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

Optimal. Leaf size=462 \[ \frac {\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{16 e^5 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{8 e^4 (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {4 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.63, antiderivative size = 462, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {810, 812, 843, 621, 206, 724} \begin {gather*} \frac {\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 ) \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^5 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac {\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 (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{8 e^4 (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {4 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((64*c^3*d^3 + b^3*e^3 + 4*b*c*e^2*(4*b*d - 5*a*e) - 16*c^2*d*e*(5*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])/(8*e^4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)) - ((16*c^2*d^3 -
b*e^2*(b*d - 4*a*e) - 4*c*d*e*(3*b*d - a*e) + 3*e*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x)*(a + b*x + c*
x^2)^(3/2))/(12*e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (4*c^(3/2)*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqr
t[c]*Sqrt[a + b*x + c*x^2])])/e^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*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^5*(c*d^2 - b*d*e + a*e^2)^(3/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

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 724

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 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((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))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), 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 812

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 843

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 {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx &=-\frac {\left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {\int \frac {\left (\frac {1}{2} \left (12 b^2 c d e+16 a c^2 d e+b^3 e^2-4 b c \left (4 c d^2+5 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)^2} \, dx}{4 e^2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {\left (64 c^3 d^3+b^3 e^3+4 b c e^2 (4 b d-5 a e)-16 c^2 d e (5 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}}{8 e^4 \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\int \frac {\frac {1}{2} \left (-16 b^3 c d e^2-b^4 e^3-64 b c^2 d \left (c d^2+2 a e^2\right )+16 a c^2 e \left (4 c d^2+3 a e^2\right )+8 b^2 c e \left (10 c d^2+3 a e^2\right )\right )-32 c^2 (2 c d-b e) \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^4 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {\left (64 c^3 d^3+b^3 e^3+4 b c e^2 (4 b d-5 a e)-16 c^2 d e (5 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}}{8 e^4 \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {\left (4 c^2 (2 c d-b e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{e^5}+\frac {\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 ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{16 e^5 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {\left (64 c^3 d^3+b^3 e^3+4 b c e^2 (4 b d-5 a e)-16 c^2 d e (5 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}}{8 e^4 \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {\left (8 c^2 (2 c d-b e)\right ) \operatorname {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^5}-\frac {\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 ) \operatorname {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 )}{8 e^5 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {\left (64 c^3 d^3+b^3 e^3+4 b c e^2 (4 b d-5 a e)-16 c^2 d e (5 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}}{8 e^4 \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {4 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^5}+\frac {\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 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^5 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 2.96, size = 485, normalized size = 1.05 \begin {gather*} \frac {\frac {2 e \sqrt {a+x (b+c x)} \left (2 c e^2 \left (-4 a^2 e^2 (d+3 e x)-2 a b e \left (9 d^2+20 d e x+23 e^2 x^2\right )+b^2 d \left (24 d^2+63 d e x+55 e^2 x^2\right )\right )+b e^3 \left (-8 a^2 e^2+2 a b e (d-7 e x)+b^2 \left (3 d^2+8 d e x-3 e^2 x^2\right )\right )-8 c^2 e \left (b d \left (30 d^3+76 d^2 e x+57 d e^2 x^2+6 e^3 x^3\right )-a e \left (20 d^3+51 d^2 e x+41 d e^2 x^2+6 e^3 x^3\right )\right )+16 c^3 d^2 \left (12 d^3+30 d^2 e x+22 d e^2 x^2+3 e^3 x^3\right )\right )}{(d+e x)^3 \left (e (a e-b d)+c d^2\right )}-\frac {3 \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 ) \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}-192 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{48 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*e*Sqrt[a + x*(b + c*x)]*(16*c^3*d^2*(12*d^3 + 30*d^2*e*x + 22*d*e^2*x^2 + 3*e^3*x^3) + b*e^3*(-8*a^2*e^2 +
2*a*b*e*(d - 7*e*x) + b^2*(3*d^2 + 8*d*e*x - 3*e^2*x^2)) + 2*c*e^2*(-4*a^2*e^2*(d + 3*e*x) - 2*a*b*e*(9*d^2 +
20*d*e*x + 23*e^2*x^2) + b^2*d*(24*d^2 + 63*d*e*x + 55*e^2*x^2)) - 8*c^2*e*(-(a*e*(20*d^3 + 51*d^2*e*x + 41*d
*e^2*x^2 + 6*e^3*x^3)) + b*d*(30*d^3 + 76*d^2*e*x + 57*d*e^2*x^2 + 6*e^3*x^3))))/((c*d^2 + e*(-(b*d) + a*e))*(
d + e*x)^3) - 192*c^(3/2)*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - (3*(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*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)]
)])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(48*e^5)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 180.06, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

________________________________________________________________________________________

fricas [F(-1)]  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((2*c*x+b)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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((2*c*x+b)*(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,x):;OUTPUT:Eval
uation time: 42.28Unable to divide, perhaps due to rounding error%%%{%%{[-1,0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,9,
0,0]%%%}+%%%{%%%{8,[1]%%%},[7,0,8,1,0]%%%}+%%%{%%{[-4,0]:[1,0,%%%{-1,[1]%%%}]%%},[6,1,8,1,0]%%%}+%%%{%%{[4,0]:
[1,0,%%%{-1,[1]%%%}]%%},[6,0,9,0,1]%%%}+%%%{%%{[%%%{-24,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,7,2,0]%%%}+%%%
{%%%{24,[1]%%%},[5,1,7,2,0]%%%}+%%%{%%%{-24,[1]%%%},[5,0,8,1,1]%%%}+%%%{%%%{32,[2]%%%},[5,0,6,3,0]%%%}+%%%{%%{
[-6,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,2,7,2,0]%%%}+%%%{%%{[12,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,1,8,1,1]%%%}+%%%{%%{[%
%%{-48,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,1,6,3,0]%%%}+%%%{%%{[-6,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,9,0,2]%%%
}+%%%{%%{[%%%{48,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,7,2,1]%%%}+%%%{%%{[%%%{-16,[2]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[4,0,5,4,0]%%%}+%%%{%%%{24,[1]%%%},[3,2,6,3,0]%%%}+%%%{%%%{-48,[1]%%%},[3,1,7,2,1]%%%}+%%%{%%%{32,[2]
%%%},[3,1,5,4,0]%%%}+%%%{%%%{24,[1]%%%},[3,0,8,1,2]%%%}+%%%{%%%{-32,[2]%%%},[3,0,6,3,1]%%%}+%%%{%%{[-4,0]:[1,0
,%%%{-1,[1]%%%}]%%},[2,3,6,3,0]%%%}+%%%{%%{[12,0]:[1,0,%%%{-1,[1]%%%}]%%},[2,2,7,2,1]%%%}+%%%{%%{[%%%{-24,[1]%
%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,2,5,4,0]%%%}+%%%{%%{[-12,0]:[1,0,%%%{-1,[1]%%%}]%%},[2,1,8,1,2]%%%}+%%%{%%{[
%%%{48,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,1,6,3,1]%%%}+%%%{%%{[4,0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,9,0,3]%%%}
+%%%{%%{[%%%{-24,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,7,2,2]%%%}+%%%{%%%{8,[1]%%%},[1,3,5,4,0]%%%}+%%%{%%%{
-24,[1]%%%},[1,2,6,3,1]%%%}+%%%{%%%{24,[1]%%%},[1,1,7,2,2]%%%}+%%%{%%%{-8,[1]%%%},[1,0,8,1,3]%%%}+%%%{%%{[-1,0
]:[1,0,%%%{-1,[1]%%%}]%%},[0,4,5,4,0]%%%}+%%%{%%{[4,0]:[1,0,%%%{-1,[1]%%%}]%%},[0,3,6,3,1]%%%}+%%%{%%{[-6,0]:[
1,0,%%%{-1,[1]%%%}]%%},[0,2,7,2,2]%%%}+%%%{%%{[4,0]:[1,0,%%%{-1,[1]%%%}]%%},[0,1,8,1,3]%%%}+%%%{%%{[-1,0]:[1,0
,%%%{-1,[1]%%%}]%%},[0,0,9,0,4]%%%} / %%%{%%%{1,[2]%%%},[8,0,4,0,0]%%%}+%%%{%%{poly1[%%%{-8,[2]%%%},0]:[1,0,%%
%{-1,[1]%%%}]%%},[7,0,3,1,0]%%%}+%%%{%%%{4,[2]%%%},[6,1,3,1,0]%%%}+%%%{%%%{-4,[2]%%%},[6,0,4,0,1]%%%}+%%%{%%%{
24,[3]%%%},[6,0,2,2,0]%%%}+%%%{%%{[%%%{-24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1,2,2,0]%%%}+%%%{%%{poly1[%%%
{24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,3,1,1]%%%}+%%%{%%{poly1[%%%{-32,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%}
,[5,0,1,3,0]%%%}+%%%{%%%{6,[2]%%%},[4,2,2,2,0]%%%}+%%%{%%%{-12,[2]%%%},[4,1,3,1,1]%%%}+%%%{%%%{48,[3]%%%},[4,1
,1,3,0]%%%}+%%%{%%%{6,[2]%%%},[4,0,4,0,2]%%%}+%%%{%%%{-48,[3]%%%},[4,0,2,2,1]%%%}+%%%{%%%{16,[4]%%%},[4,0,0,4,
0]%%%}+%%%{%%{poly1[%%%{-24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,2,1,3,0]%%%}+%%%{%%{[%%%{48,[2]%%%},0]:[1,0,
%%%{-1,[1]%%%}]%%},[3,1,2,2,1]%%%}+%%%{%%{[%%%{-32,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,0,4,0]%%%}+%%%{%%{p
oly1[%%%{-24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,3,1,2]%%%}+%%%{%%{poly1[%%%{32,[3]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[3,0,1,3,1]%%%}+%%%{%%%{4,[2]%%%},[2,3,1,3,0]%%%}+%%%{%%%{-12,[2]%%%},[2,2,2,2,1]%%%}+%%%{%%%{24,[3]%
%%},[2,2,0,4,0]%%%}+%%%{%%%{12,[2]%%%},[2,1,3,1,2]%%%}+%%%{%%%{-48,[3]%%%},[2,1,1,3,1]%%%}+%%%{%%%{-4,[2]%%%},
[2,0,4,0,3]%%%}+%%%{%%%{24,[3]%%%},[2,0,2,2,2]%%%}+%%%{%%{[%%%{-8,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,3,0,4,
0]%%%}+%%%{%%{poly1[%%%{24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,1,3,1]%%%}+%%%{%%{[%%%{-24,[2]%%%},0]:[1,0,
%%%{-1,[1]%%%}]%%},[1,1,2,2,2]%%%}+%%%{%%{poly1[%%%{8,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,3,1,3]%%%}+%%%{%
%%{1,[2]%%%},[0,4,0,4,0]%%%}+%%%{%%%{-4,[2]%%%},[0,3,1,3,1]%%%}+%%%{%%%{6,[2]%%%},[0,2,2,2,2]%%%}+%%%{%%%{-4,[
2]%%%},[0,1,3,1,3]%%%}+%%%{%%%{1,[2]%%%},[0,0,4,0,4]%%%} Error: Bad Argument Value

________________________________________________________________________________________

maple [B]  time = 0.09, size = 15982, normalized size = 34.59 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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((2*c*x+b)*(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 details)Is a*e^2-b*d*e                            +c*d^2 zero or nonzero?

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________