3.1623 \(\int \frac{(b+2 c x) (d+e x)^{7/2}}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=543 \[ \frac{7 e \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b d \sqrt{b^2-4 a c}+3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{7 e \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{7 e^2 \sqrt{d+e x} (2 c d-b e)}{4 c \left (b^2-4 a c\right )}-\frac{7 e (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

(7*e^2*(2*c*d - b*e)*Sqrt[d + e*x])/(4*c*(b^2 - 4*a*c)) - (d + e*x)^(7/2)/(2*(a + b*x + c*x^2)^2) - (7*e*(d +
e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (7*e*(8*c^3*d^3 + b^2*(b - S
qrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 2*c*e^2*(b^2*d - b*Sqrt[b^2 - 4*a*c]
*d - 4*a*b*e + 3*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]*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (7*e*(8*c^3*d^3
+ b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 2*c*e^2*(b^2*d + b*Sqrt
[b^2 - 4*a*c]*d - 4*a*b*e - 3*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]*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

________________________________________________________________________________________

Rubi [A]  time = 5.72279, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {768, 738, 824, 826, 1166, 208} \[ \frac{7 e \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b d \sqrt{b^2-4 a c}+3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{7 e \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{7 e^2 \sqrt{d+e x} (2 c d-b e)}{4 c \left (b^2-4 a c\right )}-\frac{7 e (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*e^2*(2*c*d - b*e)*Sqrt[d + e*x])/(4*c*(b^2 - 4*a*c)) - (d + e*x)^(7/2)/(2*(a + b*x + c*x^2)^2) - (7*e*(d +
e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (7*e*(8*c^3*d^3 + b^2*(b - S
qrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 2*c*e^2*(b^2*d - b*Sqrt[b^2 - 4*a*c]
*d - 4*a*b*e + 3*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]*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (7*e*(8*c^3*d^3
+ b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 2*c*e^2*(b^2*d + b*Sqrt
[b^2 - 4*a*c]*d - 4*a*b*e - 3*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]*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 768

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] - Dist[(e*g*m)/(2*c*(p + 1)), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]

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 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), 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] && FractionQ[m] && GtQ[m, 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{(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}+\frac{1}{4} (7 e) \int \frac{(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac{7 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(7 e) \int \frac{\sqrt{d+e x} \left (\frac{1}{2} \left (4 c d^2-5 b d e+6 a e^2\right )-\frac{1}{2} e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{4 \left (b^2-4 a c\right )}\\ &=\frac{7 e^2 (2 c d-b e) \sqrt{d+e x}}{4 c \left (b^2-4 a c\right )}-\frac{(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac{7 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(7 e) \int \frac{\frac{1}{2} \left (4 c^2 d^3-a b e^3-c d e (5 b d-8 a e)\right )+\frac{1}{2} e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x}{\sqrt{d+e x} \left (a+b x+c x^2\right )} \, dx}{4 c \left (b^2-4 a c\right )}\\ &=\frac{7 e^2 (2 c d-b e) \sqrt{d+e x}}{4 c \left (b^2-4 a c\right )}-\frac{(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac{7 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(7 e) \operatorname{Subst}\left (\int \frac{\frac{1}{2} e \left (4 c^2 d^3-a b e^3-c d e (5 b d-8 a e)\right )-\frac{1}{2} d e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )+\frac{1}{2} e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 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 )}{2 c \left (b^2-4 a c\right )}\\ &=\frac{7 e^2 (2 c d-b e) \sqrt{d+e x}}{4 c \left (b^2-4 a c\right )}-\frac{(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac{7 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (7 e \left (8 c^3 d^3+b^2 \left (b+\sqrt{b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt{b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt{b^2-4 a c} d-4 a b e-3 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 c \left (b^2-4 a c\right )^{3/2}}-\frac{\left (7 e \left (8 c^3 d^3+b^2 \left (b-\sqrt{b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt{b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+3 a \sqrt{b^2-4 a c} e-b \left (\sqrt{b^2-4 a c} d+4 a e\right )\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 )^{3/2}}\\ &=\frac{7 e^2 (2 c d-b e) \sqrt{d+e x}}{4 c \left (b^2-4 a c\right )}-\frac{(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac{7 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{7 e \left (8 c^3 d^3+b^2 \left (b-\sqrt{b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt{b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d-b \sqrt{b^2-4 a c} d-4 a b e+3 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} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}-\frac{7 e \left (8 c^3 d^3+b^2 \left (b+\sqrt{b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt{b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt{b^2-4 a c} d-4 a b e-3 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} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\\ \end{align*}

Mathematica [B]  time = 6.77418, size = 1403, normalized size = 2.58 \[ -\frac{\left (-2 a c (2 c d-b e)+b \left (-e b^2+c d b+2 a c e\right )+c (b (2 c d-b e)-2 c (b d-2 a e)) x\right ) (d+e x)^{9/2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac{-\frac{\left (-\frac{3}{2} a c \left (b^2-4 a c\right ) (2 c d-b e) e^2-\frac{1}{2} \left (b^2-4 a c\right ) (7 c d-5 b e) \left (-e b^2+c d b+2 a c e\right ) e+c \left (-\frac{3}{2} c \left (b^2-4 a c\right ) (b d-2 a e) e^2-\frac{1}{2} \left (b^2-4 a c\right ) (7 c d-5 b e) (2 c d-b e) e\right ) x\right ) (d+e x)^{9/2}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac{\frac{1}{2} \left (b^2-4 a c\right ) e^2 \left (14 c^2 d^2+5 b^2 e^2-2 c e (7 b d+3 a e)\right ) (d+e x)^{7/2}+\frac{2 \left (\frac{49}{4} c \left (b^2-4 a c\right ) e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right ) (d+e x)^{5/2}+\frac{2 \left (\frac{245}{4} c^2 \left (b^2-4 a c\right ) e^2 (d+e x)^{3/2} \left (c d^2-e (b d-a e)\right )^2+\frac{2 \left (\frac{735}{16} c^2 \left (b^2-4 a c\right ) e^2 (2 c d-b e) \sqrt{d+e x} \left (c d^2-e (b d-a e)\right )^2+\frac{4 \left (\frac{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e} \left (-\frac{735}{64} \left (b^2-4 a c\right ) e^2 \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2 c^3-\frac{\frac{735}{64} \left (b^2-4 a c\right ) e^2 (b e-2 c d) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2 c^3+2 \left (\frac{735}{64} c^3 \left (b^2-4 a c\right ) d e^2 \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2-\frac{735}{64} c^3 \left (b^2-4 a c\right ) e^2 \left (4 c^2 d^3-c e (5 b d-8 a e) d-a b e^3\right ) \left (c d^2-e (b d-a e)\right )^2\right ) c}{\sqrt{b^2-4 a c} e}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e}}\right )}{\sqrt{2} \sqrt{c} \left (-2 c d+b e+\sqrt{b^2-4 a c} e\right )}+\frac{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e} \left (\frac{\frac{735}{64} \left (b^2-4 a c\right ) e^2 (b e-2 c d) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2 c^3+2 \left (\frac{735}{64} c^3 \left (b^2-4 a c\right ) d e^2 \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2-\frac{735}{64} c^3 \left (b^2-4 a c\right ) e^2 \left (4 c^2 d^3-c e (5 b d-8 a e) d-a b e^3\right ) \left (c d^2-e (b d-a e)\right )^2\right ) c}{\sqrt{b^2-4 a c} e}-\frac{735}{64} c^3 \left (b^2-4 a c\right ) e^2 \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e}}\right )}{\sqrt{2} \sqrt{c} \left (-2 c d+b e-\sqrt{b^2-4 a c} e\right )}\right )}{c}\right )}{3 c}\right )}{5 c}\right )}{7 c}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d + e*x)^(9/2)*(-2*a*c*(2*c*d - b*e) + b*(b*c*d - b^2*e + 2*a*c*e) + c*(-2*c*(b*d - 2*a*e) + b*(2*c*d - b*e
))*x))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (-(((d + e*x)^(9/2)*((-3*a*c*(b^2 - 4*a
*c)*e^2*(2*c*d - b*e))/2 - ((b^2 - 4*a*c)*e*(7*c*d - 5*b*e)*(b*c*d - b^2*e + 2*a*c*e))/2 + c*((-3*c*(b^2 - 4*a
*c)*e^2*(b*d - 2*a*e))/2 - ((b^2 - 4*a*c)*e*(7*c*d - 5*b*e)*(2*c*d - b*e))/2)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*
e + a*e^2)*(a + b*x + c*x^2))) - (((b^2 - 4*a*c)*e^2*(14*c^2*d^2 + 5*b^2*e^2 - 2*c*e*(7*b*d + 3*a*e))*(d + e*x
)^(7/2))/2 + (2*((49*c*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))*(d + e*x)^(5/2))/4 + (2*((245*c
^2*(b^2 - 4*a*c)*e^2*(c*d^2 - e*(b*d - a*e))^2*(d + e*x)^(3/2))/4 + (2*((735*c^2*(b^2 - 4*a*c)*e^2*(2*c*d - b*
e)*(c*d^2 - e*(b*d - a*e))^2*Sqrt[d + e*x])/16 + (4*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*((-735*c^3*(b^2
- 4*a*c)*e^2*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 - ((735*c^3*(b^2 - 4*a*
c)*e^2*(-2*c*d + b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 + 2*c*((-735*c
^3*(b^2 - 4*a*c)*e^2*(4*c^2*d^3 - a*b*e^3 - c*d*e*(5*b*d - 8*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 + (735*c^3*(b
^2 - 4*a*c)*d*e^2*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64))/(Sqrt[b^2 - 4*a*
c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*
c*d + b*e + Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*((-735*c^3*(b^2 - 4*a*c)*e^2*(2*c
^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 + ((735*c^3*(b^2 - 4*a*c)*e^2*(-2*c*d +
b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 + 2*c*((-735*c^3*(b^2 - 4*a*c)*
e^2*(4*c^2*d^3 - a*b*e^3 - c*d*e*(5*b*d - 8*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 + (735*c^3*(b^2 - 4*a*c)*d*e^2
*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(S
qrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e - Sqrt[
b^2 - 4*a*c]*e))))/c))/(3*c)))/(5*c)))/(7*c))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(2*(b^2 - 4*a*c)*(c*d^2
- b*d*e + a*e^2))

________________________________________________________________________________________

Maple [B]  time = 0.061, size = 3503, normalized size = 6.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(5/2)*a*b-21/4*e^4/(4*a*c-b^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+21/4*e^4/(4*a*c-b^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-7*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1
/2)*b^2*d^3+14*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)*b^2*d^2-35/4*e^4/(c*e^2*x^2+b*e^2*x+a
*e^2)^2/(4*a*c-b^2)*(e*x+d)^(5/2)*b^2*d-11/2*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(7/2)*a*c+7/2
*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(7/2)*c^2*d^2+7/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-
b^2)/c*(e*x+d)^(5/2)*b^3-21/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*c^2*(e*x+d)^(5/2)*d^3+21/2*e^2/(c*e^
2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a*c-b^2)*(e*x+d)^(3/2)*d^4-7/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a*c-b^2)*(
e*x+d)^(1/2)*d^5-7/2*e^6/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*a^2*d+7/2*e^4/(c*e^2*x^2+b*e^2*
x+a*e^2)^2/(4*a*c-b^2)*c*(e*x+d)^(5/2)*a*d-7*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/(4*a*c-b^2)*(e*x+d)^(1/2)*a*d^3
+7/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/c/(4*a*c-b^2)*(e*x+d)^(1/2)*b^3*d^2-7/2*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/c
/(4*a*c-b^2)*(e*x+d)^(3/2)*b^3*d+7*e^5/(4*a*c-b^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*b+7/4
*e^3/(4*a*c-b^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+35/4*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/(4*a*c-b^2)*(e*x+d)^(1/
2)*b*d^4+7/2*e^6/(c*e^2*x^2+b*e^2*x+a*e^2)^2/c/(4*a*c-b^2)*(e*x+d)^(3/2)*a*b^2+7*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)
^2*c/(4*a*c-b^2)*(e*x+d)^(3/2)*a*d^2+7/8*e^4/(4*a*c-b^2)/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^2-21*e^3/(c*e^2*x^2+b*
e^2*x+a*e^2)^2*c/(4*a*c-b^2)*(e*x+d)^(3/2)*b*d^3+7/4*e^7/(c*e^2*x^2+b*e^2*x+a*e^2)^2/c/(4*a*c-b^2)*(e*x+d)^(1/
2)*a^2*b-7/2*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(7/2)*b*c*d-7/4*e^2/(4*a*c-b^2)*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))*d^2-7/4*e^4/(4*a*c-b^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^2*d-7/8*e^4/(4*a*c-
b^2)/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^2-7/4*e^3/(4*a*c-b^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+7/4*e^2/(4*a*c-b^2)*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))*d^2+63/4*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*c*(e*x+d)^(5/2)*b*d^2-14*e^4/(4*a*c-b^
2)*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*d-14*e^4/(4*a*c-b^2)*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*d+21/2*e^3/(4*a*c-b^2)*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*d^2+21/2*e^
3/(4*a*c-b^2)*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*d^2-7/2*e^6/(c*e^2*x^2+b*e^2*x+a*e^2)^2/
(4*a*c-b^2)*(e*x+d)^(3/2)*a^2+9/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(7/2)*b^2-7*e^2/(4*a*c-b
^2)*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))*d^3-7/2*e^6/(c*e^2*x^2+b*e^2*x+a*e^2)^2/c/(4*a*c-b^2)
*(e*x+d)^(1/2)*a*b^2*d-7/8*e^5/(4*a*c-b^2)/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^3-7/4*e^4/(
4*a*c-b^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^2*d-7*e^2/(4*a*c-b^2)*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))*d^3+7*e^5/(4*a*c-b^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*b-7/
8*e^5/(4*a*c-b^2)/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^3-7*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*
a*c-b^2)*(e*x+d)^(3/2)*a*b*d+21/2*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*a*b*d^2

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

[Out]

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

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Fricas [B]  time = 3.86029, size = 12097, normalized size = 22.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(7*sqrt(1/2)*(a^2*b^2*c - 4*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*x^3 + (b^4*c - 2*a
*b^2*c^2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*x)*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c
^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5
- 15*a*b^3*c + 60*a^2*b*c^2)*e^7 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^1
0 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c +
81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^
2*c^5 - 64*a^3*c^6))*log(343/2*sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^6 - 15*(b^5*c^2 - 8*a*
b^3*c^3 + 16*a^2*b*c^4)*d^2*e^7 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^8 + (b^7 - 17*a*b^
5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^9 - (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*
(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640
*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 +
10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^
8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 3
6*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^7 + (b^6
*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6
*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c
^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 343*(80*c^5*d^6*
e^6 - 240*b*c^4*d^5*e^7 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^8 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^9 - 6*(6*b^4*c -
47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^10 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^11 + (5*a*b^4 - 81*a^2*b^2*c
+ 324*a^3*c^2)*e^12)*sqrt(e*x + d)) - 7*sqrt(1/2)*(a^2*b^2*c - 4*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c
^2 - 4*a*b*c^3)*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*x)*sqrt((32*c^5*d^5*e^
2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b
^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^7 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5
- 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*
c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(
b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(-343/2*sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^
2*c^5)*d^3*e^6 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^7 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 +
96*a^3*c^4)*d*e^8 + (b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^9 - (8*(b^6*c^5 - 12*a*b^4*c^6 + 48
*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 2
4*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11
+ 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6
*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 +
12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15
*a*b^3*c + 60*a^2*b*c^2)*e^7 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 -
50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a
^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^
5 - 64*a^3*c^6)) + 343*(80*c^5*d^6*e^6 - 240*b*c^4*d^5*e^7 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^8 + 2*(b^3*c^2 -
404*a*b*c^3)*d^3*e^9 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^10 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^
2)*d*e^11 + (5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*e^12)*sqrt(e*x + d)) + 7*sqrt(1/2)*(a^2*b^2*c - 4*a^3*c^2 +
(b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c -
4*a^2*b*c^2)*x)*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*
a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^7 - (b^6*c
^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a
*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7
+ 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(343/2*sqrt(1/2)*
(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^6 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^7 + 3*(b^6*
c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^8 + (b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^9
+ (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6
- 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25
*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 1
8*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5*e^2 -
80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c
^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^7 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64
*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)
*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*
c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 343*(80*c^5*d^6*e^6 - 240*b*c^4*d^5*e^7 + (199*b^2*c^3 +
404*a*c^4)*d^4*e^8 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^9 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^10 - (
5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^11 + (5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*e^12)*sqrt(e*x + d)) - 7*
sqrt(1/2)*(a^2*b^2*c - 4*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*x^3 + (b^4*c - 2*a*b^2*c^
2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*x)*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12
*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*
b^3*c + 60*a^2*b*c^2)*e^7 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*
b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*
c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 -
64*a^3*c^6))*log(-343/2*sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^6 - 15*(b^5*c^2 - 8*a*b^3*c^
3 + 16*a^2*b*c^4)*d^2*e^7 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^8 + (b^7 - 17*a*b^5*c +
88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^9 + (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c
^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b
^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b
^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64
*a^3*c^9)))*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*
c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^7 - (b^6*c^3 -
12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3
)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 4
8*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 343*(80*c^5*d^6*e^6 -
240*b*c^4*d^5*e^7 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^8 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^9 - 6*(6*b^4*c - 47*a*
b^2*c^2 - 108*a^2*c^3)*d^2*e^10 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^11 + (5*a*b^4 - 81*a^2*b^2*c + 324
*a^3*c^2)*e^12)*sqrt(e*x + d)) - 2*(7*a*b*c*d^2*e - 28*a^2*c*d*e^2 + 7*a^2*b*e^3 + 2*(b^2*c - 4*a*c^2)*d^3 + (
14*c^3*d^2*e - 14*b*c^2*d*e^2 + (9*b^2*c - 22*a*c^2)*e^3)*x^3 + (21*b*c^2*d^2*e - 4*(2*b^2*c + 13*a*c^2)*d*e^2
+ 7*(b^3 - a*b*c)*e^3)*x^2 - (42*a*b*c*d*e^2 - (13*b^2*c - 10*a*c^2)*d^2*e - 14*(a*b^2 - a^2*c)*e^3)*x)*sqrt(
e*x + d))/(a^2*b^2*c - 4*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*x^3 + (b^4*c - 2*a*b^2*c^
2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^2*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((2*c*x+b)*(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((2*c*x+b)*(e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

Timed out