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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.485906, antiderivative size = 432, normalized size of antiderivative = 1., 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 = {832, 779, 612, 621, 206} \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{15360 c^5}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{49152 c^6}+\frac{\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{131072 c^7}-\frac{\left (b^2-4 a c\right )^3 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{262144 c^{15/2}}-\frac{\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{40320 c^4}-\frac{x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{180 c^2}+\frac{B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c} \]

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 832

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

Rule 779

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

Rule 612

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

Rule 621

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 206

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

Rubi steps

\begin{align*} \int 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 ) \operatorname{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*}

Mathematica [A]  time = 0.786134, size = 315, normalized size = 0.73 \[ \frac{\frac{\left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{393216 c^{13/2}}+\frac{(a+x (b+c x))^{7/2} \left (44 b c (41 a B-70 A c x)-8 a c^2 (160 A+189 B x)+22 b^2 c (90 A+91 B x)-1287 b^3 B\right )}{4032 c^3}+\frac{x^2 (a+x (b+c x))^{7/2} (20 A c-13 b B)}{18 c}+B x^3 (a+x (b+c x))^{7/2}}{10 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-13*b*B + 20*A*c)*x^2*(a + x*(b + c*x))^(7/2))/(18*c) + B*x^3*(a + x*(b + c*x))^(7/2) + ((a + x*(b + c*x))^
(7/2)*(-1287*b^3*B + 22*b^2*c*(90*A + 91*B*x) - 8*a*c^2*(160*A + 189*B*x) + 44*b*c*(41*a*B - 70*A*c*x)))/(4032
*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)*(256*c^(5/2)*(b + 2*c*x)*(a
+ x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sq
rt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]
))))/(393216*c^(13/2)))/(10*c)

________________________________________________________________________________________

Maple [B]  time = 0.011, size = 1549, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

-11/384*A*b^3/c^3*(c*x^2+b*x+a)^(5/2)*x+55/6144*A*b^5/c^4*(c*x^2+b*x+a)^(3/2)*x+1/10*B*x^3*(c*x^2+b*x+a)^(7/2)
/c+209/6144*B*b^4/c^4*(c*x^2+b*x+a)^(3/2)*x*a-11/320*B*b^2/c^3*a*(c*x^2+b*x+a)^(5/2)*x-11/512*B*b^6/c^5*(c*x^2
+b*x+a)^(1/2)*x*a-35/768*A*b^3/c^3*(c*x^2+b*x+a)^(3/2)*x*a+15/256*A*b/c^2*a^3*(c*x^2+b*x+a)^(1/2)*x-23/512*B*b
^2/c^3*a^2*(c*x^2+b*x+a)^(3/2)*x-9/128*B*b^2/c^3*a^3*(c*x^2+b*x+a)^(1/2)*x+139/2048*B*b^4/c^4*(c*x^2+b*x+a)^(1
/2)*x*a^2+125/4096*A*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x*a-85/1024*A*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x*a^2+5/128*A*b/c^2
*a^2*(c*x^2+b*x+a)^(3/2)*x+1/32*A*b/c^2*a*(c*x^2+b*x+a)^(5/2)*x+1/9*A*x^2*(c*x^2+b*x+a)^(7/2)/c-2/63*A*a/c^2*(
c*x^2+b*x+a)^(7/2)+55/65536*A*b^9/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+11/224*A*b^2/c^3*(c*x^2
+b*x+a)^(7/2)-11/768*A*b^4/c^4*(c*x^2+b*x+a)^(5/2)+55/12288*A*b^6/c^5*(c*x^2+b*x+a)^(3/2)-55/32768*A*b^8/c^6*(
c*x^2+b*x+a)^(1/2)+3/256*B*a^5/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-143/262144*B*b^10/c^(15/2)*
ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-143/4480*B*b^3/c^4*(c*x^2+b*x+a)^(7/2)+143/15360*B*b^5/c^5*(c*x^2+
b*x+a)^(5/2)-143/49152*B*b^7/c^6*(c*x^2+b*x+a)^(3/2)+143/131072*B*b^9/c^7*(c*x^2+b*x+a)^(1/2)-143/24576*B*b^6/
c^5*(c*x^2+b*x+a)^(3/2)*x+209/12288*B*b^5/c^5*(c*x^2+b*x+a)^(3/2)*a+143/65536*B*b^8/c^6*(c*x^2+b*x+a)^(1/2)*x+
139/4096*B*b^5/c^5*(c*x^2+b*x+a)^(1/2)*a^2-11/1024*B*b^7/c^6*(c*x^2+b*x+a)^(1/2)*a+143/2880*B*b^2/c^3*x*(c*x^2
+b*x+a)^(7/2)-13/180*B*b/c^2*x^2*(c*x^2+b*x+a)^(7/2)+451/10080*B*b/c^3*a*(c*x^2+b*x+a)^(7/2)+175/2048*B*b^4/c^
(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+495/65536*B*b^8/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*a-315/8192*B*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-75/1024*B*b^2/c^(7/2)*a
^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/160*B*a^2/c^2*(c*x^2+b*x+a)^(5/2)*x+1/320*B*a^2/c^3*(c*x^2+b*
x+a)^(5/2)*b+1/128*B*a^3/c^2*(c*x^2+b*x+a)^(3/2)*x+1/256*B*a^3/c^3*(c*x^2+b*x+a)^(3/2)*b+3/256*B*a^4/c^2*(c*x^
2+b*x+a)^(1/2)*x+3/512*B*a^4/c^3*(c*x^2+b*x+a)^(1/2)*b-3/80*B*a/c^2*x*(c*x^2+b*x+a)^(7/2)-11/640*B*b^3/c^4*a*(
c*x^2+b*x+a)^(5/2)-35/1536*A*b^4/c^4*(c*x^2+b*x+a)^(3/2)*a-55/16384*A*b^7/c^5*(c*x^2+b*x+a)^(1/2)*x-85/2048*A*
b^4/c^4*(c*x^2+b*x+a)^(1/2)*a^2+125/8192*A*b^6/c^5*(c*x^2+b*x+a)^(1/2)*a-11/144*A*b/c^2*x*(c*x^2+b*x+a)^(7/2)-
45/4096*A*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+105/2048*A*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))*a^2+15/256*A*b/c^(5/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-25/256*A*b^3/
c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+1/64*A*b^2/c^3*a*(c*x^2+b*x+a)^(5/2)+5/256*A*b^2/c^3*a
^2*(c*x^2+b*x+a)^(3/2)+15/512*A*b^2/c^3*a^3*(c*x^2+b*x+a)^(1/2)-23/1024*B*b^3/c^4*a^2*(c*x^2+b*x+a)^(3/2)-9/25
6*B*b^3/c^4*a^3*(c*x^2+b*x+a)^(1/2)+143/7680*B*b^4/c^4*(c*x^2+b*x+a)^(5/2)*x

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 3.27203, size = 3761, normalized size = 8.71 \begin{align*} \text{result too large to display} \end{align*}

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]

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [A]  time = 1.40182, size = 1038, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

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)