3.9.47 \(\int x^4 (A+B x) (a+b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=455 \[ \frac {\left (a+b x+c x^2\right )^{5/2} \left (2048 a^2 B c^2-10 c x \left (504 a A c^2-748 a b B c-594 A b^2 c+429 b^3 B\right )+6696 a A b c^2-7524 a b^2 B c-4158 A b^3 c+3003 b^4 B\right )}{80640 c^5}-\frac {\left (b^2-4 a c\right )^2 \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{15/2}}+\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right )}{32768 c^7}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right )}{12288 c^6}+\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{2016 c^3}-\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c} \]

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Rubi [A]  time = 0.63, antiderivative size = 455, 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 {\left (a+b x+c x^2\right )^{5/2} \left (2048 a^2 B c^2-10 c x \left (504 a A c^2-748 a b B c-594 A b^2 c+429 b^3 B\right )+6696 a A b c^2-7524 a b^2 B c-4158 A b^3 c+3003 b^4 B\right )}{80640 c^5}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right )}{12288 c^6}+\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right )}{32768 c^7}-\frac {\left (b^2-4 a c\right )^2 \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{15/2}}+\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{2016 c^3}-\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]

[Out]

((b^2 - 4*a*c)*(143*b^5*B - 198*A*b^4*c - 440*a*b^3*B*c + 432*a*A*b^2*c^2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^3)*(b
+ 2*c*x)*Sqrt[a + b*x + c*x^2])/(32768*c^7) - ((143*b^5*B - 198*A*b^4*c - 440*a*b^3*B*c + 432*a*A*b^2*c^2 + 2
40*a^2*b*B*c^2 - 96*a^2*A*c^3)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(12288*c^6) + ((143*b^2*B - 198*A*b*c - 12
8*a*B*c)*x^2*(a + b*x + c*x^2)^(5/2))/(2016*c^3) - ((13*b*B - 18*A*c)*x^3*(a + b*x + c*x^2)^(5/2))/(144*c^2) +
(B*x^4*(a + b*x + c*x^2)^(5/2))/(9*c) + ((3003*b^4*B - 4158*A*b^3*c - 7524*a*b^2*B*c + 6696*a*A*b*c^2 + 2048*
a^2*B*c^2 - 10*c*(429*b^3*B - 594*A*b^2*c - 748*a*b*B*c + 504*a*A*c^2)*x)*(a + b*x + c*x^2)^(5/2))/(80640*c^5)
- ((b^2 - 4*a*c)^2*(143*b^5*B - 198*A*b^4*c - 440*a*b^3*B*c + 432*a*A*b^2*c^2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^
3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(65536*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^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\int x^3 \left (-4 a B-\frac {1}{2} (13 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{9 c}\\ &=-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\int x^2 \left (\frac {3}{2} a (13 b B-18 A c)+\frac {1}{4} \left (143 b^2 B-198 A b c-128 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{72 c^2}\\ &=\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\int x \left (-\frac {1}{2} a \left (143 b^2 B-198 A b c-128 a B c\right )-\frac {3}{8} \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{504 c^3}\\ &=\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{1536 c^5}\\ &=-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^6}+\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}+\frac {\left (\left (b^2-4 a c\right ) \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{8192 c^6}\\ &=\frac {\left (b^2-4 a c\right ) \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^7}-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^6}+\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}-\frac {\left (\left (b^2-4 a c\right )^2 \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{65536 c^7}\\ &=\frac {\left (b^2-4 a c\right ) \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^7}-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^6}+\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}-\frac {\left (\left (b^2-4 a c\right )^2 \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\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 )}{32768 c^7}\\ &=\frac {\left (b^2-4 a c\right ) \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^7}-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^6}+\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}-\frac {\left (b^2-4 a c\right )^2 \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{15/2}}\\ \end {align*}

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Mathematica [A]  time = 0.82, size = 338, normalized size = 0.74 \begin {gather*} \frac {\frac {3 \left (96 a^2 A c^3-240 a^2 b B c^2-432 a A b^2 c^2+440 a b^3 B c+198 A b^4 c-143 b^5 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 )}{65536 c^{13/2}}+\frac {x^2 (a+x (b+c x))^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{224 c^2}+\frac {(a+x (b+c x))^{5/2} \left (396 b^2 c (15 A c x-19 a B)+8 a b c^2 (837 A+935 B x)+16 a c^2 (128 a B-315 A c x)-66 b^3 c (63 A+65 B x)+3003 b^4 B\right )}{8960 c^4}+\frac {x^3 (a+x (b+c x))^{5/2} (18 A c-13 b B)}{16 c}+B x^4 (a+x (b+c x))^{5/2}}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]

[Out]

(((143*b^2*B - 198*A*b*c - 128*a*B*c)*x^2*(a + x*(b + c*x))^(5/2))/(224*c^2) + ((-13*b*B + 18*A*c)*x^3*(a + x*
(b + c*x))^(5/2))/(16*c) + B*x^4*(a + x*(b + c*x))^(5/2) + ((a + x*(b + c*x))^(5/2)*(3003*b^4*B - 66*b^3*c*(63
*A + 65*B*x) + 8*a*b*c^2*(837*A + 935*B*x) + 16*a*c^2*(128*a*B - 315*A*c*x) + 396*b^2*c*(-19*a*B + 15*A*c*x)))
/(8960*c^4) + (3*(-143*b^5*B + 198*A*b^4*c + 440*a*b^3*B*c - 432*a*A*b^2*c^2 - 240*a^2*b*B*c^2 + 96*a^2*A*c^3)
*(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*Arc
Tanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(65536*c^(13/2)))/(9*c)

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IntegrateAlgebraic [A]  time = 3.50, size = 662, normalized size = 1.45 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (262144 a^4 B c^4+1058688 a^3 A b c^4-241920 a^3 A c^5 x-1467072 a^3 b^2 B c^3+473728 a^3 b B c^4 x-131072 a^3 B c^5 x^2-1469664 a^2 A b^3 c^3+680256 a^2 A b^2 c^4 x-347904 a^2 A b c^5 x^2+161280 a^2 A c^6 x^3+1383984 a^2 b^4 B c^2-677664 a^2 b^3 B c^3 x+378240 a^2 b^2 B c^4 x^2-206592 a^2 b B c^5 x^3+98304 a^2 B c^6 x^4+551880 a A b^5 c^2-323568 a A b^4 c^3 x+224640 a A b^3 c^4 x^2-163584 a A b^2 c^5 x^3+119808 a A b c^6 x^4+1935360 a A c^7 x^5-438900 a b^6 B c+260568 a b^5 B c^2 x-183744 a b^4 B c^3 x^2+136576 a b^3 B c^4 x^3-102912 a b^2 B c^5 x^4+76800 a b B c^6 x^5+1638400 a B c^7 x^6-62370 A b^7 c+41580 A b^6 c^2 x-33264 A b^5 c^3 x^2+28512 A b^4 c^4 x^3-25344 A b^3 c^5 x^4+23040 A b^2 c^6 x^5+1566720 A b c^7 x^6+1290240 A c^8 x^7+45045 b^8 B-30030 b^7 B c x+24024 b^6 B c^2 x^2-20592 b^5 B c^3 x^3+18304 b^4 B c^4 x^4-16640 b^3 B c^5 x^5+15360 b^2 B c^6 x^6+1361920 b B c^7 x^7+1146880 B c^8 x^8\right )}{10321920 c^7}+\frac {\left (-1536 a^4 A c^5+3840 a^4 b B c^4+7680 a^3 A b^2 c^4-8960 a^3 b^3 B c^3-6720 a^2 A b^4 c^3+6048 a^2 b^5 B c^2+2016 a A b^6 c^2-1584 a b^7 B c-198 A b^8 c+143 b^9 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{65536 c^{15/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^4*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]

[Out]

(Sqrt[a + b*x + c*x^2]*(45045*b^8*B - 62370*A*b^7*c - 438900*a*b^6*B*c + 551880*a*A*b^5*c^2 + 1383984*a^2*b^4*
B*c^2 - 1469664*a^2*A*b^3*c^3 - 1467072*a^3*b^2*B*c^3 + 1058688*a^3*A*b*c^4 + 262144*a^4*B*c^4 - 30030*b^7*B*c
*x + 41580*A*b^6*c^2*x + 260568*a*b^5*B*c^2*x - 323568*a*A*b^4*c^3*x - 677664*a^2*b^3*B*c^3*x + 680256*a^2*A*b
^2*c^4*x + 473728*a^3*b*B*c^4*x - 241920*a^3*A*c^5*x + 24024*b^6*B*c^2*x^2 - 33264*A*b^5*c^3*x^2 - 183744*a*b^
4*B*c^3*x^2 + 224640*a*A*b^3*c^4*x^2 + 378240*a^2*b^2*B*c^4*x^2 - 347904*a^2*A*b*c^5*x^2 - 131072*a^3*B*c^5*x^
2 - 20592*b^5*B*c^3*x^3 + 28512*A*b^4*c^4*x^3 + 136576*a*b^3*B*c^4*x^3 - 163584*a*A*b^2*c^5*x^3 - 206592*a^2*b
*B*c^5*x^3 + 161280*a^2*A*c^6*x^3 + 18304*b^4*B*c^4*x^4 - 25344*A*b^3*c^5*x^4 - 102912*a*b^2*B*c^5*x^4 + 11980
8*a*A*b*c^6*x^4 + 98304*a^2*B*c^6*x^4 - 16640*b^3*B*c^5*x^5 + 23040*A*b^2*c^6*x^5 + 76800*a*b*B*c^6*x^5 + 1935
360*a*A*c^7*x^5 + 15360*b^2*B*c^6*x^6 + 1566720*A*b*c^7*x^6 + 1638400*a*B*c^7*x^6 + 1361920*b*B*c^7*x^7 + 1290
240*A*c^8*x^7 + 1146880*B*c^8*x^8))/(10321920*c^7) + ((143*b^9*B - 198*A*b^8*c - 1584*a*b^7*B*c + 2016*a*A*b^6
*c^2 + 6048*a^2*b^5*B*c^2 - 6720*a^2*A*b^4*c^3 - 8960*a^3*b^3*B*c^3 + 7680*a^3*A*b^2*c^4 + 3840*a^4*b*B*c^4 -
1536*a^4*A*c^5)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(65536*c^(15/2))

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fricas [A]  time = 0.72, size = 1263, normalized size = 2.78

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/41287680*(315*(143*B*b^9 - 1536*A*a^4*c^5 + 3840*(B*a^4*b + 2*A*a^3*b^2)*c^4 - 2240*(4*B*a^3*b^3 + 3*A*a^2
*b^4)*c^3 + 2016*(3*B*a^2*b^5 + A*a*b^6)*c^2 - 198*(8*B*a*b^7 + A*b^8)*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*(1146880*B*c^9*x^8 + 45045*B*b^8*c + 71680*(19*B
*b*c^8 + 18*A*c^9)*x^7 + 5120*(3*B*b^2*c^7 + 2*(160*B*a + 153*A*b)*c^8)*x^6 + 128*(2048*B*a^4 + 8271*A*a^3*b)*
c^5 - 1280*(13*B*b^3*c^6 - 1512*A*a*c^8 - 6*(10*B*a*b + 3*A*b^2)*c^7)*x^5 - 2592*(566*B*a^3*b^2 + 567*A*a^2*b^
3)*c^4 + 128*(143*B*b^4*c^5 + 24*(32*B*a^2 + 39*A*a*b)*c^7 - 6*(134*B*a*b^2 + 33*A*b^3)*c^6)*x^4 + 504*(2746*B
*a^2*b^4 + 1095*A*a*b^5)*c^3 - 16*(1287*B*b^5*c^4 - 10080*A*a^2*c^7 + 48*(269*B*a^2*b + 213*A*a*b^2)*c^6 - 22*
(388*B*a*b^3 + 81*A*b^4)*c^5)*x^3 - 2310*(190*B*a*b^6 + 27*A*b^7)*c^2 + 8*(3003*B*b^6*c^3 - 32*(512*B*a^3 + 13
59*A*a^2*b)*c^6 + 240*(197*B*a^2*b^2 + 117*A*a*b^3)*c^5 - 198*(116*B*a*b^4 + 21*A*b^5)*c^4)*x^2 - 2*(15015*B*b
^7*c^2 + 120960*A*a^3*c^6 - 32*(7402*B*a^3*b + 10629*A*a^2*b^2)*c^5 + 72*(4706*B*a^2*b^3 + 2247*A*a*b^4)*c^4 -
1386*(94*B*a*b^5 + 15*A*b^6)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^8, 1/20643840*(315*(143*B*b^9 - 1536*A*a^4*c^5
+ 3840*(B*a^4*b + 2*A*a^3*b^2)*c^4 - 2240*(4*B*a^3*b^3 + 3*A*a^2*b^4)*c^3 + 2016*(3*B*a^2*b^5 + A*a*b^6)*c^2 -
198*(8*B*a*b^7 + A*b^8)*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*(1146880*B*c^9*x^8 + 45045*B*b^8*c + 71680*(19*B*b*c^8 + 18*A*c^9)*x^7 + 5120*(3*B*b^2*c^7 + 2*(160*
B*a + 153*A*b)*c^8)*x^6 + 128*(2048*B*a^4 + 8271*A*a^3*b)*c^5 - 1280*(13*B*b^3*c^6 - 1512*A*a*c^8 - 6*(10*B*a*
b + 3*A*b^2)*c^7)*x^5 - 2592*(566*B*a^3*b^2 + 567*A*a^2*b^3)*c^4 + 128*(143*B*b^4*c^5 + 24*(32*B*a^2 + 39*A*a*
b)*c^7 - 6*(134*B*a*b^2 + 33*A*b^3)*c^6)*x^4 + 504*(2746*B*a^2*b^4 + 1095*A*a*b^5)*c^3 - 16*(1287*B*b^5*c^4 -
10080*A*a^2*c^7 + 48*(269*B*a^2*b + 213*A*a*b^2)*c^6 - 22*(388*B*a*b^3 + 81*A*b^4)*c^5)*x^3 - 2310*(190*B*a*b^
6 + 27*A*b^7)*c^2 + 8*(3003*B*b^6*c^3 - 32*(512*B*a^3 + 1359*A*a^2*b)*c^6 + 240*(197*B*a^2*b^2 + 117*A*a*b^3)*
c^5 - 198*(116*B*a*b^4 + 21*A*b^5)*c^4)*x^2 - 2*(15015*B*b^7*c^2 + 120960*A*a^3*c^6 - 32*(7402*B*a^3*b + 10629
*A*a^2*b^2)*c^5 + 72*(4706*B*a^2*b^3 + 2247*A*a*b^4)*c^4 - 1386*(94*B*a*b^5 + 15*A*b^6)*c^3)*x)*sqrt(c*x^2 + b
*x + a))/c^8]

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giac [A]  time = 0.32, size = 639, normalized size = 1.40 \begin {gather*} \frac {1}{10321920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, {\left (16 \, B c x + \frac {19 \, B b c^{8} + 18 \, A c^{9}}{c^{8}}\right )} x + \frac {3 \, B b^{2} c^{7} + 320 \, B a c^{8} + 306 \, A b c^{8}}{c^{8}}\right )} x - \frac {13 \, B b^{3} c^{6} - 60 \, B a b c^{7} - 18 \, A b^{2} c^{7} - 1512 \, A a c^{8}}{c^{8}}\right )} x + \frac {143 \, B b^{4} c^{5} - 804 \, B a b^{2} c^{6} - 198 \, A b^{3} c^{6} + 768 \, B a^{2} c^{7} + 936 \, A a b c^{7}}{c^{8}}\right )} x - \frac {1287 \, B b^{5} c^{4} - 8536 \, B a b^{3} c^{5} - 1782 \, A b^{4} c^{5} + 12912 \, B a^{2} b c^{6} + 10224 \, A a b^{2} c^{6} - 10080 \, A a^{2} c^{7}}{c^{8}}\right )} x + \frac {3003 \, B b^{6} c^{3} - 22968 \, B a b^{4} c^{4} - 4158 \, A b^{5} c^{4} + 47280 \, B a^{2} b^{2} c^{5} + 28080 \, A a b^{3} c^{5} - 16384 \, B a^{3} c^{6} - 43488 \, A a^{2} b c^{6}}{c^{8}}\right )} x - \frac {15015 \, B b^{7} c^{2} - 130284 \, B a b^{5} c^{3} - 20790 \, A b^{6} c^{3} + 338832 \, B a^{2} b^{3} c^{4} + 161784 \, A a b^{4} c^{4} - 236864 \, B a^{3} b c^{5} - 340128 \, A a^{2} b^{2} c^{5} + 120960 \, A a^{3} c^{6}}{c^{8}}\right )} x + \frac {45045 \, B b^{8} c - 438900 \, B a b^{6} c^{2} - 62370 \, A b^{7} c^{2} + 1383984 \, B a^{2} b^{4} c^{3} + 551880 \, A a b^{5} c^{3} - 1467072 \, B a^{3} b^{2} c^{4} - 1469664 \, A a^{2} b^{3} c^{4} + 262144 \, B a^{4} c^{5} + 1058688 \, A a^{3} b c^{5}}{c^{8}}\right )} + \frac {{\left (143 \, B b^{9} - 1584 \, B a b^{7} c - 198 \, A b^{8} c + 6048 \, B a^{2} b^{5} c^{2} + 2016 \, A a b^{6} c^{2} - 8960 \, B a^{3} b^{3} c^{3} - 6720 \, A a^{2} b^{4} c^{3} + 3840 \, B a^{4} b c^{4} + 7680 \, A a^{3} b^{2} c^{4} - 1536 \, A a^{4} c^{5}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{65536 \, c^{\frac {15}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

1/10321920*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*(14*(16*B*c*x + (19*B*b*c^8 + 18*A*c^9)/c^8)*x + (3*B*b^2*
c^7 + 320*B*a*c^8 + 306*A*b*c^8)/c^8)*x - (13*B*b^3*c^6 - 60*B*a*b*c^7 - 18*A*b^2*c^7 - 1512*A*a*c^8)/c^8)*x +
(143*B*b^4*c^5 - 804*B*a*b^2*c^6 - 198*A*b^3*c^6 + 768*B*a^2*c^7 + 936*A*a*b*c^7)/c^8)*x - (1287*B*b^5*c^4 -
8536*B*a*b^3*c^5 - 1782*A*b^4*c^5 + 12912*B*a^2*b*c^6 + 10224*A*a*b^2*c^6 - 10080*A*a^2*c^7)/c^8)*x + (3003*B*
b^6*c^3 - 22968*B*a*b^4*c^4 - 4158*A*b^5*c^4 + 47280*B*a^2*b^2*c^5 + 28080*A*a*b^3*c^5 - 16384*B*a^3*c^6 - 434
88*A*a^2*b*c^6)/c^8)*x - (15015*B*b^7*c^2 - 130284*B*a*b^5*c^3 - 20790*A*b^6*c^3 + 338832*B*a^2*b^3*c^4 + 1617
84*A*a*b^4*c^4 - 236864*B*a^3*b*c^5 - 340128*A*a^2*b^2*c^5 + 120960*A*a^3*c^6)/c^8)*x + (45045*B*b^8*c - 43890
0*B*a*b^6*c^2 - 62370*A*b^7*c^2 + 1383984*B*a^2*b^4*c^3 + 551880*A*a*b^5*c^3 - 1467072*B*a^3*b^2*c^4 - 1469664
*A*a^2*b^3*c^4 + 262144*B*a^4*c^5 + 1058688*A*a^3*b*c^5)/c^8) + 1/65536*(143*B*b^9 - 1584*B*a*b^7*c - 198*A*b^
8*c + 6048*B*a^2*b^5*c^2 + 2016*A*a*b^6*c^2 - 8960*B*a^3*b^3*c^3 - 6720*A*a^2*b^4*c^3 + 3840*B*a^4*b*c^4 + 768
0*A*a^3*b^2*c^4 - 1536*A*a^4*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 = 0.06, size = 1311, normalized size = 2.88

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)*(c*x^2+b*x+a)^(3/2),x)

[Out]

-253/4096*B*b^5/c^5*(c*x^2+b*x+a)^(1/2)*x*a-15/256*B*b/c^3*a^3*(c*x^2+b*x+a)^(1/2)*x-9/128*A*b^2/c^3*a*(c*x^2+
b*x+a)^(3/2)*x-57/512*A*b^2/c^3*a^2*(c*x^2+b*x+a)^(1/2)*x+153/2048*A*b^4/c^4*(c*x^2+b*x+a)^(1/2)*x*a-5/128*B*b
/c^3*a^2*(c*x^2+b*x+a)^(3/2)*x+125/1024*B*b^3/c^4*a^2*(c*x^2+b*x+a)^(1/2)*x+55/768*B*b^3/c^4*a*(c*x^2+b*x+a)^(
3/2)*x+187/2016*B*b/c^3*a*x*(c*x^2+b*x+a)^(5/2)+1/8*A*x^3*(c*x^2+b*x+a)^(5/2)/c+99/32768*A*b^8/c^(13/2)*ln((c*
x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/128*A*a^4/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-33/640*A
*b^3/c^4*(c*x^2+b*x+a)^(5/2)+33/2048*A*b^5/c^5*(c*x^2+b*x+a)^(3/2)-99/16384*A*b^7/c^6*(c*x^2+b*x+a)^(1/2)+8/31
5*B*a^2/c^3*(c*x^2+b*x+a)^(5/2)-143/65536*B*b^9/c^(15/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+143/3840*
B*b^4/c^5*(c*x^2+b*x+a)^(5/2)-143/12288*B*b^6/c^6*(c*x^2+b*x+a)^(3/2)+143/32768*B*b^8/c^7*(c*x^2+b*x+a)^(1/2)+
3/256*A*a^3/c^3*(c*x^2+b*x+a)^(1/2)*b+3/128*A*a^3/c^2*(c*x^2+b*x+a)^(1/2)*x-15/128*A*b^2/c^(7/2)*a^3*ln((c*x+1
/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-63/2048*A*b^6/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-143/61
44*B*b^5/c^5*(c*x^2+b*x+a)^(3/2)*x-99/8192*A*b^6/c^5*(c*x^2+b*x+a)^(1/2)*x+153/4096*A*b^5/c^5*(c*x^2+b*x+a)^(1
/2)*a-9/256*A*b^3/c^4*a*(c*x^2+b*x+a)^(3/2)-57/1024*A*b^3/c^4*a^2*(c*x^2+b*x+a)^(1/2)+1/64*A*a^2/c^2*(c*x^2+b*
x+a)^(3/2)*x+1/128*A*a^2/c^3*(c*x^2+b*x+a)^(3/2)*b+105/1024*A*b^4/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))*a^2-11/112*A*b/c^2*x^2*(c*x^2+b*x+a)^(5/2)+93/1120*A*b/c^3*a*(c*x^2+b*x+a)^(5/2)+33/448*A*b^2/c^3*x*(c
*x^2+b*x+a)^(5/2)+33/1024*A*b^4/c^4*(c*x^2+b*x+a)^(3/2)*x-4/63*B*a/c^2*x^2*(c*x^2+b*x+a)^(5/2)-15/256*B*b/c^(7
/2)*a^4*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+35/256*B*b^3/c^(9/2)*a^3*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x
+a)^(1/2))+99/4096*B*b^7/c^(13/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-189/2048*B*b^5/c^(11/2)*ln((c*
x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+143/16384*B*b^7/c^6*(c*x^2+b*x+a)^(1/2)*x-253/8192*B*b^6/c^6*(c*x^2+
b*x+a)^(1/2)*a+55/1536*B*b^4/c^5*a*(c*x^2+b*x+a)^(3/2)+125/2048*B*b^4/c^5*a^2*(c*x^2+b*x+a)^(1/2)-13/144*B*b/c
^2*x^3*(c*x^2+b*x+a)^(5/2)-5/256*B*b^2/c^4*a^2*(c*x^2+b*x+a)^(3/2)-15/512*B*b^2/c^4*a^3*(c*x^2+b*x+a)^(1/2)+14
3/2016*B*b^2/c^3*x^2*(c*x^2+b*x+a)^(5/2)-209/2240*B*b^2/c^4*a*(c*x^2+b*x+a)^(5/2)-143/2688*B*b^3/c^4*x*(c*x^2+
b*x+a)^(5/2)-1/16*A*a/c^2*x*(c*x^2+b*x+a)^(5/2)+1/9*B*x^4*(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^4*(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^4\,\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^4*(A + B*x)*(a + b*x + c*x^2)^(3/2),x)

[Out]

int(x^4*(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^{4} \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**4*(B*x+A)*(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(x**4*(A + B*x)*(a + b*x + c*x**2)**(3/2), x)

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