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

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

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Rubi [A]  time = 0.49, 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 = {832, 779, 612, 621, 206} \begin {gather*} \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} \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 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 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 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 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*}

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Mathematica [A]  time = 0.76, size = 315, normalized size = 0.73 \begin {gather*} \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} \end {gather*}

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)

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IntegrateAlgebraic [A]  time = 5.17, size = 804, normalized size = 1.86 \begin {gather*} \frac {\sqrt {c x^2+b x+a} \left (45045 B b^9-69300 A c b^8-30030 B c x b^8+24024 B c^2 x^2 b^7-563640 a B c b^7+46200 A c^2 x b^7-20592 B c^3 x^3 b^6+814800 a A c^2 b^6-36960 A c^3 x^2 b^6+343728 a B c^2 x b^6+18304 B c^4 x^4 b^5+31680 A c^4 x^3 b^5+2487744 a^2 B c^2 b^5-250272 a B c^3 x^2 b^5-493920 a A c^3 x b^5-16640 B c^5 x^5 b^4-28160 A c^5 x^4 b^4-3245760 a^2 A c^3 b^4+193600 a B c^4 x^3 b^4+357120 a A c^4 x^2 b^4-1324800 a^2 B c^3 x b^4+15360 B c^6 x^6 b^3+25600 A c^6 x^5 b^3-153600 a B c^5 x^4 b^3-4406400 a^3 B c^3 b^3-273920 a A c^5 x^3 b^3+827520 a^2 B c^4 x^2 b^3+1687680 a^2 A c^4 x b^3+5490688 B c^7 x^7 b^2+6328320 A c^7 x^6 b^2+122880 a B c^6 x^5 b^2+4688640 a^3 A c^4 b^2+215040 a A c^6 x^4 b^2-533760 a^2 B c^5 x^3 b^2-1021440 a^2 A c^5 x^2 b^2+1834240 a^3 B c^4 x b^2+9404416 B c^8 x^8 b+10608640 A c^8 x^7 b+13029376 a B c^7 x^6 b+15421440 a A c^7 x^5 b+2379520 a^4 B c^4 b+337920 a^2 B c^6 x^4 b+629760 a^2 A c^6 x^3 b-826880 a^3 B c^5 x^2 b-1763840 a^3 A c^5 x b+4128768 B c^9 x^9+4587520 A c^9 x^8+10838016 a B c^8 x^7+12451840 a A c^8 x^6-1310720 a^4 A c^5+7999488 a^2 B c^7 x^5+9830400 a^2 A c^7 x^4+322560 a^3 B c^6 x^3+655360 a^3 A c^6 x^2-483840 a^4 B c^5 x\right )}{41287680 c^7}+\frac {\left (143 B b^{10}-220 A c b^9-1980 a B c b^8+2880 a A c^2 b^7+10080 a^2 B c^2 b^6-13440 a^2 A c^3 b^5-22400 a^3 B c^3 b^4+25600 a^3 A c^4 b^3+19200 a^4 B c^4 b^2-15360 a^4 A c^5 b-3072 a^5 B c^5\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x+a}\right )}{262144 c^{15/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(45045*b^9*B - 69300*A*b^8*c - 563640*a*b^7*B*c + 814800*a*A*b^6*c^2 + 2487744*a^2*b^5*
B*c^2 - 3245760*a^2*A*b^4*c^3 - 4406400*a^3*b^3*B*c^3 + 4688640*a^3*A*b^2*c^4 + 2379520*a^4*b*B*c^4 - 1310720*
a^4*A*c^5 - 30030*b^8*B*c*x + 46200*A*b^7*c^2*x + 343728*a*b^6*B*c^2*x - 493920*a*A*b^5*c^3*x - 1324800*a^2*b^
4*B*c^3*x + 1687680*a^2*A*b^3*c^4*x + 1834240*a^3*b^2*B*c^4*x - 1763840*a^3*A*b*c^5*x - 483840*a^4*B*c^5*x + 2
4024*b^7*B*c^2*x^2 - 36960*A*b^6*c^3*x^2 - 250272*a*b^5*B*c^3*x^2 + 357120*a*A*b^4*c^4*x^2 + 827520*a^2*b^3*B*
c^4*x^2 - 1021440*a^2*A*b^2*c^5*x^2 - 826880*a^3*b*B*c^5*x^2 + 655360*a^3*A*c^6*x^2 - 20592*b^6*B*c^3*x^3 + 31
680*A*b^5*c^4*x^3 + 193600*a*b^4*B*c^4*x^3 - 273920*a*A*b^3*c^5*x^3 - 533760*a^2*b^2*B*c^5*x^3 + 629760*a^2*A*
b*c^6*x^3 + 322560*a^3*B*c^6*x^3 + 18304*b^5*B*c^4*x^4 - 28160*A*b^4*c^5*x^4 - 153600*a*b^3*B*c^5*x^4 + 215040
*a*A*b^2*c^6*x^4 + 337920*a^2*b*B*c^6*x^4 + 9830400*a^2*A*c^7*x^4 - 16640*b^4*B*c^5*x^5 + 25600*A*b^3*c^6*x^5
+ 122880*a*b^2*B*c^6*x^5 + 15421440*a*A*b*c^7*x^5 + 7999488*a^2*B*c^7*x^5 + 15360*b^3*B*c^6*x^6 + 6328320*A*b^
2*c^7*x^6 + 13029376*a*b*B*c^7*x^6 + 12451840*a*A*c^8*x^6 + 5490688*b^2*B*c^7*x^7 + 10608640*A*b*c^8*x^7 + 108
38016*a*B*c^8*x^7 + 9404416*b*B*c^8*x^8 + 4587520*A*c^9*x^8 + 4128768*B*c^9*x^9))/(41287680*c^7) + ((143*b^10*
B - 220*A*b^9*c - 1980*a*b^8*B*c + 2880*a*A*b^7*c^2 + 10080*a^2*b^6*B*c^2 - 13440*a^2*A*b^5*c^3 - 22400*a^3*b^
4*B*c^3 + 25600*a^3*A*b^3*c^4 + 19200*a^4*b^2*B*c^4 - 15360*a^4*A*b*c^5 - 3072*a^5*B*c^5)*Log[b + 2*c*x - 2*Sq
rt[c]*Sqrt[a + b*x + c*x^2]])/(262144*c^(15/2))

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fricas [A]  time = 0.76, size = 1511, normalized size = 3.50

result too large to display

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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giac [A]  time = 0.32, 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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maple [B]  time = 0.07, size = 1549, normalized size = 3.59

result too large to display

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]

125/4096*A*b^5/c^4*(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+5/128*A*b/c^2*a^2*(c*x^2+b*x+a)^(3/2)*x-85/1024*A*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x*a^2-11/320*B*b^
2/c^3*a*(c*x^2+b*x+a)^(5/2)*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+209/6144*B*b^4/c^4*(c*x^2+b*x+a)^(3/2)*x*a-11/512*B*b^6/c^5*(c*x^2+b*x+a)^(1/2)*x*a+139/2048*B*b^4/c^4*(c
*x^2+b*x+a)^(1/2)*x*a^2+1/32*A*b/c^2*a*(c*x^2+b*x+a)^(5/2)*x+139/4096*B*b^5/c^5*(c*x^2+b*x+a)^(1/2)*a^2-11/102
4*B*b^7/c^6*(c*x^2+b*x+a)^(1/2)*a-45/4096*A*b^7/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+15/256*
A*b/c^(5/2)*a^4*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-25/256*A*b^3/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))*a^3-55/16384*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+55/6144*A*b^5/c^4*(c*x^2+b*x+a)^(3/2)*x-35/1536*A*b^4/c^4*(c*x^2+b*x+a)^(3/2)
*a-11/384*A*b^3/c^3*(c*x^2+b*x+a)^(5/2)*x+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)+105/2048*A*b^5/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))*a^2-11/144*A*b/c^2*x*(c*x^2+b*x+a)^(7/2)+143/7680*B*b^4/c^4*(c*x^2+b*x+a)^(5/2)*x-11/640*B*b^3/c^4*a*(c
*x^2+b*x+a)^(5/2)-23/1024*B*b^3/c^4*a^2*(c*x^2+b*x+a)^(3/2)-9/256*B*b^3/c^4*a^3*(c*x^2+b*x+a)^(1/2)+143/2880*B
*b^2/c^3*x*(c*x^2+b*x+a)^(7/2)+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+3/2
56*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-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+451/10080*B*b/c^3*a*(
c*x^2+b*x+a)^(7/2)+3/256*B*a^5/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-143/262144*B*b^10/c^(15/2)*
ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+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)-143/4480*B*b^3/c^4*(c*x^2+b*x+a)^(7/2)-2/63*A*a/c^2*(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)+11/224*A*b^2/c^3*(c*x^2+b*x+a)^(7/2)+1/9*A*x^2*(c*x^2+b*x+a)^(7/2)/c+55/65536*A*b^9/c^(1
3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-13/180*B*b/c^2*x^2*(c*x^2+b*x+a)^(7/2)+1/320*B*a^2/c^3*(c*x^2
+b*x+a)^(5/2)*b-75/1024*B*b^2/c^(7/2)*a^4*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+175/2048*B*b^4/c^(9/2)*l
n((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-315/8192*B*b^6/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))*a^2+1/160*B*a^2/c^2*(c*x^2+b*x+a)^(5/2)*x-3/80*B*a/c^2*x*(c*x^2+b*x+a)^(7/2)+495/65536*B*b^8/c^(13/2)*ln(
(c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/10*B*x^3*(c*x^2+b*x+a)^(7/2)/c

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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 details)Is 4*a*c-b^2 positive, negative or zero?

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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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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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