3.10.36 \(\int x^3 (A+B x) (a+b x+c x^2)^{5/2} \, dx\) [936]

Optimal. Leaf size=432 \[ \frac {\left (b^2-4 a c\right )^2 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{131072 c^7}-\frac {\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}-\frac {\left (b^2-4 a c\right )^3 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{262144 c^{15/2}} \]

[Out]

-1/49152*(-4*a*c+b^2)*(240*A*a*b*c^2-220*A*b^3*c+48*B*a^2*c^2-264*B*a*b^2*c+143*B*b^4)*(2*c*x+b)*(c*x^2+b*x+a)
^(3/2)/c^6+1/15360*(240*A*a*b*c^2-220*A*b^3*c+48*B*a^2*c^2-264*B*a*b^2*c+143*B*b^4)*(2*c*x+b)*(c*x^2+b*x+a)^(5
/2)/c^5-1/180*(-20*A*c+13*B*b)*x^2*(c*x^2+b*x+a)^(7/2)/c^2+1/10*B*x^3*(c*x^2+b*x+a)^(7/2)/c-1/40320*(1287*b^3*
B-1980*A*b^2*c-1804*a*b*B*c+1280*a*A*c^2-14*c*(-220*A*b*c-108*B*a*c+143*B*b^2)*x)*(c*x^2+b*x+a)^(7/2)/c^4-1/26
2144*(-4*a*c+b^2)^3*(240*A*a*b*c^2-220*A*b^3*c+48*B*a^2*c^2-264*B*a*b^2*c+143*B*b^4)*arctanh(1/2*(2*c*x+b)/c^(
1/2)/(c*x^2+b*x+a)^(1/2))/c^(15/2)+1/131072*(-4*a*c+b^2)^2*(240*A*a*b*c^2-220*A*b^3*c+48*B*a^2*c^2-264*B*a*b^2
*c+143*B*b^4)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^7

________________________________________________________________________________________

Rubi [A]
time = 0.32, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {846, 793, 626, 635, 212} \begin {gather*} -\frac {\left (b^2-4 a c\right )^3 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{262144 c^{15/2}}+\frac {\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{131072 c^7}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{49152 c^6}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{15360 c^5}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{40320 c^4}-\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

((b^2 - 4*a*c)^2*(143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^2 + 48*a^2*B*c^2)*(b + 2*c*x)*Sqrt[a +
b*x + c*x^2])/(131072*c^7) - ((b^2 - 4*a*c)*(143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^2 + 48*a^2
*B*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(49152*c^6) + ((143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A
*b*c^2 + 48*a^2*B*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(15360*c^5) - ((13*b*B - 20*A*c)*x^2*(a + b*x + c*
x^2)^(7/2))/(180*c^2) + (B*x^3*(a + b*x + c*x^2)^(7/2))/(10*c) - ((1287*b^3*B - 1980*A*b^2*c - 1804*a*b*B*c +
1280*a*A*c^2 - 14*c*(143*b^2*B - 220*A*b*c - 108*a*B*c)*x)*(a + b*x + c*x^2)^(7/2))/(40320*c^4) - ((b^2 - 4*a*
c)^3*(143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^2 + 48*a^2*B*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*S
qrt[a + b*x + c*x^2])])/(262144*c^(15/2))

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 x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}+\frac {\int x^2 \left (-3 a B-\frac {1}{2} (13 b B-20 A c) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{10 c}\\ &=-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}+\frac {\int x \left (a (13 b B-20 A c)+\frac {1}{4} \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{90 c^2}\\ &=-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{1280 c^4}\\ &=\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}-\frac {\left (\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{6144 c^5}\\ &=-\frac {\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}+\frac {\left (\left (b^2-4 a c\right )^2 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{32768 c^6}\\ &=\frac {\left (b^2-4 a c\right )^2 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{131072 c^7}-\frac {\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}-\frac {\left (\left (b^2-4 a c\right )^3 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{262144 c^7}\\ &=\frac {\left (b^2-4 a c\right )^2 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{131072 c^7}-\frac {\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}-\frac {\left (\left (b^2-4 a c\right )^3 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\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 )}{131072 c^7}\\ &=\frac {\left (b^2-4 a c\right )^2 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{131072 c^7}-\frac {\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}-\frac {\left (b^2-4 a c\right )^3 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{262144 c^{15/2}}\\ \end {align*}

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Mathematica [A]
time = 3.29, size = 585, normalized size = 1.35 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (45045 b^9 B-2310 b^8 c (30 A+13 B x)+1848 b^7 c (-305 a B+c x (25 A+13 B x))-640 b^3 c^3 \left (6885 a^3 B-8 c^3 x^5 (5 A+3 B x)+4 a c^2 x^3 (107 A+60 B x)-3 a^2 c x (879 A+431 B x)\right )-320 b^4 c^3 \left (4 c^2 x^4 (22 A+13 B x)+207 a^2 (49 A+20 B x)-a c x^2 (1116 A+605 B x)\right )+32 b^5 c^2 \left (77742 a^2 B+22 c^2 x^3 (45 A+26 B x)-9 a c x (1715 A+869 B x)\right )+48 b^6 c^2 \left (-11 c x^2 (70 A+39 B x)+7 a (2425 A+1023 B x)\right )+512 c^5 \left (896 c^4 x^8 (10 A+9 B x)+10 a^3 c x^2 (128 A+63 B x)-5 a^4 (512 A+189 B x)+24 a^2 c^2 x^4 (800 A+651 B x)+16 a c^3 x^6 (1520 A+1323 B x)\right )+256 b^2 c^4 \left (120 a c^2 x^4 (7 A+4 B x)-15 a^2 c x^2 (266 A+139 B x)+5 a^3 (3663 A+1433 B x)+8 c^3 x^6 (3090 A+2681 B x)\right )+256 b c^4 \left (9295 a^4 B+60 a^2 c^2 x^3 (41 A+22 B x)+224 c^4 x^7 (185 A+164 B x)-10 a^3 c x (689 A+323 B x)+16 a c^3 x^5 (3765 A+3181 B x)\right )\right )+315 \left (b^2-4 a c\right )^3 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{82575360 c^{15/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(45045*b^9*B - 2310*b^8*c*(30*A + 13*B*x) + 1848*b^7*c*(-305*a*B + c*x*(25*A
+ 13*B*x)) - 640*b^3*c^3*(6885*a^3*B - 8*c^3*x^5*(5*A + 3*B*x) + 4*a*c^2*x^3*(107*A + 60*B*x) - 3*a^2*c*x*(879
*A + 431*B*x)) - 320*b^4*c^3*(4*c^2*x^4*(22*A + 13*B*x) + 207*a^2*(49*A + 20*B*x) - a*c*x^2*(1116*A + 605*B*x)
) + 32*b^5*c^2*(77742*a^2*B + 22*c^2*x^3*(45*A + 26*B*x) - 9*a*c*x*(1715*A + 869*B*x)) + 48*b^6*c^2*(-11*c*x^2
*(70*A + 39*B*x) + 7*a*(2425*A + 1023*B*x)) + 512*c^5*(896*c^4*x^8*(10*A + 9*B*x) + 10*a^3*c*x^2*(128*A + 63*B
*x) - 5*a^4*(512*A + 189*B*x) + 24*a^2*c^2*x^4*(800*A + 651*B*x) + 16*a*c^3*x^6*(1520*A + 1323*B*x)) + 256*b^2
*c^4*(120*a*c^2*x^4*(7*A + 4*B*x) - 15*a^2*c*x^2*(266*A + 139*B*x) + 5*a^3*(3663*A + 1433*B*x) + 8*c^3*x^6*(30
90*A + 2681*B*x)) + 256*b*c^4*(9295*a^4*B + 60*a^2*c^2*x^3*(41*A + 22*B*x) + 224*c^4*x^7*(185*A + 164*B*x) - 1
0*a^3*c*x*(689*A + 323*B*x) + 16*a*c^3*x^5*(3765*A + 3181*B*x))) + 315*(b^2 - 4*a*c)^3*(143*b^4*B - 220*A*b^3*
c - 264*a*b^2*B*c + 240*a*A*b*c^2 + 48*a^2*B*c^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(82575360*
c^(15/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1453\) vs. \(2(398)=796\).
time = 0.76, size = 1454, normalized size = 3.37

method	result	size
risch	\(-\frac {\left (-4128768 B \,c^{9} x^{9}-4587520 A \,c^{9} x^{8}-9404416 B b \,c^{8} x^{8}-10608640 A b \,c^{8} x^{7}-10838016 B a \,c^{8} x^{7}-5490688 B \,b^{2} c^{7} x^{7}-12451840 A a \,c^{8} x^{6}-6328320 A \,b^{2} c^{7} x^{6}-13029376 B a b \,c^{7} x^{6}-15360 B \,b^{3} c^{6} x^{6}-15421440 A a b \,c^{7} x^{5}-25600 A \,b^{3} c^{6} x^{5}-7999488 B \,a^{2} c^{7} x^{5}-122880 B a \,b^{2} c^{6} x^{5}+16640 B \,b^{4} c^{5} x^{5}-9830400 A \,a^{2} c^{7} x^{4}-215040 A a \,b^{2} c^{6} x^{4}+28160 A \,b^{4} c^{5} x^{4}-337920 B \,a^{2} b \,c^{6} x^{4}+153600 B a \,b^{3} c^{5} x^{4}-18304 B \,b^{5} c^{4} x^{4}-629760 A \,a^{2} b \,c^{6} x^{3}+273920 A a \,b^{3} c^{5} x^{3}-31680 A \,b^{5} c^{4} x^{3}-322560 B \,a^{3} c^{6} x^{3}+533760 B \,a^{2} b^{2} c^{5} x^{3}-193600 B a \,b^{4} c^{4} x^{3}+20592 B \,b^{6} c^{3} x^{3}-655360 A \,a^{3} c^{6} x^{2}+1021440 A \,a^{2} b^{2} c^{5} x^{2}-357120 A a \,b^{4} c^{4} x^{2}+36960 A \,b^{6} c^{3} x^{2}+826880 B \,a^{3} b \,c^{5} x^{2}-827520 B \,a^{2} b^{3} c^{4} x^{2}+250272 B a \,b^{5} c^{3} x^{2}-24024 B \,b^{7} c^{2} x^{2}+1763840 A \,a^{3} b \,c^{5} x -1687680 A \,a^{2} b^{3} c^{4} x +493920 A a \,b^{5} c^{3} x -46200 A \,b^{7} c^{2} x +483840 B \,a^{4} c^{5} x -1834240 B \,a^{3} b^{2} c^{4} x +1324800 B \,a^{2} b^{4} c^{3} x -343728 B a \,b^{6} c^{2} x +30030 B \,b^{8} c x +1310720 A \,a^{4} c^{5}-4688640 A \,a^{3} b^{2} c^{4}+3245760 A \,a^{2} b^{4} c^{3}-814800 A a \,b^{6} c^{2}+69300 A \,b^{8} c -2379520 B \,a^{4} b \,c^{4}+4406400 B \,a^{3} b^{3} c^{3}-2487744 B \,a^{2} b^{5} c^{2}+563640 B a \,b^{7} c -45045 B \,b^{9}\right ) \sqrt {c \,x^{2}+b x +a}}{41287680 c^{7}}+\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) A \,a^{4} b}{256 c^{\frac {5}{2}}}-\frac {25 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) A \,a^{3} b^{3}}{256 c^{\frac {7}{2}}}+\frac {105 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) A \,a^{2} b^{5}}{2048 c^{\frac {9}{2}}}-\frac {45 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) A a \,b^{7}}{4096 c^{\frac {11}{2}}}+\frac {55 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) A \,b^{9}}{65536 c^{\frac {13}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) B \,a^{5}}{256 c^{\frac {5}{2}}}-\frac {75 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) B \,a^{4} b^{2}}{1024 c^{\frac {7}{2}}}+\frac {175 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) B \,a^{3} b^{4}}{2048 c^{\frac {9}{2}}}-\frac {315 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) B \,a^{2} b^{6}}{8192 c^{\frac {11}{2}}}+\frac {495 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) B a \,b^{8}}{65536 c^{\frac {13}{2}}}-\frac {143 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) B \,b^{10}}{262144 c^{\frac {15}{2}}}\)	\(1045\)
default	\(\text {Expression too large to display}\)	\(1454\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

B*(1/10*x^3*(c*x^2+b*x+a)^(7/2)/c-13/20*b/c*(1/9*x^2*(c*x^2+b*x+a)^(7/2)/c-11/18*b/c*(1/8*x*(c*x^2+b*x+a)^(7/2
)/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/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))))))-1/8*a/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^
2)/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))))))-2/9*a/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(
2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/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)))))))-3/10
*a/c*(1/8*x*(c*x^2+b*x+a)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5
/2)/c+5/24*(4*a*c-b^2)/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))))))-1/8*a/c*(1/12*(2*c*x+b)*(c*x^2
+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/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)))))))+A*(1/9*x^2*(c*x^2+b
*x+a)^(7/2)/c-11/18*b/c*(1/8*x*(c*x^2+b*x+a)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+
b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/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))))))-1/8*a/c*(1/
12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/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
/9*a/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/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)))))))

________________________________________________________________________________________

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(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),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 [A]
time = 4.04, size = 1511, normalized size = 3.50 \begin {gather*} \left [-\frac {315 \, {\left (143 \, B b^{10} - 3072 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} c^{5} + 6400 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} c^{4} - 4480 \, {\left (5 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5}\right )} c^{3} + 1440 \, {\left (7 \, B a^{2} b^{6} + 2 \, A a b^{7}\right )} c^{2} - 220 \, {\left (9 \, B a b^{8} + A b^{9}\right )} c\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 (4128768 \, B c^{10} x^{9} + 45045 \, B b^{9} c - 1310720 \, A a^{4} c^{6} + 229376 \, {\left (41 \, B b c^{9} + 20 \, A c^{10}\right )} x^{8} + 14336 \, {\left (383 \, B b^{2} c^{8} + 4 \, {\left (189 \, B a + 185 \, A b\right )} c^{9}\right )} x^{7} + 1024 \, {\left (15 \, B b^{3} c^{7} + 12160 \, A a c^{9} + 4 \, {\left (3181 \, B a b + 1545 \, A b^{2}\right )} c^{8}\right )} x^{6} + 14080 \, {\left (169 \, B a^{4} b + 333 \, A a^{3} b^{2}\right )} c^{5} - 256 \, {\left (65 \, B b^{4} c^{6} - 48 \, {\left (651 \, B a^{2} + 1255 \, A a b\right )} c^{8} - 20 \, {\left (24 \, B a b^{2} + 5 \, A b^{3}\right )} c^{7}\right )} x^{5} - 2880 \, {\left (1530 \, B a^{3} b^{3} + 1127 \, A a^{2} b^{4}\right )} c^{4} + 128 \, {\left (143 \, B b^{5} c^{5} + 76800 \, A a^{2} c^{8} + 240 \, {\left (11 \, B a^{2} b + 7 \, A a b^{2}\right )} c^{7} - 20 \, {\left (60 \, B a b^{3} + 11 \, A b^{4}\right )} c^{6}\right )} x^{4} + 336 \, {\left (7404 \, B a^{2} b^{5} + 2425 \, A a b^{6}\right )} c^{3} - 16 \, {\left (1287 \, B b^{6} c^{4} - 960 \, {\left (21 \, B a^{3} + 41 \, A a^{2} b\right )} c^{7} + 80 \, {\left (417 \, B a^{2} b^{2} + 214 \, A a b^{3}\right )} c^{6} - 220 \, {\left (55 \, B a b^{4} + 9 \, A b^{5}\right )} c^{5}\right )} x^{3} - 4620 \, {\left (122 \, B a b^{7} + 15 \, A b^{8}\right )} c^{2} + 8 \, {\left (3003 \, B b^{7} c^{3} + 81920 \, A a^{3} c^{7} - 6080 \, {\left (17 \, B a^{3} b + 21 \, A a^{2} b^{2}\right )} c^{6} + 240 \, {\left (431 \, B a^{2} b^{3} + 186 \, A a b^{4}\right )} c^{5} - 132 \, {\left (237 \, B a b^{5} + 35 \, A b^{6}\right )} c^{4}\right )} x^{2} - 2 \, {\left (15015 \, B b^{8} c^{2} + 1280 \, {\left (189 \, B a^{4} + 689 \, A a^{3} b\right )} c^{6} - 320 \, {\left (2866 \, B a^{3} b^{2} + 2637 \, A a^{2} b^{3}\right )} c^{5} + 720 \, {\left (920 \, B a^{2} b^{4} + 343 \, A a b^{5}\right )} c^{4} - 924 \, {\left (186 \, B a b^{6} + 25 \, A b^{7}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{165150720 \, c^{8}}, \frac {315 \, {\left (143 \, B b^{10} - 3072 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} c^{5} + 6400 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} c^{4} - 4480 \, {\left (5 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5}\right )} c^{3} + 1440 \, {\left (7 \, B a^{2} b^{6} + 2 \, A a b^{7}\right )} c^{2} - 220 \, {\left (9 \, B a b^{8} + A b^{9}\right )} c\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 (4128768 \, B c^{10} x^{9} + 45045 \, B b^{9} c - 1310720 \, A a^{4} c^{6} + 229376 \, {\left (41 \, B b c^{9} + 20 \, A c^{10}\right )} x^{8} + 14336 \, {\left (383 \, B b^{2} c^{8} + 4 \, {\left (189 \, B a + 185 \, A b\right )} c^{9}\right )} x^{7} + 1024 \, {\left (15 \, B b^{3} c^{7} + 12160 \, A a c^{9} + 4 \, {\left (3181 \, B a b + 1545 \, A b^{2}\right )} c^{8}\right )} x^{6} + 14080 \, {\left (169 \, B a^{4} b + 333 \, A a^{3} b^{2}\right )} c^{5} - 256 \, {\left (65 \, B b^{4} c^{6} - 48 \, {\left (651 \, B a^{2} + 1255 \, A a b\right )} c^{8} - 20 \, {\left (24 \, B a b^{2} + 5 \, A b^{3}\right )} c^{7}\right )} x^{5} - 2880 \, {\left (1530 \, B a^{3} b^{3} + 1127 \, A a^{2} b^{4}\right )} c^{4} + 128 \, {\left (143 \, B b^{5} c^{5} + 76800 \, A a^{2} c^{8} + 240 \, {\left (11 \, B a^{2} b + 7 \, A a b^{2}\right )} c^{7} - 20 \, {\left (60 \, B a b^{3} + 11 \, A b^{4}\right )} c^{6}\right )} x^{4} + 336 \, {\left (7404 \, B a^{2} b^{5} + 2425 \, A a b^{6}\right )} c^{3} - 16 \, {\left (1287 \, B b^{6} c^{4} - 960 \, {\left (21 \, B a^{3} + 41 \, A a^{2} b\right )} c^{7} + 80 \, {\left (417 \, B a^{2} b^{2} + 214 \, A a b^{3}\right )} c^{6} - 220 \, {\left (55 \, B a b^{4} + 9 \, A b^{5}\right )} c^{5}\right )} x^{3} - 4620 \, {\left (122 \, B a b^{7} + 15 \, A b^{8}\right )} c^{2} + 8 \, {\left (3003 \, B b^{7} c^{3} + 81920 \, A a^{3} c^{7} - 6080 \, {\left (17 \, B a^{3} b + 21 \, A a^{2} b^{2}\right )} c^{6} + 240 \, {\left (431 \, B a^{2} b^{3} + 186 \, A a b^{4}\right )} c^{5} - 132 \, {\left (237 \, B a b^{5} + 35 \, A b^{6}\right )} c^{4}\right )} x^{2} - 2 \, {\left (15015 \, B b^{8} c^{2} + 1280 \, {\left (189 \, B a^{4} + 689 \, A a^{3} b\right )} c^{6} - 320 \, {\left (2866 \, B a^{3} b^{2} + 2637 \, A a^{2} b^{3}\right )} c^{5} + 720 \, {\left (920 \, B a^{2} b^{4} + 343 \, A a b^{5}\right )} c^{4} - 924 \, {\left (186 \, B a b^{6} + 25 \, A b^{7}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{82575360 \, c^{8}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/165150720*(315*(143*B*b^10 - 3072*(B*a^5 + 5*A*a^4*b)*c^5 + 6400*(3*B*a^4*b^2 + 4*A*a^3*b^3)*c^4 - 4480*(5
*B*a^3*b^4 + 3*A*a^2*b^5)*c^3 + 1440*(7*B*a^2*b^6 + 2*A*a*b^7)*c^2 - 220*(9*B*a*b^8 + A*b^9)*c)*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*(4128768*B*c^10*x^9 + 4504
5*B*b^9*c - 1310720*A*a^4*c^6 + 229376*(41*B*b*c^9 + 20*A*c^10)*x^8 + 14336*(383*B*b^2*c^8 + 4*(189*B*a + 185*
A*b)*c^9)*x^7 + 1024*(15*B*b^3*c^7 + 12160*A*a*c^9 + 4*(3181*B*a*b + 1545*A*b^2)*c^8)*x^6 + 14080*(169*B*a^4*b
+ 333*A*a^3*b^2)*c^5 - 256*(65*B*b^4*c^6 - 48*(651*B*a^2 + 1255*A*a*b)*c^8 - 20*(24*B*a*b^2 + 5*A*b^3)*c^7)*x
^5 - 2880*(1530*B*a^3*b^3 + 1127*A*a^2*b^4)*c^4 + 128*(143*B*b^5*c^5 + 76800*A*a^2*c^8 + 240*(11*B*a^2*b + 7*A
*a*b^2)*c^7 - 20*(60*B*a*b^3 + 11*A*b^4)*c^6)*x^4 + 336*(7404*B*a^2*b^5 + 2425*A*a*b^6)*c^3 - 16*(1287*B*b^6*c
^4 - 960*(21*B*a^3 + 41*A*a^2*b)*c^7 + 80*(417*B*a^2*b^2 + 214*A*a*b^3)*c^6 - 220*(55*B*a*b^4 + 9*A*b^5)*c^5)*
x^3 - 4620*(122*B*a*b^7 + 15*A*b^8)*c^2 + 8*(3003*B*b^7*c^3 + 81920*A*a^3*c^7 - 6080*(17*B*a^3*b + 21*A*a^2*b^
2)*c^6 + 240*(431*B*a^2*b^3 + 186*A*a*b^4)*c^5 - 132*(237*B*a*b^5 + 35*A*b^6)*c^4)*x^2 - 2*(15015*B*b^8*c^2 +
1280*(189*B*a^4 + 689*A*a^3*b)*c^6 - 320*(2866*B*a^3*b^2 + 2637*A*a^2*b^3)*c^5 + 720*(920*B*a^2*b^4 + 343*A*a*
b^5)*c^4 - 924*(186*B*a*b^6 + 25*A*b^7)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^8, 1/82575360*(315*(143*B*b^10 - 3072
*(B*a^5 + 5*A*a^4*b)*c^5 + 6400*(3*B*a^4*b^2 + 4*A*a^3*b^3)*c^4 - 4480*(5*B*a^3*b^4 + 3*A*a^2*b^5)*c^3 + 1440*
(7*B*a^2*b^6 + 2*A*a*b^7)*c^2 - 220*(9*B*a*b^8 + A*b^9)*c)*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*(4128768*B*c^10*x^9 + 45045*B*b^9*c - 1310720*A*a^4*c^6 + 229376*(41*
B*b*c^9 + 20*A*c^10)*x^8 + 14336*(383*B*b^2*c^8 + 4*(189*B*a + 185*A*b)*c^9)*x^7 + 1024*(15*B*b^3*c^7 + 12160*
A*a*c^9 + 4*(3181*B*a*b + 1545*A*b^2)*c^8)*x^6 + 14080*(169*B*a^4*b + 333*A*a^3*b^2)*c^5 - 256*(65*B*b^4*c^6 -
48*(651*B*a^2 + 1255*A*a*b)*c^8 - 20*(24*B*a*b^2 + 5*A*b^3)*c^7)*x^5 - 2880*(1530*B*a^3*b^3 + 1127*A*a^2*b^4)
*c^4 + 128*(143*B*b^5*c^5 + 76800*A*a^2*c^8 + 240*(11*B*a^2*b + 7*A*a*b^2)*c^7 - 20*(60*B*a*b^3 + 11*A*b^4)*c^
6)*x^4 + 336*(7404*B*a^2*b^5 + 2425*A*a*b^6)*c^3 - 16*(1287*B*b^6*c^4 - 960*(21*B*a^3 + 41*A*a^2*b)*c^7 + 80*(
417*B*a^2*b^2 + 214*A*a*b^3)*c^6 - 220*(55*B*a*b^4 + 9*A*b^5)*c^5)*x^3 - 4620*(122*B*a*b^7 + 15*A*b^8)*c^2 + 8
*(3003*B*b^7*c^3 + 81920*A*a^3*c^7 - 6080*(17*B*a^3*b + 21*A*a^2*b^2)*c^6 + 240*(431*B*a^2*b^3 + 186*A*a*b^4)*
c^5 - 132*(237*B*a*b^5 + 35*A*b^6)*c^4)*x^2 - 2*(15015*B*b^8*c^2 + 1280*(189*B*a^4 + 689*A*a^3*b)*c^6 - 320*(2
866*B*a^3*b^2 + 2637*A*a^2*b^3)*c^5 + 720*(920*B*a^2*b^4 + 343*A*a*b^5)*c^4 - 924*(186*B*a*b^6 + 25*A*b^7)*c^3
)*x)*sqrt(c*x^2 + b*x + a))/c^8]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(x**3*(A + B*x)*(a + b*x + c*x**2)**(5/2), x)

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Giac [A]
time = 1.31, size = 769, normalized size = 1.78 \begin {gather*} \frac {1}{41287680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, {\left (18 \, B c^{2} x + \frac {41 \, B b c^{10} + 20 \, A c^{11}}{c^{9}}\right )} x + \frac {383 \, B b^{2} c^{9} + 756 \, B a c^{10} + 740 \, A b c^{10}}{c^{9}}\right )} x + \frac {15 \, B b^{3} c^{8} + 12724 \, B a b c^{9} + 6180 \, A b^{2} c^{9} + 12160 \, A a c^{10}}{c^{9}}\right )} x - \frac {65 \, B b^{4} c^{7} - 480 \, B a b^{2} c^{8} - 100 \, A b^{3} c^{8} - 31248 \, B a^{2} c^{9} - 60240 \, A a b c^{9}}{c^{9}}\right )} x + \frac {143 \, B b^{5} c^{6} - 1200 \, B a b^{3} c^{7} - 220 \, A b^{4} c^{7} + 2640 \, B a^{2} b c^{8} + 1680 \, A a b^{2} c^{8} + 76800 \, A a^{2} c^{9}}{c^{9}}\right )} x - \frac {1287 \, B b^{6} c^{5} - 12100 \, B a b^{4} c^{6} - 1980 \, A b^{5} c^{6} + 33360 \, B a^{2} b^{2} c^{7} + 17120 \, A a b^{3} c^{7} - 20160 \, B a^{3} c^{8} - 39360 \, A a^{2} b c^{8}}{c^{9}}\right )} x + \frac {3003 \, B b^{7} c^{4} - 31284 \, B a b^{5} c^{5} - 4620 \, A b^{6} c^{5} + 103440 \, B a^{2} b^{3} c^{6} + 44640 \, A a b^{4} c^{6} - 103360 \, B a^{3} b c^{7} - 127680 \, A a^{2} b^{2} c^{7} + 81920 \, A a^{3} c^{8}}{c^{9}}\right )} x - \frac {15015 \, B b^{8} c^{3} - 171864 \, B a b^{6} c^{4} - 23100 \, A b^{7} c^{4} + 662400 \, B a^{2} b^{4} c^{5} + 246960 \, A a b^{5} c^{5} - 917120 \, B a^{3} b^{2} c^{6} - 843840 \, A a^{2} b^{3} c^{6} + 241920 \, B a^{4} c^{7} + 881920 \, A a^{3} b c^{7}}{c^{9}}\right )} x + \frac {45045 \, B b^{9} c^{2} - 563640 \, B a b^{7} c^{3} - 69300 \, A b^{8} c^{3} + 2487744 \, B a^{2} b^{5} c^{4} + 814800 \, A a b^{6} c^{4} - 4406400 \, B a^{3} b^{3} c^{5} - 3245760 \, A a^{2} b^{4} c^{5} + 2379520 \, B a^{4} b c^{6} + 4688640 \, A a^{3} b^{2} c^{6} - 1310720 \, A a^{4} c^{7}}{c^{9}}\right )} + \frac {{\left (143 \, B b^{10} - 1980 \, B a b^{8} c - 220 \, A b^{9} c + 10080 \, B a^{2} b^{6} c^{2} + 2880 \, A a b^{7} c^{2} - 22400 \, B a^{3} b^{4} c^{3} - 13440 \, A a^{2} b^{5} c^{3} + 19200 \, B a^{4} b^{2} c^{4} + 25600 \, A a^{3} b^{3} c^{4} - 3072 \, B a^{5} c^{5} - 15360 \, A a^{4} b c^{5}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{262144 \, c^{\frac {15}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/41287680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*(18*B*c^2*x + (41*B*b*c^10 + 20*A*c^11)/c^9)*x + (3
83*B*b^2*c^9 + 756*B*a*c^10 + 740*A*b*c^10)/c^9)*x + (15*B*b^3*c^8 + 12724*B*a*b*c^9 + 6180*A*b^2*c^9 + 12160*
A*a*c^10)/c^9)*x - (65*B*b^4*c^7 - 480*B*a*b^2*c^8 - 100*A*b^3*c^8 - 31248*B*a^2*c^9 - 60240*A*a*b*c^9)/c^9)*x
+ (143*B*b^5*c^6 - 1200*B*a*b^3*c^7 - 220*A*b^4*c^7 + 2640*B*a^2*b*c^8 + 1680*A*a*b^2*c^8 + 76800*A*a^2*c^9)/
c^9)*x - (1287*B*b^6*c^5 - 12100*B*a*b^4*c^6 - 1980*A*b^5*c^6 + 33360*B*a^2*b^2*c^7 + 17120*A*a*b^3*c^7 - 2016
0*B*a^3*c^8 - 39360*A*a^2*b*c^8)/c^9)*x + (3003*B*b^7*c^4 - 31284*B*a*b^5*c^5 - 4620*A*b^6*c^5 + 103440*B*a^2*
b^3*c^6 + 44640*A*a*b^4*c^6 - 103360*B*a^3*b*c^7 - 127680*A*a^2*b^2*c^7 + 81920*A*a^3*c^8)/c^9)*x - (15015*B*b
^8*c^3 - 171864*B*a*b^6*c^4 - 23100*A*b^7*c^4 + 662400*B*a^2*b^4*c^5 + 246960*A*a*b^5*c^5 - 917120*B*a^3*b^2*c
^6 - 843840*A*a^2*b^3*c^6 + 241920*B*a^4*c^7 + 881920*A*a^3*b*c^7)/c^9)*x + (45045*B*b^9*c^2 - 563640*B*a*b^7*
c^3 - 69300*A*b^8*c^3 + 2487744*B*a^2*b^5*c^4 + 814800*A*a*b^6*c^4 - 4406400*B*a^3*b^3*c^5 - 3245760*A*a^2*b^4
*c^5 + 2379520*B*a^4*b*c^6 + 4688640*A*a^3*b^2*c^6 - 1310720*A*a^4*c^7)/c^9) + 1/262144*(143*B*b^10 - 1980*B*a
*b^8*c - 220*A*b^9*c + 10080*B*a^2*b^6*c^2 + 2880*A*a*b^7*c^2 - 22400*B*a^3*b^4*c^3 - 13440*A*a^2*b^5*c^3 + 19
200*B*a^4*b^2*c^4 + 25600*A*a^3*b^3*c^4 - 3072*B*a^5*c^5 - 15360*A*a^4*b*c^5)*log(abs(-2*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*sqrt(c) - b))/c^(15/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(A + B*x)*(a + b*x + c*x^2)^(5/2),x)

[Out]

int(x^3*(A + B*x)*(a + b*x + c*x^2)^(5/2), x)

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