3.16.57 \(\int (b+2 c x) (d+e x)^3 (a+b x+c x^2)^{3/2} \, dx\) [1557]

Optimal. Leaf size=379 \[ -\frac {3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac {3 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{11/2}} \]

[Out]

1/1024*(-4*a*c+b^2)*e*(32*c^2*d^2+9*b^2*e^2-4*c*e*(a*e+8*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^4+3/56*(-b*e+2*
c*d)*(e*x+d)^2*(c*x^2+b*x+a)^(5/2)/c+1/4*(e*x+d)^3*(c*x^2+b*x+a)^(5/2)+1/2240*(96*c^3*d^3-63*b^3*e^3+4*b*c*e^2
*(61*a*e+56*b*d)-8*c^2*d*e*(96*a*e+13*b*d)+10*c*e*(8*c^2*d^2+9*b^2*e^2-4*c*e*(7*a*e+2*b*d))*x)*(c*x^2+b*x+a)^(
5/2)/c^3+3/16384*(-4*a*c+b^2)^3*e*(32*c^2*d^2+9*b^2*e^2-4*c*e*(a*e+8*b*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^
2+b*x+a)^(1/2))/c^(11/2)-3/8192*(-4*a*c+b^2)^2*e*(32*c^2*d^2+9*b^2*e^2-4*c*e*(a*e+8*b*d))*(2*c*x+b)*(c*x^2+b*x
+a)^(1/2)/c^5

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Rubi [A]
time = 0.33, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {846, 793, 626, 635, 212} \begin {gather*} \frac {3 e \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{11/2}}-\frac {3 e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{8192 c^5}+\frac {e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{1024 c^4}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (10 c e x \left (-4 c e (7 a e+2 b d)+9 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (96 a e+13 b d)+4 b c e^2 (61 a e+56 b d)-63 b^3 e^3+96 c^3 d^3\right )}{2240 c^3}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {3 (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{56 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(b^2 - 4*a*c)^2*e*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*
c^5) + ((b^2 - 4*a*c)*e*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(1
024*c^4) + (3*(2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(56*c) + ((d + e*x)^3*(a + b*x + c*x^2)^(5/2)
)/4 + ((96*c^3*d^3 - 63*b^3*e^3 + 4*b*c*e^2*(56*b*d + 61*a*e) - 8*c^2*d*e*(13*b*d + 96*a*e) + 10*c*e*(8*c^2*d^
2 + 9*b^2*e^2 - 4*c*e*(2*b*d + 7*a*e))*x)*(a + b*x + c*x^2)^(5/2))/(2240*c^3) + (3*(b^2 - 4*a*c)^3*e*(32*c^2*d
^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16384*c^(11/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

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 635

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 793

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 846

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 (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\int (d+e x)^2 (3 c (b d-2 a e)+3 c (2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2} \, dx}{8 c}\\ &=\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\int (d+e x) \left (\frac {3}{2} c \left (5 b^2 d e-36 a c d e+4 b \left (c d^2+a e^2\right )\right )+\frac {3}{2} c \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{56 c^2}\\ &=\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac {\left (\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{128 c^3}\\ &=\frac {\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}-\frac {\left (3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{2048 c^4}\\ &=-\frac {3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac {\left (3 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16384 c^5}\\ &=-\frac {3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac {\left (3 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8192 c^5}\\ &=-\frac {3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac {3 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 3.24, size = 613, normalized size = 1.62 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-945 b^7 e^3+210 b^6 c e^2 (16 d+3 e x)-192 a^3 c^3 e^2 (512 c d-221 b e+70 c e x)-128 b^3 c^4 e x^2 \left (14 d^2+12 d e x+3 e^2 x^2\right )-56 b^5 c^2 e \left (60 d^2+40 d e x+9 e^2 x^2\right )+16 b^4 c^3 e x \left (140 d^2+112 d e x+27 e^2 x^2\right )+2048 c^7 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+1024 b c^6 x^3 \left (224 d^3+532 d^2 e x+440 d e^2 x^2+125 e^3 x^3\right )+256 b^2 c^5 x^2 \left (448 d^3+966 d^2 e x+752 d e^2 x^2+205 e^3 x^3\right )+16 a^2 c^2 \left (-2359 b^3 e^3+6 b^2 c e^2 (1232 d+199 e x)-24 b c^2 e \left (308 d^2+152 d e x+29 e^2 x^2\right )+16 c^3 \left (448 d^3+420 d^2 e x+192 d e^2 x^2+35 e^3 x^3\right )\right )+4 a c \left (2625 b^5 e^3-14 b^4 c e^2 (640 d+113 e x)-96 b^2 c^3 e x \left (56 d^2+40 d e x+9 e^2 x^2\right )+16 b^3 c^2 e \left (560 d^2+336 d e x+71 e^2 x^2\right )+256 c^5 x^2 \left (224 d^3+490 d^2 e x+384 d e^2 x^2+105 e^3 x^3\right )+128 b c^4 x \left (448 d^3+798 d^2 e x+556 d e^2 x^2+141 e^3 x^3\right )\right )\right )-105 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{573440 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-945*b^7*e^3 + 210*b^6*c*e^2*(16*d + 3*e*x) - 192*a^3*c^3*e^2*(512*c*d - 221
*b*e + 70*c*e*x) - 128*b^3*c^4*e*x^2*(14*d^2 + 12*d*e*x + 3*e^2*x^2) - 56*b^5*c^2*e*(60*d^2 + 40*d*e*x + 9*e^2
*x^2) + 16*b^4*c^3*e*x*(140*d^2 + 112*d*e*x + 27*e^2*x^2) + 2048*c^7*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2
+ 35*e^3*x^3) + 1024*b*c^6*x^3*(224*d^3 + 532*d^2*e*x + 440*d*e^2*x^2 + 125*e^3*x^3) + 256*b^2*c^5*x^2*(448*d
^3 + 966*d^2*e*x + 752*d*e^2*x^2 + 205*e^3*x^3) + 16*a^2*c^2*(-2359*b^3*e^3 + 6*b^2*c*e^2*(1232*d + 199*e*x) -
24*b*c^2*e*(308*d^2 + 152*d*e*x + 29*e^2*x^2) + 16*c^3*(448*d^3 + 420*d^2*e*x + 192*d*e^2*x^2 + 35*e^3*x^3))
+ 4*a*c*(2625*b^5*e^3 - 14*b^4*c*e^2*(640*d + 113*e*x) - 96*b^2*c^3*e*x*(56*d^2 + 40*d*e*x + 9*e^2*x^2) + 16*b
^3*c^2*e*(560*d^2 + 336*d*e*x + 71*e^2*x^2) + 256*c^5*x^2*(224*d^3 + 490*d^2*e*x + 384*d*e^2*x^2 + 105*e^3*x^3
) + 128*b*c^4*x*(448*d^3 + 798*d^2*e*x + 556*d*e^2*x^2 + 141*e^3*x^3))) - 105*(b^2 - 4*a*c)^3*e*(32*c^2*d^2 +
9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(573440*c^(11/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1684\) vs. \(2(349)=698\).
time = 1.07, size = 1685, normalized size = 4.45

method	result	size
risch	\(\frac {\left (71680 c^{7} e^{3} x^{7}+128000 b \,c^{6} e^{3} x^{6}+245760 c^{7} d \,e^{2} x^{6}+107520 a \,c^{6} e^{3} x^{5}+52480 b^{2} c^{5} e^{3} x^{5}+450560 b \,c^{6} d \,e^{2} x^{5}+286720 c^{7} d^{2} e \,x^{5}+72192 a b \,c^{5} e^{3} x^{4}+393216 a \,c^{6} d \,e^{2} x^{4}-384 b^{3} c^{4} e^{3} x^{4}+192512 b^{2} c^{5} d \,e^{2} x^{4}+544768 b \,c^{6} d^{2} e \,x^{4}+114688 d^{3} c^{7} x^{4}+8960 a^{2} c^{5} e^{3} x^{3}-3456 a \,b^{2} c^{4} e^{3} x^{3}+284672 a b \,c^{5} d \,e^{2} x^{3}+501760 a \,c^{6} d^{2} e \,x^{3}+432 b^{4} c^{3} e^{3} x^{3}-1536 b^{3} c^{4} d \,e^{2} x^{3}+247296 b^{2} c^{5} d^{2} e \,x^{3}+229376 b \,c^{6} d^{3} x^{3}-11136 a^{2} b \,c^{4} e^{3} x^{2}+49152 a^{2} c^{5} d \,e^{2} x^{2}+4544 a \,b^{3} c^{3} e^{3} x^{2}-15360 a \,b^{2} c^{4} d \,e^{2} x^{2}+408576 a b \,c^{5} d^{2} e \,x^{2}+229376 a \,c^{6} d^{3} x^{2}-504 b^{5} c^{2} e^{3} x^{2}+1792 b^{4} c^{3} d \,e^{2} x^{2}-1792 b^{3} c^{4} d^{2} e \,x^{2}+114688 b^{2} c^{5} d^{3} x^{2}-13440 a^{3} c^{4} e^{3} x +19104 a^{2} b^{2} c^{3} e^{3} x -58368 a^{2} b \,c^{4} d \,e^{2} x +107520 a^{2} c^{5} d^{2} e x -6328 a \,b^{4} c^{2} e^{3} x +21504 a \,b^{3} c^{3} d \,e^{2} x -21504 a \,b^{2} c^{4} d^{2} e x +229376 a b \,c^{5} d^{3} x +630 b^{6} c \,e^{3} x -2240 b^{5} c^{2} d \,e^{2} x +2240 b^{4} c^{3} d^{2} e x +42432 a^{3} b \,c^{3} e^{3}-98304 a^{3} c^{4} d \,e^{2}-37744 a^{2} b^{3} c^{2} e^{3}+118272 a^{2} b^{2} c^{3} d \,e^{2}-118272 a^{2} b \,c^{4} d^{2} e +114688 a^{2} c^{5} d^{3}+10500 a \,b^{5} c \,e^{3}-35840 a \,b^{4} c^{2} d \,e^{2}+35840 a \,b^{3} c^{3} d^{2} e -945 b^{7} e^{3}+3360 b^{6} c d \,e^{2}-3360 b^{5} c^{2} d^{2} e \right ) \sqrt {c \,x^{2}+b x +a}}{286720 c^{5}}+\frac {3 e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{4}}{64 c^{\frac {3}{2}}}-\frac {9 e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3} b^{2}}{64 c^{\frac {5}{2}}}+\frac {3 e^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3} b d}{8 c^{\frac {3}{2}}}-\frac {3 e \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3} d^{2}}{8 \sqrt {c}}+\frac {45 e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{4}}{512 c^{\frac {7}{2}}}-\frac {9 e^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{3} d}{32 c^{\frac {5}{2}}}+\frac {9 e \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{2} d^{2}}{32 c^{\frac {3}{2}}}-\frac {21 e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{6}}{1024 c^{\frac {9}{2}}}+\frac {9 e^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{5} d}{128 c^{\frac {7}{2}}}-\frac {9 e \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{4} d^{2}}{128 c^{\frac {5}{2}}}+\frac {27 e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{8}}{16384 c^{\frac {11}{2}}}-\frac {3 e^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{7} d}{512 c^{\frac {9}{2}}}+\frac {3 e \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{6} d^{2}}{512 c^{\frac {7}{2}}}\)	\(1219\)
default	\(\text {Expression too large to display}\)	\(1685\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c*e^3*(1/8*x^3*(c*x^2+b*x+a)^(5/2)/c-11/16*b/c*(1/7*x^2*(c*x^2+b*x+a)^(5/2)/c-9/14*b/c*(1/6*x*(c*x^2+b*x+a)^
(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1
/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*
a/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^
2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-2/7*a/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c
*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))-3/8*a/c*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x+a)
^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+
1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3
/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))))))+(b*e^3+6*c*d*e^2)*(1/7*x^2*(c*x^2+b*x+a)^(5/2)/c-9/14*b/c*(1/6*x*(c*x^2+b*x+a)^(5/2)/c
-7/12*b/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c
*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/
8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3
/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-2/7*a/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(
c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b
+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))+(3*b*d*e^2+6*c*d^2*e)*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(c*x^2
+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(
1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)*(c*x^2+b*
x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2)))))+(3*b*d^2*e+2*c*d^3)*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x
+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2)))))+b*d^3*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x
^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 774 vs. \(2 (361) = 722\).
time = 4.38, size = 1551, normalized size = 4.09 \begin {gather*} \left [\frac {105 \, {\left (32 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} e - 32 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d e^{2} + {\left (9 \, b^{8} - 112 \, a b^{6} c + 480 \, a^{2} b^{4} c^{2} - 768 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} e^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (114688 \, c^{8} d^{3} x^{4} + 229376 \, b c^{7} d^{3} x^{3} + 229376 \, a b c^{6} d^{3} x + 114688 \, a^{2} c^{6} d^{3} + 114688 \, {\left (b^{2} c^{6} + 2 \, a c^{7}\right )} d^{3} x^{2} + {\left (71680 \, c^{8} x^{7} + 128000 \, b c^{7} x^{6} - 945 \, b^{7} c + 10500 \, a b^{5} c^{2} - 37744 \, a^{2} b^{3} c^{3} + 42432 \, a^{3} b c^{4} + 1280 \, {\left (41 \, b^{2} c^{6} + 84 \, a c^{7}\right )} x^{5} - 384 \, {\left (b^{3} c^{5} - 188 \, a b c^{6}\right )} x^{4} + 16 \, {\left (27 \, b^{4} c^{4} - 216 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{3} - 8 \, {\left (63 \, b^{5} c^{3} - 568 \, a b^{3} c^{4} + 1392 \, a^{2} b c^{5}\right )} x^{2} + 2 \, {\left (315 \, b^{6} c^{2} - 3164 \, a b^{4} c^{3} + 9552 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x\right )} e^{3} + 32 \, {\left (7680 \, c^{8} d x^{6} + 14080 \, b c^{7} d x^{5} + 128 \, {\left (47 \, b^{2} c^{6} + 96 \, a c^{7}\right )} d x^{4} - 16 \, {\left (3 \, b^{3} c^{5} - 556 \, a b c^{6}\right )} d x^{3} + 8 \, {\left (7 \, b^{4} c^{4} - 60 \, a b^{2} c^{5} + 192 \, a^{2} c^{6}\right )} d x^{2} - 2 \, {\left (35 \, b^{5} c^{3} - 336 \, a b^{3} c^{4} + 912 \, a^{2} b c^{5}\right )} d x + {\left (105 \, b^{6} c^{2} - 1120 \, a b^{4} c^{3} + 3696 \, a^{2} b^{2} c^{4} - 3072 \, a^{3} c^{5}\right )} d\right )} e^{2} + 224 \, {\left (1280 \, c^{8} d^{2} x^{5} + 2432 \, b c^{7} d^{2} x^{4} + 16 \, {\left (69 \, b^{2} c^{6} + 140 \, a c^{7}\right )} d^{2} x^{3} - 8 \, {\left (b^{3} c^{5} - 228 \, a b c^{6}\right )} d^{2} x^{2} + 2 \, {\left (5 \, b^{4} c^{4} - 48 \, a b^{2} c^{5} + 240 \, a^{2} c^{6}\right )} d^{2} x - {\left (15 \, b^{5} c^{3} - 160 \, a b^{3} c^{4} + 528 \, a^{2} b c^{5}\right )} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{1146880 \, c^{6}}, -\frac {105 \, {\left (32 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} e - 32 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d e^{2} + {\left (9 \, b^{8} - 112 \, a b^{6} c + 480 \, a^{2} b^{4} c^{2} - 768 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (114688 \, c^{8} d^{3} x^{4} + 229376 \, b c^{7} d^{3} x^{3} + 229376 \, a b c^{6} d^{3} x + 114688 \, a^{2} c^{6} d^{3} + 114688 \, {\left (b^{2} c^{6} + 2 \, a c^{7}\right )} d^{3} x^{2} + {\left (71680 \, c^{8} x^{7} + 128000 \, b c^{7} x^{6} - 945 \, b^{7} c + 10500 \, a b^{5} c^{2} - 37744 \, a^{2} b^{3} c^{3} + 42432 \, a^{3} b c^{4} + 1280 \, {\left (41 \, b^{2} c^{6} + 84 \, a c^{7}\right )} x^{5} - 384 \, {\left (b^{3} c^{5} - 188 \, a b c^{6}\right )} x^{4} + 16 \, {\left (27 \, b^{4} c^{4} - 216 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{3} - 8 \, {\left (63 \, b^{5} c^{3} - 568 \, a b^{3} c^{4} + 1392 \, a^{2} b c^{5}\right )} x^{2} + 2 \, {\left (315 \, b^{6} c^{2} - 3164 \, a b^{4} c^{3} + 9552 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x\right )} e^{3} + 32 \, {\left (7680 \, c^{8} d x^{6} + 14080 \, b c^{7} d x^{5} + 128 \, {\left (47 \, b^{2} c^{6} + 96 \, a c^{7}\right )} d x^{4} - 16 \, {\left (3 \, b^{3} c^{5} - 556 \, a b c^{6}\right )} d x^{3} + 8 \, {\left (7 \, b^{4} c^{4} - 60 \, a b^{2} c^{5} + 192 \, a^{2} c^{6}\right )} d x^{2} - 2 \, {\left (35 \, b^{5} c^{3} - 336 \, a b^{3} c^{4} + 912 \, a^{2} b c^{5}\right )} d x + {\left (105 \, b^{6} c^{2} - 1120 \, a b^{4} c^{3} + 3696 \, a^{2} b^{2} c^{4} - 3072 \, a^{3} c^{5}\right )} d\right )} e^{2} + 224 \, {\left (1280 \, c^{8} d^{2} x^{5} + 2432 \, b c^{7} d^{2} x^{4} + 16 \, {\left (69 \, b^{2} c^{6} + 140 \, a c^{7}\right )} d^{2} x^{3} - 8 \, {\left (b^{3} c^{5} - 228 \, a b c^{6}\right )} d^{2} x^{2} + 2 \, {\left (5 \, b^{4} c^{4} - 48 \, a b^{2} c^{5} + 240 \, a^{2} c^{6}\right )} d^{2} x - {\left (15 \, b^{5} c^{3} - 160 \, a b^{3} c^{4} + 528 \, a^{2} b c^{5}\right )} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{573440 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/1146880*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e - 32*(b^7*c - 12*a*b^5*c^2 +
48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e^2 + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768*a^3*b^2*c^3 + 256*a^4*c^4)
*e^3)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(11468
8*c^8*d^3*x^4 + 229376*b*c^7*d^3*x^3 + 229376*a*b*c^6*d^3*x + 114688*a^2*c^6*d^3 + 114688*(b^2*c^6 + 2*a*c^7)*
d^3*x^2 + (71680*c^8*x^7 + 128000*b*c^7*x^6 - 945*b^7*c + 10500*a*b^5*c^2 - 37744*a^2*b^3*c^3 + 42432*a^3*b*c^
4 + 1280*(41*b^2*c^6 + 84*a*c^7)*x^5 - 384*(b^3*c^5 - 188*a*b*c^6)*x^4 + 16*(27*b^4*c^4 - 216*a*b^2*c^5 + 560*
a^2*c^6)*x^3 - 8*(63*b^5*c^3 - 568*a*b^3*c^4 + 1392*a^2*b*c^5)*x^2 + 2*(315*b^6*c^2 - 3164*a*b^4*c^3 + 9552*a^
2*b^2*c^4 - 6720*a^3*c^5)*x)*e^3 + 32*(7680*c^8*d*x^6 + 14080*b*c^7*d*x^5 + 128*(47*b^2*c^6 + 96*a*c^7)*d*x^4
- 16*(3*b^3*c^5 - 556*a*b*c^6)*d*x^3 + 8*(7*b^4*c^4 - 60*a*b^2*c^5 + 192*a^2*c^6)*d*x^2 - 2*(35*b^5*c^3 - 336*
a*b^3*c^4 + 912*a^2*b*c^5)*d*x + (105*b^6*c^2 - 1120*a*b^4*c^3 + 3696*a^2*b^2*c^4 - 3072*a^3*c^5)*d)*e^2 + 224
*(1280*c^8*d^2*x^5 + 2432*b*c^7*d^2*x^4 + 16*(69*b^2*c^6 + 140*a*c^7)*d^2*x^3 - 8*(b^3*c^5 - 228*a*b*c^6)*d^2*
x^2 + 2*(5*b^4*c^4 - 48*a*b^2*c^5 + 240*a^2*c^6)*d^2*x - (15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^5)*d^2)*e)*
sqrt(c*x^2 + b*x + a))/c^6, -1/573440*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e -
32*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e^2 + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768
*a^3*b^2*c^3 + 256*a^4*c^4)*e^3)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c
*x + a*c)) - 2*(114688*c^8*d^3*x^4 + 229376*b*c^7*d^3*x^3 + 229376*a*b*c^6*d^3*x + 114688*a^2*c^6*d^3 + 114688
*(b^2*c^6 + 2*a*c^7)*d^3*x^2 + (71680*c^8*x^7 + 128000*b*c^7*x^6 - 945*b^7*c + 10500*a*b^5*c^2 - 37744*a^2*b^3
*c^3 + 42432*a^3*b*c^4 + 1280*(41*b^2*c^6 + 84*a*c^7)*x^5 - 384*(b^3*c^5 - 188*a*b*c^6)*x^4 + 16*(27*b^4*c^4 -
216*a*b^2*c^5 + 560*a^2*c^6)*x^3 - 8*(63*b^5*c^3 - 568*a*b^3*c^4 + 1392*a^2*b*c^5)*x^2 + 2*(315*b^6*c^2 - 316
4*a*b^4*c^3 + 9552*a^2*b^2*c^4 - 6720*a^3*c^5)*x)*e^3 + 32*(7680*c^8*d*x^6 + 14080*b*c^7*d*x^5 + 128*(47*b^2*c
^6 + 96*a*c^7)*d*x^4 - 16*(3*b^3*c^5 - 556*a*b*c^6)*d*x^3 + 8*(7*b^4*c^4 - 60*a*b^2*c^5 + 192*a^2*c^6)*d*x^2 -
2*(35*b^5*c^3 - 336*a*b^3*c^4 + 912*a^2*b*c^5)*d*x + (105*b^6*c^2 - 1120*a*b^4*c^3 + 3696*a^2*b^2*c^4 - 3072*
a^3*c^5)*d)*e^2 + 224*(1280*c^8*d^2*x^5 + 2432*b*c^7*d^2*x^4 + 16*(69*b^2*c^6 + 140*a*c^7)*d^2*x^3 - 8*(b^3*c^
5 - 228*a*b*c^6)*d^2*x^2 + 2*(5*b^4*c^4 - 48*a*b^2*c^5 + 240*a^2*c^6)*d^2*x - (15*b^5*c^3 - 160*a*b^3*c^4 + 52
8*a^2*b*c^5)*d^2)*e)*sqrt(c*x^2 + b*x + a))/c^6]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b + 2 c x\right ) \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b + 2*c*x)*(d + e*x)**3*(a + b*x + c*x**2)**(3/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 856 vs. \(2 (361) = 722\).
time = 4.67, size = 856, normalized size = 2.26 \begin {gather*} \frac {1}{286720} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, c^{2} x e^{3} + \frac {48 \, c^{9} d e^{2} + 25 \, b c^{8} e^{3}}{c^{7}}\right )} x + \frac {224 \, c^{9} d^{2} e + 352 \, b c^{8} d e^{2} + 41 \, b^{2} c^{7} e^{3} + 84 \, a c^{8} e^{3}}{c^{7}}\right )} x + \frac {896 \, c^{9} d^{3} + 4256 \, b c^{8} d^{2} e + 1504 \, b^{2} c^{7} d e^{2} + 3072 \, a c^{8} d e^{2} - 3 \, b^{3} c^{6} e^{3} + 564 \, a b c^{7} e^{3}}{c^{7}}\right )} x + \frac {14336 \, b c^{8} d^{3} + 15456 \, b^{2} c^{7} d^{2} e + 31360 \, a c^{8} d^{2} e - 96 \, b^{3} c^{6} d e^{2} + 17792 \, a b c^{7} d e^{2} + 27 \, b^{4} c^{5} e^{3} - 216 \, a b^{2} c^{6} e^{3} + 560 \, a^{2} c^{7} e^{3}}{c^{7}}\right )} x + \frac {14336 \, b^{2} c^{7} d^{3} + 28672 \, a c^{8} d^{3} - 224 \, b^{3} c^{6} d^{2} e + 51072 \, a b c^{7} d^{2} e + 224 \, b^{4} c^{5} d e^{2} - 1920 \, a b^{2} c^{6} d e^{2} + 6144 \, a^{2} c^{7} d e^{2} - 63 \, b^{5} c^{4} e^{3} + 568 \, a b^{3} c^{5} e^{3} - 1392 \, a^{2} b c^{6} e^{3}}{c^{7}}\right )} x + \frac {114688 \, a b c^{7} d^{3} + 1120 \, b^{4} c^{5} d^{2} e - 10752 \, a b^{2} c^{6} d^{2} e + 53760 \, a^{2} c^{7} d^{2} e - 1120 \, b^{5} c^{4} d e^{2} + 10752 \, a b^{3} c^{5} d e^{2} - 29184 \, a^{2} b c^{6} d e^{2} + 315 \, b^{6} c^{3} e^{3} - 3164 \, a b^{4} c^{4} e^{3} + 9552 \, a^{2} b^{2} c^{5} e^{3} - 6720 \, a^{3} c^{6} e^{3}}{c^{7}}\right )} x + \frac {114688 \, a^{2} c^{7} d^{3} - 3360 \, b^{5} c^{4} d^{2} e + 35840 \, a b^{3} c^{5} d^{2} e - 118272 \, a^{2} b c^{6} d^{2} e + 3360 \, b^{6} c^{3} d e^{2} - 35840 \, a b^{4} c^{4} d e^{2} + 118272 \, a^{2} b^{2} c^{5} d e^{2} - 98304 \, a^{3} c^{6} d e^{2} - 945 \, b^{7} c^{2} e^{3} + 10500 \, a b^{5} c^{3} e^{3} - 37744 \, a^{2} b^{3} c^{4} e^{3} + 42432 \, a^{3} b c^{5} e^{3}}{c^{7}}\right )} - \frac {3 \, {\left (32 \, b^{6} c^{2} d^{2} e - 384 \, a b^{4} c^{3} d^{2} e + 1536 \, a^{2} b^{2} c^{4} d^{2} e - 2048 \, a^{3} c^{5} d^{2} e - 32 \, b^{7} c d e^{2} + 384 \, a b^{5} c^{2} d e^{2} - 1536 \, a^{2} b^{3} c^{3} d e^{2} + 2048 \, a^{3} b c^{4} d e^{2} + 9 \, b^{8} e^{3} - 112 \, a b^{6} c e^{3} + 480 \, a^{2} b^{4} c^{2} e^{3} - 768 \, a^{3} b^{2} c^{3} e^{3} + 256 \, a^{4} c^{4} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16384 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/286720*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*(14*c^2*x*e^3 + (48*c^9*d*e^2 + 25*b*c^8*e^3)/c^7)*x + (224*
c^9*d^2*e + 352*b*c^8*d*e^2 + 41*b^2*c^7*e^3 + 84*a*c^8*e^3)/c^7)*x + (896*c^9*d^3 + 4256*b*c^8*d^2*e + 1504*b
^2*c^7*d*e^2 + 3072*a*c^8*d*e^2 - 3*b^3*c^6*e^3 + 564*a*b*c^7*e^3)/c^7)*x + (14336*b*c^8*d^3 + 15456*b^2*c^7*d
^2*e + 31360*a*c^8*d^2*e - 96*b^3*c^6*d*e^2 + 17792*a*b*c^7*d*e^2 + 27*b^4*c^5*e^3 - 216*a*b^2*c^6*e^3 + 560*a
^2*c^7*e^3)/c^7)*x + (14336*b^2*c^7*d^3 + 28672*a*c^8*d^3 - 224*b^3*c^6*d^2*e + 51072*a*b*c^7*d^2*e + 224*b^4*
c^5*d*e^2 - 1920*a*b^2*c^6*d*e^2 + 6144*a^2*c^7*d*e^2 - 63*b^5*c^4*e^3 + 568*a*b^3*c^5*e^3 - 1392*a^2*b*c^6*e^
3)/c^7)*x + (114688*a*b*c^7*d^3 + 1120*b^4*c^5*d^2*e - 10752*a*b^2*c^6*d^2*e + 53760*a^2*c^7*d^2*e - 1120*b^5*
c^4*d*e^2 + 10752*a*b^3*c^5*d*e^2 - 29184*a^2*b*c^6*d*e^2 + 315*b^6*c^3*e^3 - 3164*a*b^4*c^4*e^3 + 9552*a^2*b^
2*c^5*e^3 - 6720*a^3*c^6*e^3)/c^7)*x + (114688*a^2*c^7*d^3 - 3360*b^5*c^4*d^2*e + 35840*a*b^3*c^5*d^2*e - 1182
72*a^2*b*c^6*d^2*e + 3360*b^6*c^3*d*e^2 - 35840*a*b^4*c^4*d*e^2 + 118272*a^2*b^2*c^5*d*e^2 - 98304*a^3*c^6*d*e
^2 - 945*b^7*c^2*e^3 + 10500*a*b^5*c^3*e^3 - 37744*a^2*b^3*c^4*e^3 + 42432*a^3*b*c^5*e^3)/c^7) - 3/16384*(32*b
^6*c^2*d^2*e - 384*a*b^4*c^3*d^2*e + 1536*a^2*b^2*c^4*d^2*e - 2048*a^3*c^5*d^2*e - 32*b^7*c*d*e^2 + 384*a*b^5*
c^2*d*e^2 - 1536*a^2*b^3*c^3*d*e^2 + 2048*a^3*b*c^4*d*e^2 + 9*b^8*e^3 - 112*a*b^6*c*e^3 + 480*a^2*b^4*c^2*e^3
- 768*a^3*b^2*c^3*e^3 + 256*a^4*c^4*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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