f+e^2)/(e*f)^(1/2))*a*e^3*f^2+4*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1
/2))*b*d*e^2*f^2-12*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*b*e^4*f-ln(1/2
*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*c*d^2*e*f^2-6*ln(1/2*(2*e*f*x+2*((e*x+d)
*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*c*d*e^3*f+15*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)
^(1/2)+d*f+e^2)/(e*f)^(1/2))*c*e^5+8*b*e*f^2*x*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+2*c*d*f^2*x*((e*x+d)*(f*x+e
))^(1/2)*(e*f)^(1/2)-10*c*e^2*f*x*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)-16*a*e*f^2*((e*x+d)*(f*x+e))^(1/2)*(e*f)
^(1/2)+24*b*e^2*f*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+2*c*d*e*f*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)-30*c*e^3*(
(e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2))/(e*f)^(1/2)/e/((e*x+d)*(f*x+e))^(1/2)/f^3/(f*x+e)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.75 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.33 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{(e+f x)^{3/2}} \, dx=\left [\frac {{\left (15 \, c e^{5} - {\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f^{2} - 6 \, {\left (c d e^{3} + 2 \, b e^{4}\right )} f + {\left (15 \, c e^{4} f - {\left (c d^{2} - 4 \, b d e - 8 \, a e^{2}\right )} f^{3} - 6 \, {\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2}\right )} x\right )} \sqrt {e f} \log \left (8 \, e^{2} f^{2} x^{2} + e^{4} + 6 \, d e^{2} f + d^{2} f^{2} + 4 \, {\left (2 \, e f x + e^{2} + d f\right )} \sqrt {e f} \sqrt {e x + d} \sqrt {f x + e} + 8 \, {\left (e^{3} f + d e f^{2}\right )} x\right ) + 4 \, {\left (2 \, c e^{2} f^{3} x^{2} - 15 \, c e^{4} f - 8 \, a e^{2} f^{3} + {\left (c d e^{2} + 12 \, b e^{3}\right )} f^{2} - {\left (5 \, c e^{3} f^{2} - {\left (c d e + 4 \, b e^{2}\right )} f^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {f x + e}}{16 \, {\left (e^{2} f^{5} x + e^{3} f^{4}\right )}}, -\frac {{\left (15 \, c e^{5} - {\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f^{2} - 6 \, {\left (c d e^{3} + 2 \, b e^{4}\right )} f + {\left (15 \, c e^{4} f - {\left (c d^{2} - 4 \, b d e - 8 \, a e^{2}\right )} f^{3} - 6 \, {\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2}\right )} x\right )} \sqrt {-e f} \arctan \left (\frac {{\left (2 \, e f x + e^{2} + d f\right )} \sqrt {-e f} \sqrt {e x + d} \sqrt {f x + e}}{2 \, {\left (e^{2} f^{2} x^{2} + d e^{2} f + {\left (e^{3} f + d e f^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, c e^{2} f^{3} x^{2} - 15 \, c e^{4} f - 8 \, a e^{2} f^{3} + {\left (c d e^{2} + 12 \, b e^{3}\right )} f^{2} - {\left (5 \, c e^{3} f^{2} - {\left (c d e + 4 \, b e^{2}\right )} f^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {f x + e}}{8 \, {\left (e^{2} f^{5} x + e^{3} f^{4}\right )}}\right ] \]

[In]

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

[Out]

[1/16*((15*c*e^5 - (c*d^2*e - 4*b*d*e^2 - 8*a*e^3)*f^2 - 6*(c*d*e^3 + 2*b*e^4)*f + (15*c*e^4*f - (c*d^2 - 4*b*
d*e - 8*a*e^2)*f^3 - 6*(c*d*e^2 + 2*b*e^3)*f^2)*x)*sqrt(e*f)*log(8*e^2*f^2*x^2 + e^4 + 6*d*e^2*f + d^2*f^2 + 4
*(2*e*f*x + e^2 + d*f)*sqrt(e*f)*sqrt(e*x + d)*sqrt(f*x + e) + 8*(e^3*f + d*e*f^2)*x) + 4*(2*c*e^2*f^3*x^2 - 1
5*c*e^4*f - 8*a*e^2*f^3 + (c*d*e^2 + 12*b*e^3)*f^2 - (5*c*e^3*f^2 - (c*d*e + 4*b*e^2)*f^3)*x)*sqrt(e*x + d)*sq
rt(f*x + e))/(e^2*f^5*x + e^3*f^4), -1/8*((15*c*e^5 - (c*d^2*e - 4*b*d*e^2 - 8*a*e^3)*f^2 - 6*(c*d*e^3 + 2*b*e
^4)*f + (15*c*e^4*f - (c*d^2 - 4*b*d*e - 8*a*e^2)*f^3 - 6*(c*d*e^2 + 2*b*e^3)*f^2)*x)*sqrt(-e*f)*arctan(1/2*(2
*e*f*x + e^2 + d*f)*sqrt(-e*f)*sqrt(e*x + d)*sqrt(f*x + e)/(e^2*f^2*x^2 + d*e^2*f + (e^3*f + d*e*f^2)*x)) - 2*
(2*c*e^2*f^3*x^2 - 15*c*e^4*f - 8*a*e^2*f^3 + (c*d*e^2 + 12*b*e^3)*f^2 - (5*c*e^3*f^2 - (c*d*e + 4*b*e^2)*f^3)
*x)*sqrt(e*x + d)*sqrt(f*x + e))/(e^2*f^5*x + e^3*f^4)]

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(d + e*x)*(a + b*x + c*x**2)/(e + f*x)**(3/2), x)

Maxima [F(-2)]

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

[In]

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

none

Time = 0.34 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{(e+f x)^{3/2}} \, dx=\frac {\sqrt {e x + d} {\left ({\left (e x + d\right )} {\left (\frac {2 \, {\left (e x + d\right )} c}{f {\left | e \right |}} - \frac {5 \, c e^{4} f^{3} + 3 \, c d e^{2} f^{4} - 4 \, b e^{3} f^{4}}{e^{2} f^{5} {\left | e \right |}}\right )} - \frac {15 \, c e^{6} f^{2} - 6 \, c d e^{4} f^{3} - 12 \, b e^{5} f^{3} - c d^{2} e^{2} f^{4} + 4 \, b d e^{3} f^{4} + 8 \, a e^{4} f^{4}}{e^{2} f^{5} {\left | e \right |}}\right )}}{4 \, \sqrt {e^{3} + {\left (e x + d\right )} e f - d e f}} - \frac {{\left (15 \, c e^{4} - 6 \, c d e^{2} f - 12 \, b e^{3} f - c d^{2} f^{2} + 4 \, b d e f^{2} + 8 \, a e^{2} f^{2}\right )} \log \left ({\left | -\sqrt {e f} \sqrt {e x + d} + \sqrt {e^{3} + {\left (e x + d\right )} e f - d e f} \right |}\right )}{4 \, \sqrt {e f} f^{3} {\left | e \right |}} \]

[In]

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

[Out]

1/4*sqrt(e*x + d)*((e*x + d)*(2*(e*x + d)*c/(f*abs(e)) - (5*c*e^4*f^3 + 3*c*d*e^2*f^4 - 4*b*e^3*f^4)/(e^2*f^5*
abs(e))) - (15*c*e^6*f^2 - 6*c*d*e^4*f^3 - 12*b*e^5*f^3 - c*d^2*e^2*f^4 + 4*b*d*e^3*f^4 + 8*a*e^4*f^4)/(e^2*f^
5*abs(e)))/sqrt(e^3 + (e*x + d)*e*f - d*e*f) - 1/4*(15*c*e^4 - 6*c*d*e^2*f - 12*b*e^3*f - c*d^2*f^2 + 4*b*d*e*
f^2 + 8*a*e^2*f^2)*log(abs(-sqrt(e*f)*sqrt(e*x + d) + sqrt(e^3 + (e*x + d)*e*f - d*e*f)))/(sqrt(e*f)*f^3*abs(e
))

Mupad [F(-1)]

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

[In]

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

[Out]

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