3.1569 \(\int \frac{(b+2 c x) (a+b x+c x^2)^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=627 \[ -\frac{\sqrt{a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{256 c^2 e^6}+\frac{\left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{5/2} e^7}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{96 c e^4}-\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^7}-\frac{\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2} \]

[Out]

-((512*c^5*d^5 + b^5*e^5 - 128*c^4*d^3*e*(11*b*d - 8*a*e) + 8*b^3*c*e^4*(b*d - 2*a*e) + 32*c^3*d*e^2*(40*b^2*d
^2 - 55*a*b*d*e + 16*a^2*e^2) - 8*b*c^2*e^3*(49*b^2*d^2 - 92*a*b*d*e + 42*a^2*e^2) - 2*c*e*(16*c^2*d^2 - b^2*e
^2 - 4*c*e*(4*b*d - 5*a*e))*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x)*Sqrt[a + b*x + c*x^2])/(256*c^2*e^6
) - ((64*c^3*d^3 - b^3*e^3 + 4*b*c*e^2*(14*b*d - 13*a*e) - 8*c^2*d*e*(15*b*d - 8*a*e) - 2*c*e*(24*c^2*d^2 + b^
2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(96*c*e^4) - ((12*c*d - 11*b*e - 10*c*e*x)*(a + b*x
+ c*x^2)^(5/2))/(30*e^2) + ((1024*c^6*d^6 + b^6*e^6 + 4*b^4*c*e^5*(2*b*d - 5*a*e) - 512*c^5*d^4*e*(6*b*d - 5*
a*e) - 320*c^3*e^3*(b*d - a*e)^2*(4*b*d - a*e) + 640*c^4*d^2*e^2*(5*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2) + 40*b^2*
c^2*e^4*(3*b^2*d^2 - 8*a*b*d*e + 6*a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(512*c^(5
/2)*e^7) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(5/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2
- b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e^7

________________________________________________________________________________________

Rubi [A]  time = 1.14364, antiderivative size = 627, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {814, 843, 621, 206, 724} \[ -\frac{\sqrt{a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{256 c^2 e^6}+\frac{\left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{5/2} e^7}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{96 c e^4}-\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^7}-\frac{\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((512*c^5*d^5 + b^5*e^5 - 128*c^4*d^3*e*(11*b*d - 8*a*e) + 8*b^3*c*e^4*(b*d - 2*a*e) + 32*c^3*d*e^2*(40*b^2*d
^2 - 55*a*b*d*e + 16*a^2*e^2) - 8*b*c^2*e^3*(49*b^2*d^2 - 92*a*b*d*e + 42*a^2*e^2) - 2*c*e*(16*c^2*d^2 - b^2*e
^2 - 4*c*e*(4*b*d - 5*a*e))*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x)*Sqrt[a + b*x + c*x^2])/(256*c^2*e^6
) - ((64*c^3*d^3 - b^3*e^3 + 4*b*c*e^2*(14*b*d - 13*a*e) - 8*c^2*d*e*(15*b*d - 8*a*e) - 2*c*e*(24*c^2*d^2 + b^
2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(96*c*e^4) - ((12*c*d - 11*b*e - 10*c*e*x)*(a + b*x
+ c*x^2)^(5/2))/(30*e^2) + ((1024*c^6*d^6 + b^6*e^6 + 4*b^4*c*e^5*(2*b*d - 5*a*e) - 512*c^5*d^4*e*(6*b*d - 5*
a*e) - 320*c^3*e^3*(b*d - a*e)^2*(4*b*d - a*e) + 640*c^4*d^2*e^2*(5*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2) + 40*b^2*
c^2*e^4*(3*b^2*d^2 - 8*a*b*d*e + 6*a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(512*c^(5
/2)*e^7) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(5/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2
- b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e^7

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

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

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])

Rule 724

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

Rubi steps

\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx &=-\frac{(12 c d-11 b e-10 c e x) \left (a+b x+c x^2\right )^{5/2}}{30 e^2}-\frac{\int \frac{\left (c \left (11 b^2 d e+4 a c d e-12 b \left (c d^2+a e^2\right )\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{12 c e^2}\\ &=-\frac{\left (64 c^3 d^3-b^3 e^3+4 b c e^2 (14 b d-13 a e)-8 c^2 d e (15 b d-8 a e)-2 c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 c e^4}-\frac{(12 c d-11 b e-10 c e x) \left (a+b x+c x^2\right )^{5/2}}{30 e^2}+\frac{\int \frac{\left (\frac{1}{2} c \left (d \left (8 b c d-3 b^2 e-4 a c e\right ) \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right )+8 c e (b d-2 a e) \left (11 b^2 d e+4 a c d e-12 b \left (c d^2+a e^2\right )\right )\right )+\frac{3}{2} c \left (8 c^2 d^2-8 b c d e+b^2 e^2+4 a c e^2\right ) \left (16 c^2 d^2-16 b c d e-b^2 e^2+20 a c e^2\right ) x\right ) \sqrt{a+b x+c x^2}}{d+e x} \, dx}{96 c^2 e^4}\\ &=-\frac{\left (512 c^5 d^5+b^5 e^5-128 c^4 d^3 e (11 b d-8 a e)+8 b^3 c e^4 (b d-2 a e)+32 c^3 d e^2 \left (40 b^2 d^2-55 a b d e+16 a^2 e^2\right )-8 b c^2 e^3 \left (49 b^2 d^2-92 a b d e+42 a^2 e^2\right )-2 c e \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{256 c^2 e^6}-\frac{\left (64 c^3 d^3-b^3 e^3+4 b c e^2 (14 b d-13 a e)-8 c^2 d e (15 b d-8 a e)-2 c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 c e^4}-\frac{(12 c d-11 b e-10 c e x) \left (a+b x+c x^2\right )^{5/2}}{30 e^2}-\frac{\int \frac{-\frac{1}{4} c \left (3 d \left (4 b c d-b^2 e-4 a c e\right ) \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right )-4 c e (b d-2 a e) \left (d \left (8 b c d-3 b^2 e-4 a c e\right ) \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right )+8 c e (b d-2 a e) \left (11 b^2 d e+4 a c d e-12 b \left (c d^2+a e^2\right )\right )\right )\right )-\frac{3}{4} c \left (1024 c^6 d^6+b^6 e^6+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+640 c^4 d^2 e^2 \left (5 b^2 d^2-8 a b d e+3 a^2 e^2\right )+40 b^2 c^2 e^4 \left (3 b^2 d^2-8 a b d e+6 a^2 e^2\right )\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{384 c^3 e^6}\\ &=-\frac{\left (512 c^5 d^5+b^5 e^5-128 c^4 d^3 e (11 b d-8 a e)+8 b^3 c e^4 (b d-2 a e)+32 c^3 d e^2 \left (40 b^2 d^2-55 a b d e+16 a^2 e^2\right )-8 b c^2 e^3 \left (49 b^2 d^2-92 a b d e+42 a^2 e^2\right )-2 c e \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{256 c^2 e^6}-\frac{\left (64 c^3 d^3-b^3 e^3+4 b c e^2 (14 b d-13 a e)-8 c^2 d e (15 b d-8 a e)-2 c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 c e^4}-\frac{(12 c d-11 b e-10 c e x) \left (a+b x+c x^2\right )^{5/2}}{30 e^2}-\frac{\left ((2 c d-b e) \left (c d^2-b d e+a e^2\right )^3\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{e^7}+\frac{\left (1024 c^6 d^6+b^6 e^6+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+640 c^4 d^2 e^2 \left (5 b^2 d^2-8 a b d e+3 a^2 e^2\right )+40 b^2 c^2 e^4 \left (3 b^2 d^2-8 a b d e+6 a^2 e^2\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{512 c^2 e^7}\\ &=-\frac{\left (512 c^5 d^5+b^5 e^5-128 c^4 d^3 e (11 b d-8 a e)+8 b^3 c e^4 (b d-2 a e)+32 c^3 d e^2 \left (40 b^2 d^2-55 a b d e+16 a^2 e^2\right )-8 b c^2 e^3 \left (49 b^2 d^2-92 a b d e+42 a^2 e^2\right )-2 c e \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{256 c^2 e^6}-\frac{\left (64 c^3 d^3-b^3 e^3+4 b c e^2 (14 b d-13 a e)-8 c^2 d e (15 b d-8 a e)-2 c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 c e^4}-\frac{(12 c d-11 b e-10 c e x) \left (a+b x+c x^2\right )^{5/2}}{30 e^2}+\frac{\left (2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{e^7}+\frac{\left (1024 c^6 d^6+b^6 e^6+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+640 c^4 d^2 e^2 \left (5 b^2 d^2-8 a b d e+3 a^2 e^2\right )+40 b^2 c^2 e^4 \left (3 b^2 d^2-8 a b d e+6 a^2 e^2\right )\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 )}{256 c^2 e^7}\\ &=-\frac{\left (512 c^5 d^5+b^5 e^5-128 c^4 d^3 e (11 b d-8 a e)+8 b^3 c e^4 (b d-2 a e)+32 c^3 d e^2 \left (40 b^2 d^2-55 a b d e+16 a^2 e^2\right )-8 b c^2 e^3 \left (49 b^2 d^2-92 a b d e+42 a^2 e^2\right )-2 c e \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{256 c^2 e^6}-\frac{\left (64 c^3 d^3-b^3 e^3+4 b c e^2 (14 b d-13 a e)-8 c^2 d e (15 b d-8 a e)-2 c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 c e^4}-\frac{(12 c d-11 b e-10 c e x) \left (a+b x+c x^2\right )^{5/2}}{30 e^2}+\frac{\left (1024 c^6 d^6+b^6 e^6+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+640 c^4 d^2 e^2 \left (5 b^2 d^2-8 a b d e+3 a^2 e^2\right )+40 b^2 c^2 e^4 \left (3 b^2 d^2-8 a b d e+6 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{5/2} e^7}-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{e^7}\\ \end{align*}

Mathematica [A]  time = 1.66025, size = 607, normalized size = 0.97 \[ \frac{-2 c e \sqrt{a+x (b+c x)} \left (16 c^3 e^2 \left (2 a^2 e^2 (16 d-5 e x)+2 a b d e (14 e x-55 d)+b^2 d^2 (80 d-17 e x)\right )-8 b c^2 e^3 \left (42 a^2 e^2+4 a b e (e x-23 d)+b^2 d (49 d-2 e x)\right )+2 b^3 c e^4 (-8 a e+4 b d+b e x)-64 c^4 d^2 e (a e (7 e x-16 d)+2 b d (11 d-4 e x))+b^5 e^5+256 c^5 d^4 (2 d-e x)\right )+\sqrt{c} \left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)+320 c^3 e^3 (b d-a e)^2 (a e-4 b d)+b^6 e^6+1024 c^6 d^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )+512 c^3 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{512 c^3 e^7}+\frac{(a+x (b+c x))^{3/2} \left (8 c^2 e (a e (5 e x-8 d)+3 b d (5 d-2 e x))+2 b c e^2 (26 a e-28 b d+b e x)+b^3 e^3+16 c^3 d^2 (3 e x-4 d)\right )}{96 c e^4}+\frac{(a+x (b+c x))^{5/2} (11 b e-12 c d+10 c e x)}{30 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*c*d + 11*b*e + 10*c*e*x)*(a + x*(b + c*x))^(5/2))/(30*e^2) + ((a + x*(b + c*x))^(3/2)*(b^3*e^3 + 16*c^3*
d^2*(-4*d + 3*e*x) + 2*b*c*e^2*(-28*b*d + 26*a*e + b*e*x) + 8*c^2*e*(3*b*d*(5*d - 2*e*x) + a*e*(-8*d + 5*e*x))
))/(96*c*e^4) + (-2*c*e*Sqrt[a + x*(b + c*x)]*(b^5*e^5 + 256*c^5*d^4*(2*d - e*x) + 2*b^3*c*e^4*(4*b*d - 8*a*e
+ b*e*x) - 8*b*c^2*e^3*(42*a^2*e^2 + b^2*d*(49*d - 2*e*x) + 4*a*b*e*(-23*d + e*x)) - 64*c^4*d^2*e*(2*b*d*(11*d
- 4*e*x) + a*e*(-16*d + 7*e*x)) + 16*c^3*e^2*(b^2*d^2*(80*d - 17*e*x) + 2*a^2*e^2*(16*d - 5*e*x) + 2*a*b*d*e*
(-55*d + 14*e*x))) + Sqrt[c]*(1024*c^6*d^6 + b^6*e^6 + 4*b^4*c*e^5*(2*b*d - 5*a*e) - 512*c^5*d^4*e*(6*b*d - 5*
a*e) + 320*c^3*e^3*(b*d - a*e)^2*(-4*b*d + a*e) + 640*c^4*d^2*e^2*(5*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2) + 40*b^2
*c^2*e^4*(3*b^2*d^2 - 8*a*b*d*e + 6*a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 512*c^3
*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(
-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(512*c^3*e^7)

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Maple [B]  time = 0.012, size = 6077, normalized size = 9.7 \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)*(c*x^2+b*x+a)^(5/2)/(e*x+d),x)

[Out]

result too large to display

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError