\(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^2} \, dx\) [2360]

Optimal result

Integrand size = 22, antiderivative size = 388 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {5 \left (64 c^3 d^3-b^3 e^3+4 b c e^2 (12 b d-11 a e)-16 c^2 d e (7 b d-4 a e)-2 c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c e^5}-\frac {5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{3/2} e^6}-\frac {5 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 e^6} \]

[Out]

-5/24*(-6*c*e*x-7*b*e+8*c*d)*(c*x^2+b*x+a)^(3/2)/e^3-(c*x^2+b*x+a)^(5/2)/e/(e*x+d)+5/128*(128*c^4*d^4-b^4*e^4-
8*b^2*c*e^3*(-3*a*e+2*b*d)-64*c^3*d^2*e*(-3*a*e+4*b*d)+48*c^2*e^2*(a^2*e^2-4*a*b*d*e+3*b^2*d^2))*arctanh(1/2*(
2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/e^6-5/2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^(3/2)*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^6-5/64*(64*c^3*d^3-b^3*e^3+4*b*c*e^2*(-
11*a*e+12*b*d)-16*c^2*d*e*(-4*a*e+7*b*d)-2*c*e*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4*b*d))*x)*(c*x^2+b*x+a)^(1/2
)/c/e^5

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {746, 828, 857, 635, 212, 738} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {5 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right )}{128 c^{3/2} e^6}-\frac {5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 e^6}-\frac {5 \sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-4 a e)+4 b c e^2 (12 b d-11 a e)-b^3 e^3+64 c^3 d^3\right )}{64 c e^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)} \]

[In]

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

[Out]

(-5*(64*c^3*d^3 - b^3*e^3 + 4*b*c*e^2*(12*b*d - 11*a*e) - 16*c^2*d*e*(7*b*d - 4*a*e) - 2*c*e*(16*c^2*d^2 + b^2
*e^2 - 4*c*e*(4*b*d - 3*a*e))*x)*Sqrt[a + b*x + c*x^2])/(64*c*e^5) - (5*(8*c*d - 7*b*e - 6*c*e*x)*(a + b*x + c
*x^2)^(3/2))/(24*e^3) - (a + b*x + c*x^2)^(5/2)/(e*(d + e*x)) + (5*(128*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b*d
- 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(3/2)*e^6) - (5*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(3/2)*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])])/(2*e^6)

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 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*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] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 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 )^{5/2}}{e (d+e x)}+\frac {5 \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{2 e} \\ & = -\frac {5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}-\frac {5 \int \frac {\left (c \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{d+e x} \, dx}{16 c e^3} \\ & = -\frac {5 \left (64 c^3 d^3-b^3 e^3+4 b c e^2 (12 b d-11 a e)-16 c^2 d e (7 b d-4 a e)-2 c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c e^5}-\frac {5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \int \frac {\frac {1}{2} c \left (d \left (4 b c d-b^2 e-4 a c e\right ) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )+4 c e (b d-2 a e) \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )\right )\right )+\frac {1}{2} c \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{64 c^2 e^5} \\ & = -\frac {5 \left (64 c^3 d^3-b^3 e^3+4 b c e^2 (12 b d-11 a e)-16 c^2 d e (7 b d-4 a e)-2 c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c e^5}-\frac {5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}-\frac {\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2 e^6}+\frac {\left (5 \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c e^6} \\ & = -\frac {5 \left (64 c^3 d^3-b^3 e^3+4 b c e^2 (12 b d-11 a e)-16 c^2 d e (7 b d-4 a e)-2 c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c e^5}-\frac {5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac {\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2\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 )}{e^6}+\frac {\left (5 \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right )\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 )}{64 c e^6} \\ & = -\frac {5 \left (64 c^3 d^3-b^3 e^3+4 b c e^2 (12 b d-11 a e)-16 c^2 d e (7 b d-4 a e)-2 c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c e^5}-\frac {5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{3/2} e^6}-\frac {5 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^{3/2} \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 )}{2 e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.73 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {5 (-8 c d+7 b e+6 c e x) (a+x (b+c x))^{3/2}}{24 e^3}-\frac {(a+x (b+c x))^{5/2}}{e (d+e x)}+\frac {5 \left (\left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {c} \left (e \sqrt {a+x (b+c x)} \left (b^3 e^3+32 c^3 d^2 (-2 d+e x)+2 b c e^2 (-24 b d+22 a e+b e x)+8 c^2 e (2 b d (7 d-2 e x)+a e (-8 d+3 e x))\right )+32 c (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^{3/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )\right )}{128 c^{3/2} e^6} \]

[In]

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

[Out]

(5*(-8*c*d + 7*b*e + 6*c*e*x)*(a + x*(b + c*x))^(3/2))/(24*e^3) - (a + x*(b + c*x))^(5/2)/(e*(d + e*x)) + (5*(
(128*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 -
4*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*
x)]*(b^3*e^3 + 32*c^3*d^2*(-2*d + e*x) + 2*b*c*e^2*(-24*b*d + 22*a*e + b*e*x) + 8*c^2*e*(2*b*d*(7*d - 2*e*x) +
a*e*(-8*d + 3*e*x))) + 32*c*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*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)])])))/(128*c^(3/2)*e^6)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(915\) vs. \(2(358)=716\).

Time = 0.36 (sec) , antiderivative size = 916, normalized size of antiderivative = 2.36

method	result	size
risch	\(\frac {\left (48 c^{3} x^{3} e^{3}+136 b \,c^{2} e^{3} x^{2}-128 c^{3} d \,e^{2} x^{2}+216 a \,c^{2} e^{3} x +118 b^{2} c \,e^{3} x -416 b \,c^{2} d \,e^{2} x +288 c^{3} d^{2} e x +556 a b c \,e^{3}-896 a \,c^{2} d \,e^{2}+15 b^{3} e^{3}-528 b^{2} c d \,e^{2}+1296 b \,c^{2} d^{2} e -768 c^{3} d^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{192 c \,e^{5}}+\frac {\frac {5 \left (48 a^{2} c^{2} e^{4}+24 a \,b^{2} e^{4} c -192 a b \,c^{2} d \,e^{3}+192 a \,c^{3} d^{2} e^{2}-b^{4} e^{4}-16 b^{3} c d \,e^{3}+144 b^{2} c^{2} d^{2} e^{2}-256 b \,c^{3} d^{3} e +128 c^{4} d^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}-\frac {384 c \left (a^{2} b \,e^{5}-2 a^{2} c d \,e^{4}-2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 a \,c^{2} d^{3} e^{2}+b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 c^{3} d^{5}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {128 c \left (a^{3} e^{6}-3 d \,a^{2} b \,e^{5}+3 a^{2} c \,d^{2} e^{4}+3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 a \,c^{2} d^{4} e^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 d^{5} b \,c^{2} e +c^{3} d^{6}\right ) \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (b e -2 c d \right ) e \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}}{128 e^{5} c}\)	\(916\)
default	\(\text {Expression too large to display}\)	\(1642\)

[In]

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

[Out]

1/192/c*(48*c^3*e^3*x^3+136*b*c^2*e^3*x^2-128*c^3*d*e^2*x^2+216*a*c^2*e^3*x+118*b^2*c*e^3*x-416*b*c^2*d*e^2*x+
288*c^3*d^2*e*x+556*a*b*c*e^3-896*a*c^2*d*e^2+15*b^3*e^3-528*b^2*c*d*e^2+1296*b*c^2*d^2*e-768*c^3*d^3)*(c*x^2+
b*x+a)^(1/2)/e^5+1/128/e^5/c*(5*(48*a^2*c^2*e^4+24*a*b^2*c*e^4-192*a*b*c^2*d*e^3+192*a*c^3*d^2*e^2-b^4*e^4-16*
b^3*c*d*e^3+144*b^2*c^2*d^2*e^2-256*b*c^3*d^3*e+128*c^4*d^4)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(
1/2)-384*c/e^2*(a^2*b*e^5-2*a^2*c*d*e^4-2*a*b^2*d*e^4+6*a*b*c*d^2*e^3-4*a*c^2*d^3*e^2+b^3*d^2*e^3-4*b^2*c*d^3*
e^2+5*b*c^2*d^4*e-2*c^3*d^5)/((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
))+128*c*(a^3*e^6-3*a^2*b*d*e^5+3*a^2*c*d^2*e^4+3*a*b^2*d^2*e^4-6*a*b*c*d^3*e^3+3*a*c^2*d^4*e^2-b^3*d^3*e^3+3*
b^2*c*d^4*e^2-3*b*c^2*d^5*e+c^3*d^6)/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))))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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