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

Optimal. Leaf size=307 \[ -\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^4}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 e^4 \left (c d^2-b d e+a e^2\right )^{3/2}} \]

[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]
time = 0.25, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {746, 824, 857, 635, 212, 738} \begin {gather*} -\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 ) \tanh ^{-1}\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 {\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 {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 {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx &=-\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]
time = 11.05, size = 421, normalized size = 1.37 \begin {gather*} \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 \tanh ^{-1}\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 ) \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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] Leaf count of result is larger than twice the leaf count of optimal. \(3084\) vs. \(2(281)=562\).
time = 0.80, size = 3085, normalized size = 10.05

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

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(c*d^2-%e*b*d+%e^2*a>0)', see `
assume?` for

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 14.4Unable to divide, perhaps due to rounding error%%%{%%{[-1,0]:[1,0,%%%{-1,[1]%%%}]%%},[
8,0,0,8,0]%

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________