3.16.62 \(\int \frac {(b+2 c x) (a+b x+c x^2)^{3/2}}{(d+e x)^2} \, dx\) [1562]

Optimal. Leaf size=303 \[ \frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{2 e^4}+\frac {(8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {(2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {c} e^5}+\frac {\sqrt {c d^2-b d e+a e^2} \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 e^5} \]

[Out]

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

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Rubi [A]
time = 0.32, antiderivative size = 303, 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 = {826, 828, 857, 635, 212, 738} \begin {gather*} -\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {c} e^5}+\frac {\sqrt {a e^2-b d e+c d^2} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 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 )}{2 e^5}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^4}+\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2053\) vs. \(2(303)=606\).
time = 11.65, size = 2053, normalized size = 6.78 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b + 2*c*x)*(a + x*(b + c*x))*(2*a*Sqrt[c] + 2*b*Sqrt[c]*x + 2*c^(3/2)*x^2 - b*Sqrt[a + x*(b + c*x)] - 2*c*x*
Sqrt[a + x*(b + c*x)]))/(d*e*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])) + (x*(b + 2*c*x)*(a + x*(b + c*x))
*(-2*a*Sqrt[c] - 2*b*Sqrt[c]*x - 2*c^(3/2)*x^2 + b*Sqrt[a + x*(b + c*x)] + 2*c*x*Sqrt[a + x*(b + c*x)]))/(d*(d
+ e*x)*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])) - (16*c^3*d^4*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*
(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/(e^5*Sqrt[-(c*d^2) + e*(b*d - a*e)]) - (32*b*c^2*d^3*ArcTan[(-(Sq
rt[c]*(d + e*x)) + e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/(e^4*Sqrt[-(c*d^2) + e*(b*d - a*e
)]) + (a*(3*b^2 + 4*a*c)*ArcTan[(-(Sqrt[c]*(d + e*x)) + e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)
]])/(e*Sqrt[-(c*d^2) + e*(b*d - a*e)]) + (8*c^(5/2)*d^3*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/e^5
- (c^(3/2)*d^2*(3*b^2 + b*(38*c*x - 22*Sqrt[c]*Sqrt[a + x*(b + c*x)]) + 32*c*(a + c*x^2 - Sqrt[c]*x*Sqrt[a + x
*(b + c*x)]) + 24*b*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*
x)]]))/(2*e^4*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])) + (8*c*(19*b^2 + 20*a*c)*d^2*(b^2 + b*(8*c*x - 4*
Sqrt[c]*Sqrt[a + x*(b + c*x)]) + 4*c*(a + 2*c*x^2 - 2*Sqrt[c]*x*Sqrt[a + x*(b + c*x)]))*ArcTan[(-(Sqrt[c]*(d +
e*x)) + e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]] + Sqrt[c]*d*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(
-5*b^4 + 316*a*b^2*c + 64*a^2*c^2 + 280*b^3*c*x + 896*a*b*c^2*x + 1048*b^2*c^2*x^2 + 384*a*c^3*x^2 + 1024*b*c^
3*x^3 + 256*c^4*x^4 - 60*b^3*Sqrt[c]*Sqrt[a + x*(b + c*x)] - 384*a*b*c^(3/2)*Sqrt[a + x*(b + c*x)] - 632*b^2*c
^(3/2)*x*Sqrt[a + x*(b + c*x)] - 256*a*c^(5/2)*x*Sqrt[a + x*(b + c*x)] - 896*b*c^(5/2)*x^2*Sqrt[a + x*(b + c*x
)] - 256*c^(7/2)*x^3*Sqrt[a + x*(b + c*x)] + 12*(3*b^2 + 4*a*c)*(b^2 + b*(8*c*x - 4*Sqrt[c]*Sqrt[a + x*(b + c*
x)]) + 4*c*(a + 2*c*x^2 - 2*Sqrt[c]*x*Sqrt[a + x*(b + c*x)]))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]
))/(8*e^3*Sqrt[-(c*d^2) + e*(b*d - a*e)]*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])^2) + (-24*b*Sqrt[c]*(3*
b^2 + 20*a*c)*d*(b^3 - 6*b^2*(-3*c*x + Sqrt[c]*Sqrt[a + x*(b + c*x)]) + 4*b*c*(3*a + 12*c*x^2 - 8*Sqrt[c]*x*Sq
rt[a + x*(b + c*x)]) - 8*(-3*a*c^2*x - 4*c^3*x^3 + a*c^(3/2)*Sqrt[a + x*(b + c*x)] + 4*c^(5/2)*x^2*Sqrt[a + x*
(b + c*x)]))*ArcTan[(-(Sqrt[c]*(d + e*x)) + e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]] - Sqrt[-(
c*d^2) + b*d*e - a*e^2]*(-7*b^6 - 18*b^5*(-13*c*x + Sqrt[c]*Sqrt[a + x*(b + c*x)]) + b^4*c*(285*a + 2616*c*x^2
- 968*Sqrt[c]*x*Sqrt[a + x*(b + c*x)]) + b^3*(3738*a*c^2*x + 8256*c^3*x^3 - 830*a*c^(3/2)*Sqrt[a + x*(b + c*x
)] - 4736*c^(5/2)*x^2*Sqrt[a + x*(b + c*x)]) + 12*b^2*c^2*(105*a^2 + 996*a*c*x^2 + 960*c^2*x^4 - 424*a*Sqrt[c]
*x*Sqrt[a + x*(b + c*x)] - 704*c^(3/2)*x^3*Sqrt[a + x*(b + c*x)]) + 128*c^3*(7*a + 4*c*x^2)*(a^2 + 5*a*c*x^2 +
4*c^2*x^4 - 3*a*Sqrt[c]*x*Sqrt[a + x*(b + c*x)] - 4*c^(3/2)*x^3*Sqrt[a + x*(b + c*x)]) - 8*b*(-960*c^5*x^5 +
832*c^(9/2)*x^4*Sqrt[a + x*(b + c*x)] + 3*a^2*(-233*c^3*x + 59*c^(5/2)*Sqrt[a + x*(b + c*x)]) + 12*a*(-151*c^4
*x^3 + 95*c^(7/2)*x^2*Sqrt[a + x*(b + c*x)])) + 6*b*(b^2 + 12*a*c)*(b^3 - 6*b^2*(-3*c*x + Sqrt[c]*Sqrt[a + x*(
b + c*x)]) + 4*b*c*(3*a + 12*c*x^2 - 8*Sqrt[c]*x*Sqrt[a + x*(b + c*x)]) - 8*(-3*a*c^2*x - 4*c^3*x^3 + a*c^(3/2
)*Sqrt[a + x*(b + c*x)] + 4*c^(5/2)*x^2*Sqrt[a + x*(b + c*x)]))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)
]]))/(24*Sqrt[c]*e^2*Sqrt[-(c*d^2) + e*(b*d - a*e)]*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1730\) vs. \(2(275)=550\).
time = 1.23, size = 1731, normalized size = 5.71

method	result	size
default	\(\text {Expression too large to display}\)	\(1731\)
risch	\(\text {Expression too large to display}\)	\(4567\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

1/e^3*(b*e-2*c*d)*(-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)))))+2*c/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))))

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

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Fricas [A]
time = 198.19, size = 2177, normalized size = 7.18 \begin {gather*} \text {Too large to display} \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)^2,x, algorithm="fricas")

[Out]

[-1/24*(3*(32*c^3*d^4 - (b^3 + 12*a*b*c)*x*e^4 + (6*(3*b^2*c + 4*a*c^2)*d*x - (b^3 + 12*a*b*c)*d)*e^3 - 6*(8*b
*c^2*d^2*x - (3*b^2*c + 4*a*c^2)*d^2)*e^2 + 16*(2*c^3*d^3*x - 3*b*c^2*d^3)*e)*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) - 6*(16*c^3*d^3 + (3*b^2*c + 4*a*c^2)*x*e^3 - (1
6*b*c^2*d*x - (3*b^2*c + 4*a*c^2)*d)*e^2 + 16*(c^3*d^2*x - b*c^2*d^2)*e)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c
^2*d^2*x^2 + 8*b*c*d^2*x + (b^2 + 4*a*c)*d^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*s
qrt(c*x^2 + b*x + a) + (8*a*b*x + (b^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*
d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)) - 4*(48*c^3*d^3*e + (4*c^3*x^3 + 10*b*c^2*x^2 - 6*a*b*c + (9*b^2*c + 16*a*c
^2)*x)*e^4 - (8*c^3*d*x^2 + 32*b*c^2*d*x - (15*b^2*c + 28*a*c^2)*d)*e^3 + 12*(2*c^3*d^2*x - 5*b*c^2*d^2)*e^2)*
sqrt(c*x^2 + b*x + a))/(c*x*e^6 + c*d*e^5), 1/12*(3*(32*c^3*d^4 - (b^3 + 12*a*b*c)*x*e^4 + (6*(3*b^2*c + 4*a*c
^2)*d*x - (b^3 + 12*a*b*c)*d)*e^3 - 6*(8*b*c^2*d^2*x - (3*b^2*c + 4*a*c^2)*d^2)*e^2 + 16*(2*c^3*d^3*x - 3*b*c^
2*d^3)*e)*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^3 + (3*b^2*c + 4*a*c^2)*x*e^3 - (16*b*c^2*d*x - (3*b^2*c + 4*a*c^2)*d)*e^2 + 16*(c^3*d^2*x - b*c^2*d^2)*e)*s
qrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b^2 + 4*a*c)*d^2 + 4*sqrt(c*d^2 - b*d*e + a*e^
2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b*x + (b^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*
c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(48*c^3*d^3*e + (4*c^3*x^3 + 10*b*c
^2*x^2 - 6*a*b*c + (9*b^2*c + 16*a*c^2)*x)*e^4 - (8*c^3*d*x^2 + 32*b*c^2*d*x - (15*b^2*c + 28*a*c^2)*d)*e^3 +
12*(2*c^3*d^2*x - 5*b*c^2*d^2)*e^2)*sqrt(c*x^2 + b*x + a))/(c*x*e^6 + c*d*e^5), 1/24*(12*(16*c^3*d^3 + (3*b^2*
c + 4*a*c^2)*x*e^3 - (16*b*c^2*d*x - (3*b^2*c + 4*a*c^2)*d)*e^2 + 16*(c^3*d^2*x - b*c^2*d^2)*e)*sqrt(-c*d^2 +
b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/
(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x^2 + b^2*d*x + a*b*d)*e)) - 3*(32*c
^3*d^4 - (b^3 + 12*a*b*c)*x*e^4 + (6*(3*b^2*c + 4*a*c^2)*d*x - (b^3 + 12*a*b*c)*d)*e^3 - 6*(8*b*c^2*d^2*x - (3
*b^2*c + 4*a*c^2)*d^2)*e^2 + 16*(2*c^3*d^3*x - 3*b*c^2*d^3)*e)*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) + 4*(48*c^3*d^3*e + (4*c^3*x^3 + 10*b*c^2*x^2 - 6*a*b*c + (9*b^
2*c + 16*a*c^2)*x)*e^4 - (8*c^3*d*x^2 + 32*b*c^2*d*x - (15*b^2*c + 28*a*c^2)*d)*e^3 + 12*(2*c^3*d^2*x - 5*b*c^
2*d^2)*e^2)*sqrt(c*x^2 + b*x + a))/(c*x*e^6 + c*d*e^5), 1/12*(6*(16*c^3*d^3 + (3*b^2*c + 4*a*c^2)*x*e^3 - (16*
b*c^2*d*x - (3*b^2*c + 4*a*c^2)*d)*e^2 + 16*(c^3*d^2*x - b*c^2*d^2)*e)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/
2*sqrt(-c*d^2 + b*d*e - a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^2*x^2 + b*c*d^2*x
+ a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x^2 + b^2*d*x + a*b*d)*e)) + 3*(32*c^3*d^4 - (b^3 + 12*a*b*c)
*x*e^4 + (6*(3*b^2*c + 4*a*c^2)*d*x - (b^3 + 12*a*b*c)*d)*e^3 - 6*(8*b*c^2*d^2*x - (3*b^2*c + 4*a*c^2)*d^2)*e^
2 + 16*(2*c^3*d^3*x - 3*b*c^2*d^3)*e)*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*(48*c^3*d^3*e + (4*c^3*x^3 + 10*b*c^2*x^2 - 6*a*b*c + (9*b^2*c + 16*a*c^2)*x)*e^4 - (8*c^3
*d*x^2 + 32*b*c^2*d*x - (15*b^2*c + 28*a*c^2)*d)*e^3 + 12*(2*c^3*d^2*x - 5*b*c^2*d^2)*e^2)*sqrt(c*x^2 + b*x +
a))/(c*x*e^6 + c*d*e^5)]

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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 )^{2}}\, 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)**2,x)

[Out]

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

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

[Out]

Timed out

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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 )}^2} \,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)^2,x)

[Out]

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

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