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

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

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Rubi [A]  time = 0.55, antiderivative size = 404, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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 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])

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*}

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Mathematica [A]  time = 1.08, size = 422, normalized size = 1.04 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (-210 b^4 c e^2 (2 A e+6 B d+B e x)+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 (60 d^3+75 d^2 e x+42 d e^2 x^2+9 e^3 x^3\right )\right )+64 b c^4 \left (3 A \left (10 d^3+10 d^2 e x+5 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+15 d^2 e x+9 d e^2 x^2+2 e^3 x^3\right )\right )+128 c^5 x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+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 (-28 b^3 c e^2 (A e+3 B d)+120 b^2 c^2 d e (A e+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}} \end {gather*}

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))

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IntegrateAlgebraic [A]  time = 2.14, size = 559, normalized size = 1.38 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-420 A b^4 c e^3+1800 A b^3 c^2 d e^2+280 A b^3 c^2 e^3 x-2880 A b^2 c^3 d^2 e-1200 A b^2 c^3 d e^2 x-224 A b^2 c^3 e^3 x^2+1920 A b c^4 d^3+1920 A b c^4 d^2 e x+960 A b c^4 d e^2 x^2+192 A b c^4 e^3 x^3+3840 A c^5 d^3 x+7680 A c^5 d^2 e x^2+5760 A c^5 d e^2 x^3+1536 A c^5 e^3 x^4+315 b^5 B e^3-1260 b^4 B c d e^2-210 b^4 B c e^3 x+1800 b^3 B c^2 d^2 e+840 b^3 B c^2 d e^2 x+168 b^3 B c^2 e^3 x^2-960 b^2 B c^3 d^3-1200 b^2 B c^3 d^2 e x-672 b^2 B c^3 d e^2 x^2-144 b^2 B c^3 e^3 x^3+640 b B c^4 d^3 x+960 b B c^4 d^2 e x^2+576 b B c^4 d e^2 x^3+128 b B c^4 e^3 x^4+2560 B c^5 d^3 x^2+5760 B c^5 d^2 e x^3+4608 B c^5 d e^2 x^4+1280 B c^5 e^3 x^5\right )}{7680 c^5}+\frac {\log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (-28 A b^5 c e^3+120 A b^4 c^2 d e^2-192 A b^3 c^3 d^2 e+128 A b^2 c^4 d^3+21 b^6 B e^3-84 b^5 B c d e^2+120 b^4 B c^2 d^2 e-64 b^3 B c^3 d^3\right )}{1024 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(-960*b^2*B*c^3*d^3 + 1920*A*b*c^4*d^3 + 1800*b^3*B*c^2*d^2*e - 2880*A*b^2*c^3*d^2*e - 1260
*b^4*B*c*d*e^2 + 1800*A*b^3*c^2*d*e^2 + 315*b^5*B*e^3 - 420*A*b^4*c*e^3 + 640*b*B*c^4*d^3*x + 3840*A*c^5*d^3*x
- 1200*b^2*B*c^3*d^2*e*x + 1920*A*b*c^4*d^2*e*x + 840*b^3*B*c^2*d*e^2*x - 1200*A*b^2*c^3*d*e^2*x - 210*b^4*B*
c*e^3*x + 280*A*b^3*c^2*e^3*x + 2560*B*c^5*d^3*x^2 + 960*b*B*c^4*d^2*e*x^2 + 7680*A*c^5*d^2*e*x^2 - 672*b^2*B*
c^3*d*e^2*x^2 + 960*A*b*c^4*d*e^2*x^2 + 168*b^3*B*c^2*e^3*x^2 - 224*A*b^2*c^3*e^3*x^2 + 5760*B*c^5*d^2*e*x^3 +
576*b*B*c^4*d*e^2*x^3 + 5760*A*c^5*d*e^2*x^3 - 144*b^2*B*c^3*e^3*x^3 + 192*A*b*c^4*e^3*x^3 + 4608*B*c^5*d*e^2
*x^4 + 128*b*B*c^4*e^3*x^4 + 1536*A*c^5*e^3*x^4 + 1280*B*c^5*e^3*x^5))/(7680*c^5) + ((-64*b^3*B*c^3*d^3 + 128*
A*b^2*c^4*d^3 + 120*b^4*B*c^2*d^2*e - 192*A*b^3*c^3*d^2*e - 84*b^5*B*c*d*e^2 + 120*A*b^4*c^2*d*e^2 + 21*b^6*B*
e^3 - 28*A*b^5*c*e^3)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*x^2]])/(1024*c^(11/2))

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fricas [A]  time = 0.49, size = 1022, normalized size = 2.53

result too large to display

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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giac [A]  time = 0.23, size = 521, normalized size = 1.29 \begin {gather*} \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 {gather*}

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)

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maple [B]  time = 0.06, size = 1027, normalized size = 2.54

result too large to display

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]

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

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maxima [A]  time = 0.76, size = 760, normalized size = 1.88 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B e^{3} x^{3}}{6 \, c} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b e^{3} x^{2}}{20 \, c^{2}} + \frac {1}{2} \, \sqrt {c x^{2} + b x} A d^{3} x + \frac {21 \, \sqrt {c x^{2} + b x} B b^{4} e^{3} x}{256 \, c^{4}} + \frac {21 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{2} e^{3} x}{160 \, c^{3}} - \frac {A b^{2} d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {3}{2}}} - \frac {21 \, B b^{6} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {11}{2}}} + \frac {\sqrt {c x^{2} + b x} A b d^{3}}{4 \, c} + \frac {21 \, \sqrt {c x^{2} + b x} B b^{5} e^{3}}{512 \, c^{5}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} e^{3}}{64 \, c^{4}} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} x^{2}}{5 \, c} - \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} b^{3} x}{64 \, c^{3}} - \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b x}{40 \, c^{2}} + \frac {15 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {c x^{2} + b x} b^{2} x}{32 \, c^{2}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} x}{4 \, c} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt {c x^{2} + b x} b x}{4 \, c} + \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {9}{2}}} - \frac {15 \, {\left (B d^{2} e + A d e^{2}\right )} b^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {7}{2}}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {5}{2}}} - \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} b^{4}}{128 \, c^{4}} + \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}}{48 \, c^{3}} + \frac {15 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {c x^{2} + b x} b^{3}}{64 \, c^{3}} - \frac {5 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b}{8 \, c^{2}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt {c x^{2} + b x} b^{2}}{8 \, c^{2}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{3 \, c} \end {gather*}

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]

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

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mupad [B]  time = 3.84, size = 827, normalized size = 2.05 \begin {gather*} A\,d^3\,\sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )-\frac {7\,A\,b\,e^3\,\left (\frac {x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,b\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}\right )}{10\,c}+\frac {A\,e^3\,x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}}{5\,c}+\frac {B\,e^3\,x^3\,{\left (c\,x^2+b\,x\right )}^{3/2}}{6\,c}-\frac {A\,b^2\,d^3\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}}+\frac {B\,b^3\,d^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {B\,d^3\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}+\frac {3\,B\,b\,e^3\,\left (\frac {7\,b\,\left (\frac {x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,b\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}\right )}{10\,c}-\frac {x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}}{5\,c}\right )}{4\,c}+\frac {3\,A\,b^3\,d^2\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {A\,d^2\,e\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{8\,c^2}+\frac {3\,A\,d\,e^2\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}+\frac {3\,B\,d^2\,e\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {15\,A\,b\,d\,e^2\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}-\frac {15\,B\,b\,d^2\,e\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}-\frac {21\,B\,b\,d\,e^2\,\left (\frac {x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,b\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}\right )}{10\,c}+\frac {3\,B\,d\,e^2\,x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}}{5\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*d^3*(b*x + c*x^2)^(1/2)*(x/2 + b/(4*c)) - (7*A*b*e^3*((x*(b*x + c*x^2)^(3/2))/(4*c) - (5*b*((b^3*log((b + 2*
c*x)/c^(1/2) + 2*(b*x + c*x^2)^(1/2)))/(16*c^(5/2)) + ((b*x + c*x^2)^(1/2)*(8*c^2*x^2 - 3*b^2 + 2*b*c*x))/(24*
c^2)))/(8*c)))/(10*c) + (A*e^3*x^2*(b*x + c*x^2)^(3/2))/(5*c) + (B*e^3*x^3*(b*x + c*x^2)^(3/2))/(6*c) - (A*b^2
*d^3*log((b/2 + c*x)/c^(1/2) + (b*x + c*x^2)^(1/2)))/(8*c^(3/2)) + (B*b^3*d^3*log((b + 2*c*x)/c^(1/2) + 2*(b*x
+ c*x^2)^(1/2)))/(16*c^(5/2)) + (B*d^3*(b*x + c*x^2)^(1/2)*(8*c^2*x^2 - 3*b^2 + 2*b*c*x))/(24*c^2) + (3*B*b*e
^3*((7*b*((x*(b*x + c*x^2)^(3/2))/(4*c) - (5*b*((b^3*log((b + 2*c*x)/c^(1/2) + 2*(b*x + c*x^2)^(1/2)))/(16*c^(
5/2)) + ((b*x + c*x^2)^(1/2)*(8*c^2*x^2 - 3*b^2 + 2*b*c*x))/(24*c^2)))/(8*c)))/(10*c) - (x^2*(b*x + c*x^2)^(3/
2))/(5*c)))/(4*c) + (3*A*b^3*d^2*e*log((b + 2*c*x)/c^(1/2) + 2*(b*x + c*x^2)^(1/2)))/(16*c^(5/2)) + (A*d^2*e*(
b*x + c*x^2)^(1/2)*(8*c^2*x^2 - 3*b^2 + 2*b*c*x))/(8*c^2) + (3*A*d*e^2*x*(b*x + c*x^2)^(3/2))/(4*c) + (3*B*d^2
*e*x*(b*x + c*x^2)^(3/2))/(4*c) - (15*A*b*d*e^2*((b^3*log((b + 2*c*x)/c^(1/2) + 2*(b*x + c*x^2)^(1/2)))/(16*c^
(5/2)) + ((b*x + c*x^2)^(1/2)*(8*c^2*x^2 - 3*b^2 + 2*b*c*x))/(24*c^2)))/(8*c) - (15*B*b*d^2*e*((b^3*log((b + 2
*c*x)/c^(1/2) + 2*(b*x + c*x^2)^(1/2)))/(16*c^(5/2)) + ((b*x + c*x^2)^(1/2)*(8*c^2*x^2 - 3*b^2 + 2*b*c*x))/(24
*c^2)))/(8*c) - (21*B*b*d*e^2*((x*(b*x + c*x^2)^(3/2))/(4*c) - (5*b*((b^3*log((b + 2*c*x)/c^(1/2) + 2*(b*x + c
*x^2)^(1/2)))/(16*c^(5/2)) + ((b*x + c*x^2)^(1/2)*(8*c^2*x^2 - 3*b^2 + 2*b*c*x))/(24*c^2)))/(8*c)))/(10*c) + (
3*B*d*e^2*x^2*(b*x + c*x^2)^(3/2))/(5*c)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x \left (b + c x\right )} \left (A + B x\right ) \left (d + e x\right )^{3}\, dx \end {gather*}

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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