\(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^4} \, dx\) [2350]

Optimal result

Integrand size = 22, antiderivative size = 307 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=-\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^4}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 e^4 \left (c d^2-b d e+a e^2\right )^{3/2}} \]

[Out]

-1/3*(c*x^2+b*x+a)^(3/2)/e/(e*x+d)^3+c^(3/2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^4-1/16*(-b*e
+2*c*d)*(8*c^2*d^2-b^2*e^2-4*c*e*(-3*a*e+2*b*d))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1
/2)/(c*x^2+b*x+a)^(1/2))/e^4/(a*e^2-b*d*e+c*d^2)^(3/2)-1/8*(8*c^2*d^3-b*e^2*(-2*a*e+b*d)-2*c*d*e*(-2*a*e+3*b*d
)+e*(12*c^2*d^2+b^2*e^2-4*c*e*(-2*a*e+3*b*d))*x)*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {746, 824, 857, 635, 212, 738} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=-\frac {(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{16 e^4 \left (a e^2-b d e+c d^2\right )^{3/2}}+\frac {c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^4}-\frac {\sqrt {a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{8 e^3 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3} \]

[In]

Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^4,x]

[Out]

-1/8*((8*c^2*d^3 - b*e^2*(b*d - 2*a*e) - 2*c*d*e*(3*b*d - 2*a*e) + e*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*
a*e))*x)*Sqrt[a + b*x + c*x^2])/(e^3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (a + b*x + c*x^2)^(3/2)/(3*e*(d +
e*x)^3) + (c^(3/2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e^4 - ((2*c*d - b*e)*(8*c^2*d^2 - b
^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a
+ b*x + c*x^2])])/(16*e^4*(c*d^2 - b*d*e + a*e^2)^(3/2))

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 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 746

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 1))), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ
[2*c*d - b*e, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2)
)*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d -
b*e)*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*
x + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m +
1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m +
1) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3,
0]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx}{2 e} \\ & = -\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}-\frac {\int \frac {\frac {1}{2} \left (6 b^2 c d e+8 a c^2 d e+b^3 e^2-4 b c \left (2 c d^2+3 a e^2\right )\right )-8 c^2 \left (c d^2-b d e+a e^2\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 e^3 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {c^2 \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{e^4}-\frac {\left (8 c^2 d \left (c d^2-b d e+a e^2\right )+\frac {1}{2} e \left (6 b^2 c d e+8 a c^2 d e+b^3 e^2-4 b c \left (2 c d^2+3 a e^2\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 e^4 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {\left (2 c^2\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{e^4}+\frac {\left (8 c^2 d \left (c d^2-b d e+a e^2\right )+\frac {1}{2} e \left (6 b^2 c d e+8 a c^2 d e+b^3 e^2-4 b c \left (2 c d^2+3 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{4 e^4 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^4}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 e^4 \left (c d^2-b d e+a e^2\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.03 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {-\frac {2 (a+x (b+c x))^{3/2}}{(d+e x)^3}+\frac {3 (2 c d-b e) (a+x (b+c x))^{3/2}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {3 \left (\frac {2 \left (-4 c^2 d^2+b^2 e^2+4 c e (b d-2 a e)\right ) (a+x (b+c x))^{3/2}}{d+e x}+\frac {2 \sqrt {a+x (b+c x)} \left (-b^3 e^3+4 c^3 d^2 (-2 d+e x)-b c e^2 (5 b d-10 a e+b e x)+2 c^2 e (b d (7 d-2 e x)+2 a e (-3 d+2 e x))\right )}{e^2}+\frac {16 c^{3/2} \left (c d^2+e (-b d+a e)\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+(2 c d-b e) \sqrt {c d^2+e (-b d+a e)} \left (8 c^2 d^2-b^2 e^2+4 c e (-2 b d+3 a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{e^3}\right )}{8 \left (c d^2+e (-b d+a e)\right )^2}}{6 e} \]

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^4,x]

[Out]

((-2*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 + (3*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(2*(c*d^2 + e*(-(b*d) +
a*e))*(d + e*x)^2) + (3*((2*(-4*c^2*d^2 + b^2*e^2 + 4*c*e*(b*d - 2*a*e))*(a + x*(b + c*x))^(3/2))/(d + e*x) +
(2*Sqrt[a + x*(b + c*x)]*(-(b^3*e^3) + 4*c^3*d^2*(-2*d + e*x) - b*c*e^2*(5*b*d - 10*a*e + b*e*x) + 2*c^2*e*(b*
d*(7*d - 2*e*x) + 2*a*e*(-3*d + 2*e*x))))/e^2 + (16*c^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^2*ArcTanh[(b + 2*c*x)/(
2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + (2*c*d - b*e)*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*(8*c^2*d^2 - b^2*e^2 + 4*c*e*
(-2*b*d + 3*a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b +
c*x)])])/e^3))/(8*(c*d^2 + e*(-(b*d) + a*e))^2))/(6*e)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3084\) vs. \(2(281)=562\).

Time = 0.45 (sec) , antiderivative size = 3085, normalized size of antiderivative = 10.05

method	result	size
default	\(\text {Expression too large to display}\)	\(3085\)

[In]

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/e^4*(-1/3/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5/2
)-1/6*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+
d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5/2)+1/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*
((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5/2)+3/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1/3*(
(x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2*(b*e-2*c*d)/e*(1/4*(2*c*(x+d/e)+(b*e-2*c*
d)/e)/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*
e-2*c*d)^2/e^2)/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d
*e+c*d^2)/e^2)^(1/2)))+(a*e^2-b*d*e+c*d^2)/e^2*(((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1
/2)+1/2*(b*e-2*c*d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e
+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/
e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*
d^2)/e^2)^(1/2))/(x+d/e))))+4*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/8*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*((x+d/e)^2*c+(b*e-2
*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/c*(1/4*(2*
c*(x+d/e)+(b*e-2*c*d)/e)/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b
*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e
*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))))+3/2*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d
/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2*(b*e-2*c*d)/e*(1/4*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*((x+d/e)^2*c+(b*e-2*c*
d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/c^(3/2)*ln((1/
2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))+(a*e^2-
b*d*e+c*d^2)/e^2*(((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)/e*ln((1/2*
(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a
*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))))+2
/3*c/(a*e^2-b*d*e+c*d^2)*e^2*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d
*e+c*d^2)/e^2)^(5/2)+3/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1/3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*
e+c*d^2)/e^2)^(3/2)+1/2*(b*e-2*c*d)/e*(1/4*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/c^(3/2)*ln((1/2*(b*e-2*c*d)/e
+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))+(a*e^2-b*d*e+c*d^2)/e^
2*(((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)/e*ln((1/2*(b*e-2*c*d)/e+c
*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^
2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d
^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))))+4*c/(a*e^2-b*d*e+
c*d^2)*e^2*(1/8*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2
)+3/16*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/c*(1/4*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*((x+d/e)^2*c+(b*e-
2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/c^(3/2)*ln
((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1206 vs. \(2 (281) = 562\).

Time = 186.22 (sec) , antiderivative size = 4911, normalized size of antiderivative = 16.00 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[1/96*(48*(c^3*d^7 - 2*b*c^2*d^6*e - 2*a*b*c*d^4*e^3 + a^2*c*d^3*e^4 + (b^2*c + 2*a*c^2)*d^5*e^2 + (c^3*d^4*e^
3 - 2*b*c^2*d^3*e^4 - 2*a*b*c*d*e^6 + a^2*c*e^7 + (b^2*c + 2*a*c^2)*d^2*e^5)*x^3 + 3*(c^3*d^5*e^2 - 2*b*c^2*d^
4*e^3 - 2*a*b*c*d^2*e^5 + a^2*c*d*e^6 + (b^2*c + 2*a*c^2)*d^3*e^4)*x^2 + 3*(c^3*d^6*e - 2*b*c^2*d^5*e^2 - 2*a*
b*c*d^3*e^4 + a^2*c*d^2*e^5 + (b^2*c + 2*a*c^2)*d^4*e^3)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*
x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 3*(16*c^3*d^6 - 24*b*c^2*d^5*e + 6*(b^2*c + 4*a*c^2)*d^4*e^2 + (
b^3 - 12*a*b*c)*d^3*e^3 + (16*c^3*d^3*e^3 - 24*b*c^2*d^2*e^4 + 6*(b^2*c + 4*a*c^2)*d*e^5 + (b^3 - 12*a*b*c)*e^
6)*x^3 + 3*(16*c^3*d^4*e^2 - 24*b*c^2*d^3*e^3 + 6*(b^2*c + 4*a*c^2)*d^2*e^4 + (b^3 - 12*a*b*c)*d*e^5)*x^2 + 3*
(16*c^3*d^5*e - 24*b*c^2*d^4*e^2 + 6*(b^2*c + 4*a*c^2)*d^3*e^3 + (b^3 - 12*a*b*c)*d^2*e^4)*x)*sqrt(c*d^2 - b*d
*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 -
4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^
2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 4*(24*c^3*d^6*e - 42*b*c^2*d^5*e^2 - 10*a^2*b*d*e^6 +
8*a^3*e^7 + (15*b^2*c + 44*a*c^2)*d^4*e^3 + (3*b^3 - 40*a*b*c)*d^3*e^4 - (a*b^2 - 28*a^2*c)*d^2*e^5 + (44*c^3
*d^4*e^3 - 88*b*c^2*d^3*e^4 + (47*b^2*c + 76*a*c^2)*d^2*e^5 - (3*b^3 + 76*a*b*c)*d*e^6 + (3*a*b^2 + 32*a^2*c)*
e^7)*x^2 + 2*(30*c^3*d^5*e^2 - 53*b*c^2*d^4*e^3 + 7*a^2*b*e^7 + (19*b^2*c + 48*a*c^2)*d^3*e^4 + 2*(2*b^3 - 17*
a*b*c)*d^2*e^5 - (11*a*b^2 - 18*a^2*c)*d*e^6)*x)*sqrt(c*x^2 + b*x + a))/(c^2*d^7*e^4 - 2*b*c*d^6*e^5 - 2*a*b*d
^4*e^7 + a^2*d^3*e^8 + (b^2 + 2*a*c)*d^5*e^6 + (c^2*d^4*e^7 - 2*b*c*d^3*e^8 - 2*a*b*d*e^10 + a^2*e^11 + (b^2 +
2*a*c)*d^2*e^9)*x^3 + 3*(c^2*d^5*e^6 - 2*b*c*d^4*e^7 - 2*a*b*d^2*e^9 + a^2*d*e^10 + (b^2 + 2*a*c)*d^3*e^8)*x^
2 + 3*(c^2*d^6*e^5 - 2*b*c*d^5*e^6 - 2*a*b*d^3*e^8 + a^2*d^2*e^9 + (b^2 + 2*a*c)*d^4*e^7)*x), -1/48*(3*(16*c^3
*d^6 - 24*b*c^2*d^5*e + 6*(b^2*c + 4*a*c^2)*d^4*e^2 + (b^3 - 12*a*b*c)*d^3*e^3 + (16*c^3*d^3*e^3 - 24*b*c^2*d^
2*e^4 + 6*(b^2*c + 4*a*c^2)*d*e^5 + (b^3 - 12*a*b*c)*e^6)*x^3 + 3*(16*c^3*d^4*e^2 - 24*b*c^2*d^3*e^3 + 6*(b^2*
c + 4*a*c^2)*d^2*e^4 + (b^3 - 12*a*b*c)*d*e^5)*x^2 + 3*(16*c^3*d^5*e - 24*b*c^2*d^4*e^2 + 6*(b^2*c + 4*a*c^2)*
d^3*e^3 + (b^3 - 12*a*b*c)*d^2*e^4)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*s
qrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e
^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) - 24*(c^3*d^7 - 2*b*c^2*d^6*e - 2*a*b*c*d^4*e^3 + a^2*c*d^3*e^4 +
(b^2*c + 2*a*c^2)*d^5*e^2 + (c^3*d^4*e^3 - 2*b*c^2*d^3*e^4 - 2*a*b*c*d*e^6 + a^2*c*e^7 + (b^2*c + 2*a*c^2)*d^2
*e^5)*x^3 + 3*(c^3*d^5*e^2 - 2*b*c^2*d^4*e^3 - 2*a*b*c*d^2*e^5 + a^2*c*d*e^6 + (b^2*c + 2*a*c^2)*d^3*e^4)*x^2
+ 3*(c^3*d^6*e - 2*b*c^2*d^5*e^2 - 2*a*b*c*d^3*e^4 + a^2*c*d^2*e^5 + (b^2*c + 2*a*c^2)*d^4*e^3)*x)*sqrt(c)*log
(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 2*(24*c^3*d^6*e - 42*b*c^
2*d^5*e^2 - 10*a^2*b*d*e^6 + 8*a^3*e^7 + (15*b^2*c + 44*a*c^2)*d^4*e^3 + (3*b^3 - 40*a*b*c)*d^3*e^4 - (a*b^2 -
28*a^2*c)*d^2*e^5 + (44*c^3*d^4*e^3 - 88*b*c^2*d^3*e^4 + (47*b^2*c + 76*a*c^2)*d^2*e^5 - (3*b^3 + 76*a*b*c)*d
*e^6 + (3*a*b^2 + 32*a^2*c)*e^7)*x^2 + 2*(30*c^3*d^5*e^2 - 53*b*c^2*d^4*e^3 + 7*a^2*b*e^7 + (19*b^2*c + 48*a*c
^2)*d^3*e^4 + 2*(2*b^3 - 17*a*b*c)*d^2*e^5 - (11*a*b^2 - 18*a^2*c)*d*e^6)*x)*sqrt(c*x^2 + b*x + a))/(c^2*d^7*e
^4 - 2*b*c*d^6*e^5 - 2*a*b*d^4*e^7 + a^2*d^3*e^8 + (b^2 + 2*a*c)*d^5*e^6 + (c^2*d^4*e^7 - 2*b*c*d^3*e^8 - 2*a*
b*d*e^10 + a^2*e^11 + (b^2 + 2*a*c)*d^2*e^9)*x^3 + 3*(c^2*d^5*e^6 - 2*b*c*d^4*e^7 - 2*a*b*d^2*e^9 + a^2*d*e^10
+ (b^2 + 2*a*c)*d^3*e^8)*x^2 + 3*(c^2*d^6*e^5 - 2*b*c*d^5*e^6 - 2*a*b*d^3*e^8 + a^2*d^2*e^9 + (b^2 + 2*a*c)*d
^4*e^7)*x), -1/96*(96*(c^3*d^7 - 2*b*c^2*d^6*e - 2*a*b*c*d^4*e^3 + a^2*c*d^3*e^4 + (b^2*c + 2*a*c^2)*d^5*e^2 +
(c^3*d^4*e^3 - 2*b*c^2*d^3*e^4 - 2*a*b*c*d*e^6 + a^2*c*e^7 + (b^2*c + 2*a*c^2)*d^2*e^5)*x^3 + 3*(c^3*d^5*e^2
- 2*b*c^2*d^4*e^3 - 2*a*b*c*d^2*e^5 + a^2*c*d*e^6 + (b^2*c + 2*a*c^2)*d^3*e^4)*x^2 + 3*(c^3*d^6*e - 2*b*c^2*d^
5*e^2 - 2*a*b*c*d^3*e^4 + a^2*c*d^2*e^5 + (b^2*c + 2*a*c^2)*d^4*e^3)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x +
a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 3*(16*c^3*d^6 - 24*b*c^2*d^5*e + 6*(b^2*c + 4*a*c^2)*d^4*e
^2 + (b^3 - 12*a*b*c)*d^3*e^3 + (16*c^3*d^3*e^3 - 24*b*c^2*d^2*e^4 + 6*(b^2*c + 4*a*c^2)*d*e^5 + (b^3 - 12*a*b
*c)*e^6)*x^3 + 3*(16*c^3*d^4*e^2 - 24*b*c^2*d^3*e^3 + 6*(b^2*c + 4*a*c^2)*d^2*e^4 + (b^3 - 12*a*b*c)*d*e^5)*x^
2 + 3*(16*c^3*d^5*e - 24*b*c^2*d^4*e^2 + 6*(b^2*c + 4*a*c^2)*d^3*e^3 + (b^3 - 12*a*b*c)*d^2*e^4)*x)*sqrt(c*d^2
- b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)
*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*
a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 4*(24*c^3*d^6*e - 42*b*c^2*d^5*e^2 - 10*a^2*b*d
*e^6 + 8*a^3*e^7 + (15*b^2*c + 44*a*c^2)*d^4*e^3 + (3*b^3 - 40*a*b*c)*d^3*e^4 - (a*b^2 - 28*a^2*c)*d^2*e^5 + (
44*c^3*d^4*e^3 - 88*b*c^2*d^3*e^4 + (47*b^2*c + 76*a*c^2)*d^2*e^5 - (3*b^3 + 76*a*b*c)*d*e^6 + (3*a*b^2 + 32*a
^2*c)*e^7)*x^2 + 2*(30*c^3*d^5*e^2 - 53*b*c^2*d^4*e^3 + 7*a^2*b*e^7 + (19*b^2*c + 48*a*c^2)*d^3*e^4 + 2*(2*b^3
- 17*a*b*c)*d^2*e^5 - (11*a*b^2 - 18*a^2*c)*d*e^6)*x)*sqrt(c*x^2 + b*x + a))/(c^2*d^7*e^4 - 2*b*c*d^6*e^5 - 2
*a*b*d^4*e^7 + a^2*d^3*e^8 + (b^2 + 2*a*c)*d^5*e^6 + (c^2*d^4*e^7 - 2*b*c*d^3*e^8 - 2*a*b*d*e^10 + a^2*e^11 +
(b^2 + 2*a*c)*d^2*e^9)*x^3 + 3*(c^2*d^5*e^6 - 2*b*c*d^4*e^7 - 2*a*b*d^2*e^9 + a^2*d*e^10 + (b^2 + 2*a*c)*d^3*e
^8)*x^2 + 3*(c^2*d^6*e^5 - 2*b*c*d^5*e^6 - 2*a*b*d^3*e^8 + a^2*d^2*e^9 + (b^2 + 2*a*c)*d^4*e^7)*x), -1/48*(3*(
16*c^3*d^6 - 24*b*c^2*d^5*e + 6*(b^2*c + 4*a*c^2)*d^4*e^2 + (b^3 - 12*a*b*c)*d^3*e^3 + (16*c^3*d^3*e^3 - 24*b*
c^2*d^2*e^4 + 6*(b^2*c + 4*a*c^2)*d*e^5 + (b^3 - 12*a*b*c)*e^6)*x^3 + 3*(16*c^3*d^4*e^2 - 24*b*c^2*d^3*e^3 + 6
*(b^2*c + 4*a*c^2)*d^2*e^4 + (b^3 - 12*a*b*c)*d*e^5)*x^2 + 3*(16*c^3*d^5*e - 24*b*c^2*d^4*e^2 + 6*(b^2*c + 4*a
*c^2)*d^3*e^3 + (b^3 - 12*a*b*c)*d^2*e^4)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*
e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e +
a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) + 48*(c^3*d^7 - 2*b*c^2*d^6*e - 2*a*b*c*d^4*e^3 + a^2*c*d^3*
e^4 + (b^2*c + 2*a*c^2)*d^5*e^2 + (c^3*d^4*e^3 - 2*b*c^2*d^3*e^4 - 2*a*b*c*d*e^6 + a^2*c*e^7 + (b^2*c + 2*a*c^
2)*d^2*e^5)*x^3 + 3*(c^3*d^5*e^2 - 2*b*c^2*d^4*e^3 - 2*a*b*c*d^2*e^5 + a^2*c*d*e^6 + (b^2*c + 2*a*c^2)*d^3*e^4
)*x^2 + 3*(c^3*d^6*e - 2*b*c^2*d^5*e^2 - 2*a*b*c*d^3*e^4 + a^2*c*d^2*e^5 + (b^2*c + 2*a*c^2)*d^4*e^3)*x)*sqrt(
-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(24*c^3*d^6*e - 42*b*c^
2*d^5*e^2 - 10*a^2*b*d*e^6 + 8*a^3*e^7 + (15*b^2*c + 44*a*c^2)*d^4*e^3 + (3*b^3 - 40*a*b*c)*d^3*e^4 - (a*b^2 -
28*a^2*c)*d^2*e^5 + (44*c^3*d^4*e^3 - 88*b*c^2*d^3*e^4 + (47*b^2*c + 76*a*c^2)*d^2*e^5 - (3*b^3 + 76*a*b*c)*d
*e^6 + (3*a*b^2 + 32*a^2*c)*e^7)*x^2 + 2*(30*c^3*d^5*e^2 - 53*b*c^2*d^4*e^3 + 7*a^2*b*e^7 + (19*b^2*c + 48*a*c
^2)*d^3*e^4 + 2*(2*b^3 - 17*a*b*c)*d^2*e^5 - (11*a*b^2 - 18*a^2*c)*d*e^6)*x)*sqrt(c*x^2 + b*x + a))/(c^2*d^7*e
^4 - 2*b*c*d^6*e^5 - 2*a*b*d^4*e^7 + a^2*d^3*e^8 + (b^2 + 2*a*c)*d^5*e^6 + (c^2*d^4*e^7 - 2*b*c*d^3*e^8 - 2*a*
b*d*e^10 + a^2*e^11 + (b^2 + 2*a*c)*d^2*e^9)*x^3 + 3*(c^2*d^5*e^6 - 2*b*c*d^4*e^7 - 2*a*b*d^2*e^9 + a^2*d*e^10
+ (b^2 + 2*a*c)*d^3*e^8)*x^2 + 3*(c^2*d^6*e^5 - 2*b*c*d^5*e^6 - 2*a*b*d^3*e^8 + a^2*d^2*e^9 + (b^2 + 2*a*c)*d
^4*e^7)*x)]

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{4}}\, dx \]

[In]

integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**4,x)

[Out]

Integral((a + b*x + c*x**2)**(3/2)/(d + e*x)**4, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^4,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(a*e^2-b*d*e>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2059 vs. \(2 (281) = 562\).

Time = 4.24 (sec) , antiderivative size = 2059, normalized size of antiderivative = 6.71 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-1/8*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + 24*a*c^2*d*e^2 + b^3*e^3 - 12*a*b*c*e^3)*arctan(-((sqrt(c)
*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c*d^2*e^4 - b*d*e^5 + a*e^6)*sqrt(-
c*d^2 + b*d*e - a*e^2)) - c^(3/2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/e^4 - 1/24*(144
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^3*d^3*e^2 - 216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^2*d^2*e^3 +
78*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c*d*e^4 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^2*d*e^4
- 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*e^5 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c*e^5 + 432*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(7/2)*d^4*e - 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(5/2)*d^3*e
^2 + 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(3/2)*d^2*e^3 + 216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*
a*c^(5/2)*d^2*e^3 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*sqrt(c)*d*e^4 + 84*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^4*a*b*c^(3/2)*d*e^4 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*sqrt(c)*e^5 - 96*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^4*a^2*c^(3/2)*e^5 + 352*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^4*d^5 - 16*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^3*b*c^3*d^4*e - 420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^2*d^3*e^2 - 272*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^3*a*c^3*d^3*e^2 + 106*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c*d^2*e^3 + 840*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^2*d^2*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*d*e^4 - 144
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c*d*e^4 - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^2*d*e^4
- 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*e^5 + 528*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(7/2)*d^5
- 516*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(5/2)*d^4*e - 624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c
^(7/2)*d^4*e - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(3/2)*d^3*e^2 + 840*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*a*b*c^(5/2)*d^3*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*sqrt(c)*d^2*e^3 + 144*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(3/2)*d^2*e^3 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(5/2)*d^2*e^
3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3*sqrt(c)*d*e^4 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a
^2*b*c^(3/2)*d*e^4 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(3/2)*e^5 + 264*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*b^2*c^3*d^5 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c^2*d^4*e - 624*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a*b*c^3*d^4*e + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c*d^3*e^2 + 852*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a*b^2*c^2*d^3*e^2 + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c^3*d^3*e^2 + 3*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*b^5*d^2*e^3 - 90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c*d^2*e^3 - 864*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))*a^2*b*c^2*d^2*e^3 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*d*e^4 + 90*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*a^2*b^2*c*d*e^4 + 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c^2*d*e^4 + 3*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*a^2*b^3*e^5 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c*e^5 + 44*b^3*c^(5/2)*d^5 - 4
4*b^4*c^(3/2)*d^4*e - 156*a*b^2*c^(5/2)*d^4*e + 3*b^5*sqrt(c)*d^3*e^2 + 182*a*b^3*c^(3/2)*d^3*e^2 + 192*a^2*b*
c^(5/2)*d^3*e^2 - 6*a*b^4*sqrt(c)*d^2*e^3 - 294*a^2*b^2*c^(3/2)*d^2*e^3 - 88*a^3*c^(5/2)*d^2*e^3 + 3*a^2*b^3*s
qrt(c)*d*e^4 + 220*a^3*b*c^(3/2)*d*e^4 - 64*a^4*c^(3/2)*e^5)/((c*d^2*e^4 - b*d*e^5 + a*e^6)*((sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^4} \,d x \]

[In]

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^4,x)

[Out]

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^4, x)