3.1176 \(\int \frac{(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=323 \[ -\frac{\sqrt{b x+c x^2} \left (2 c e x (-6 A c e-b B e+8 B c d)+6 A c e (4 c d-3 b e)-B \left (b^2 e^2-28 b c d e+32 c^2 d^2\right )\right )}{8 c e^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 b c e (4 B d-3 A e) (2 c d-b e)-\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) (-6 A c e-b B e+8 B c d)\right )}{8 c^{3/2} e^5}+\frac{\left (b x+c x^2\right )^{3/2} (-3 A e+4 B d+B e x)}{3 e^2 (d+e x)}+\frac{\sqrt{d} \sqrt{c d-b e} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^5} \]

[Out]

-((6*A*c*e*(4*c*d - 3*b*e) - B*(32*c^2*d^2 - 28*b*c*d*e + b^2*e^2) + 2*c*e*(8*B*c*d - b*B*e - 6*A*c*e)*x)*Sqrt
[b*x + c*x^2])/(8*c*e^4) + ((4*B*d - 3*A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(3*e^2*(d + e*x)) + ((4*b*c*e*(4*B*d
- 3*A*e)*(2*c*d - b*e) - (8*B*c*d - b*B*e - 6*A*c*e)*(8*c^2*d^2 - 4*b*c*d*e - b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sq
rt[b*x + c*x^2]])/(8*c^(3/2)*e^5) + (Sqrt[d]*Sqrt[c*d - b*e]*(B*d*(8*c*d - 5*b*e) - 3*A*e*(2*c*d - b*e))*ArcTa
nh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^5)

________________________________________________________________________________________

Rubi [A]  time = 0.424148, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {812, 814, 843, 620, 206, 724} \[ -\frac{\sqrt{b x+c x^2} \left (2 c e x (-6 A c e-b B e+8 B c d)+6 A c e (4 c d-3 b e)-B \left (b^2 e^2-28 b c d e+32 c^2 d^2\right )\right )}{8 c e^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 b c e (4 B d-3 A e) (2 c d-b e)-\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) (-6 A c e-b B e+8 B c d)\right )}{8 c^{3/2} e^5}+\frac{\left (b x+c x^2\right )^{3/2} (-3 A e+4 B d+B e x)}{3 e^2 (d+e x)}+\frac{\sqrt{d} \sqrt{c d-b e} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((6*A*c*e*(4*c*d - 3*b*e) - B*(32*c^2*d^2 - 28*b*c*d*e + b^2*e^2) + 2*c*e*(8*B*c*d - b*B*e - 6*A*c*e)*x)*Sqrt
[b*x + c*x^2])/(8*c*e^4) + ((4*B*d - 3*A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(3*e^2*(d + e*x)) + ((4*b*c*e*(4*B*d
- 3*A*e)*(2*c*d - b*e) - (8*B*c*d - b*B*e - 6*A*c*e)*(8*c^2*d^2 - 4*b*c*d*e - b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sq
rt[b*x + c*x^2]])/(8*c^(3/2)*e^5) + (Sqrt[d]*Sqrt[c*d - b*e]*(B*d*(8*c*d - 5*b*e) - 3*A*e*(2*c*d - b*e))*ArcTa
nh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^5)

Rule 812

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

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 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

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{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx &=\frac{(4 B d-3 A e+B e x) \left (b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac{\int \frac{(b (4 B d-3 A e)+(8 B c d-b B e-6 A c e) x) \sqrt{b x+c x^2}}{d+e x} \, dx}{2 e^2}\\ &=-\frac{\left (6 A c e (4 c d-3 b e)-B \left (32 c^2 d^2-28 b c d e+b^2 e^2\right )+2 c e (8 B c d-b B e-6 A c e) x\right ) \sqrt{b x+c x^2}}{8 c e^4}+\frac{(4 B d-3 A e+B e x) \left (b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}+\frac{\int \frac{\frac{1}{2} b d \left (6 A c e (4 c d-3 b e)-B \left (32 c^2 d^2-28 b c d e+b^2 e^2\right )\right )+\frac{1}{2} \left (4 b c e (4 B d-3 A e) (2 c d-b e)-(8 B c d-b B e-6 A c e) \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{8 c e^4}\\ &=-\frac{\left (6 A c e (4 c d-3 b e)-B \left (32 c^2 d^2-28 b c d e+b^2 e^2\right )+2 c e (8 B c d-b B e-6 A c e) x\right ) \sqrt{b x+c x^2}}{8 c e^4}+\frac{(4 B d-3 A e+B e x) \left (b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}+\frac{(d (c d-b e) (B d (8 c d-5 b e)-3 A e (2 c d-b e))) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{2 e^5}+\frac{\left (4 b c e (4 B d-3 A e) (2 c d-b e)-(8 B c d-b B e-6 A c e) \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{16 c e^5}\\ &=-\frac{\left (6 A c e (4 c d-3 b e)-B \left (32 c^2 d^2-28 b c d e+b^2 e^2\right )+2 c e (8 B c d-b B e-6 A c e) x\right ) \sqrt{b x+c x^2}}{8 c e^4}+\frac{(4 B d-3 A e+B e x) \left (b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac{(d (c d-b e) (B d (8 c d-5 b e)-3 A e (2 c d-b e))) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{e^5}+\frac{\left (4 b c e (4 B d-3 A e) (2 c d-b e)-(8 B c d-b B e-6 A c e) \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{8 c e^5}\\ &=-\frac{\left (6 A c e (4 c d-3 b e)-B \left (32 c^2 d^2-28 b c d e+b^2 e^2\right )+2 c e (8 B c d-b B e-6 A c e) x\right ) \sqrt{b x+c x^2}}{8 c e^4}+\frac{(4 B d-3 A e+B e x) \left (b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}+\frac{\left (4 b c e (4 B d-3 A e) (2 c d-b e)-(8 B c d-b B e-6 A c e) \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{3/2} e^5}+\frac{\sqrt{d} \sqrt{c d-b e} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{2 e^5}\\ \end{align*}

Mathematica [B]  time = 5.68241, size = 1059, normalized size = 3.28 \[ \frac{(x (b+c x))^{3/2} \left (\frac{e (5 b B d-2 A c d-3 A b e) \left (e^3 \sqrt{b e-c d} (b+c x)^2 \left (b c x \sqrt{\frac{c x}{b}+1} \left (3 b^2+14 c x b+8 c^2 x^2\right )-3 b^{7/2} \sqrt{c} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )-6 b c^{3/2} d \sqrt{x} \left (b \sqrt{b e-c d} \left (b \sqrt{c} \sqrt{x} (5 b+2 c x) \left (\frac{c x}{b}+1\right )^{5/2}+3 \sqrt{b} (b+c x)^2 \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right ) e^2-4 c d \sqrt{b e-c d} (b+c x) \left (b \sqrt{c} \sqrt{x} \left (\frac{c x}{b}+1\right )^{3/2}+\sqrt{b} (b+c x) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right ) e+8 b \sqrt{c} \sqrt{d} (c d-b e) \left (\frac{c x}{b}+1\right )^{3/2} \left (\sqrt{b} \sqrt{c} \sqrt{d} \sqrt{b e-c d} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )+(b e-c d) \sqrt{b+c x} \tan ^{-1}\left (\frac{\sqrt{b e-c d} \sqrt{x}}{\sqrt{d} \sqrt{b+c x}}\right )\right )\right )\right )+(B d-A e) \left (3 e^4 \sqrt{b e-c d} (b+c x)^2 \left (3 \sqrt{c} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right ) b^{9/2}+c x \sqrt{\frac{c x}{b}+1} \left (-3 b^3+2 c x b^2+24 c^2 x^2 b+16 c^3 x^3\right ) b\right )-8 c d \left (e^3 \sqrt{b e-c d} (b+c x)^2 \left (b c x \sqrt{\frac{c x}{b}+1} \left (3 b^2+14 c x b+8 c^2 x^2\right )-3 b^{7/2} \sqrt{c} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )-6 b c^{3/2} d \sqrt{x} \left (b \sqrt{b e-c d} \left (b \sqrt{c} \sqrt{x} (5 b+2 c x) \left (\frac{c x}{b}+1\right )^{5/2}+3 \sqrt{b} (b+c x)^2 \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right ) e^2-4 c d \sqrt{b e-c d} (b+c x) \left (b \sqrt{c} \sqrt{x} \left (\frac{c x}{b}+1\right )^{3/2}+\sqrt{b} (b+c x) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right ) e+8 b \sqrt{c} \sqrt{d} (c d-b e) \left (\frac{c x}{b}+1\right )^{3/2} \left (\sqrt{b} \sqrt{c} \sqrt{d} \sqrt{b e-c d} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )+(b e-c d) \sqrt{b+c x} \tan ^{-1}\left (\frac{\sqrt{b e-c d} \sqrt{x}}{\sqrt{d} \sqrt{b+c x}}\right )\right )\right )\right )\right )}{48 b c^2 e^5 \sqrt{b e-c d} \sqrt{x} (b+c x)^3 \sqrt{\frac{c x}{b}+1}}-\frac{(B d-A e) x^{5/2} (b+c x)}{d+e x}\right )}{d (b e-c d) x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((x*(b + c*x))^(3/2)*(-(((B*d - A*e)*x^(5/2)*(b + c*x))/(d + e*x)) + (e*(5*b*B*d - 2*A*c*d - 3*A*b*e)*(e^3*Sqr
t[-(c*d) + b*e]*(b + c*x)^2*(b*c*x*Sqrt[1 + (c*x)/b]*(3*b^2 + 14*b*c*x + 8*c^2*x^2) - 3*b^(7/2)*Sqrt[c]*Sqrt[x
]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]) - 6*b*c^(3/2)*d*Sqrt[x]*(-4*c*d*e*Sqrt[-(c*d) + b*e]*(b + c*x)*(b*Sqrt[c
]*Sqrt[x]*(1 + (c*x)/b)^(3/2) + Sqrt[b]*(b + c*x)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]) + b*e^2*Sqrt[-(c*d) + b*
e]*(b*Sqrt[c]*Sqrt[x]*(5*b + 2*c*x)*(1 + (c*x)/b)^(5/2) + 3*Sqrt[b]*(b + c*x)^2*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt
[b]]) + 8*b*Sqrt[c]*Sqrt[d]*(c*d - b*e)*(1 + (c*x)/b)^(3/2)*(Sqrt[b]*Sqrt[c]*Sqrt[d]*Sqrt[-(c*d) + b*e]*Sqrt[1
+ (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]] + (-(c*d) + b*e)*Sqrt[b + c*x]*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[
x])/(Sqrt[d]*Sqrt[b + c*x])]))) + (B*d - A*e)*(3*e^4*Sqrt[-(c*d) + b*e]*(b + c*x)^2*(b*c*x*Sqrt[1 + (c*x)/b]*(
-3*b^3 + 2*b^2*c*x + 24*b*c^2*x^2 + 16*c^3*x^3) + 3*b^(9/2)*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]
) - 8*c*d*(e^3*Sqrt[-(c*d) + b*e]*(b + c*x)^2*(b*c*x*Sqrt[1 + (c*x)/b]*(3*b^2 + 14*b*c*x + 8*c^2*x^2) - 3*b^(7
/2)*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]) - 6*b*c^(3/2)*d*Sqrt[x]*(-4*c*d*e*Sqrt[-(c*d) + b*e]*(
b + c*x)*(b*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3/2) + Sqrt[b]*(b + c*x)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]) + b*e^
2*Sqrt[-(c*d) + b*e]*(b*Sqrt[c]*Sqrt[x]*(5*b + 2*c*x)*(1 + (c*x)/b)^(5/2) + 3*Sqrt[b]*(b + c*x)^2*ArcSinh[(Sqr
t[c]*Sqrt[x])/Sqrt[b]]) + 8*b*Sqrt[c]*Sqrt[d]*(c*d - b*e)*(1 + (c*x)/b)^(3/2)*(Sqrt[b]*Sqrt[c]*Sqrt[d]*Sqrt[-(
c*d) + b*e]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]] + (-(c*d) + b*e)*Sqrt[b + c*x]*ArcTan[(Sqrt[-
(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])))))/(48*b*c^2*e^5*Sqrt[-(c*d) + b*e]*Sqrt[x]*(b + c*x)^3*Sqrt[
1 + (c*x)/b])))/(d*(-(c*d) + b*e)*x^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.011, size = 4283, normalized size = 13.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^2,x)

[Out]

1/3*B/e^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)-27/8/e^2*d/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e
+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b^2*A-21/4/e^2*d/(b*e-c
*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*c*A-3/8/e^2/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(x
+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^3*B*d+3/2/e^2*d/(b*e-c*d
)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*
c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^3*A+3/2/e/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d
/e)-d*(b*e-c*d)/e^2)^(1/2)*x*b*c*A-3/2/e^2*d/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/
2)*x*c^2*A+2*B/e^5*d^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e
^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b*c-6/e^4*d^3/(b*e-c*d)*ln((1/2*
(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*b*B-3/e^5*
d^4/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)
*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*c^3*A+3/e^6*d^5/(b*e-c*d)/(-d*(b*e-c*d)/e
^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(
x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*c^3*B+21/4/e^3*d^2/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e
-c*d)/e^2)^(1/2)*b*c*B-3/2/e^3*d^2/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+
d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^3*B+3/2/
e^3*d^2/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*B+27/8/e^3*d^2/(b*e-c*d)*ln(
(1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b^2*B
+6/e^3*d^2/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e
^2)^(1/2))*c^(3/2)*b*A-6/e^3*d^2/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/
e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^2*c*A+6/e^
4*d^3/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/
2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^2*c*B-3/2/e^2/(b*e-c*d)*((x+d/e)^2*c+
(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*b*c*B*d+15/2/e^4*d^3/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*
d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d
)/e^2)^(1/2))/(x+d/e))*b*c^2*A-15/2/e^5*d^4/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c
*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b
*c^2*B+3/e^5*d^4/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-
c*d)/e^2)^(1/2))*c^(5/2)*B-1/e^2/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*c*B*d-9/4
/e^2/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2*B*d-c/d/(b*e-c*d)*((x+d/e)^2*c+(b
*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*x*A+1/e*c/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/
e^2)^(3/2)*x*B+3/e^3*d^2/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^2*A-3/e^4*d^3/(
b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^2*B+3/8/e/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e
+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^3*A-3/e^4*d^3/(b*e-c*
d)*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(5/2)
*A-1/2*B/e^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c*d-3/8*B/e^3*d*ln((1/2*(b*e-2*c*d)/e
+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^2+3/2*B/e^4*d^2*ln((1
/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b-B/e^4
*d^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)
^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^2-B/e^6*d^4/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b
*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^
2)^(1/2))/(x+d/e))*c^2+1/e/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*c*A+1/d/(b*e-c*
d)/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(5/2)*A-1/d/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/
e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*b*A-1/e/(b*e-c*d)/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)
^(5/2)*B+1/4*B/e^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*b+1/8*B/e^2/c*((x+d/e)^2*c+(b*e
-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2-5/4*B/e^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/
2)*b*d-1/16*B/e^2/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c
*d)/e^2)^(1/2))*b^3+B/e^4*d^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c-B/e^5*d^3*ln((1/2*(b
*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)+9/4/e/(b*e-c
*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2*A+1/e/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(
x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*b*B

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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((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 22.6489, size = 4340, normalized size = 13.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 + 2*A*c^3)*d^3*e + 12*(B*b^2*c + 4*A*b*c^2)*d^2*e^2 + (B*b^3 - 6*A*b^2
*c)*d*e^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^2 + 2*A*c^3)*d^2*e^2 + 12*(B*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A
*b^2*c)*e^4)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 24*(8*B*c^3*d^3 + 3*A*b*c^2*d*e^2 - (5*
B*b*c^2 + 6*A*c^3)*d^2*e + (8*B*c^3*d^2*e + 3*A*b*c^2*e^3 - (5*B*b*c^2 + 6*A*c^3)*d*e^2)*x)*sqrt(c*d^2 - b*d*e
)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(8*B*c^3*e^4*x^3 + 96*B
*c^3*d^3*e - 12*(7*B*b*c^2 + 6*A*c^3)*d^2*e^2 + 3*(B*b^2*c + 18*A*b*c^2)*d*e^3 - 2*(8*B*c^3*d*e^3 - (7*B*b*c^2
+ 6*A*c^3)*e^4)*x^2 + (48*B*c^3*d^2*e^2 - 2*(23*B*b*c^2 + 18*A*c^3)*d*e^3 + 3*(B*b^2*c + 10*A*b*c^2)*e^4)*x)*
sqrt(c*x^2 + b*x))/(c^2*e^6*x + c^2*d*e^5), 1/48*(48*(8*B*c^3*d^3 + 3*A*b*c^2*d*e^2 - (5*B*b*c^2 + 6*A*c^3)*d^
2*e + (8*B*c^3*d^2*e + 3*A*b*c^2*e^3 - (5*B*b*c^2 + 6*A*c^3)*d*e^2)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^
2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - 3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 + 2*A*c^3)*d^3*e + 12*(B*b^2*c
+ 4*A*b*c^2)*d^2*e^2 + (B*b^3 - 6*A*b^2*c)*d*e^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^2 + 2*A*c^3)*d^2*e^2 + 12*(B
*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A*b^2*c)*e^4)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) +
2*(8*B*c^3*e^4*x^3 + 96*B*c^3*d^3*e - 12*(7*B*b*c^2 + 6*A*c^3)*d^2*e^2 + 3*(B*b^2*c + 18*A*b*c^2)*d*e^3 - 2*(
8*B*c^3*d*e^3 - (7*B*b*c^2 + 6*A*c^3)*e^4)*x^2 + (48*B*c^3*d^2*e^2 - 2*(23*B*b*c^2 + 18*A*c^3)*d*e^3 + 3*(B*b^
2*c + 10*A*b*c^2)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*e^6*x + c^2*d*e^5), 1/24*(3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 +
2*A*c^3)*d^3*e + 12*(B*b^2*c + 4*A*b*c^2)*d^2*e^2 + (B*b^3 - 6*A*b^2*c)*d*e^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^
2 + 2*A*c^3)*d^2*e^2 + 12*(B*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A*b^2*c)*e^4)*x)*sqrt(-c)*arctan(sqrt(c*x^2
+ b*x)*sqrt(-c)/(c*x)) + 12*(8*B*c^3*d^3 + 3*A*b*c^2*d*e^2 - (5*B*b*c^2 + 6*A*c^3)*d^2*e + (8*B*c^3*d^2*e + 3
*A*b*c^2*e^3 - (5*B*b*c^2 + 6*A*c^3)*d*e^2)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 -
b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + (8*B*c^3*e^4*x^3 + 96*B*c^3*d^3*e - 12*(7*B*b*c^2 + 6*A*c^3)*d^2*e^2 +
3*(B*b^2*c + 18*A*b*c^2)*d*e^3 - 2*(8*B*c^3*d*e^3 - (7*B*b*c^2 + 6*A*c^3)*e^4)*x^2 + (48*B*c^3*d^2*e^2 - 2*(2
3*B*b*c^2 + 18*A*c^3)*d*e^3 + 3*(B*b^2*c + 10*A*b*c^2)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*e^6*x + c^2*d*e^5), 1/2
4*(24*(8*B*c^3*d^3 + 3*A*b*c^2*d*e^2 - (5*B*b*c^2 + 6*A*c^3)*d^2*e + (8*B*c^3*d^2*e + 3*A*b*c^2*e^3 - (5*B*b*c
^2 + 6*A*c^3)*d*e^2)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) +
3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 + 2*A*c^3)*d^3*e + 12*(B*b^2*c + 4*A*b*c^2)*d^2*e^2 + (B*b^3 - 6*A*b^2*c)*d*e
^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^2 + 2*A*c^3)*d^2*e^2 + 12*(B*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A*b^2*c)
*e^4)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (8*B*c^3*e^4*x^3 + 96*B*c^3*d^3*e - 12*(7*B*b*c^2
+ 6*A*c^3)*d^2*e^2 + 3*(B*b^2*c + 18*A*b*c^2)*d*e^3 - 2*(8*B*c^3*d*e^3 - (7*B*b*c^2 + 6*A*c^3)*e^4)*x^2 + (48
*B*c^3*d^2*e^2 - 2*(23*B*b*c^2 + 18*A*c^3)*d*e^3 + 3*(B*b^2*c + 10*A*b*c^2)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*e^
6*x + c^2*d*e^5)]

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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((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**2,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((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^2,x, algorithm="giac")

[Out]

Timed out