### 3.2303 $$\int \frac{(d+e x)^{5/2}}{(a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=577 $-\frac{3 \sqrt{d+e x} \left (-x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 b \left (a e^2+c d^2\right )+4 a c d e+3 b^2 d e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (-8 c^2 d e \left (-d \sqrt{b^2-4 a c}-3 a e+6 b d\right )+2 c e^2 \left (-4 b d \sqrt{b^2-4 a c}+2 a e \sqrt{b^2-4 a c}-6 a b e+9 b^2 d\right )-b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+32 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{3 \left (-8 c^2 d e \left (d \sqrt{b^2-4 a c}-3 a e+6 b d\right )+2 c e^2 \left (4 b d \sqrt{b^2-4 a c}-2 a e \sqrt{b^2-4 a c}-6 a b e+9 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+32 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}$

[Out]

-((d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*Sqrt[d + e*x]*(3
*b^2*d*e + 4*a*c*d*e - 4*b*(c*d^2 + a*e^2) - (8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x))/(4*(b^2 - 4*a*c)^
2*(a + b*x + c*x^2)) - (3*(32*c^3*d^3 - b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*c]
*d - 3*a*e) + 2*c*e^2*(9*b^2*d - 4*b*Sqrt[b^2 - 4*a*c]*d - 6*a*b*e + 2*a*Sqrt[b^2 - 4*a*c]*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]*Sqrt[c]*(b^2 - 4*a*c)^(5/2)*Sqrt
[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (3*(32*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(6*b*d + S
qrt[b^2 - 4*a*c]*d - 3*a*e) + 2*c*e^2*(9*b^2*d + 4*b*Sqrt[b^2 - 4*a*c]*d - 6*a*b*e - 2*a*Sqrt[b^2 - 4*a*c]*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]*Sqrt[c]*(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 = 5.05684, antiderivative size = 577, 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, 820, 826, 1166, 208} $-\frac{3 \sqrt{d+e x} \left (-x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 b \left (a e^2+c d^2\right )+4 a c d e+3 b^2 d e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (-8 c^2 d e \left (-d \sqrt{b^2-4 a c}-3 a e+6 b d\right )+2 c e^2 \left (-4 b d \sqrt{b^2-4 a c}+2 a e \sqrt{b^2-4 a c}-6 a b e+9 b^2 d\right )-b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+32 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{3 \left (-8 c^2 d e \left (d \sqrt{b^2-4 a c}-3 a e+6 b d\right )+2 c e^2 \left (4 b d \sqrt{b^2-4 a c}-2 a e \sqrt{b^2-4 a c}-6 a b e+9 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+32 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(5/2)/(a + b*x + c*x^2)^3,x]

[Out]

-((d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*Sqrt[d + e*x]*(3
*b^2*d*e + 4*a*c*d*e - 4*b*(c*d^2 + a*e^2) - (8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x))/(4*(b^2 - 4*a*c)^
2*(a + b*x + c*x^2)) - (3*(32*c^3*d^3 - b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*c]
*d - 3*a*e) + 2*c*e^2*(9*b^2*d - 4*b*Sqrt[b^2 - 4*a*c]*d - 6*a*b*e + 2*a*Sqrt[b^2 - 4*a*c]*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]*Sqrt[c]*(b^2 - 4*a*c)^(5/2)*Sqrt
[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (3*(32*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(6*b*d + S
qrt[b^2 - 4*a*c]*d - 3*a*e) + 2*c*e^2*(9*b^2*d + 4*b*Sqrt[b^2 - 4*a*c]*d - 6*a*b*e - 2*a*Sqrt[b^2 - 4*a*c]*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]*Sqrt[c]*(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 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*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, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])

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)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^{3/2} (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{\sqrt{d+e x} \left (\frac{3}{2} \left (4 c d^2-3 b d e+2 a e^2\right )+\frac{3}{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)^{3/2} (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{3 \sqrt{d+e x} \left (3 b^2 d e+4 a c d e-4 b \left (c d^2+a e^2\right )-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} \left (16 c^2 d^3+b e^2 (5 b d-4 a e)-4 c d e (5 b d-3 a e)\right )+\frac{3}{4} e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x}{\sqrt{d+e x} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2}\\ &=-\frac{(d+e x)^{3/2} (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{3 \sqrt{d+e x} \left (3 b^2 d e+4 a c d e-4 b \left (c d^2+a e^2\right )-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{4} e \left (16 c^2 d^3+b e^2 (5 b d-4 a e)-4 c d e (5 b d-3 a e)\right )-\frac{3}{4} d e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right )+\frac{3}{4} e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-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 )}{\left (b^2-4 a c\right )^2}\\ &=-\frac{(d+e x)^{3/2} (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{3 \sqrt{d+e x} \left (3 b^2 d e+4 a c d e-4 b \left (c d^2+a e^2\right )-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (3 \left (32 c^3 d^3-b^2 \left (b+\sqrt{b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d+\sqrt{b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d+4 b \sqrt{b^2-4 a c} d-6 a b e-2 a \sqrt{b^2-4 a c} 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 \left (b^2-4 a c\right )^{5/2}}+\frac{\left (3 \left (32 c^3 d^3-b^2 \left (b-\sqrt{b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d-\sqrt{b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d-4 b \sqrt{b^2-4 a c} d-6 a b e+2 a \sqrt{b^2-4 a c} 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 \left (b^2-4 a c\right )^{5/2}}\\ &=-\frac{(d+e x)^{3/2} (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{3 \sqrt{d+e x} \left (3 b^2 d e+4 a c d e-4 b \left (c d^2+a e^2\right )-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (32 c^3 d^3-b^2 \left (b-\sqrt{b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d-\sqrt{b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d-4 b \sqrt{b^2-4 a c} d-6 a b e+2 a \sqrt{b^2-4 a c} e\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} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}+\frac{3 \left (32 c^3 d^3-b^2 \left (b+\sqrt{b^2-4 a c}\right ) e^3-8 c^2 d e \left (6 b d+\sqrt{b^2-4 a c} d-3 a e\right )+2 c e^2 \left (9 b^2 d+4 b \sqrt{b^2-4 a c} d-6 a b e-2 a \sqrt{b^2-4 a c} e\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} \sqrt{c} \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 [A]  time = 5.05538, size = 506, normalized size = 0.88 $\frac{\sqrt{d+e x} \left (4 b \left (3 a^2 e^2+a c \left (5 d^2-9 d e x+4 e^2 x^2\right )+3 c^2 d x^2 (3 d-2 e x)\right )+4 c \left (-a^2 e (7 d+e x)+a c x \left (10 d^2+d e x+3 e^2 x^2\right )+6 c^2 d^2 x^3\right )+b^2 \left (a e (19 e x-5 d)+c x \left (8 d^2-37 d e x+3 e^2 x^2\right )\right )+b^3 \left (-2 d^2-9 d e x+5 e^2 x^2\right )\right )}{4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac{3 \sqrt{2 e \left (\sqrt{b^2-4 a c}-b\right )+4 c d} \left (4 c e \left (-d \sqrt{b^2-4 a c}+a e-4 b d\right )+b e^2 \left (2 \sqrt{b^2-4 a c}+3 b\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{8 \sqrt{c} \left (b^2-4 a c\right )^{5/2}}+\frac{3 \sqrt{4 c d-2 e \left (\sqrt{b^2-4 a c}+b\right )} \left (4 c e \left (d \sqrt{b^2-4 a c}+a e-4 b d\right )+b e^2 \left (3 b-2 \sqrt{b^2-4 a c}\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{8 \sqrt{c} \left (b^2-4 a c\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^3,x]

[Out]

(Sqrt[d + e*x]*(b^3*(-2*d^2 - 9*d*e*x + 5*e^2*x^2) + b^2*(a*e*(-5*d + 19*e*x) + c*x*(8*d^2 - 37*d*e*x + 3*e^2*
x^2)) + 4*c*(6*c^2*d^2*x^3 - a^2*e*(7*d + e*x) + a*c*x*(10*d^2 + d*e*x + 3*e^2*x^2)) + 4*b*(3*a^2*e^2 + 3*c^2*
d*x^2*(3*d - 2*e*x) + a*c*(5*d^2 - 9*d*e*x + 4*e^2*x^2))))/(4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (3*Sqrt[4
*c*d + 2*(-b + Sqrt[b^2 - 4*a*c])*e]*(16*c^2*d^2 + b*(3*b + 2*Sqrt[b^2 - 4*a*c])*e^2 + 4*c*e*(-4*b*d - Sqrt[b^
2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(8*Sqrt
[c]*(b^2 - 4*a*c)^(5/2)) + (3*Sqrt[4*c*d - 2*(b + Sqrt[b^2 - 4*a*c])*e]*(16*c^2*d^2 + b*(3*b - 2*Sqrt[b^2 - 4*
a*c])*e^2 + 4*c*e*(-4*b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (
b + Sqrt[b^2 - 4*a*c])*e]])/(8*Sqrt[c]*(b^2 - 4*a*c)^(5/2))

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Maple [B]  time = 0.287, size = 3925, normalized size = 6.8 \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)^(5/2)/(c*x^2+b*x+a)^3,x)

[Out]

-17*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)*a*b*c*d-9*e^3/(16*a^2*c^2-8*a*b^2
*c+b^4)*c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(
1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*d-27/4*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-e^
2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/(
(-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*d+18*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c
+b^4)*(e*x+d)^(1/2)*a*b*c*d^2-12*e/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d
+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1
/2))*d^3-3*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x
+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d-12*e/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-
e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)
/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^3+3*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((-b*e+2*c*d+
(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(
1/2))*b*d-27/4*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2)
)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*d+18*e^2/
(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)
*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d^2+9/2*e^4/(16*a^2*c^2-8*a*
b^2*c+b^4)*c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^
(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b+18*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-e
^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/
((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d^2+9/2*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-e^2*(4*a*c-b^2))
^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2
*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b-9*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e
-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))
*c)^(1/2))*a*d-12*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*b^2*c*d^3+15*e^2/(c
*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*b*c^2*d^4+3/8*e^4/(16*a^2*c^2-8*a*b^2*c+b^4
)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/
2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^3+3/8*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)/(-e^2*(4*a*c-b^2))^(
1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^
2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^3+3/2*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2)
)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a+3*e/(16*a^2
*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/
((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^2-3/2*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((-b*e+2*c*d+
(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(
1/2))*a-3*e/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*
x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^2-6*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^
2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(7/2)*b*d+4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e
*x+d)^(5/2)*a*b*c-8*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(5/2)*c^2*a*d-23/2*e^3/
(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(5/2)*b^2*c*d+27*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^
2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(5/2)*b*c^2*d^2+17*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+
b^4)*(e*x+d)^(3/2)*a*c^2*d^2+91/4*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)*b^2
*c*d^2-6*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*a^2*c*d-6*e^5/(c*e^2*x^2+b*e
^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*a*b^2*d-36*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-
8*a*b^2*c+b^4)*(e*x+d)^(3/2)*b*c^2*d^3-12*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(
1/2)*a*c^2*d^3+5/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(5/2)*b^3-19/4*e^4/(c*e^
2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)*b^3*d+18*e/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2
*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)*c^3*d^4+3*e^6/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)
^(1/2)*a^2*b+3*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*b^3*d^2-6*e/(c*e^2*x^2
+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*c^3*d^5+3/8*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*2^(1/2)/
((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(
1/2))*c)^(1/2))*b^2-3/8*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)
*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2+3*e^3/(c*e^2*x^2+b*e^2*x
+a*e^2)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(7/2)*a-e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c
+b^4)*(e*x+d)^(3/2)*a^2*c+19/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)*a*b^2+
3/4*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(7/2)*b^2+6*e/(c*e^2*x^2+b*e^2*x+a*e^
2)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(7/2)*d^2-18*e/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b
^4)*(e*x+d)^(5/2)*c^3*d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{5}{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)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(5/2)/(c*x^2 + b*x + a)^3, x)

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Fricas [B]  time = 4.53833, size = 11310, normalized size = 19.6 \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)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

-1/8*(3*sqrt(1/2)*(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((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*
e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + (b^10*c - 20*a*
b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*
c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*
b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6))*log(27*sqrt(1/2)*(4*(b^6*c - 12*a*b^4*c^2 + 48*a
^2*b^2*c^3 - 64*a^3*c^4)*d*e^6 - 2*(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^7 + sqrt(e^10/(b^10*c^
2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))*(16*(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)*d^2 - 16*(b^11*c^2 - 20*a*b^9*c
^3 + 160*a^2*b^7*c^4 - 640*a^3*b^5*c^5 + 1280*a^4*b^3*c^6 - 1024*a^5*b*c^7)*d*e + (3*b^12*c - 56*a*b^10*c^2 +
400*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 1280*a^4*b^4*c^5 + 2048*a^5*b^2*c^6 - 4096*a^6*c^7)*e^2))*sqrt((512*c^5*d
^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c
+ 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + (b^10*c - 20*a*b^8*c^2 + 160*a^2*
b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*
c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*
b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)) + 27*(256*c^4*d^4*e^5 - 512*b*c^3*d^3*e^6 + 48*(7*b^2*c^2 + 4*a*c^
3)*d^2*e^7 - 16*(5*b^3*c + 12*a*b*c^2)*d*e^8 + (5*b^4 + 40*a*b^2*c + 16*a^2*c^2)*e^9)*sqrt(e*x + d)) - 3*sqrt(
1/2)*(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((512*
c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b
^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + (b^10*c - 20*a*b^8*c^2 + 160
*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2
*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640
*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6))*log(-27*sqrt(1/2)*(4*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 -
64*a^3*c^4)*d*e^6 - 2*(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^7 + sqrt(e^10/(b^10*c^2 - 20*a*b^8
*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))*(16*(b^10*c^3 - 20*a*b^8*c^4 + 16
0*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)*d^2 - 16*(b^11*c^2 - 20*a*b^9*c^3 + 160*a^2
*b^7*c^4 - 640*a^3*b^5*c^5 + 1280*a^4*b^3*c^6 - 1024*a^5*b*c^7)*d*e + (3*b^12*c - 56*a*b^10*c^2 + 400*a^2*b^8*
c^3 - 1280*a^3*b^6*c^4 + 1280*a^4*b^4*c^5 + 2048*a^5*b^2*c^6 - 4096*a^6*c^7)*e^2))*sqrt((512*c^5*d^5 - 1280*b*
c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c
^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 64
0*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^
3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 12
80*a^4*b^2*c^5 - 1024*a^5*c^6)) + 27*(256*c^4*d^4*e^5 - 512*b*c^3*d^3*e^6 + 48*(7*b^2*c^2 + 4*a*c^3)*d^2*e^7 -
16*(5*b^3*c + 12*a*b*c^2)*d*e^8 + (5*b^4 + 40*a*b^2*c + 16*a^2*c^2)*e^9)*sqrt(e*x + d)) + 3*sqrt(1/2)*(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((512*c^5*d^5 - 12
80*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*
b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 - (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3
- 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 6
40*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4
+ 1280*a^4*b^2*c^5 - 1024*a^5*c^6))*log(27*sqrt(1/2)*(4*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*
d*e^6 - 2*(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^7 - sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^
2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))*(16*(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)*d^2 - 16*(b^11*c^2 - 20*a*b^9*c^3 + 160*a^2*b^7*c^4 - 64
0*a^3*b^5*c^5 + 1280*a^4*b^3*c^6 - 1024*a^5*b*c^7)*d*e + (3*b^12*c - 56*a*b^10*c^2 + 400*a^2*b^8*c^3 - 1280*a^
3*b^6*c^4 + 1280*a^4*b^4*c^5 + 2048*a^5*b^2*c^6 - 4096*a^6*c^7)*e^2))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 1
60*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c
^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 - (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4
+ 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1
280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^
5 - 1024*a^5*c^6)) + 27*(256*c^4*d^4*e^5 - 512*b*c^3*d^3*e^6 + 48*(7*b^2*c^2 + 4*a*c^3)*d^2*e^7 - 16*(5*b^3*c
+ 12*a*b*c^2)*d*e^8 + (5*b^4 + 40*a*b^2*c + 16*a^2*c^2)*e^9)*sqrt(e*x + d)) - 3*sqrt(1/2)*(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((512*c^5*d^5 - 1280*b*c^4*d^4*
e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*
a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 - (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^
4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^
5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b
^2*c^5 - 1024*a^5*c^6))*log(-27*sqrt(1/2)*(4*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^6 - 2*(b
^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^7 - sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 -
640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))*(16*(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)*d^2 - 16*(b^11*c^2 - 20*a*b^9*c^3 + 160*a^2*b^7*c^4 - 640*a^3*b^5*c^
5 + 1280*a^4*b^3*c^6 - 1024*a^5*b*c^7)*d*e + (3*b^12*c - 56*a*b^10*c^2 + 400*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 +
1280*a^4*b^4*c^5 + 2048*a^5*b^2*c^6 - 4096*a^6*c^7)*e^2))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^
3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 -
(b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 - (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*
b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*
c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5
*c^6)) + 27*(256*c^4*d^4*e^5 - 512*b*c^3*d^3*e^6 + 48*(7*b^2*c^2 + 4*a*c^3)*d^2*e^7 - 16*(5*b^3*c + 12*a*b*c^2
)*d*e^8 + (5*b^4 + 40*a*b^2*c + 16*a^2*c^2)*e^9)*sqrt(e*x + d)) - 2*(12*a^2*b*e^2 + 3*(8*c^3*d^2 - 8*b*c^2*d*e
+ (b^2*c + 4*a*c^2)*e^2)*x^3 - 2*(b^3 - 10*a*b*c)*d^2 - (5*a*b^2 + 28*a^2*c)*d*e + (36*b*c^2*d^2 - (37*b^2*c
- 4*a*c^2)*d*e + (5*b^3 + 16*a*b*c)*e^2)*x^2 + (8*(b^2*c + 5*a*c^2)*d^2 - 9*(b^3 + 4*a*b*c)*d*e + (19*a*b^2 -
4*a^2*c)*e^2)*x)*sqrt(e*x + d))/(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)

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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)**(5/2)/(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)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

Timed out