3.24.92 \(\int \frac {(d+e x)^4}{(a+b x+c x^2)^{5/2}} \, dx\) [2392]

Optimal. Leaf size=295 \[ -\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (4 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (c d^2+3 a e^2\right )-b^2 \left (5 c d^2 e-a e^3\right )+(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )^2}+\frac {e^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.46, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {752, 832, 654, 635, 212} \begin {gather*} -\frac {2 e \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )^2}+\frac {4 (d+e x) \left (x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\left (b^2 \left (5 c d^2 e-a e^3\right )\right )+4 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {e^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4/(a + b*x + c*x^2)^(5/2),x]

[Out]

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

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 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, 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] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 832

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 + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(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] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps

\begin {align*} \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {(d+e x)^2 \left (4 c d^2-e (5 b d-6 a e)-e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (4 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (c d^2+3 a e^2\right )-b^2 \left (5 c d^2 e-a e^3\right )+(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {4 \int \frac {\frac {1}{2} \left (b^3 d e^3-24 a^2 c e^4+4 b c d e \left (2 c d^2+5 a e^2\right )-2 b^2 \left (6 c d^2 e^2-a e^4\right )\right )+\frac {1}{2} e (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) x}{\sqrt {a+b x+c x^2}} \, dx}{3 c \left (b^2-4 a c\right )^2}\\ &=-\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (4 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (c d^2+3 a e^2\right )-b^2 \left (5 c d^2 e-a e^3\right )+(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )^2}+\frac {e^4 \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{c^2}\\ &=-\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (4 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (c d^2+3 a e^2\right )-b^2 \left (5 c d^2 e-a e^3\right )+(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )^2}+\frac {\left (2 e^4\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 )}{c^2}\\ &=-\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (4 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (c d^2+3 a e^2\right )-b^2 \left (5 c d^2 e-a e^3\right )+(2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )^2}+\frac {e^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 2.08, size = 400, normalized size = 1.36 \begin {gather*} -\frac {2 \left (3 b^5 e^4 x^2+2 b^4 e^4 x \left (3 a+2 c x^2\right )+b^3 \left (3 a^2 e^4-18 a c e^4 x^2+c^2 d \left (d^3+12 d^2 e x-18 d e^2 x^2-4 e^3 x^3\right )\right )-4 b c \left (5 a^3 e^4+2 c^3 d^3 x^2 (3 d-4 e x)+12 a^2 c d e^2 (d-2 e x)+3 a c^2 d \left (d^3-4 d^2 e x+6 d e^2 x^2-4 e^3 x^3\right )\right )+8 c^2 \left (-2 c^3 d^4 x^3+a^3 e^3 (8 d+3 e x)-3 a c^2 d^2 x \left (d^2+2 e^2 x^2\right )+4 a^2 c e \left (d^3+3 d e^2 x^2+e^3 x^3\right )\right )-2 b^2 c \left (21 a^2 e^4 x+3 c^2 d^2 x \left (d^2-8 d e x+2 e^2 x^2\right )+2 a c e \left (-2 d^3+18 d^2 e x-6 d e^2 x^2+7 e^3 x^3\right )\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}-\frac {e^4 \log \left (c^2 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*(3*b^5*e^4*x^2 + 2*b^4*e^4*x*(3*a + 2*c*x^2) + b^3*(3*a^2*e^4 - 18*a*c*e^4*x^2 + c^2*d*(d^3 + 12*d^2*e*x -
18*d*e^2*x^2 - 4*e^3*x^3)) - 4*b*c*(5*a^3*e^4 + 2*c^3*d^3*x^2*(3*d - 4*e*x) + 12*a^2*c*d*e^2*(d - 2*e*x) + 3*
a*c^2*d*(d^3 - 4*d^2*e*x + 6*d*e^2*x^2 - 4*e^3*x^3)) + 8*c^2*(-2*c^3*d^4*x^3 + a^3*e^3*(8*d + 3*e*x) - 3*a*c^2
*d^2*x*(d^2 + 2*e^2*x^2) + 4*a^2*c*e*(d^3 + 3*d*e^2*x^2 + e^3*x^3)) - 2*b^2*c*(21*a^2*e^4*x + 3*c^2*d^2*x*(d^2
- 8*d*e*x + 2*e^2*x^2) + 2*a*c*e*(-2*d^3 + 18*d^2*e*x - 6*d*e^2*x^2 + 7*e^3*x^3))))/(3*c^2*(b^2 - 4*a*c)^2*(a
+ x*(b + c*x))^(3/2)) - (e^4*Log[c^2*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1131\) vs. \(2(275)=550\).
time = 0.79, size = 1132, normalized size = 3.84

method	result	size
default	\(\text {Expression too large to display}\)	\(1132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

e^4*(-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-
1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^
2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*
(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*
x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+1/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(
c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))))+4*d*e^3*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b
*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a
)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^
(1/2)))+2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c
-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+6*d^2*e^2*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)
^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/
2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)
))+4*d^3*e*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^
2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+d^4*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(
2*c*x+b)/(c*x^2+b*x+a)^(1/2))

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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((e*x+d)^4/(c*x^2+b*x+a)^(5/2),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(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 739 vs. \(2 (282) = 564\).
time = 5.18, size = 1481, normalized size = 5.02 \begin {gather*} \left [\frac {3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )} \sqrt {c} e^{4} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (16 \, c^{6} d^{4} x^{3} + 24 \, b c^{5} d^{4} x^{2} + 6 \, {\left (b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x - {\left (b^{3} c^{3} - 12 \, a b c^{4}\right )} d^{4} - {\left (3 \, a^{2} b^{3} c - 20 \, a^{3} b c^{2} + 4 \, {\left (b^{4} c^{2} - 7 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} x^{3} + 3 \, {\left (b^{5} c - 6 \, a b^{3} c^{2}\right )} x^{2} + 6 \, {\left (a b^{4} c - 7 \, a^{2} b^{2} c^{2} + 4 \, a^{3} c^{3}\right )} x\right )} e^{4} - 4 \, {\left (24 \, a^{2} b c^{3} d x + 16 \, a^{3} c^{3} d - {\left (b^{3} c^{3} - 12 \, a b c^{4}\right )} d x^{3} + 6 \, {\left (a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d x^{2}\right )} e^{3} + 6 \, {\left (12 \, a b^{2} c^{3} d^{2} x + 8 \, a^{2} b c^{3} d^{2} + 2 \, {\left (b^{2} c^{4} + 4 \, a c^{5}\right )} d^{2} x^{3} + 3 \, {\left (b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{2} x^{2}\right )} e^{2} - 4 \, {\left (8 \, b c^{5} d^{3} x^{3} + 12 \, b^{2} c^{4} d^{3} x^{2} + 3 \, {\left (b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{3} x + 2 \, {\left (a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d^{3}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{6 \, {\left (a^{2} b^{4} c^{3} - 8 \, a^{3} b^{2} c^{4} + 16 \, a^{4} c^{5} + {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} x^{4} + 2 \, {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} x^{3} + {\left (b^{6} c^{3} - 6 \, a b^{4} c^{4} + 32 \, a^{3} c^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{3} - 8 \, a^{2} b^{3} c^{4} + 16 \, a^{3} b c^{5}\right )} x\right )}}, -\frac {3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) e^{4} - 2 \, {\left (16 \, c^{6} d^{4} x^{3} + 24 \, b c^{5} d^{4} x^{2} + 6 \, {\left (b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x - {\left (b^{3} c^{3} - 12 \, a b c^{4}\right )} d^{4} - {\left (3 \, a^{2} b^{3} c - 20 \, a^{3} b c^{2} + 4 \, {\left (b^{4} c^{2} - 7 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} x^{3} + 3 \, {\left (b^{5} c - 6 \, a b^{3} c^{2}\right )} x^{2} + 6 \, {\left (a b^{4} c - 7 \, a^{2} b^{2} c^{2} + 4 \, a^{3} c^{3}\right )} x\right )} e^{4} - 4 \, {\left (24 \, a^{2} b c^{3} d x + 16 \, a^{3} c^{3} d - {\left (b^{3} c^{3} - 12 \, a b c^{4}\right )} d x^{3} + 6 \, {\left (a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d x^{2}\right )} e^{3} + 6 \, {\left (12 \, a b^{2} c^{3} d^{2} x + 8 \, a^{2} b c^{3} d^{2} + 2 \, {\left (b^{2} c^{4} + 4 \, a c^{5}\right )} d^{2} x^{3} + 3 \, {\left (b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{2} x^{2}\right )} e^{2} - 4 \, {\left (8 \, b c^{5} d^{3} x^{3} + 12 \, b^{2} c^{4} d^{3} x^{2} + 3 \, {\left (b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{3} x + 2 \, {\left (a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d^{3}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} c^{3} - 8 \, a^{3} b^{2} c^{4} + 16 \, a^{4} c^{5} + {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} x^{4} + 2 \, {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} x^{3} + {\left (b^{6} c^{3} - 6 \, a b^{4} c^{4} + 32 \, a^{3} c^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{3} - 8 \, a^{2} b^{3} c^{4} + 16 \, a^{3} b c^{5}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^
2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqrt(c)
*e^4*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*(16*c^6*d^4*x^3
+ 24*b*c^5*d^4*x^2 + 6*(b^2*c^4 + 4*a*c^5)*d^4*x - (b^3*c^3 - 12*a*b*c^4)*d^4 - (3*a^2*b^3*c - 20*a^3*b*c^2 +
4*(b^4*c^2 - 7*a*b^2*c^3 + 8*a^2*c^4)*x^3 + 3*(b^5*c - 6*a*b^3*c^2)*x^2 + 6*(a*b^4*c - 7*a^2*b^2*c^2 + 4*a^3*
c^3)*x)*e^4 - 4*(24*a^2*b*c^3*d*x + 16*a^3*c^3*d - (b^3*c^3 - 12*a*b*c^4)*d*x^3 + 6*(a*b^2*c^3 + 4*a^2*c^4)*d*
x^2)*e^3 + 6*(12*a*b^2*c^3*d^2*x + 8*a^2*b*c^3*d^2 + 2*(b^2*c^4 + 4*a*c^5)*d^2*x^3 + 3*(b^3*c^3 + 4*a*b*c^4)*d
^2*x^2)*e^2 - 4*(8*b*c^5*d^3*x^3 + 12*b^2*c^4*d^3*x^2 + 3*(b^3*c^3 + 4*a*b*c^4)*d^3*x + 2*(a*b^2*c^3 + 4*a^2*c
^4)*d^3)*e)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2
*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5
*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x), -1/3*(3*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3
+ 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5
- 8*a^2*b^3*c + 16*a^3*b*c^2)*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))*e^4 - 2*(16*c^6*d^4*x^3 + 24*b*c^5*d^4*x^2 + 6*(b^2*c^4 + 4*a*c^5)*d^4*x - (b^3*c^3 - 12*a*b*c^4)*
d^4 - (3*a^2*b^3*c - 20*a^3*b*c^2 + 4*(b^4*c^2 - 7*a*b^2*c^3 + 8*a^2*c^4)*x^3 + 3*(b^5*c - 6*a*b^3*c^2)*x^2 +
6*(a*b^4*c - 7*a^2*b^2*c^2 + 4*a^3*c^3)*x)*e^4 - 4*(24*a^2*b*c^3*d*x + 16*a^3*c^3*d - (b^3*c^3 - 12*a*b*c^4)*d
*x^3 + 6*(a*b^2*c^3 + 4*a^2*c^4)*d*x^2)*e^3 + 6*(12*a*b^2*c^3*d^2*x + 8*a^2*b*c^3*d^2 + 2*(b^2*c^4 + 4*a*c^5)*
d^2*x^3 + 3*(b^3*c^3 + 4*a*b*c^4)*d^2*x^2)*e^2 - 4*(8*b*c^5*d^3*x^3 + 12*b^2*c^4*d^3*x^2 + 3*(b^3*c^3 + 4*a*b*
c^4)*d^3*x + 2*(a*b^2*c^3 + 4*a^2*c^4)*d^3)*e)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^
5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b
^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x)]

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Sympy [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((e*x+d)**4/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [A]
time = 0.64, size = 549, normalized size = 1.86 \begin {gather*} \frac {2 \, {\left ({\left ({\left (\frac {4 \, {\left (4 \, c^{5} d^{4} - 8 \, b c^{4} d^{3} e + 3 \, b^{2} c^{3} d^{2} e^{2} + 12 \, a c^{4} d^{2} e^{2} + b^{3} c^{2} d e^{3} - 12 \, a b c^{3} d e^{3} - b^{4} c e^{4} + 7 \, a b^{2} c^{2} e^{4} - 8 \, a^{2} c^{3} e^{4}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac {3 \, {\left (8 \, b c^{4} d^{4} - 16 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} + 24 \, a b c^{3} d^{2} e^{2} - 8 \, a b^{2} c^{2} d e^{3} - 32 \, a^{2} c^{3} d e^{3} - b^{5} e^{4} + 6 \, a b^{3} c e^{4}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac {6 \, {\left (b^{2} c^{3} d^{4} + 4 \, a c^{4} d^{4} - 2 \, b^{3} c^{2} d^{3} e - 8 \, a b c^{3} d^{3} e + 12 \, a b^{2} c^{2} d^{2} e^{2} - 16 \, a^{2} b c^{2} d e^{3} - a b^{4} e^{4} + 7 \, a^{2} b^{2} c e^{4} - 4 \, a^{3} c^{2} e^{4}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac {b^{3} c^{2} d^{4} - 12 \, a b c^{3} d^{4} + 8 \, a b^{2} c^{2} d^{3} e + 32 \, a^{2} c^{3} d^{3} e - 48 \, a^{2} b c^{2} d^{2} e^{2} + 64 \, a^{3} c^{2} d e^{3} + 3 \, a^{2} b^{3} e^{4} - 20 \, a^{3} b c e^{4}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} - \frac {e^{4} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

2/3*(((4*(4*c^5*d^4 - 8*b*c^4*d^3*e + 3*b^2*c^3*d^2*e^2 + 12*a*c^4*d^2*e^2 + b^3*c^2*d*e^3 - 12*a*b*c^3*d*e^3
- b^4*c*e^4 + 7*a*b^2*c^2*e^4 - 8*a^2*c^3*e^4)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 3*(8*b*c^4*d^4 - 16*b^
2*c^3*d^3*e + 6*b^3*c^2*d^2*e^2 + 24*a*b*c^3*d^2*e^2 - 8*a*b^2*c^2*d*e^3 - 32*a^2*c^3*d*e^3 - b^5*e^4 + 6*a*b^
3*c*e^4)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + 6*(b^2*c^3*d^4 + 4*a*c^4*d^4 - 2*b^3*c^2*d^3*e - 8*a*b*c^3*
d^3*e + 12*a*b^2*c^2*d^2*e^2 - 16*a^2*b*c^2*d*e^3 - a*b^4*e^4 + 7*a^2*b^2*c*e^4 - 4*a^3*c^2*e^4)/(b^4*c^2 - 8*
a*b^2*c^3 + 16*a^2*c^4))*x - (b^3*c^2*d^4 - 12*a*b*c^3*d^4 + 8*a*b^2*c^2*d^3*e + 32*a^2*c^3*d^3*e - 48*a^2*b*c
^2*d^2*e^2 + 64*a^3*c^2*d*e^3 + 3*a^2*b^3*e^4 - 20*a^3*b*c*e^4)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 +
b*x + a)^(3/2) - e^4*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4/(a + b*x + c*x^2)^(5/2),x)

[Out]

int((d + e*x)^4/(a + b*x + c*x^2)^(5/2), x)

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