3.2302 \(\int \frac{(d+e x)^{7/2}}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=751 \[ \frac{\sqrt{d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )+b^2 \left (-\left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (2 c^2 e^2 \left (-2 b d \left (9 d \sqrt{b^2-4 a c}+38 a e\right )+4 a e \left (4 d \sqrt{b^2-4 a c}+5 a e\right )+53 b^2 d^2\right )-8 c^3 d^2 e \left (-3 d \sqrt{b^2-4 a c}-19 a e+24 b d\right )-2 b c e^3 \left (-5 b d \sqrt{b^2-4 a c}+8 a e \sqrt{b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (b-\sqrt{b^2-4 a c}\right )+96 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\left (2 c^2 e^2 \left (2 b d \left (9 d \sqrt{b^2-4 a c}-38 a e\right )-4 a e \left (4 d \sqrt{b^2-4 a c}-5 a e\right )+53 b^2 d^2\right )-8 c^3 d^2 e \left (3 d \sqrt{b^2-4 a c}-19 a e+24 b d\right )-2 b c e^3 \left (5 b d \sqrt{b^2-4 a c}-8 a e \sqrt{b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (\sqrt{b^2-4 a c}+b\right )+96 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

-((d + e*x)^(5/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (Sqrt[d + e*x]*(12*
b*c*d*(c*d^2 + 3*a*e^2) - 4*a*c*e*(7*c*d^2 + 5*a*e^2) - b^2*(11*c*d^2*e + a*e^3) + (2*c*d - b*e)*(12*c^2*d^2 +
b^2*e^2 - 4*c*e*(3*b*d - 2*a*e))*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - ((96*c^4*d^4 - b^3*(b - Sqrt[b
^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(24*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 19*a*e) - 2*b*c*e^3*(5*b^2*d - 5*b*Sqrt[b^2 -
4*a*c]*d - 9*a*b*e + 8*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(53*b^2*d^2 + 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d + 5*a*e)
- 2*b*d*(9*Sqrt[b^2 - 4*a*c]*d + 38*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2
- 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + ((96*c^4*d^4
- b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(24*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 19*a*e) - 2*b*c*e^3*(5*b^2*
d + 5*b*Sqrt[b^2 - 4*a*c]*d - 9*a*b*e - 8*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(53*b^2*d^2 + 2*b*d*(9*Sqrt[b^2 -
4*a*c]*d - 38*a*e) - 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d - 5*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c
*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*
e])

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Rubi [A]  time = 13.5123, antiderivative size = 751, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {738, 818, 826, 1166, 208} \[ \frac{\sqrt{d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )+b^2 \left (-\left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (2 c^2 e^2 \left (-2 b d \left (9 d \sqrt{b^2-4 a c}+38 a e\right )+4 a e \left (4 d \sqrt{b^2-4 a c}+5 a e\right )+53 b^2 d^2\right )-8 c^3 d^2 e \left (-3 d \sqrt{b^2-4 a c}-19 a e+24 b d\right )-2 b c e^3 \left (-5 b d \sqrt{b^2-4 a c}+8 a e \sqrt{b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (b-\sqrt{b^2-4 a c}\right )+96 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\left (2 c^2 e^2 \left (2 b d \left (9 d \sqrt{b^2-4 a c}-38 a e\right )-4 a e \left (4 d \sqrt{b^2-4 a c}-5 a e\right )+53 b^2 d^2\right )-8 c^3 d^2 e \left (3 d \sqrt{b^2-4 a c}-19 a e+24 b d\right )-2 b c e^3 \left (5 b d \sqrt{b^2-4 a c}-8 a e \sqrt{b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (\sqrt{b^2-4 a c}+b\right )+96 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]

[Out]

-((d + e*x)^(5/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (Sqrt[d + e*x]*(12*
b*c*d*(c*d^2 + 3*a*e^2) - 4*a*c*e*(7*c*d^2 + 5*a*e^2) - b^2*(11*c*d^2*e + a*e^3) + (2*c*d - b*e)*(12*c^2*d^2 +
b^2*e^2 - 4*c*e*(3*b*d - 2*a*e))*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - ((96*c^4*d^4 - b^3*(b - Sqrt[b
^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(24*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 19*a*e) - 2*b*c*e^3*(5*b^2*d - 5*b*Sqrt[b^2 -
4*a*c]*d - 9*a*b*e + 8*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(53*b^2*d^2 + 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d + 5*a*e)
- 2*b*d*(9*Sqrt[b^2 - 4*a*c]*d + 38*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2
- 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + ((96*c^4*d^4
- b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(24*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 19*a*e) - 2*b*c*e^3*(5*b^2*
d + 5*b*Sqrt[b^2 - 4*a*c]*d - 9*a*b*e - 8*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(53*b^2*d^2 + 2*b*d*(9*Sqrt[b^2 -
4*a*c]*d - 38*a*e) - 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d - 5*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c
*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*
e])

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 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*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]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{(d+e x)^{3/2} \left (\frac{1}{2} \left (12 c d^2-11 b d e+10 a e^2\right )+\frac{1}{2} e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\int \frac{\frac{1}{4} \left (-48 c^3 d^4-a b^2 e^4+4 c^2 d^2 e (21 b d-19 a e)-5 c e^2 \left (7 b^2 d^2-12 a b d e+4 a^2 e^2\right )\right )-\frac{1}{4} e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x}{\sqrt{d+e x} \left (a+b x+c x^2\right )} \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} d e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right )+\frac{1}{4} e \left (-48 c^3 d^4-a b^2 e^4+4 c^2 d^2 e (21 b d-19 a e)-5 c e^2 \left (7 b^2 d^2-12 a b d e+4 a^2 e^2\right )\right )-\frac{1}{4} e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{c \left (b^2-4 a c\right )^2}\\ &=-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (96 c^4 d^4-b^3 \left (b+\sqrt{b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d+3 \sqrt{b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d+5 b \sqrt{b^2-4 a c} d-9 a b e-8 a \sqrt{b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+2 b d \left (9 \sqrt{b^2-4 a c} d-38 a e\right )-4 a e \left (4 \sqrt{b^2-4 a c} d-5 a e\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \sqrt{b^2-4 a c} e+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{8 c \left (b^2-4 a c\right )^{5/2}}+\frac{\left (96 c^4 d^4-b^3 \left (b-\sqrt{b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d-3 \sqrt{b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d-5 b \sqrt{b^2-4 a c} d-9 a b e+8 a \sqrt{b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+4 a e \left (4 \sqrt{b^2-4 a c} d+5 a e\right )-2 b d \left (9 \sqrt{b^2-4 a c} d+38 a e\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \sqrt{b^2-4 a c} e+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{8 c \left (b^2-4 a c\right )^{5/2}}\\ &=-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (96 c^4 d^4-b^3 \left (b-\sqrt{b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d-3 \sqrt{b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d-5 b \sqrt{b^2-4 a c} d-9 a b e+8 a \sqrt{b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+4 a e \left (4 \sqrt{b^2-4 a c} d+5 a e\right )-2 b d \left (9 \sqrt{b^2-4 a c} d+38 a e\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}+\frac{\left (96 c^4 d^4-b^3 \left (b+\sqrt{b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d+3 \sqrt{b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d+5 b \sqrt{b^2-4 a c} d-9 a b e-8 a \sqrt{b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+2 b d \left (9 \sqrt{b^2-4 a c} d-38 a e\right )-4 a e \left (4 \sqrt{b^2-4 a c} d-5 a e\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\\ \end{align*}

Mathematica [B]  time = 6.69539, size = 17950, normalized size = 23.9 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]

[Out]

Result too large to show

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Maple [B]  time = 0.31, size = 5849, normalized size = 7.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x + a)^3, x)

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Fricas [B]  time = 15.9702, size = 19941, normalized size = 26.55 \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)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

-1/8*(sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^
2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16
*a^3*b*c^3)*x)*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4
+ 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 -
880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c +
280*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a
^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 +
42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^
6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 64
0*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*log(1/2*sqrt(1/2)*(504*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*
c^6 - 64*a^3*c^7)*d^4*e^6 - 1008*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^7 + 3*(167*b^8
*c^2 - 1664*a*b^6*c^3 + 3936*a^2*b^4*c^4 + 5632*a^3*b^2*c^5 - 21760*a^4*c^6)*d^2*e^8 + 3*(b^9*c - 352*a*b^7*c^
2 + 4128*a^2*b^5*c^3 - 16384*a^3*b^3*c^4 + 21760*a^4*b*c^5)*d*e^9 - (b^10 - 17*a*b^8*c - 392*a^2*b^6*c^2 + 569
6*a^3*b^4*c^3 - 23680*a^4*b^2*c^4 + 32000*a^5*c^5)*e^10 - (96*(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640
*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*d^3 - 144*(b^11*c^5 - 20*a*b^9*c^6 + 160*a^2*b^7*c^7 - 640*a
^3*b^5*c^8 + 1280*a^4*b^3*c^9 - 1024*a^5*b*c^10)*d^2*e + 2*(23*b^12*c^4 - 408*a*b^10*c^5 + 2640*a^2*b^8*c^6 -
6400*a^3*b^6*c^7 - 3840*a^4*b^4*c^8 + 43008*a^5*b^2*c^9 - 53248*a^6*c^10)*d*e^2 + (b^13*c^3 - 72*a*b^11*c^4 +
1200*a^2*b^9*c^5 - 8960*a^3*b^7*c^6 + 34560*a^4*b^5*c^7 - 67584*a^5*b^3*c^8 + 53248*a^6*b*c^9)*e^3)*sqrt((441*
c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^
4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^
2*c^10 - 1024*a^5*c^11)))*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*
(7*b^3*c^4 + 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a
*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 3
5*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c
^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)
*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 +
160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b
^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) + (48384*c^7*d^8*e^5 - 193536*b*c^6*d^7*e^6 + 432
*(683*b^2*c^5 + 404*a*c^6)*d^6*e^7 - 432*(481*b^3*c^4 + 1212*a*b*c^5)*d^5*e^8 + 9*(6841*b^4*c^3 + 60712*a*b^2*
c^4 + 24016*a^2*c^5)*d^4*e^9 - 18*(145*b^5*c^2 + 12232*a*b^3*c^3 + 24016*a^2*b*c^4)*d^3*e^10 - 2*(518*b^6*c -
10131*a*b^4*c^2 - 124608*a^2*b^2*c^3 - 50000*a^3*c^4)*d^2*e^11 - (35*b^7 - 2562*a*b^5*c + 33072*a^2*b^3*c^2 +
100000*a^3*b*c^3)*d*e^12 + (35*a*b^6 - 1491*a^2*b^4*c + 15000*a^3*b^2*c^2 + 10000*a^4*c^3)*e^13)*sqrt(e*x + d)
) - sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2
- 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a
^3*b*c^3)*x)*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4 +
20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 - 88
0*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 2
80*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4
*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 4
2*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*
c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*
a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*log(-1/2*sqrt(1/2)*(504*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c
^6 - 64*a^3*c^7)*d^4*e^6 - 1008*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^7 + 3*(167*b^8*
c^2 - 1664*a*b^6*c^3 + 3936*a^2*b^4*c^4 + 5632*a^3*b^2*c^5 - 21760*a^4*c^6)*d^2*e^8 + 3*(b^9*c - 352*a*b^7*c^2
+ 4128*a^2*b^5*c^3 - 16384*a^3*b^3*c^4 + 21760*a^4*b*c^5)*d*e^9 - (b^10 - 17*a*b^8*c - 392*a^2*b^6*c^2 + 5696
*a^3*b^4*c^3 - 23680*a^4*b^2*c^4 + 32000*a^5*c^5)*e^10 - (96*(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*
a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*d^3 - 144*(b^11*c^5 - 20*a*b^9*c^6 + 160*a^2*b^7*c^7 - 640*a^
3*b^5*c^8 + 1280*a^4*b^3*c^9 - 1024*a^5*b*c^10)*d^2*e + 2*(23*b^12*c^4 - 408*a*b^10*c^5 + 2640*a^2*b^8*c^6 - 6
400*a^3*b^6*c^7 - 3840*a^4*b^4*c^8 + 43008*a^5*b^2*c^9 - 53248*a^6*c^10)*d*e^2 + (b^13*c^3 - 72*a*b^11*c^4 + 1
200*a^2*b^9*c^5 - 8960*a^3*b^7*c^6 + 34560*a^4*b^5*c^7 - 67584*a^5*b^3*c^8 + 53248*a^6*b*c^9)*e^3)*sqrt((441*c
^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4
- 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2
*c^10 - 1024*a^5*c^11)))*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(
7*b^3*c^4 + 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a*
b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35
*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^
6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*
d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 +
160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^
6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) + (48384*c^7*d^8*e^5 - 193536*b*c^6*d^7*e^6 + 432*
(683*b^2*c^5 + 404*a*c^6)*d^6*e^7 - 432*(481*b^3*c^4 + 1212*a*b*c^5)*d^5*e^8 + 9*(6841*b^4*c^3 + 60712*a*b^2*c
^4 + 24016*a^2*c^5)*d^4*e^9 - 18*(145*b^5*c^2 + 12232*a*b^3*c^3 + 24016*a^2*b*c^4)*d^3*e^10 - 2*(518*b^6*c - 1
0131*a*b^4*c^2 - 124608*a^2*b^2*c^3 - 50000*a^3*c^4)*d^2*e^11 - (35*b^7 - 2562*a*b^5*c + 33072*a^2*b^3*c^2 + 1
00000*a^3*b*c^3)*d*e^12 + (35*a*b^6 - 1491*a^2*b^4*c + 15000*a^3*b^2*c^2 + 10000*a^4*c^3)*e^13)*sqrt(e*x + d))
+ sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2 -
8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^
3*b*c^3)*x)*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4 + 2
0*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 - 880
*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 28
0*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 - (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*
b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42
*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c
^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a
^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*log(1/2*sqrt(1/2)*(504*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6
- 64*a^3*c^7)*d^4*e^6 - 1008*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^7 + 3*(167*b^8*c^
2 - 1664*a*b^6*c^3 + 3936*a^2*b^4*c^4 + 5632*a^3*b^2*c^5 - 21760*a^4*c^6)*d^2*e^8 + 3*(b^9*c - 352*a*b^7*c^2 +
4128*a^2*b^5*c^3 - 16384*a^3*b^3*c^4 + 21760*a^4*b*c^5)*d*e^9 - (b^10 - 17*a*b^8*c - 392*a^2*b^6*c^2 + 5696*a
^3*b^4*c^3 - 23680*a^4*b^2*c^4 + 32000*a^5*c^5)*e^10 + (96*(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^
3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*d^3 - 144*(b^11*c^5 - 20*a*b^9*c^6 + 160*a^2*b^7*c^7 - 640*a^3*
b^5*c^8 + 1280*a^4*b^3*c^9 - 1024*a^5*b*c^10)*d^2*e + 2*(23*b^12*c^4 - 408*a*b^10*c^5 + 2640*a^2*b^8*c^6 - 640
0*a^3*b^6*c^7 - 3840*a^4*b^4*c^8 + 43008*a^5*b^2*c^9 - 53248*a^6*c^10)*d*e^2 + (b^13*c^3 - 72*a*b^11*c^4 + 120
0*a^2*b^9*c^5 - 8960*a^3*b^7*c^6 + 34560*a^4*b^5*c^7 - 67584*a^5*b^3*c^8 + 53248*a^6*b*c^9)*e^3)*sqrt((441*c^4
*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 -
50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c
^10 - 1024*a^5*c^11)))*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*
b^3*c^4 + 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a*b^
3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a
*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 - (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6
+ 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^
2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 16
0*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*
c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) + (48384*c^7*d^8*e^5 - 193536*b*c^6*d^7*e^6 + 432*(6
83*b^2*c^5 + 404*a*c^6)*d^6*e^7 - 432*(481*b^3*c^4 + 1212*a*b*c^5)*d^5*e^8 + 9*(6841*b^4*c^3 + 60712*a*b^2*c^4
+ 24016*a^2*c^5)*d^4*e^9 - 18*(145*b^5*c^2 + 12232*a*b^3*c^3 + 24016*a^2*b*c^4)*d^3*e^10 - 2*(518*b^6*c - 101
31*a*b^4*c^2 - 124608*a^2*b^2*c^3 - 50000*a^3*c^4)*d^2*e^11 - (35*b^7 - 2562*a*b^5*c + 33072*a^2*b^3*c^2 + 100
000*a^3*b*c^3)*d*e^12 + (35*a*b^6 - 1491*a^2*b^4*c + 15000*a^3*b^2*c^2 + 10000*a^4*c^3)*e^13)*sqrt(e*x + d)) -
sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8
*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*
b*c^3)*x)*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4 + 20*
a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 - 880*a
^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 280*
a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 - (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^
2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(
b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8
- 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3
*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*log(-1/2*sqrt(1/2)*(504*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6
- 64*a^3*c^7)*d^4*e^6 - 1008*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^7 + 3*(167*b^8*c^2
- 1664*a*b^6*c^3 + 3936*a^2*b^4*c^4 + 5632*a^3*b^2*c^5 - 21760*a^4*c^6)*d^2*e^8 + 3*(b^9*c - 352*a*b^7*c^2 +
4128*a^2*b^5*c^3 - 16384*a^3*b^3*c^4 + 21760*a^4*b*c^5)*d*e^9 - (b^10 - 17*a*b^8*c - 392*a^2*b^6*c^2 + 5696*a^
3*b^4*c^3 - 23680*a^4*b^2*c^4 + 32000*a^5*c^5)*e^10 + (96*(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3
*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*d^3 - 144*(b^11*c^5 - 20*a*b^9*c^6 + 160*a^2*b^7*c^7 - 640*a^3*b
^5*c^8 + 1280*a^4*b^3*c^9 - 1024*a^5*b*c^10)*d^2*e + 2*(23*b^12*c^4 - 408*a*b^10*c^5 + 2640*a^2*b^8*c^6 - 6400
*a^3*b^6*c^7 - 3840*a^4*b^4*c^8 + 43008*a^5*b^2*c^9 - 53248*a^6*c^10)*d*e^2 + (b^13*c^3 - 72*a*b^11*c^4 + 1200
*a^2*b^9*c^5 - 8960*a^3*b^7*c^6 + 34560*a^4*b^5*c^7 - 67584*a^5*b^3*c^8 + 53248*a^6*b*c^9)*e^3)*sqrt((441*c^4*
d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 -
50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^
10 - 1024*a^5*c^11)))*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b
^3*c^4 + 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a*b^3
*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*
b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 - (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 +
1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2
*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160
*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c
^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) + (48384*c^7*d^8*e^5 - 193536*b*c^6*d^7*e^6 + 432*(68
3*b^2*c^5 + 404*a*c^6)*d^6*e^7 - 432*(481*b^3*c^4 + 1212*a*b*c^5)*d^5*e^8 + 9*(6841*b^4*c^3 + 60712*a*b^2*c^4
+ 24016*a^2*c^5)*d^4*e^9 - 18*(145*b^5*c^2 + 12232*a*b^3*c^3 + 24016*a^2*b*c^4)*d^3*e^10 - 2*(518*b^6*c - 1013
1*a*b^4*c^2 - 124608*a^2*b^2*c^3 - 50000*a^3*c^4)*d^2*e^11 - (35*b^7 - 2562*a*b^5*c + 33072*a^2*b^3*c^2 + 1000
00*a^3*b*c^3)*d*e^12 + (35*a*b^6 - 1491*a^2*b^4*c + 15000*a^3*b^2*c^2 + 10000*a^4*c^3)*e^13)*sqrt(e*x + d)) -
2*(36*a^2*b*c*d*e^2 - 2*(b^3*c - 10*a*b*c^2)*d^3 - (7*a*b^2*c + 44*a^2*c^2)*d^2*e - (a^2*b^2 + 20*a^3*c)*e^3 +
(24*c^4*d^3 - 36*b*c^3*d^2*e + 2*(5*b^2*c^2 + 16*a*c^3)*d*e^2 + (b^3*c - 16*a*b*c^2)*e^3)*x^3 + (36*b*c^3*d^3
- (55*b^2*c^2 - 4*a*c^3)*d^2*e + 4*(4*b^3*c + 11*a*b*c^2)*d*e^2 - (b^4 + 5*a*b^2*c + 36*a^2*c^2)*e^3)*x^2 + (
8*(b^2*c^2 + 5*a*c^3)*d^3 - (13*b^3*c + 56*a*b*c^2)*d^2*e + 2*(29*a*b^2*c - 8*a^2*c^2)*d*e^2 - 2*(a*b^3 + 14*a
^2*b*c)*e^3)*x)*sqrt(e*x + d))/(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*
x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a
^2*b^3*c^2 + 16*a^3*b*c^3)*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)**(7/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Giac [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)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

Timed out