3.935 \(\int x^4 (A+B x) (a+b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=543 \[ \frac{\left (a+b x+c x^2\right )^{7/2} \left (10240 a^2 B c^2-14 c x \left (2376 a A c^2-3380 a b B c-3146 A b^2 c+2145 b^3 B\right )+39688 a A b c^2-42900 a b^2 B c-28314 A b^3 c+19305 b^4 B\right )}{887040 c^5}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{30720 c^6}+\frac{\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{98304 c^7}-\frac{\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{262144 c^8}+\frac{\left (b^2-4 a c\right )^3 \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{524288 c^{17/2}}+\frac{x^2 \left (a+b x+c x^2\right )^{7/2} \left (-160 a B c-286 A b c+195 b^2 B\right )}{3960 c^3}-\frac{x^3 \left (a+b x+c x^2\right )^{7/2} (15 b B-22 A c)}{220 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c} \]

[Out]

-((b^2 - 4*a*c)^2*(195*b^5*B - 286*A*b^4*c - 520*a*b^3*B*c + 528*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])/(262144*c^8) + ((b^2 - 4*a*c)*(195*b^5*B - 286*A*b^4*c - 520*a*b^3*B*c + 5
28*a*A*b^2*c^2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^3)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(98304*c^7) - ((195*b^5*
B - 286*A*b^4*c - 520*a*b^3*B*c + 528*a*A*b^2*c^2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^3)*(b + 2*c*x)*(a + b*x + c*x
^2)^(5/2))/(30720*c^6) + ((195*b^2*B - 286*A*b*c - 160*a*B*c)*x^2*(a + b*x + c*x^2)^(7/2))/(3960*c^3) - ((15*b
*B - 22*A*c)*x^3*(a + b*x + c*x^2)^(7/2))/(220*c^2) + (B*x^4*(a + b*x + c*x^2)^(7/2))/(11*c) + ((19305*b^4*B -
28314*A*b^3*c - 42900*a*b^2*B*c + 39688*a*A*b*c^2 + 10240*a^2*B*c^2 - 14*c*(2145*b^3*B - 3146*A*b^2*c - 3380*
a*b*B*c + 2376*a*A*c^2)*x)*(a + b*x + c*x^2)^(7/2))/(887040*c^5) + ((b^2 - 4*a*c)^3*(195*b^5*B - 286*A*b^4*c -
520*a*b^3*B*c + 528*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])])/(524288*c^(17/2))

________________________________________________________________________________________

Rubi [A]  time = 0.718192, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{\left (a+b x+c x^2\right )^{7/2} \left (10240 a^2 B c^2-14 c x \left (2376 a A c^2-3380 a b B c-3146 A b^2 c+2145 b^3 B\right )+39688 a A b c^2-42900 a b^2 B c-28314 A b^3 c+19305 b^4 B\right )}{887040 c^5}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{30720 c^6}+\frac{\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{98304 c^7}-\frac{\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{262144 c^8}+\frac{\left (b^2-4 a c\right )^3 \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{524288 c^{17/2}}+\frac{x^2 \left (a+b x+c x^2\right )^{7/2} \left (-160 a B c-286 A b c+195 b^2 B\right )}{3960 c^3}-\frac{x^3 \left (a+b x+c x^2\right )^{7/2} (15 b B-22 A c)}{220 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

-((b^2 - 4*a*c)^2*(195*b^5*B - 286*A*b^4*c - 520*a*b^3*B*c + 528*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])/(262144*c^8) + ((b^2 - 4*a*c)*(195*b^5*B - 286*A*b^4*c - 520*a*b^3*B*c + 5
28*a*A*b^2*c^2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^3)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(98304*c^7) - ((195*b^5*
B - 286*A*b^4*c - 520*a*b^3*B*c + 528*a*A*b^2*c^2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^3)*(b + 2*c*x)*(a + b*x + c*x
^2)^(5/2))/(30720*c^6) + ((195*b^2*B - 286*A*b*c - 160*a*B*c)*x^2*(a + b*x + c*x^2)^(7/2))/(3960*c^3) - ((15*b
*B - 22*A*c)*x^3*(a + b*x + c*x^2)^(7/2))/(220*c^2) + (B*x^4*(a + b*x + c*x^2)^(7/2))/(11*c) + ((19305*b^4*B -
28314*A*b^3*c - 42900*a*b^2*B*c + 39688*a*A*b*c^2 + 10240*a^2*B*c^2 - 14*c*(2145*b^3*B - 3146*A*b^2*c - 3380*
a*b*B*c + 2376*a*A*c^2)*x)*(a + b*x + c*x^2)^(7/2))/(887040*c^5) + ((b^2 - 4*a*c)^3*(195*b^5*B - 286*A*b^4*c -
520*a*b^3*B*c + 528*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])])/(524288*c^(17/2))

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^4 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac{\int x^3 \left (-4 a B-\frac{1}{2} (15 b B-22 A c) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{11 c}\\ &=-\frac{(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac{\int x^2 \left (\frac{3}{2} a (15 b B-22 A c)+\frac{1}{4} \left (195 b^2 B-286 A b c-160 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{110 c^2}\\ &=\frac{\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac{(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac{\int x \left (-\frac{1}{2} a \left (195 b^2 B-286 A b c-160 a B c\right )-\frac{1}{8} \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{990 c^3}\\ &=\frac{\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac{(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac{\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}-\frac{\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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 )^{5/2} \, dx}{2560 c^5}\\ &=-\frac{\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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 )^{5/2}}{30720 c^6}+\frac{\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac{(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac{\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}+\frac{\left (\left (b^2-4 a c\right ) \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12288 c^6}\\ &=\frac{\left (b^2-4 a c\right ) \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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}}{98304 c^7}-\frac{\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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 )^{5/2}}{30720 c^6}+\frac{\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac{(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac{\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}-\frac{\left (\left (b^2-4 a c\right )^2 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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}{65536 c^7}\\ &=-\frac{\left (b^2-4 a c\right )^2 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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}}{262144 c^8}+\frac{\left (b^2-4 a c\right ) \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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}}{98304 c^7}-\frac{\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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 )^{5/2}}{30720 c^6}+\frac{\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac{(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac{\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}+\frac{\left (\left (b^2-4 a c\right )^3 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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}{524288 c^8}\\ &=-\frac{\left (b^2-4 a c\right )^2 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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}}{262144 c^8}+\frac{\left (b^2-4 a c\right ) \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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}}{98304 c^7}-\frac{\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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 )^{5/2}}{30720 c^6}+\frac{\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac{(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac{\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}+\frac{\left (\left (b^2-4 a c\right )^3 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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 )}{262144 c^8}\\ &=-\frac{\left (b^2-4 a c\right )^2 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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}}{262144 c^8}+\frac{\left (b^2-4 a c\right ) \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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}}{98304 c^7}-\frac{\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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 )^{5/2}}{30720 c^6}+\frac{\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac{(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac{\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}+\frac{\left (b^2-4 a c\right )^3 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 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 )}{524288 c^{17/2}}\\ \end{align*}

Mathematica [A]  time = 1.02085, size = 386, normalized size = 0.71 \[ \frac{\frac{11 \left (96 a^2 A c^3-240 a^2 b B c^2-528 a A b^2 c^2+520 a b^3 B c+286 A b^4 c-195 b^5 B\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (11 c x^2-20 a\right )+32 b c^2 x \left (13 a+8 c x^2\right )-40 b^3 c x+15 b^4\right )-15 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{7864320 c^{15/2}}+\frac{x^2 (a+x (b+c x))^{7/2} \left (-160 a B c-286 A b c+195 b^2 B\right )}{360 c^2}+\frac{(a+x (b+c x))^{7/2} \left (572 b^2 c (77 A c x-75 a B)+8 a b c^2 (4961 A+5915 B x)+16 a c^2 (640 a B-2079 A c x)-858 b^3 c (33 A+35 B x)+19305 b^4 B\right )}{80640 c^4}+\frac{x^3 (a+x (b+c x))^{7/2} (22 A c-15 b B)}{20 c}+B x^4 (a+x (b+c x))^{7/2}}{11 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(((195*b^2*B - 286*A*b*c - 160*a*B*c)*x^2*(a + x*(b + c*x))^(7/2))/(360*c^2) + ((-15*b*B + 22*A*c)*x^3*(a + x*
(b + c*x))^(7/2))/(20*c) + B*x^4*(a + x*(b + c*x))^(7/2) + ((a + x*(b + c*x))^(7/2)*(19305*b^4*B - 858*b^3*c*(
33*A + 35*B*x) + 8*a*b*c^2*(4961*A + 5915*B*x) + 16*a*c^2*(640*a*B - 2079*A*c*x) + 572*b^2*c*(-75*a*B + 77*A*c
*x)))/(80640*c^4) + (11*(-195*b^5*B + 286*A*b^4*c + 520*a*b^3*B*c - 528*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)]*(15*b^4 - 40*b^3*c*x + 32*b*c^2*x*(13*a + 8*c*x^2) + 8*b^
2*c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c^2*x^4)) - 15*(b^2 - 4*a*c)^3*ArcTanh[(b + 2*c*x)/(2
*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(7864320*c^(15/2)))/(11*c)

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Maple [B]  time = 0.018, size = 1848, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)

[Out]

1/11*B*x^4*(c*x^2+b*x+a)^(7/2)/c-235/4096*B*b^5/c^5*(c*x^2+b*x+a)^(1/2)*x*a^2+65/4096*B*b^7/c^6*(c*x^2+b*x+a)^
(1/2)*x*a+169/3168*B*b/c^3*a*x*(c*x^2+b*x+a)^(7/2)+13/384*B*b^3/c^4*a*(c*x^2+b*x+a)^(5/2)*x-325/12288*B*b^5/c^
5*(c*x^2+b*x+a)^(3/2)*x*a-11/512*A*b^6/c^5*(c*x^2+b*x+a)^(1/2)*x*a+145/3072*B*b^3/c^4*a^2*(c*x^2+b*x+a)^(3/2)*
x-15/512*B*b/c^3*a^4*(c*x^2+b*x+a)^(1/2)*x-5/256*B*b/c^3*a^3*(c*x^2+b*x+a)^(3/2)*x-1/64*B*b/c^3*a^2*(c*x^2+b*x
+a)^(5/2)*x+5/64*B*b^3/c^4*a^3*(c*x^2+b*x+a)^(1/2)*x-23/512*A*b^2/c^3*a^2*(c*x^2+b*x+a)^(3/2)*x-9/128*A*b^2/c^
3*a^3*(c*x^2+b*x+a)^(1/2)*x+209/6144*A*b^4/c^4*(c*x^2+b*x+a)^(3/2)*x*a-11/320*A*b^2/c^3*a*(c*x^2+b*x+a)^(5/2)*
x+139/2048*A*b^4/c^4*(c*x^2+b*x+a)^(1/2)*x*a^2+143/15360*A*b^5/c^5*(c*x^2+b*x+a)^(5/2)-143/49152*A*b^7/c^6*(c*
x^2+b*x+a)^(3/2)+143/131072*A*b^9/c^7*(c*x^2+b*x+a)^(1/2)+3/256*A*a^5/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))-143/262144*A*b^10/c^(15/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/10*A*x^3*(c*x^2+b*x+a)^(7
/2)/c-143/4480*A*b^3/c^4*(c*x^2+b*x+a)^(7/2)-13/2048*B*b^6/c^6*(c*x^2+b*x+a)^(5/2)+65/32768*B*b^8/c^7*(c*x^2+b
*x+a)^(3/2)-195/262144*B*b^10/c^8*(c*x^2+b*x+a)^(1/2)+8/693*B*a^2/c^3*(c*x^2+b*x+a)^(7/2)+195/524288*B*b^11/c^
(17/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+39/1792*B*b^4/c^5*(c*x^2+b*x+a)^(7/2)-15/1024*B*b^2/c^4*a^4
*(c*x^2+b*x+a)^(1/2)+13/768*B*b^4/c^5*a*(c*x^2+b*x+a)^(5/2)+145/6144*B*b^4/c^5*a^2*(c*x^2+b*x+a)^(3/2)+5/128*B
*b^4/c^5*a^3*(c*x^2+b*x+a)^(1/2)-13/1024*B*b^5/c^5*(c*x^2+b*x+a)^(5/2)*x+65/16384*B*b^7/c^6*(c*x^2+b*x+a)^(3/2
)*x-325/24576*B*b^6/c^6*(c*x^2+b*x+a)^(3/2)*a-195/131072*B*b^9/c^7*(c*x^2+b*x+a)^(1/2)*x-235/8192*B*b^6/c^6*(c
*x^2+b*x+a)^(1/2)*a^2+65/8192*B*b^8/c^7*(c*x^2+b*x+a)^(1/2)*a-13/384*B*b^3/c^4*x*(c*x^2+b*x+a)^(7/2)+13/264*B*
b^2/c^3*x^2*(c*x^2+b*x+a)^(7/2)-65/1344*B*b^2/c^4*a*(c*x^2+b*x+a)^(7/2)-3/44*B*b/c^2*x^3*(c*x^2+b*x+a)^(7/2)-4
/99*B*a/c^2*x^2*(c*x^2+b*x+a)^(7/2)+175/2048*B*b^3/c^(9/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-315
/4096*B*b^5/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+495/16384*B*b^7/c^(13/2)*ln((1/2*b+c*x)/c
^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+3/512*A*a^4/c^3*(c*x^2+b*x+a)^(1/2)*b-3/80*A*a/c^2*x*(c*x^2+b*x+a)^(7/2)-11/64
0*A*b^3/c^4*a*(c*x^2+b*x+a)^(5/2)-23/1024*A*b^3/c^4*a^2*(c*x^2+b*x+a)^(3/2)-9/256*A*b^3/c^4*a^3*(c*x^2+b*x+a)^
(1/2)+143/7680*A*b^4/c^4*(c*x^2+b*x+a)^(5/2)*x-143/24576*A*b^6/c^5*(c*x^2+b*x+a)^(3/2)*x+209/12288*A*b^5/c^5*(
c*x^2+b*x+a)^(3/2)*a+143/65536*A*b^8/c^6*(c*x^2+b*x+a)^(1/2)*x+139/4096*A*b^5/c^5*(c*x^2+b*x+a)^(1/2)*a^2-715/
131072*B*b^9/c^(15/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-15/512*B*b/c^(7/2)*a^5*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))-1/128*B*b^2/c^4*a^2*(c*x^2+b*x+a)^(5/2)-5/512*B*b^2/c^4*a^3*(c*x^2+b*x+a)^(3/2)-13/18
0*A*b/c^2*x^2*(c*x^2+b*x+a)^(7/2)+451/10080*A*b/c^3*a*(c*x^2+b*x+a)^(7/2)-75/1024*A*b^2/c^(7/2)*a^4*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+175/2048*A*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-315/8
192*A*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+495/65536*A*b^8/c^(13/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))*a+1/160*A*a^2/c^2*(c*x^2+b*x+a)^(5/2)*x+1/320*A*a^2/c^3*(c*x^2+b*x+a)^(5/2)*b+1/128*
A*a^3/c^2*(c*x^2+b*x+a)^(3/2)*x+1/256*A*a^3/c^3*(c*x^2+b*x+a)^(3/2)*b+3/256*A*a^4/c^2*(c*x^2+b*x+a)^(1/2)*x-11
/1024*A*b^7/c^6*(c*x^2+b*x+a)^(1/2)*a+143/2880*A*b^2/c^3*x*(c*x^2+b*x+a)^(7/2)

________________________________________________________________________________________

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^4*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 3.67637, size = 4562, normalized size = 8.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/3633315840*(3465*(195*B*b^11 + 6144*A*a^5*c^6 - 7680*(2*B*a^5*b + 5*A*a^4*b^2)*c^5 + 44800*(B*a^4*b^3 + A*a
^3*b^4)*c^4 - 20160*(2*B*a^3*b^5 + A*a^2*b^6)*c^3 + 3960*(4*B*a^2*b^7 + A*a*b^8)*c^2 - 286*(10*B*a*b^9 + A*b^1
0)*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*(82575
360*B*c^11*x^10 - 675675*B*b^10*c + 4128768*(45*B*b*c^10 + 22*A*c^11)*x^9 + 229376*(465*B*b^2*c^9 + 2*(460*B*a
+ 451*A*b)*c^10)*x^8 + 14336*(15*B*b^3*c^8 + 16632*A*a*c^10 + 2*(8650*B*a*b + 4213*A*b^2)*c^9)*x^7 + 2560*(40
96*B*a^5 + 20449*A*a^4*b)*c^6 - 1024*(225*B*b^4*c^7 - 8*(18080*B*a^2 + 34991*A*a*b)*c^9 - 30*(54*B*a*b^2 + 11*
A*b^3)*c^8)*x^6 - 42240*(1733*B*a^4*b^2 + 2295*A*a^3*b^3)*c^5 + 256*(975*B*b^5*c^6 + 687456*A*a^2*c^9 + 240*(7
1*B*a^2*b + 44*A*a*b^2)*c^8 - 10*(800*B*a*b^3 + 143*A*b^4)*c^7)*x^5 + 12672*(7265*B*a^3*b^4 + 4319*A*a^2*b^5)*
c^4 - 128*(2145*B*b^6*c^5 - 480*(64*B*a^3 + 121*A*a^2*b)*c^8 + 240*(221*B*a^2*b^2 + 110*A*a*b^3)*c^7 - 26*(760
*B*a*b^4 + 121*A*b^5)*c^6)*x^4 - 18480*(2372*B*a^2*b^6 + 671*A*a*b^7)*c^3 + 16*(19305*B*b^7*c^4 + 443520*A*a^3
*c^8 - 480*(1286*B*a^3*b + 1529*A*a^2*b^2)*c^7 + 40*(15934*B*a^2*b^3 + 6655*A*a*b^4)*c^6 - 858*(230*B*a*b^5 +
33*A*b^6)*c^5)*x^3 + 90090*(100*B*a*b^8 + 11*A*b^9)*c^2 - 8*(45045*B*b^8*c^3 + 640*(1024*B*a^4 + 3553*A*a^3*b)
*c^7 - 480*(5358*B*a^3*b^2 + 4741*A*a^2*b^3)*c^6 + 792*(2410*B*a^2*b^4 + 869*A*a*b^5)*c^5 - 858*(590*B*a*b^6 +
77*A*b^7)*c^4)*x^2 + 2*(225225*B*b^9*c^2 - 5322240*A*a^4*c^7 + 1280*(8347*B*a^4*b + 15763*A*a^3*b^2)*c^6 - 21
120*(971*B*a^3*b^3 + 690*A*a^2*b^4)*c^5 + 1584*(7520*B*a^2*b^5 + 2387*A*a*b^6)*c^4 - 30030*(92*B*a*b^7 + 11*A*
b^8)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^9, -1/1816657920*(3465*(195*B*b^11 + 6144*A*a^5*c^6 - 7680*(2*B*a^5*b +
5*A*a^4*b^2)*c^5 + 44800*(B*a^4*b^3 + A*a^3*b^4)*c^4 - 20160*(2*B*a^3*b^5 + A*a^2*b^6)*c^3 + 3960*(4*B*a^2*b^7
+ A*a*b^8)*c^2 - 286*(10*B*a*b^9 + A*b^10)*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*(82575360*B*c^11*x^10 - 675675*B*b^10*c + 4128768*(45*B*b*c^10 + 22*A*c^11)*x^9 +
229376*(465*B*b^2*c^9 + 2*(460*B*a + 451*A*b)*c^10)*x^8 + 14336*(15*B*b^3*c^8 + 16632*A*a*c^10 + 2*(8650*B*a*
b + 4213*A*b^2)*c^9)*x^7 + 2560*(4096*B*a^5 + 20449*A*a^4*b)*c^6 - 1024*(225*B*b^4*c^7 - 8*(18080*B*a^2 + 3499
1*A*a*b)*c^9 - 30*(54*B*a*b^2 + 11*A*b^3)*c^8)*x^6 - 42240*(1733*B*a^4*b^2 + 2295*A*a^3*b^3)*c^5 + 256*(975*B*
b^5*c^6 + 687456*A*a^2*c^9 + 240*(71*B*a^2*b + 44*A*a*b^2)*c^8 - 10*(800*B*a*b^3 + 143*A*b^4)*c^7)*x^5 + 12672
*(7265*B*a^3*b^4 + 4319*A*a^2*b^5)*c^4 - 128*(2145*B*b^6*c^5 - 480*(64*B*a^3 + 121*A*a^2*b)*c^8 + 240*(221*B*a
^2*b^2 + 110*A*a*b^3)*c^7 - 26*(760*B*a*b^4 + 121*A*b^5)*c^6)*x^4 - 18480*(2372*B*a^2*b^6 + 671*A*a*b^7)*c^3 +
16*(19305*B*b^7*c^4 + 443520*A*a^3*c^8 - 480*(1286*B*a^3*b + 1529*A*a^2*b^2)*c^7 + 40*(15934*B*a^2*b^3 + 6655
*A*a*b^4)*c^6 - 858*(230*B*a*b^5 + 33*A*b^6)*c^5)*x^3 + 90090*(100*B*a*b^8 + 11*A*b^9)*c^2 - 8*(45045*B*b^8*c^
3 + 640*(1024*B*a^4 + 3553*A*a^3*b)*c^7 - 480*(5358*B*a^3*b^2 + 4741*A*a^2*b^3)*c^6 + 792*(2410*B*a^2*b^4 + 86
9*A*a*b^5)*c^5 - 858*(590*B*a*b^6 + 77*A*b^7)*c^4)*x^2 + 2*(225225*B*b^9*c^2 - 5322240*A*a^4*c^7 + 1280*(8347*
B*a^4*b + 15763*A*a^3*b^2)*c^6 - 21120*(971*B*a^3*b^3 + 690*A*a^2*b^4)*c^5 + 1584*(7520*B*a^2*b^5 + 2387*A*a*b
^6)*c^4 - 30030*(92*B*a*b^7 + 11*A*b^8)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^9]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \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**4*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(x**4*(A + B*x)*(a + b*x + c*x**2)**(5/2), x)

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Giac [A]  time = 1.28481, size = 1226, normalized size = 2.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/908328960*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*(18*(20*B*c^2*x + (45*B*b*c^11 + 22*A*c^12)/c^10)*
x + (465*B*b^2*c^10 + 920*B*a*c^11 + 902*A*b*c^11)/c^10)*x + (15*B*b^3*c^9 + 17300*B*a*b*c^10 + 8426*A*b^2*c^1
0 + 16632*A*a*c^11)/c^10)*x - (225*B*b^4*c^8 - 1620*B*a*b^2*c^9 - 330*A*b^3*c^9 - 144640*B*a^2*c^10 - 279928*A
*a*b*c^10)/c^10)*x + (975*B*b^5*c^7 - 8000*B*a*b^3*c^8 - 1430*A*b^4*c^8 + 17040*B*a^2*b*c^9 + 10560*A*a*b^2*c^
9 + 687456*A*a^2*c^10)/c^10)*x - (2145*B*b^6*c^6 - 19760*B*a*b^4*c^7 - 3146*A*b^5*c^7 + 53040*B*a^2*b^2*c^8 +
26400*A*a*b^3*c^8 - 30720*B*a^3*c^9 - 58080*A*a^2*b*c^9)/c^10)*x + (19305*B*b^7*c^5 - 197340*B*a*b^5*c^6 - 283
14*A*b^6*c^6 + 637360*B*a^2*b^3*c^7 + 266200*A*a*b^4*c^7 - 617280*B*a^3*b*c^8 - 733920*A*a^2*b^2*c^8 + 443520*
A*a^3*c^9)/c^10)*x - (45045*B*b^8*c^4 - 506220*B*a*b^6*c^5 - 66066*A*b^7*c^5 + 1908720*B*a^2*b^4*c^6 + 688248*
A*a*b^5*c^6 - 2571840*B*a^3*b^2*c^7 - 2275680*A*a^2*b^3*c^7 + 655360*B*a^4*c^8 + 2273920*A*a^3*b*c^8)/c^10)*x
+ (225225*B*b^9*c^3 - 2762760*B*a*b^7*c^4 - 330330*A*b^8*c^4 + 11911680*B*a^2*b^5*c^5 + 3781008*A*a*b^6*c^5 -
20507520*B*a^3*b^3*c^6 - 14572800*A*a^2*b^4*c^6 + 10684160*B*a^4*b*c^7 + 20176640*A*a^3*b^2*c^7 - 5322240*A*a^
4*c^8)/c^10)*x - (675675*B*b^10*c^2 - 9009000*B*a*b^8*c^3 - 990990*A*b^9*c^3 + 43834560*B*a^2*b^6*c^4 + 124000
80*A*a*b^7*c^4 - 92062080*B*a^3*b^4*c^5 - 54730368*A*a^2*b^5*c^5 + 73201920*B*a^4*b^2*c^6 + 96940800*A*a^3*b^3
*c^6 - 10485760*B*a^5*c^7 - 52349440*A*a^4*b*c^7)/c^10) - 1/524288*(195*B*b^11 - 2860*B*a*b^9*c - 286*A*b^10*c
+ 15840*B*a^2*b^7*c^2 + 3960*A*a*b^8*c^2 - 40320*B*a^3*b^5*c^3 - 20160*A*a^2*b^6*c^3 + 44800*B*a^4*b^3*c^4 +
44800*A*a^3*b^4*c^4 - 15360*B*a^5*b*c^5 - 38400*A*a^4*b^2*c^5 + 6144*A*a^5*c^6)*log(abs(-2*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*sqrt(c) - b))/c^(17/2)