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

Optimal. Leaf size=269 \[ \frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{840 c^3}-\frac {\left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{1024 c^5}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]

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Rubi [A]  time = 0.25, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{840 c^3}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{384 c^4}+\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{1024 c^5}-\frac {\left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)*(9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^5) -
((9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(384*c^4) + (B*x^2*(a +
b*x + c*x^2)^(5/2))/(7*c) + ((63*b^2*B - 98*A*b*c - 48*a*B*c - 10*c*(9*b*B - 14*A*c)*x)*(a + b*x + c*x^2)^(5/
2))/(840*c^3) - ((b^2 - 4*a*c)^2*(9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c
]*Sqrt[a + b*x + c*x^2])])/(2048*c^(11/2))

Rule 206

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

Rule 612

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

Rule 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^2 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\int x \left (-2 a B-\frac {1}{2} (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{48 c^3}\\ &=-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{256 c^4}\\ &=\frac {\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (\left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^5}\\ &=\frac {\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (\left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A 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 )}{1024 c^5}\\ &=\frac {\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 206, normalized size = 0.77 \begin {gather*} \frac {\frac {(a+x (b+c x))^{5/2} \left (4 c (35 A c x-12 a B)-2 b c (49 A+45 B x)+63 b^2 B\right )}{120 c^2}-\frac {7 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{6144 c^{9/2}}+B x^2 (a+x (b+c x))^{5/2}}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*x^2*(a + x*(b + c*x))^(5/2) + ((a + x*(b + c*x))^(5/2)*(63*b^2*B - 2*b*c*(49*A + 45*B*x) + 4*c*(-12*a*B + 3
5*A*c*x)))/(120*c^2) - (7*(9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b
+ c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
x*(b + c*x)])]))/(6144*c^(9/2)))/(7*c)

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IntegrateAlgebraic [A]  time = 1.80, size = 423, normalized size = 1.57 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-6144 a^3 B c^3-18144 a^2 A b c^3+6720 a^2 A c^4 x+16464 a^2 b^2 B c^2-7008 a^2 b B c^3 x+3072 a^2 B c^4 x^2+10640 a A b^3 c^2-6048 a A b^2 c^3 x+4032 a A b c^4 x^2+31360 a A c^5 x^3-7560 a b^4 B c+4368 a b^3 B c^2 x-2976 a b^2 B c^3 x^2+2112 a b B c^4 x^3+24576 a B c^5 x^4-1470 A b^5 c+980 A b^4 c^2 x-784 A b^3 c^3 x^2+672 A b^2 c^4 x^3+23296 A b c^5 x^4+17920 A c^6 x^5+945 b^6 B-630 b^5 B c x+504 b^4 B c^2 x^2-432 b^3 B c^3 x^3+384 b^2 B c^4 x^4+19200 b B c^5 x^5+15360 B c^6 x^6\right )}{107520 c^5}+\frac {\left (128 a^3 A c^4-192 a^3 b B c^3-288 a^2 A b^2 c^3+240 a^2 b^3 B c^2+120 a A b^4 c^2-84 a b^5 B c-14 A b^6 c+9 b^7 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{2048 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(945*b^6*B - 1470*A*b^5*c - 7560*a*b^4*B*c + 10640*a*A*b^3*c^2 + 16464*a^2*b^2*B*c^2 -
18144*a^2*A*b*c^3 - 6144*a^3*B*c^3 - 630*b^5*B*c*x + 980*A*b^4*c^2*x + 4368*a*b^3*B*c^2*x - 6048*a*A*b^2*c^3*x
- 7008*a^2*b*B*c^3*x + 6720*a^2*A*c^4*x + 504*b^4*B*c^2*x^2 - 784*A*b^3*c^3*x^2 - 2976*a*b^2*B*c^3*x^2 + 4032
*a*A*b*c^4*x^2 + 3072*a^2*B*c^4*x^2 - 432*b^3*B*c^3*x^3 + 672*A*b^2*c^4*x^3 + 2112*a*b*B*c^4*x^3 + 31360*a*A*c
^5*x^3 + 384*b^2*B*c^4*x^4 + 23296*A*b*c^5*x^4 + 24576*a*B*c^5*x^4 + 19200*b*B*c^5*x^5 + 17920*A*c^6*x^5 + 153
60*B*c^6*x^6))/(107520*c^5) + ((9*b^7*B - 14*A*b^6*c - 84*a*b^5*B*c + 120*a*A*b^4*c^2 + 240*a^2*b^3*B*c^2 - 28
8*a^2*A*b^2*c^3 - 192*a^3*b*B*c^3 + 128*a^3*A*c^4)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(2048*c^(
11/2))

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fricas [A]  time = 0.59, size = 845, normalized size = 3.14 \begin {gather*} \left [\frac {105 \, {\left (9 \, B b^{7} + 128 \, A a^{3} c^{4} - 96 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 120 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 14 \, {\left (6 \, B a b^{5} + A b^{6}\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 (15360 \, B c^{7} x^{6} + 945 \, B b^{6} c + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} - 96 \, {\left (64 \, B a^{3} + 189 \, A a^{2} b\right )} c^{4} + 128 \, {\left (3 \, B b^{2} c^{5} + 2 \, {\left (96 \, B a + 91 \, A b\right )} c^{6}\right )} x^{4} + 112 \, {\left (147 \, B a^{2} b^{2} + 95 \, A a b^{3}\right )} c^{3} - 16 \, {\left (27 \, B b^{3} c^{4} - 1960 \, A a c^{6} - 6 \, {\left (22 \, B a b + 7 \, A b^{2}\right )} c^{5}\right )} x^{3} - 210 \, {\left (36 \, B a b^{4} + 7 \, A b^{5}\right )} c^{2} + 8 \, {\left (63 \, B b^{4} c^{3} + 24 \, {\left (16 \, B a^{2} + 21 \, A a b\right )} c^{5} - 2 \, {\left (186 \, B a b^{2} + 49 \, A b^{3}\right )} c^{4}\right )} x^{2} - 2 \, {\left (315 \, B b^{5} c^{2} - 3360 \, A a^{2} c^{5} + 48 \, {\left (73 \, B a^{2} b + 63 \, A a b^{2}\right )} c^{4} - 14 \, {\left (156 \, B a b^{3} + 35 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{430080 \, c^{6}}, \frac {105 \, {\left (9 \, B b^{7} + 128 \, A a^{3} c^{4} - 96 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 120 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 14 \, {\left (6 \, B a b^{5} + A b^{6}\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 (15360 \, B c^{7} x^{6} + 945 \, B b^{6} c + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} - 96 \, {\left (64 \, B a^{3} + 189 \, A a^{2} b\right )} c^{4} + 128 \, {\left (3 \, B b^{2} c^{5} + 2 \, {\left (96 \, B a + 91 \, A b\right )} c^{6}\right )} x^{4} + 112 \, {\left (147 \, B a^{2} b^{2} + 95 \, A a b^{3}\right )} c^{3} - 16 \, {\left (27 \, B b^{3} c^{4} - 1960 \, A a c^{6} - 6 \, {\left (22 \, B a b + 7 \, A b^{2}\right )} c^{5}\right )} x^{3} - 210 \, {\left (36 \, B a b^{4} + 7 \, A b^{5}\right )} c^{2} + 8 \, {\left (63 \, B b^{4} c^{3} + 24 \, {\left (16 \, B a^{2} + 21 \, A a b\right )} c^{5} - 2 \, {\left (186 \, B a b^{2} + 49 \, A b^{3}\right )} c^{4}\right )} x^{2} - 2 \, {\left (315 \, B b^{5} c^{2} - 3360 \, A a^{2} c^{5} + 48 \, {\left (73 \, B a^{2} b + 63 \, A a b^{2}\right )} c^{4} - 14 \, {\left (156 \, B a b^{3} + 35 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{215040 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/430080*(105*(9*B*b^7 + 128*A*a^3*c^4 - 96*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 120*(2*B*a^2*b^3 + A*a*b^4)*c^2 -
14*(6*B*a*b^5 + A*b^6)*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*(15360*B*c^7*x^6 + 945*B*b^6*c + 1280*(15*B*b*c^6 + 14*A*c^7)*x^5 - 96*(64*B*a^3 + 189*A*a^2*b)
*c^4 + 128*(3*B*b^2*c^5 + 2*(96*B*a + 91*A*b)*c^6)*x^4 + 112*(147*B*a^2*b^2 + 95*A*a*b^3)*c^3 - 16*(27*B*b^3*c
^4 - 1960*A*a*c^6 - 6*(22*B*a*b + 7*A*b^2)*c^5)*x^3 - 210*(36*B*a*b^4 + 7*A*b^5)*c^2 + 8*(63*B*b^4*c^3 + 24*(1
6*B*a^2 + 21*A*a*b)*c^5 - 2*(186*B*a*b^2 + 49*A*b^3)*c^4)*x^2 - 2*(315*B*b^5*c^2 - 3360*A*a^2*c^5 + 48*(73*B*a
^2*b + 63*A*a*b^2)*c^4 - 14*(156*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/215040*(105*(9*B*b^
7 + 128*A*a^3*c^4 - 96*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 120*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 14*(6*B*a*b^5 + A*b^6
)*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*(15360*B*c^7*
x^6 + 945*B*b^6*c + 1280*(15*B*b*c^6 + 14*A*c^7)*x^5 - 96*(64*B*a^3 + 189*A*a^2*b)*c^4 + 128*(3*B*b^2*c^5 + 2*
(96*B*a + 91*A*b)*c^6)*x^4 + 112*(147*B*a^2*b^2 + 95*A*a*b^3)*c^3 - 16*(27*B*b^3*c^4 - 1960*A*a*c^6 - 6*(22*B*
a*b + 7*A*b^2)*c^5)*x^3 - 210*(36*B*a*b^4 + 7*A*b^5)*c^2 + 8*(63*B*b^4*c^3 + 24*(16*B*a^2 + 21*A*a*b)*c^5 - 2*
(186*B*a*b^2 + 49*A*b^3)*c^4)*x^2 - 2*(315*B*b^5*c^2 - 3360*A*a^2*c^5 + 48*(73*B*a^2*b + 63*A*a*b^2)*c^4 - 14*
(156*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^6]

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giac [A]  time = 0.24, size = 422, normalized size = 1.57 \begin {gather*} \frac {1}{107520} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B c x + \frac {15 \, B b c^{6} + 14 \, A c^{7}}{c^{6}}\right )} x + \frac {3 \, B b^{2} c^{5} + 192 \, B a c^{6} + 182 \, A b c^{6}}{c^{6}}\right )} x - \frac {27 \, B b^{3} c^{4} - 132 \, B a b c^{5} - 42 \, A b^{2} c^{5} - 1960 \, A a c^{6}}{c^{6}}\right )} x + \frac {63 \, B b^{4} c^{3} - 372 \, B a b^{2} c^{4} - 98 \, A b^{3} c^{4} + 384 \, B a^{2} c^{5} + 504 \, A a b c^{5}}{c^{6}}\right )} x - \frac {315 \, B b^{5} c^{2} - 2184 \, B a b^{3} c^{3} - 490 \, A b^{4} c^{3} + 3504 \, B a^{2} b c^{4} + 3024 \, A a b^{2} c^{4} - 3360 \, A a^{2} c^{5}}{c^{6}}\right )} x + \frac {945 \, B b^{6} c - 7560 \, B a b^{4} c^{2} - 1470 \, A b^{5} c^{2} + 16464 \, B a^{2} b^{2} c^{3} + 10640 \, A a b^{3} c^{3} - 6144 \, B a^{3} c^{4} - 18144 \, A a^{2} b c^{4}}{c^{6}}\right )} + \frac {{\left (9 \, B b^{7} - 84 \, B a b^{5} c - 14 \, A b^{6} c + 240 \, B a^{2} b^{3} c^{2} + 120 \, A a b^{4} c^{2} - 192 \, B a^{3} b c^{3} - 288 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*B*c*x + (15*B*b*c^6 + 14*A*c^7)/c^6)*x + (3*B*b^2*c^5 + 192
*B*a*c^6 + 182*A*b*c^6)/c^6)*x - (27*B*b^3*c^4 - 132*B*a*b*c^5 - 42*A*b^2*c^5 - 1960*A*a*c^6)/c^6)*x + (63*B*b
^4*c^3 - 372*B*a*b^2*c^4 - 98*A*b^3*c^4 + 384*B*a^2*c^5 + 504*A*a*b*c^5)/c^6)*x - (315*B*b^5*c^2 - 2184*B*a*b^
3*c^3 - 490*A*b^4*c^3 + 3504*B*a^2*b*c^4 + 3024*A*a*b^2*c^4 - 3360*A*a^2*c^5)/c^6)*x + (945*B*b^6*c - 7560*B*a
*b^4*c^2 - 1470*A*b^5*c^2 + 16464*B*a^2*b^2*c^3 + 10640*A*a*b^3*c^3 - 6144*B*a^3*c^4 - 18144*A*a^2*b*c^4)/c^6)
+ 1/2048*(9*B*b^7 - 84*B*a*b^5*c - 14*A*b^6*c + 240*B*a^2*b^3*c^2 + 120*A*a*b^4*c^2 - 192*B*a^3*b*c^3 - 288*A
*a^2*b^2*c^3 + 128*A*a^3*c^4)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)

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maple [B]  time = 0.06, size = 838, normalized size = 3.12 \begin {gather*} -\frac {A \,a^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {9 A \,a^{2} b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {5}{2}}}-\frac {15 A a \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}+\frac {7 A \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {9}{2}}}+\frac {3 B \,a^{3} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {5}{2}}}-\frac {15 B \,a^{2} b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {7}{2}}}+\frac {21 B a \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {9}{2}}}-\frac {9 B \,b^{7} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {11}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A \,a^{2} x}{16 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, A a \,b^{2} x}{8 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4} x}{256 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b x}{32 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{3} x}{32 c^{3}}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, B \,b^{5} x}{512 c^{4}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A \,a^{2} b}{32 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A a \,b^{3}}{16 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a x}{24 c}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A \,b^{5}}{512 c^{4}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2} x}{96 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b^{2}}{64 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{4}}{64 c^{4}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a b x}{16 c^{2}}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, B \,b^{6}}{1024 c^{5}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3} x}{64 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B \,x^{2}}{7 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a b}{48 c^{2}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3}}{192 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A x}{6 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a \,b^{2}}{32 c^{3}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{4}}{128 c^{4}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b x}{28 c^{2}}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b}{60 c^{2}}-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B a}{35 c^{2}}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B \,b^{2}}{40 c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/32*B*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x*a+1/16*B*b/c^2*a*(c*x^2+b*x+a)^(3/2)*x+1/8*A*b^2/c^2*(c*x^2+b*x+a)^(1/2)
*x*a+3/32*B*b/c^2*a^2*(c*x^2+b*x+a)^(1/2)*x+3/40*B*b^2/c^3*(c*x^2+b*x+a)^(5/2)-3/128*B*b^4/c^4*(c*x^2+b*x+a)^(
3/2)+9/1024*B*b^6/c^5*(c*x^2+b*x+a)^(1/2)-1/16*A*a^3/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/64*
B*b^4/c^4*(c*x^2+b*x+a)^(1/2)*a+1/32*B*b^2/c^3*a*(c*x^2+b*x+a)^(3/2)+9/64*A*b^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))*a^2-1/48*A*a/c^2*(c*x^2+b*x+a)^(3/2)*b-15/256*A*b^4/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))*a-3/28*B*b/c^2*x*(c*x^2+b*x+a)^(5/2)+7/96*A*b^2/c^2*(c*x^2+b*x+a)^(3/2)*x-1/16*A*a^2/c*(c*x^2+
b*x+a)^(1/2)*x+3/64*B*b^2/c^3*a^2*(c*x^2+b*x+a)^(1/2)-15/128*B*b^3/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a
)^(1/2))*a^2-1/24*A*a/c*(c*x^2+b*x+a)^(3/2)*x+9/512*B*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x-3/64*B*b^3/c^3*(c*x^2+b*x+
a)^(3/2)*x-7/256*A*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x+1/16*A*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a-1/32*A*a^2/c^2*(c*x^2+b*
x+a)^(1/2)*b+21/512*B*b^5/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/32*B*b/c^(5/2)*a^3*ln((c*x+1
/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/6*A*x*(c*x^2+b*x+a)^(5/2)/c-7/60*A*b/c^2*(c*x^2+b*x+a)^(5/2)+7/192*A*b^3/
c^3*(c*x^2+b*x+a)^(3/2)-7/512*A*b^5/c^4*(c*x^2+b*x+a)^(1/2)+7/1024*A*b^6/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))-2/35*B*a/c^2*(c*x^2+b*x+a)^(5/2)-9/2048*B*b^7/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))+1/7*B*x^2*(c*x^2+b*x+a)^(5/2)/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^2*(B*x+A)*(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 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^2\,\left (A+B\,x\right )\,{\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(x^2*(A + B*x)*(a + b*x + c*x^2)^(3/2),x)

[Out]

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

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

[Out]

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

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