3.1162 \(\int (A+B x) (d+e x)^3 \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=404 \[ \frac{\left (b x+c x^2\right )^{3/2} \left (6 c e x \left (28 A c e (2 c d-b e)+B \left (21 b^2 e^2-36 b c d e+8 c^2 d^2\right )\right )+4 A c e \left (35 b^2 e^2-150 b c d e+192 c^2 d^2\right )+B \left (420 b^2 c d e^2-105 b^3 e^3-456 b c^2 d^2 e+64 c^3 d^3\right )\right )}{960 c^4}+\frac{(b+2 c x) \sqrt{b x+c x^2} \left (120 b^2 c^2 d e (A e+B d)-28 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+21 b^4 B e^3\right )}{512 c^5}-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (120 b^2 c^2 d e (A e+B d)-28 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+21 b^4 B e^3\right )}{512 c^{11/2}}+\frac{\left (b x+c x^2\right )^{3/2} (d+e x)^2 (4 A c e-3 b B e+2 B c d)}{20 c^2}+\frac{B \left (b x+c x^2\right )^{3/2} (d+e x)^3}{6 c} \]

[Out]

((128*A*c^4*d^3 + 21*b^4*B*e^3 + 120*b^2*c^2*d*e*(B*d + A*e) - 28*b^3*c*e^2*(3*B*d + A*e) - 64*b*c^3*d^2*(B*d
+ 3*A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^5) + ((2*B*c*d - 3*b*B*e + 4*A*c*e)*(d + e*x)^2*(b*x + c*x^2)^
(3/2))/(20*c^2) + (B*(d + e*x)^3*(b*x + c*x^2)^(3/2))/(6*c) + ((4*A*c*e*(192*c^2*d^2 - 150*b*c*d*e + 35*b^2*e^
2) + B*(64*c^3*d^3 - 456*b*c^2*d^2*e + 420*b^2*c*d*e^2 - 105*b^3*e^3) + 6*c*e*(28*A*c*e*(2*c*d - b*e) + B*(8*c
^2*d^2 - 36*b*c*d*e + 21*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(960*c^4) - (b^2*(128*A*c^4*d^3 + 21*b^4*B*e^3 + 12
0*b^2*c^2*d*e*(B*d + A*e) - 28*b^3*c*e^2*(3*B*d + A*e) - 64*b*c^3*d^2*(B*d + 3*A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[
b*x + c*x^2]])/(512*c^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.54723, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {832, 779, 612, 620, 206} \[ \frac{\left (b x+c x^2\right )^{3/2} \left (6 c e x \left (28 A c e (2 c d-b e)+B \left (21 b^2 e^2-36 b c d e+8 c^2 d^2\right )\right )+4 A c e \left (35 b^2 e^2-150 b c d e+192 c^2 d^2\right )+B \left (420 b^2 c d e^2-105 b^3 e^3-456 b c^2 d^2 e+64 c^3 d^3\right )\right )}{960 c^4}+\frac{(b+2 c x) \sqrt{b x+c x^2} \left (120 b^2 c^2 d e (A e+B d)-28 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+21 b^4 B e^3\right )}{512 c^5}-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (120 b^2 c^2 d e (A e+B d)-28 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+21 b^4 B e^3\right )}{512 c^{11/2}}+\frac{\left (b x+c x^2\right )^{3/2} (d+e x)^2 (4 A c e-3 b B e+2 B c d)}{20 c^2}+\frac{B \left (b x+c x^2\right )^{3/2} (d+e x)^3}{6 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((128*A*c^4*d^3 + 21*b^4*B*e^3 + 120*b^2*c^2*d*e*(B*d + A*e) - 28*b^3*c*e^2*(3*B*d + A*e) - 64*b*c^3*d^2*(B*d
+ 3*A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^5) + ((2*B*c*d - 3*b*B*e + 4*A*c*e)*(d + e*x)^2*(b*x + c*x^2)^
(3/2))/(20*c^2) + (B*(d + e*x)^3*(b*x + c*x^2)^(3/2))/(6*c) + ((4*A*c*e*(192*c^2*d^2 - 150*b*c*d*e + 35*b^2*e^
2) + B*(64*c^3*d^3 - 456*b*c^2*d^2*e + 420*b^2*c*d*e^2 - 105*b^3*e^3) + 6*c*e*(28*A*c*e*(2*c*d - b*e) + B*(8*c
^2*d^2 - 36*b*c*d*e + 21*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(960*c^4) - (b^2*(128*A*c^4*d^3 + 21*b^4*B*e^3 + 12
0*b^2*c^2*d*e*(B*d + A*e) - 28*b^3*c*e^2*(3*B*d + A*e) - 64*b*c^3*d^2*(B*d + 3*A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[
b*x + c*x^2]])/(512*c^(11/2))

Rule 832

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))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

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])

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^3 \sqrt{b x+c x^2} \, dx &=\frac{B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac{\int (d+e x)^2 \left (-\frac{3}{2} (b B-4 A c) d+\frac{3}{2} (2 B c d-3 b B e+4 A c e) x\right ) \sqrt{b x+c x^2} \, dx}{6 c}\\ &=\frac{(2 B c d-3 b B e+4 A c e) (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{20 c^2}+\frac{B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac{\int (d+e x) \left (-\frac{3}{4} d \left (16 b B c d-40 A c^2 d-9 b^2 B e+12 A b c e\right )+\frac{3}{4} \left (28 A c e (2 c d-b e)+B \left (8 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2} \, dx}{30 c^2}\\ &=\frac{(2 B c d-3 b B e+4 A c e) (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{20 c^2}+\frac{B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac{\left (4 A c e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2\right )+B \left (64 c^3 d^3-456 b c^2 d^2 e+420 b^2 c d e^2-105 b^3 e^3\right )+6 c e \left (28 A c e (2 c d-b e)+B \left (8 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{960 c^4}+\frac{\left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \int \sqrt{b x+c x^2} \, dx}{128 c^4}\\ &=\frac{\left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^5}+\frac{(2 B c d-3 b B e+4 A c e) (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{20 c^2}+\frac{B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac{\left (4 A c e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2\right )+B \left (64 c^3 d^3-456 b c^2 d^2 e+420 b^2 c d e^2-105 b^3 e^3\right )+6 c e \left (28 A c e (2 c d-b e)+B \left (8 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{960 c^4}-\frac{\left (b^2 \left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{1024 c^5}\\ &=\frac{\left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^5}+\frac{(2 B c d-3 b B e+4 A c e) (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{20 c^2}+\frac{B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac{\left (4 A c e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2\right )+B \left (64 c^3 d^3-456 b c^2 d^2 e+420 b^2 c d e^2-105 b^3 e^3\right )+6 c e \left (28 A c e (2 c d-b e)+B \left (8 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{960 c^4}-\frac{\left (b^2 \left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{512 c^5}\\ &=\frac{\left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^5}+\frac{(2 B c d-3 b B e+4 A c e) (d+e x)^2 \left (b x+c x^2\right )^{3/2}}{20 c^2}+\frac{B (d+e x)^3 \left (b x+c x^2\right )^{3/2}}{6 c}+\frac{\left (4 A c e \left (192 c^2 d^2-150 b c d e+35 b^2 e^2\right )+B \left (64 c^3 d^3-456 b c^2 d^2 e+420 b^2 c d e^2-105 b^3 e^3\right )+6 c e \left (28 A c e (2 c d-b e)+B \left (8 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{960 c^4}-\frac{b^2 \left (128 A c^4 d^3+21 b^4 B e^3+120 b^2 c^2 d e (B d+A e)-28 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 1.12181, size = 422, normalized size = 1.04 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (8 b^3 c^2 e \left (5 A e (45 d+7 e x)+3 B \left (75 d^2+35 d e x+7 e^2 x^2\right )\right )-16 b^2 c^3 \left (A e \left (180 d^2+75 d e x+14 e^2 x^2\right )+B \left (75 d^2 e x+60 d^3+42 d e^2 x^2+9 e^3 x^3\right )\right )-210 b^4 c e^2 (2 A e+6 B d+B e x)+64 b c^4 \left (3 A \left (10 d^2 e x+10 d^3+5 d e^2 x^2+e^3 x^3\right )+B x \left (15 d^2 e x+10 d^3+9 d e^2 x^2+2 e^3 x^3\right )\right )+128 c^5 x \left (3 A \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )\right )+315 b^5 B e^3\right )-\frac{15 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right ) \left (120 b^2 c^2 d e (A e+B d)-28 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+21 b^4 B e^3\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{7680 c^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(315*b^5*B*e^3 - 210*b^4*c*e^2*(6*B*d + 2*A*e + B*e*x) + 8*b^3*c^2*e*(5*A*e*(45*d
+ 7*e*x) + 3*B*(75*d^2 + 35*d*e*x + 7*e^2*x^2)) + 64*b*c^4*(3*A*(10*d^3 + 10*d^2*e*x + 5*d*e^2*x^2 + e^3*x^3)
+ B*x*(10*d^3 + 15*d^2*e*x + 9*d*e^2*x^2 + 2*e^3*x^3)) - 16*b^2*c^3*(A*e*(180*d^2 + 75*d*e*x + 14*e^2*x^2) + B
*(60*d^3 + 75*d^2*e*x + 42*d*e^2*x^2 + 9*e^3*x^3)) + 128*c^5*x*(3*A*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^
3*x^3) + B*x*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3))) - (15*b^(3/2)*(128*A*c^4*d^3 + 21*b^4*B*e^3 +
120*b^2*c^2*d*e*(B*d + A*e) - 28*b^3*c*e^2*(3*B*d + A*e) - 64*b*c^3*d^2*(B*d + 3*A*e))*ArcSinh[(Sqrt[c]*Sqrt[
x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(7680*c^(11/2))

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 1027, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3*(c*x^2+b*x)^(1/2),x)

[Out]

1/6*B*e^3*x^3*(c*x^2+b*x)^(3/2)/c-7/64*B*e^3*b^3/c^4*(c*x^2+b*x)^(3/2)+21/512*B*e^3*b^5/c^5*(c*x^2+b*x)^(1/2)-
21/1024*B*e^3*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+7/48*b^2/c^3*(c*x^2+b*x)^(3/2)*A*e^3+1/5*
x^2*(c*x^2+b*x)^(3/2)/c*A*e^3+7/256*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*e^3-1/8*b^2/c^2*(c
*x^2+b*x)^(1/2)*B*d^3+1/16*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d^3-7/128*b^4/c^4*(c*x^2+b*
x)^(1/2)*A*e^3+(c*x^2+b*x)^(3/2)/c*A*d^2*e-1/8*A*d^3*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/4
*A*d^3/c*(c*x^2+b*x)^(1/2)*b-3/4*b/c*x*(c*x^2+b*x)^(1/2)*A*d^2*e-21/40*b/c^2*x*(c*x^2+b*x)^(3/2)*B*d*e^2-21/64
*b^3/c^3*x*(c*x^2+b*x)^(1/2)*B*d*e^2+15/32*b^2/c^2*x*(c*x^2+b*x)^(1/2)*A*d*e^2+15/32*b^2/c^2*x*(c*x^2+b*x)^(1/
2)*B*d^2*e+1/3*(c*x^2+b*x)^(3/2)/c*B*d^3+1/2*A*d^3*x*(c*x^2+b*x)^(1/2)-21/128*b^4/c^4*(c*x^2+b*x)^(1/2)*B*d*e^
2+21/256*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d*e^2+3/4*x*(c*x^2+b*x)^(3/2)/c*A*d*e^2-3/20*
B*e^3*b/c^2*x^2*(c*x^2+b*x)^(3/2)+21/160*B*e^3*b^2/c^3*x*(c*x^2+b*x)^(3/2)+21/256*B*e^3*b^4/c^4*x*(c*x^2+b*x)^
(1/2)-3/8*b^2/c^2*(c*x^2+b*x)^(1/2)*A*d^2*e+3/16*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*d^2*e
-1/4*b/c*x*(c*x^2+b*x)^(1/2)*B*d^3+3/4*x*(c*x^2+b*x)^(3/2)/c*B*d^2*e-5/8*b/c^2*(c*x^2+b*x)^(3/2)*A*d*e^2-5/8*b
/c^2*(c*x^2+b*x)^(3/2)*B*d^2*e+15/64*b^3/c^3*(c*x^2+b*x)^(1/2)*A*d*e^2+15/64*b^3/c^3*(c*x^2+b*x)^(1/2)*B*d^2*e
-15/128*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*d*e^2-15/128*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x)^(1/2))*B*d^2*e+3/5*x^2*(c*x^2+b*x)^(3/2)/c*B*d*e^2-7/40*b/c^2*x*(c*x^2+b*x)^(3/2)*A*e^3+7/16*b^2
/c^3*(c*x^2+b*x)^(3/2)*B*d*e^2-7/64*b^3/c^3*x*(c*x^2+b*x)^(1/2)*A*e^3

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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)*(e*x+d)^3*(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7863, size = 2242, normalized size = 5.55 \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)*(e*x+d)^3*(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

[1/15360*(15*(64*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 24*(5*B*b^4*c^2 - 8*A*b^3*c^3)*d^2*e + 12*(7*B*b^5*c - 10*A*b
^4*c^2)*d*e^2 - 7*(3*B*b^6 - 4*A*b^5*c)*e^3)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(1280*B*
c^6*e^3*x^5 + 128*(36*B*c^6*d*e^2 + (B*b*c^5 + 12*A*c^6)*e^3)*x^4 - 960*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 360*(5*B
*b^3*c^3 - 8*A*b^2*c^4)*d^2*e - 180*(7*B*b^4*c^2 - 10*A*b^3*c^3)*d*e^2 + 105*(3*B*b^5*c - 4*A*b^4*c^2)*e^3 + 4
8*(120*B*c^6*d^2*e + 12*(B*b*c^5 + 10*A*c^6)*d*e^2 - (3*B*b^2*c^4 - 4*A*b*c^5)*e^3)*x^3 + 8*(320*B*c^6*d^3 + 1
20*(B*b*c^5 + 8*A*c^6)*d^2*e - 12*(7*B*b^2*c^4 - 10*A*b*c^5)*d*e^2 + 7*(3*B*b^3*c^3 - 4*A*b^2*c^4)*e^3)*x^2 +
10*(64*(B*b*c^5 + 6*A*c^6)*d^3 - 24*(5*B*b^2*c^4 - 8*A*b*c^5)*d^2*e + 12*(7*B*b^3*c^3 - 10*A*b^2*c^4)*d*e^2 -
7*(3*B*b^4*c^2 - 4*A*b^3*c^3)*e^3)*x)*sqrt(c*x^2 + b*x))/c^6, -1/7680*(15*(64*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 -
24*(5*B*b^4*c^2 - 8*A*b^3*c^3)*d^2*e + 12*(7*B*b^5*c - 10*A*b^4*c^2)*d*e^2 - 7*(3*B*b^6 - 4*A*b^5*c)*e^3)*sqrt
(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (1280*B*c^6*e^3*x^5 + 128*(36*B*c^6*d*e^2 + (B*b*c^5 + 12*A*c^
6)*e^3)*x^4 - 960*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 360*(5*B*b^3*c^3 - 8*A*b^2*c^4)*d^2*e - 180*(7*B*b^4*c^2 - 10*
A*b^3*c^3)*d*e^2 + 105*(3*B*b^5*c - 4*A*b^4*c^2)*e^3 + 48*(120*B*c^6*d^2*e + 12*(B*b*c^5 + 10*A*c^6)*d*e^2 - (
3*B*b^2*c^4 - 4*A*b*c^5)*e^3)*x^3 + 8*(320*B*c^6*d^3 + 120*(B*b*c^5 + 8*A*c^6)*d^2*e - 12*(7*B*b^2*c^4 - 10*A*
b*c^5)*d*e^2 + 7*(3*B*b^3*c^3 - 4*A*b^2*c^4)*e^3)*x^2 + 10*(64*(B*b*c^5 + 6*A*c^6)*d^3 - 24*(5*B*b^2*c^4 - 8*A
*b*c^5)*d^2*e + 12*(7*B*b^3*c^3 - 10*A*b^2*c^4)*d*e^2 - 7*(3*B*b^4*c^2 - 4*A*b^3*c^3)*e^3)*x)*sqrt(c*x^2 + b*x
))/c^6]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x \left (b + c x\right )} \left (A + B x\right ) \left (d + e x\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3*(c*x**2+b*x)**(1/2),x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)*(d + e*x)**3, x)

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Giac [A]  time = 1.30525, size = 703, normalized size = 1.74 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B x e^{3} + \frac{36 \, B c^{5} d e^{2} + B b c^{4} e^{3} + 12 \, A c^{5} e^{3}}{c^{5}}\right )} x + \frac{3 \,{\left (120 \, B c^{5} d^{2} e + 12 \, B b c^{4} d e^{2} + 120 \, A c^{5} d e^{2} - 3 \, B b^{2} c^{3} e^{3} + 4 \, A b c^{4} e^{3}\right )}}{c^{5}}\right )} x + \frac{320 \, B c^{5} d^{3} + 120 \, B b c^{4} d^{2} e + 960 \, A c^{5} d^{2} e - 84 \, B b^{2} c^{3} d e^{2} + 120 \, A b c^{4} d e^{2} + 21 \, B b^{3} c^{2} e^{3} - 28 \, A b^{2} c^{3} e^{3}}{c^{5}}\right )} x + \frac{5 \,{\left (64 \, B b c^{4} d^{3} + 384 \, A c^{5} d^{3} - 120 \, B b^{2} c^{3} d^{2} e + 192 \, A b c^{4} d^{2} e + 84 \, B b^{3} c^{2} d e^{2} - 120 \, A b^{2} c^{3} d e^{2} - 21 \, B b^{4} c e^{3} + 28 \, A b^{3} c^{2} e^{3}\right )}}{c^{5}}\right )} x - \frac{15 \,{\left (64 \, B b^{2} c^{3} d^{3} - 128 \, A b c^{4} d^{3} - 120 \, B b^{3} c^{2} d^{2} e + 192 \, A b^{2} c^{3} d^{2} e + 84 \, B b^{4} c d e^{2} - 120 \, A b^{3} c^{2} d e^{2} - 21 \, B b^{5} e^{3} + 28 \, A b^{4} c e^{3}\right )}}{c^{5}}\right )} - \frac{{\left (64 \, B b^{3} c^{3} d^{3} - 128 \, A b^{2} c^{4} d^{3} - 120 \, B b^{4} c^{2} d^{2} e + 192 \, A b^{3} c^{3} d^{2} e + 84 \, B b^{5} c d e^{2} - 120 \, A b^{4} c^{2} d e^{2} - 21 \, B b^{6} e^{3} + 28 \, A b^{5} c e^{3}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*B*x*e^3 + (36*B*c^5*d*e^2 + B*b*c^4*e^3 + 12*A*c^5*e^3)/c^5)*x + 3*(1
20*B*c^5*d^2*e + 12*B*b*c^4*d*e^2 + 120*A*c^5*d*e^2 - 3*B*b^2*c^3*e^3 + 4*A*b*c^4*e^3)/c^5)*x + (320*B*c^5*d^3
+ 120*B*b*c^4*d^2*e + 960*A*c^5*d^2*e - 84*B*b^2*c^3*d*e^2 + 120*A*b*c^4*d*e^2 + 21*B*b^3*c^2*e^3 - 28*A*b^2*
c^3*e^3)/c^5)*x + 5*(64*B*b*c^4*d^3 + 384*A*c^5*d^3 - 120*B*b^2*c^3*d^2*e + 192*A*b*c^4*d^2*e + 84*B*b^3*c^2*d
*e^2 - 120*A*b^2*c^3*d*e^2 - 21*B*b^4*c*e^3 + 28*A*b^3*c^2*e^3)/c^5)*x - 15*(64*B*b^2*c^3*d^3 - 128*A*b*c^4*d^
3 - 120*B*b^3*c^2*d^2*e + 192*A*b^2*c^3*d^2*e + 84*B*b^4*c*d*e^2 - 120*A*b^3*c^2*d*e^2 - 21*B*b^5*e^3 + 28*A*b
^4*c*e^3)/c^5) - 1/1024*(64*B*b^3*c^3*d^3 - 128*A*b^2*c^4*d^3 - 120*B*b^4*c^2*d^2*e + 192*A*b^3*c^3*d^2*e + 84
*B*b^5*c*d*e^2 - 120*A*b^4*c^2*d*e^2 - 21*B*b^6*e^3 + 28*A*b^5*c*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x
))*sqrt(c) - b))/c^(11/2)