### 3.2391 $$\int \frac{(d+e x)^5}{(a+b x+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=389 $-\frac{e \sqrt{a+b x+c x^2} \left (8 c^2 e^2 \left (-16 a^2 e^2-25 a b d e+b^2 d^2\right )+2 c e x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )+10 b^2 c e^3 (10 a e+3 b d)-16 c^3 d^2 e (7 b d-16 a e)-15 b^4 e^4+64 c^4 d^4\right )}{3 c^3 \left (b^2-4 a c\right )^2}-\frac{8 (d+e x)^2 \left (8 a^2 c e^3-x (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-2 b c d \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{7/2}}$

[Out]

(-2*(d + e*x)^4*(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)) - (8*(d + e*x)^2*(8
*a^2*c*e^3 - 2*b*c*d*(c*d^2 + 3*a*e^2) + b^2*(3*c*d^2*e - a*e^3) - (2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*
(b*d - 3*a*e))*x))/(3*c*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) - (e*(64*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(7
*b*d - 16*a*e) + 10*b^2*c*e^3*(3*b*d + 10*a*e) + 8*c^2*e^2*(b^2*d^2 - 25*a*b*d*e - 16*a^2*e^2) + 2*c*e*(2*c*d
- b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2*b*d - 7*a*e))*x)*Sqrt[a + b*x + c*x^2])/(3*c^3*(b^2 - 4*a*c)^2) + (5*
e^4*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(7/2))

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Rubi [A]  time = 0.465186, antiderivative size = 389, normalized size of antiderivative = 1., 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 = {738, 818, 779, 621, 206} $-\frac{e \sqrt{a+b x+c x^2} \left (8 c^2 e^2 \left (-16 a^2 e^2-25 a b d e+b^2 d^2\right )+2 c e x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )+10 b^2 c e^3 (10 a e+3 b d)-16 c^3 d^2 e (7 b d-16 a e)-15 b^4 e^4+64 c^4 d^4\right )}{3 c^3 \left (b^2-4 a c\right )^2}-\frac{8 (d+e x)^2 \left (8 a^2 c e^3-x (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-2 b c d \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(d + e*x)^4*(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)) - (8*(d + e*x)^2*(8
*a^2*c*e^3 - 2*b*c*d*(c*d^2 + 3*a*e^2) + b^2*(3*c*d^2*e - a*e^3) - (2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*
(b*d - 3*a*e))*x))/(3*c*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) - (e*(64*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(7
*b*d - 16*a*e) + 10*b^2*c*e^3*(3*b*d + 10*a*e) + 8*c^2*e^2*(b^2*d^2 - 25*a*b*d*e - 16*a^2*e^2) + 2*c*e*(2*c*d
- b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2*b*d - 7*a*e))*x)*Sqrt[a + b*x + c*x^2])/(3*c^3*(b^2 - 4*a*c)^2) + (5*
e^4*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(7/2))

Rule 738

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 818

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 + 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] && Ne
Q[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])

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

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 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])

Rubi steps

\begin{align*} \int \frac{(d+e x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (d+e x)^4 (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)^3 \left (2 \left (2 c d^2-3 b d e+4 a e^2\right )-2 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)^4 (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{8 (d+e x)^2 \left (8 a^2 c e^3-2 b c d \left (c d^2+3 a e^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 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{(d+e x) \left (e \left (b^3 d e^2+8 a c e \left (c d^2-4 a e^2\right )+4 b c d \left (2 c d^2+5 a e^2\right )-b^2 \left (14 c d^2 e-4 a e^3\right )\right )+e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{3 c \left (b^2-4 a c\right )^2}\\ &=-\frac{2 (d+e x)^4 (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{8 (d+e x)^2 \left (8 a^2 c e^3-2 b c d \left (c d^2+3 a e^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{e \left (64 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (7 b d-16 a e)+10 b^2 c e^3 (3 b d+10 a e)+8 c^2 e^2 \left (b^2 d^2-25 a b d e-16 a^2 e^2\right )+2 c e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}+\frac{\left (5 e^4 (2 c d-b e)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 c^3}\\ &=-\frac{2 (d+e x)^4 (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{8 (d+e x)^2 \left (8 a^2 c e^3-2 b c d \left (c d^2+3 a e^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{e \left (64 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (7 b d-16 a e)+10 b^2 c e^3 (3 b d+10 a e)+8 c^2 e^2 \left (b^2 d^2-25 a b d e-16 a^2 e^2\right )+2 c e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}+\frac{\left (5 e^4 (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 )}{c^3}\\ &=-\frac{2 (d+e x)^4 (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{8 (d+e x)^2 \left (8 a^2 c e^3-2 b c d \left (c d^2+3 a e^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{e \left (64 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (7 b d-16 a e)+10 b^2 c e^3 (3 b d+10 a e)+8 c^2 e^2 \left (b^2 d^2-25 a b d e-16 a^2 e^2\right )+2 c e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}+\frac{5 e^4 (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 c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.89061, size = 566, normalized size = 1.46 $\frac{-2 b^3 c \left (15 a^2 e^4 (d+7 e x)+2 a c e^4 x^2 (37 e x-45 d)+c^2 d^2 \left (15 d^2 e x+d^3-30 d e^2 x^2-10 e^3 x^3\right )\right )+4 b^2 c \left (3 a^2 c e^4 x (35 d+4 e x)-25 a^3 e^5+a c^2 e \left (-30 d^2 e^2 x^2+60 d^3 e x-5 d^4+70 d e^3 x^3-6 e^4 x^4\right )+c^3 d^3 x \left (3 d^2-30 d e x+10 e^2 x^2\right )\right )+b^4 e^4 \left (15 a^2 e-30 a c x (2 d+3 e x)+c^2 x^3 (3 e x-40 d)\right )+8 b c^2 \left (4 a^2 c e^2 \left (-15 d^2 e x+5 d^3+8 e^3 x^3\right )+a^3 e^4 (25 d+39 e x)+3 a c^2 d^2 \left (-5 d^2 e x+d^3+10 d e^2 x^2-10 e^3 x^3\right )+2 c^3 d^4 x^2 (3 d-5 e x)\right )+16 c^2 \left (a^2 c^2 e \left (-30 d^2 e^2 x^2-5 d^4-20 d e^3 x^3+3 e^4 x^4\right )+a^3 c e^3 \left (-20 d^2-15 d e x+12 e^2 x^2\right )+8 a^4 e^5+a c^3 d^3 x \left (3 d^2+10 e^2 x^2\right )+2 c^4 d^5 x^3\right )+10 b^5 e^4 x (3 a e+c x (2 e x-3 d))+15 b^6 e^5 x^2}{3 c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{2 c^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(15*b^6*e^5*x^2 + 10*b^5*e^4*x*(3*a*e + c*x*(-3*d + 2*e*x)) + b^4*e^4*(15*a^2*e + c^2*x^3*(-40*d + 3*e*x) - 30
*a*c*x*(2*d + 3*e*x)) - 2*b^3*c*(15*a^2*e^4*(d + 7*e*x) + 2*a*c*e^4*x^2*(-45*d + 37*e*x) + c^2*d^2*(d^3 + 15*d
^2*e*x - 30*d*e^2*x^2 - 10*e^3*x^3)) + 8*b*c^2*(2*c^3*d^4*x^2*(3*d - 5*e*x) + a^3*e^4*(25*d + 39*e*x) + 3*a*c^
2*d^2*(d^3 - 5*d^2*e*x + 10*d*e^2*x^2 - 10*e^3*x^3) + 4*a^2*c*e^2*(5*d^3 - 15*d^2*e*x + 8*e^3*x^3)) + 4*b^2*c*
(-25*a^3*e^5 + 3*a^2*c*e^4*x*(35*d + 4*e*x) + c^3*d^3*x*(3*d^2 - 30*d*e*x + 10*e^2*x^2) + a*c^2*e*(-5*d^4 + 60
*d^3*e*x - 30*d^2*e^2*x^2 + 70*d*e^3*x^3 - 6*e^4*x^4)) + 16*c^2*(8*a^4*e^5 + 2*c^4*d^5*x^3 + a*c^3*d^3*x*(3*d^
2 + 10*e^2*x^2) + a^3*c*e^3*(-20*d^2 - 15*d*e*x + 12*e^2*x^2) + a^2*c^2*e*(-5*d^4 - 30*d^2*e^2*x^2 - 20*d*e^3*
x^3 + 3*e^4*x^4)))/(3*c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) + (5*e^4*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(2*c^(7/2))

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Maple [B]  time = 0.058, size = 2395, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32/3*d^5*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+5/2*e^5*b/c^3*x/(c*x^2+b*x+a)^(1/2)+4*e^5*a/c^2*x^2/(c*x^2+b*
x+a)^(3/2)-10*d^2*e^3*b/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+20*d*e^4*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/
2)*x+5/2*d*e^4*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-5/3*d^4*e/c/(c*x^2+b*x+a)^(3/2)+2/3*d^5/(4*a*c-b^2)
/(c*x^2+b*x+a)^(3/2)*b+8/3*e^5*a^2/c^3/(c*x^2+b*x+a)^(3/2)+e^5*x^4/c/(c*x^2+b*x+a)^(3/2)-5/2*e^5*b/c^(7/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/96*e^5*b^4/c^5/(c*x^2+b*x+a)^(3/2)-5/4*e^5*b^2/c^4/(c*x^2+b*x+a)^(1
/2)+5*d*e^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/24*d*e^4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(
3/2)*x-5/3*d*e^4*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+4*e^5*a^2/c^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x
-19/12*e^5*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-80*d^2*e^3*b*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-40*d
^2*e^3*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+5/4*d*e^4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+10*d*e^4*
b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+160/3*d^3*e^2*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-80/3*d^4*e*b
*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-40/3*d^4*e*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+4/3*d^5/(4*a*c-b^2)/(c
*x^2+b*x+a)^(3/2)*x*c-10*d^2*e^3*x^2/c/(c*x^2+b*x+a)^(3/2)+5/12*d^2*e^3*b^2/c^3/(c*x^2+b*x+a)^(3/2)-20/3*d^2*e
^3*a/c^2/(c*x^2+b*x+a)^(3/2)-5*d^3*e^2*x/c/(c*x^2+b*x+a)^(3/2)+5/6*d^3*e^2*b/c^2/(c*x^2+b*x+a)^(3/2)-5/3*d*e^4
*x^3/c/(c*x^2+b*x+a)^(3/2)-5/48*d*e^4*b^3/c^4/(c*x^2+b*x+a)^(3/2)-5*d*e^4/c^2*x/(c*x^2+b*x+a)^(1/2)+5/2*d*e^4/
c^3*b/(c*x^2+b*x+a)^(1/2)+5/6*e^5*b/c^2*x^3/(c*x^2+b*x+a)^(3/2)-5/4*e^5*b^2/c^3*x^2/(c*x^2+b*x+a)^(3/2)+16/3*d
^5*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b-5/4*e^5*b^4/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+5/3*d^3*e^2*b^2/c/(4*
a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+10/3*d^3*e^2*a/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b-38/3*e^5*b^3/c^2*a/(4*a*c-b^
2)^2/(c*x^2+b*x+a)^(1/2)*x+32*e^5*a^2/c*b/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+5*d*e^4/c^2*b^2/(4*a*c-b^2)/(c*x
^2+b*x+a)^(1/2)*x+5/6*d^2*e^3*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+20/3*d^2*e^3*b^3/c/(4*a*c-b^2)^2/(c*x^
2+b*x+a)^(1/2)*x-5*d^2*e^3*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-5/16*e^5*b^3/c^4*x/(c*x^2+b*x+a)^(3/2)+5/
96*e^5*b^6/c^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+5/12*e^5*b^6/c^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-e^5*b^2/c^4*
a/(c*x^2+b*x+a)^(3/2)+5/6*d^3*e^2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+40/3*d^3*e^2*b^2/(4*a*c-b^2)^2/(c*x^
2+b*x+a)^(1/2)*x+20/3*d^3*e^2*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+20/3*d^3*e^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)
^(3/2)*x-5/6*d*e^4*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+5/3*d*e^4*b/c^3*a/(c*x^2+b*x+a)^(3/2)+80/3*d^3*e^
2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b-10/3*d^4*e*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-5/3*d^4*e*b^2/c/(4*a*c-
b^2)/(c*x^2+b*x+a)^(3/2)+5/2*d*e^4*b/c^2*x^2/(c*x^2+b*x+a)^(3/2)+5/8*d*e^4*b^2/c^3*x/(c*x^2+b*x+a)^(3/2)-5/48*
d*e^4*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+5/2*d*e^4/c^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-5/2*d^2*e^3*b/
c^2*x/(c*x^2+b*x+a)^(3/2)+5/12*d^2*e^3*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+10/3*d^2*e^3*b^4/c^2/(4*a*c-b^2
)^2/(c*x^2+b*x+a)^(1/2)-19/24*e^5*b^4/c^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-19/3*e^5*b^4/c^3*a/(4*a*c-b^2)^2/(
c*x^2+b*x+a)^(1/2)-5/2*e^5*b^3/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+16*e^5*a^2/c^2*b^2/(4*a*c-b^2)^2/(c*x^2+b
*x+a)^(1/2)+e^5*a/c^3*b*x/(c*x^2+b*x+a)^(3/2)+5/6*e^5*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+2*e^5*a^2/c^
3*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+5/48*e^5*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 15.039, size = 4651, normalized size = 11.96 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(15*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^5 + (2
*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)*x^4 + 2*(2*(b^5*c^2
- 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 - (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^3 + (2*(b^6*c - 6*a*b^4*c^
2 + 32*a^3*c^4)*d*e^4 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c
^3)*d*e^4 - (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^5)*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) - 4*(160*a^2*b*c^4*d^3*e^2 - 320*a^3*c^4*d^2*e^3 + 3*(b^4*c^3 - 8*a
*b^2*c^4 + 16*a^2*c^5)*e^5*x^4 - 2*(b^3*c^4 - 12*a*b*c^5)*d^5 - 20*(a*b^2*c^4 + 4*a^2*c^5)*d^4*e - 10*(3*a^2*b
^3*c^2 - 20*a^3*b*c^3)*d*e^4 + (15*a^2*b^4*c - 100*a^3*b^2*c^2 + 128*a^4*c^3)*e^5 + 4*(8*c^7*d^5 - 20*b*c^6*d^
4*e + 10*(b^2*c^5 + 4*a*c^6)*d^3*e^2 + 5*(b^3*c^4 - 12*a*b*c^5)*d^2*e^3 - 10*(b^4*c^3 - 7*a*b^2*c^4 + 8*a^2*c^
5)*d*e^4 + (5*b^5*c^2 - 37*a*b^3*c^3 + 64*a^2*b*c^4)*e^5)*x^3 + 3*(16*b*c^6*d^5 - 40*b^2*c^5*d^4*e + 20*(b^3*c
^4 + 4*a*b*c^5)*d^3*e^2 - 40*(a*b^2*c^4 + 4*a^2*c^5)*d^2*e^3 - 10*(b^5*c^2 - 6*a*b^3*c^3)*d*e^4 + (5*b^6*c - 3
0*a*b^4*c^2 + 16*a^2*b^2*c^3 + 64*a^3*c^4)*e^5)*x^2 + 6*(40*a*b^2*c^4*d^3*e^2 - 80*a^2*b*c^4*d^2*e^3 + 2*(b^2*
c^5 + 4*a*c^6)*d^5 - 5*(b^3*c^4 + 4*a*b*c^5)*d^4*e - 10*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 4*a^3*c^4)*d*e^4 + (5*a*b
^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^3)*e^5)*x)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c^6
+ (b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*x^4 + 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*x^3 + (b^6*c^4 - 6*a*b^
4*c^5 + 32*a^3*c^7)*x^2 + 2*(a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*x), -1/6*(15*(2*(a^2*b^4*c - 8*a^3*b^2*
c^2 + 16*a^4*c^3)*d*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^5 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)
*d*e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)*x^4 + 2*(2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 -
(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d*e^4 - (b^7 - 6*a*b^
5*c + 32*a^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 - (a*b^6 - 8*a^2*b^4*c + 16
*a^3*b^2*c^2)*e^5)*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*(160*a^2*b*c^4*d^3*e^2 - 320*a^3*c^4*d^2*e^3 + 3*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e^5*x^4 - 2*(b^3*c^4
- 12*a*b*c^5)*d^5 - 20*(a*b^2*c^4 + 4*a^2*c^5)*d^4*e - 10*(3*a^2*b^3*c^2 - 20*a^3*b*c^3)*d*e^4 + (15*a^2*b^4*
c - 100*a^3*b^2*c^2 + 128*a^4*c^3)*e^5 + 4*(8*c^7*d^5 - 20*b*c^6*d^4*e + 10*(b^2*c^5 + 4*a*c^6)*d^3*e^2 + 5*(b
^3*c^4 - 12*a*b*c^5)*d^2*e^3 - 10*(b^4*c^3 - 7*a*b^2*c^4 + 8*a^2*c^5)*d*e^4 + (5*b^5*c^2 - 37*a*b^3*c^3 + 64*a
^2*b*c^4)*e^5)*x^3 + 3*(16*b*c^6*d^5 - 40*b^2*c^5*d^4*e + 20*(b^3*c^4 + 4*a*b*c^5)*d^3*e^2 - 40*(a*b^2*c^4 + 4
*a^2*c^5)*d^2*e^3 - 10*(b^5*c^2 - 6*a*b^3*c^3)*d*e^4 + (5*b^6*c - 30*a*b^4*c^2 + 16*a^2*b^2*c^3 + 64*a^3*c^4)*
e^5)*x^2 + 6*(40*a*b^2*c^4*d^3*e^2 - 80*a^2*b*c^4*d^2*e^3 + 2*(b^2*c^5 + 4*a*c^6)*d^5 - 5*(b^3*c^4 + 4*a*b*c^5
)*d^4*e - 10*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 4*a^3*c^4)*d*e^4 + (5*a*b^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^3)*e^5)*
x)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c^6 + (b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*x^4
+ 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*x^3 + (b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*x^2 + 2*(a*b^5*c^4 - 8*
a^2*b^3*c^5 + 16*a^3*b*c^6)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.18102, size = 1064, normalized size = 2.74 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((((3*(b^4*c^2*e^5 - 8*a*b^2*c^3*e^5 + 16*a^2*c^4*e^5)*x/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5) + 4*(8*c^6*d
^5 - 20*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 + 40*a*c^5*d^3*e^2 + 5*b^3*c^3*d^2*e^3 - 60*a*b*c^4*d^2*e^3 - 10*b^4*
c^2*d*e^4 + 70*a*b^2*c^3*d*e^4 - 80*a^2*c^4*d*e^4 + 5*b^5*c*e^5 - 37*a*b^3*c^2*e^5 + 64*a^2*b*c^3*e^5)/(b^4*c^
3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x + 3*(16*b*c^5*d^5 - 40*b^2*c^4*d^4*e + 20*b^3*c^3*d^3*e^2 + 80*a*b*c^4*d^3*e^
2 - 40*a*b^2*c^3*d^2*e^3 - 160*a^2*c^4*d^2*e^3 - 10*b^5*c*d*e^4 + 60*a*b^3*c^2*d*e^4 + 5*b^6*e^5 - 30*a*b^4*c*
e^5 + 16*a^2*b^2*c^2*e^5 + 64*a^3*c^3*e^5)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x + 6*(2*b^2*c^4*d^5 + 8*a*c^
5*d^5 - 5*b^3*c^3*d^4*e - 20*a*b*c^4*d^4*e + 40*a*b^2*c^3*d^3*e^2 - 80*a^2*b*c^3*d^2*e^3 - 10*a*b^4*c*d*e^4 +
70*a^2*b^2*c^2*d*e^4 - 40*a^3*c^3*d*e^4 + 5*a*b^5*e^5 - 35*a^2*b^3*c*e^5 + 52*a^3*b*c^2*e^5)/(b^4*c^3 - 8*a*b^
2*c^4 + 16*a^2*c^5))*x - (2*b^3*c^3*d^5 - 24*a*b*c^4*d^5 + 20*a*b^2*c^3*d^4*e + 80*a^2*c^4*d^4*e - 160*a^2*b*c
^3*d^3*e^2 + 320*a^3*c^3*d^2*e^3 + 30*a^2*b^3*c*d*e^4 - 200*a^3*b*c^2*d*e^4 - 15*a^2*b^4*e^5 + 100*a^3*b^2*c*e
^5 - 128*a^4*c^2*e^5)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))/(c*x^2 + b*x + a)^(3/2) - 5/2*(2*c*d*e^4 - b*e^5)*
log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)