\(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^3} \, dx\) [2349]

Optimal result

Integrand size = 22, antiderivative size = 236 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {3 (4 c d-b e+2 c e x) \sqrt {a+b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 \sqrt {c} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 e^4}+\frac {3 \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 e^4 \sqrt {c d^2-b d e+a e^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 236, 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, 826, 857, 635, 212, 738} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {3 \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 e^4 \sqrt {a e^2-b d e+c d^2}}-\frac {3 \sqrt {c} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 e^4}+\frac {3 \sqrt {a+b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]

[In]

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

[Out]

(3*(4*c*d - b*e + 2*c*e*x)*Sqrt[a + b*x + c*x^2])/(4*e^3*(d + e*x)) - (a + b*x + c*x^2)^(3/2)/(2*e*(d + e*x)^2
) - (3*Sqrt[c]*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*e^4) + (3*(8*c^2*d^2 +
b^2*e^2 - 4*c*e*(2*b*d - 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])])/(8*e^4*Sqrt[c*d^2 - b*d*e + a*e^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 826

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

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

Mathematica [A] (verified)

Time = 10.57 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {-\frac {2 (a+x (b+c x))^{3/2}}{(d+e x)^2}+\frac {3 (2 c d-b e) (a+x (b+c x))^{3/2}}{\left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {3 \left (-\frac {2 e \sqrt {a+x (b+c x)} \left (-b^2 e^2+2 c^2 d (-2 d+e x)-c e (-5 b d+2 a e+b e x)\right )}{c d^2+e (-b d+a e)}-4 \sqrt {c} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-\frac {\left (8 c^2 d^2+b^2 e^2+4 c e (-2 b d+a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c d^2+e (-b d+a e)}}\right )}{2 e^3}}{4 e} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1176\) vs. \(2(208)=416\).

Time = 0.44 (sec) , antiderivative size = 1177, normalized size of antiderivative = 4.99

method	result	size
risch	\(\text {Expression too large to display}\)	\(1177\)
default	\(\text {Expression too large to display}\)	\(1855\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (208) = 416\).

Time = 15.09 (sec) , antiderivative size = 2711, normalized size of antiderivative = 11.49 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/16*(12*(2*c^2*d^5 - 3*b*c*d^4*e - a*b*d^2*e^3 + (b^2 + 2*a*c)*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 - a
*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 - a*b*d*e^4 + (b^2 + 2*a*c)*d^2*e^3)*x)*sqr
t(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*(8*c^2*d^4 - 8*
b*c*d^3*e + (b^2 + 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (b^2 + 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8
*b*c*d^2*e^2 + (b^2 + 4*a*c)*d*e^3)*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*(12*c^2*d^4*e - 15*b*c*d^3*e^2 - a*b*d*e^4 - 2*a^2*e^5 + (3*b^2 + 10*a*c)*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d
*e^4 + a*c*e^5)*x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 - 5*a*b*e^5 + (5*b^2 + 18*a*c)*d*e^4)*x)*sqrt(c*x^2 + b
*x + a))/(c*d^4*e^4 - b*d^3*e^5 + a*d^2*e^6 + (c*d^2*e^6 - b*d*e^7 + a*e^8)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6 + a
*d*e^7)*x), 1/16*(24*(2*c^2*d^5 - 3*b*c*d^4*e - a*b*d^2*e^3 + (b^2 + 2*a*c)*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c*d
^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 - a*b*d*e^4 + (b^2 + 2*a*c)*d^2*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*(8*c^2*d^4
- 8*b*c*d^3*e + (b^2 + 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (b^2 + 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e
- 8*b*c*d^2*e^2 + (b^2 + 4*a*c)*d*e^3)*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*(12*c^2*d^4*e - 15*b*c*d^3*e^2 - a*b*d*e^4 - 2*a^2*e^5 + (3*b^2 + 10*a*c)*d^2*e^3 + 4*(c^2*d^2*e^3 - b
*c*d*e^4 + a*c*e^5)*x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 - 5*a*b*e^5 + (5*b^2 + 18*a*c)*d*e^4)*x)*sqrt(c*x^2
+ b*x + a))/(c*d^4*e^4 - b*d^3*e^5 + a*d^2*e^6 + (c*d^2*e^6 - b*d*e^7 + a*e^8)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6
+ a*d*e^7)*x), 1/8*(3*(8*c^2*d^4 - 8*b*c*d^3*e + (b^2 + 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (b^2
+ 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + (b^2 + 4*a*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arct
an(-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)) - 6*(2*c^2*d^5 - 3*b*c*d^4*e
- a*b*d^2*e^3 + (b^2 + 2*a*c)*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x^2 +
2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 - a*b*d*e^4 + (b^2 + 2*a*c)*d^2*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*(12*c^2*d^4*e - 15*b*c*d^3*e^2 - a*b*d*e^4 - 2*a^2
*e^5 + (3*b^2 + 10*a*c)*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d*e^4 + a*c*e^5)*x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3
- 5*a*b*e^5 + (5*b^2 + 18*a*c)*d*e^4)*x)*sqrt(c*x^2 + b*x + a))/(c*d^4*e^4 - b*d^3*e^5 + a*d^2*e^6 + (c*d^2*e
^6 - b*d*e^7 + a*e^8)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6 + a*d*e^7)*x), 1/8*(3*(8*c^2*d^4 - 8*b*c*d^3*e + (b^2 + 4
*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (b^2 + 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + (b^2
+ 4*a*c)*d*e^3)*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)) + 12*(2*c^2*d^5 - 3*b*c*d^4*e - a*b*d^2*e^3 + (b^2 + 2*a*c)*d^3*e^2 + (2*c^2*d^3*e^2 -
3*b*c*d^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 - a*b*d*e^4 + (b^2 + 2*a*
c)*d^2*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*(12
*c^2*d^4*e - 15*b*c*d^3*e^2 - a*b*d*e^4 - 2*a^2*e^5 + (3*b^2 + 10*a*c)*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d*e^4 +
a*c*e^5)*x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 - 5*a*b*e^5 + (5*b^2 + 18*a*c)*d*e^4)*x)*sqrt(c*x^2 + b*x + a)
)/(c*d^4*e^4 - b*d^3*e^5 + a*d^2*e^6 + (c*d^2*e^6 - b*d*e^7 + a*e^8)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6 + a*d*e^7)
*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^3,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. 794 vs. \(2 (208) = 416\).

Time = 0.45 (sec) , antiderivative size = 794, normalized size of antiderivative = 3.36 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\sqrt {c x^{2} + b x + a} c}{e^{3}} + \frac {3 \, {\left (8 \, c^{2} d^{2} - 8 \, b c d e + b^{2} e^{2} + 4 \, a c e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{4 \, \sqrt {-c d^{2} + b d e - a e^{2}} e^{4}} + \frac {3 \, {\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, \sqrt {c} e^{4}} + \frac {24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} c^{2} d^{2} e - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b c d e^{2} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{2} e^{3} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a c e^{3} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {5}{2}} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b c^{\frac {3}{2}} d^{2} e - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} \sqrt {c} d e^{2} - 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{\frac {3}{2}} d e^{2} + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b \sqrt {c} e^{3} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{2} d^{3} - 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{2} c d^{2} e - 56 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a c^{2} d^{2} e + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} d e^{2} + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c d e^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{2} e^{3} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} c e^{3} + 10 \, b^{2} c^{\frac {3}{2}} d^{3} - 5 \, b^{3} \sqrt {c} d^{2} e - 28 \, a b c^{\frac {3}{2}} d^{2} e + 13 \, a b^{2} \sqrt {c} d e^{2} + 20 \, a^{2} c^{\frac {3}{2}} d e^{2} - 8 \, a^{2} b \sqrt {c} e^{3}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} d + b d - a e\right )}^{2} e^{4}} \]

[In]

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

[Out]

sqrt(c*x^2 + b*x + a)*c/e^3 + 3/4*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2 + 4*a*c*e^2)*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))/(sqrt(-c*d^2 + b*d*e - a*e^2)*e^4) + 3/2*(2*c^2*d
- b*c*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/(sqrt(c)*e^4) + 1/4*(24*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^3*c^2*d^2*e - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d*e^2 + 5*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^3*b^2*e^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*e^3 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*c^(5/2)*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*d^2*e - (sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*b^2*sqrt(c)*d*e^2 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)*d*e^2 + 16*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^2*a*b*sqrt(c)*e^3 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*d^3 - 28*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*b^2*c*d^2*e - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*c^2*d^2*e + 3*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*b^3*d*e^2 + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c*d*e^2 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a*b^2*e^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*e^3 + 10*b^2*c^(3/2)*d^3 - 5*b^3*sqrt(c)*d^2*e -
28*a*b*c^(3/2)*d^2*e + 13*a*b^2*sqrt(c)*d*e^2 + 20*a^2*c^(3/2)*d*e^2 - 8*a^2*b*sqrt(c)*e^3)/(((sqrt(c)*x - sqr
t(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)^2*e^4)

Mupad [F(-1)]

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

[In]

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

[Out]

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