### 3.2356 $$\int (d+e x)^2 (a+b x+c x^2)^{5/2} \, dx$$

Optimal. Leaf size=323 $\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{384 c^3}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{6144 c^4}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac{9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}$

[Out]

(5*(b^2 - 4*a*c)^2*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16384*c^
5) - (5*(b^2 - 4*a*c)*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(614
4*c^4) + ((32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(384*c^3) + (9*e
*(2*c*d - b*e)*(a + b*x + c*x^2)^(7/2))/(112*c^2) + (e*(d + e*x)*(a + b*x + c*x^2)^(7/2))/(8*c) - (5*(b^2 - 4*
a*c)^3*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/
(32768*c^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.43986, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.227, Rules used = {742, 640, 612, 621, 206} $\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{384 c^3}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{6144 c^4}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac{9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*(a + b*x + c*x^2)^(5/2),x]

[Out]

(5*(b^2 - 4*a*c)^2*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16384*c^
5) - (5*(b^2 - 4*a*c)*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(614
4*c^4) + ((32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(384*c^3) + (9*e
*(2*c*d - b*e)*(a + b*x + c*x^2)^(7/2))/(112*c^2) + (e*(d + e*x)*(a + b*x + c*x^2)^(7/2))/(8*c) - (5*(b^2 - 4*
a*c)^3*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/
(32768*c^(11/2))

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[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 (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac{\int \left (\frac{1}{2} \left (16 c d^2-2 e \left (\frac{7 b d}{2}+a e\right )\right )+\frac{9}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac{9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac{\left (-\frac{9}{2} b e (2 c d-b e)+c \left (16 c d^2-2 e \left (\frac{7 b d}{2}+a e\right )\right )\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac{\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac{9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac{\left (5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac{5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac{9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac{\left (5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{4096 c^4}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16384 c^5}-\frac{5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac{9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac{\left (5 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16384 c^5}-\frac{5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac{9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac{\left (5 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\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 )}{16384 c^5}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16384 c^5}-\frac{5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac{9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac{5 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.452208, size = 238, normalized size = 0.74 $\frac{-\frac{\left (2 c e (a e+8 b d)-\frac{9 b^2 e^2}{2}-16 c^2 d^2\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{6144 c^{9/2}}+\frac{9 e (a+x (b+c x))^{7/2} (2 c d-b e)}{14 c}+e (d+e x) (a+x (b+c x))^{7/2}}{8 c}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(5/2),x]

[Out]

((9*e*(2*c*d - b*e)*(a + x*(b + c*x))^(7/2))/(14*c) + e*(d + e*x)*(a + x*(b + c*x))^(7/2) - ((-16*c^2*d^2 - (9
*b^2*e^2)/2 + 2*c*e*(8*b*d + a*e))*(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*Sqrt[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)])]))))/(6144*c^(9/2)))/(8*c)

________________________________________________________________________________________

Maple [B]  time = 0.05, size = 1517, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(c*x^2+b*x+a)^(5/2),x)

[Out]

15/64*d*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-15/256*d*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))*a-15/1024*e^2*b^4/c^3*(c*x^2+b*x+a)^(3/2)*x+25/768*e^2*b^3/c^3*(c*x^2+b*x+a)^(3/2)*
a+45/8192*e^2*b^6/c^4*(c*x^2+b*x+a)^(1/2)*x+35/2048*e^2*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)*a-75/1024*e^2*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-5/96*d^2/c*(c*x^2+b*x+a)^(3/2)*x*b
^2-5/128*e^2*a^3/c*(c*x^2+b*x+a)^(1/2)*x-5/256*e^2*a^3/c^2*(c*x^2+b*x+a)^(1/2)*b-1/48*e^2*a/c*x*(c*x^2+b*x+a)^
(5/2)-1/96*e^2*a/c^2*(c*x^2+b*x+a)^(5/2)*b-1/12*d*e*b^2/c^2*(c*x^2+b*x+a)^(5/2)+5/192*d*e*b^4/c^3*(c*x^2+b*x+a
)^(3/2)-5/512*d*e*b^6/c^4*(c*x^2+b*x+a)^(1/2)+5/1024*d*e*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))-5/192*e^2*a^2/c*(c*x^2+b*x+a)^(3/2)*x-5/384*e^2*a^2/c^2*(c*x^2+b*x+a)^(3/2)*b+55/1024*e^2*b^3/c^3*(c*x^2+b*
x+a)^(1/2)*a^2-95/4096*e^2*b^5/c^4*(c*x^2+b*x+a)^(1/2)*a+3/64*e^2*b^2/c^2*x*(c*x^2+b*x+a)^(5/2)-5/64*d^2/c^2*(
c*x^2+b*x+a)^(1/2)*b^3*a-15/64*d^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a^2+15/256*d^2/c^(5
/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a+5/48*d^2/c*(c*x^2+b*x+a)^(3/2)*b*a+15/128*e^2*b^2/c^(5/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+2/7*d*e*(c*x^2+b*x+a)^(7/2)/c-15/2048*e^2*b^5/c^4*(c*x^2+b*x
+a)^(3/2)+45/16384*e^2*b^7/c^5*(c*x^2+b*x+a)^(1/2)+1/12*d^2/c*(c*x^2+b*x+a)^(5/2)*b+5/24*d^2*(c*x^2+b*x+a)^(3/
2)*x*a-5/192*d^2/c^2*(c*x^2+b*x+a)^(3/2)*b^3+5/16*d^2*(c*x^2+b*x+a)^(1/2)*x*a^2+5/512*d^2/c^3*(c*x^2+b*x+a)^(1
/2)*b^5+5/16*d^2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-5/1024*d^2/c^(7/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))*b^6-95/2048*e^2*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*a+25/384*e^2*b^2/c^2*(c*x^2+b*x+a)^(3/
2)*x*a+5/64*d*e*b^4/c^3*(c*x^2+b*x+a)^(1/2)*a-45/32768*e^2*b^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(
1/2))+1/8*e^2*x*(c*x^2+b*x+a)^(7/2)/c-5/128*e^2*a^4/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-9/112*
e^2*b/c^2*(c*x^2+b*x+a)^(7/2)+3/128*e^2*b^3/c^3*(c*x^2+b*x+a)^(5/2)-1/6*d*e*b/c*x*(c*x^2+b*x+a)^(5/2)+1/6*d^2*
x*(c*x^2+b*x+a)^(5/2)+5/32*d*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a+5/256*d^2/c^2*(c*x^2+b*x+a)^(1/2)*x*b^4+5/32*d^
2/c*(c*x^2+b*x+a)^(1/2)*b*a^2+5/96*d*e*b^3/c^2*(c*x^2+b*x+a)^(3/2)*x-5/48*d*e*b^2/c^2*(c*x^2+b*x+a)^(3/2)*a-5/
256*d*e*b^5/c^3*(c*x^2+b*x+a)^(1/2)*x-5/32*d*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a^2-5/32*d^2/c*(c*x^2+b*x+a)^(1/2)*
x*a*b^2+55/512*e^2*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a^2-5/16*d*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(
1/2))*a^3-5/16*d*e*b/c*(c*x^2+b*x+a)^(1/2)*x*a^2-5/24*d*e*b/c*(c*x^2+b*x+a)^(3/2)*x*a

________________________________________________________________________________________

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((e*x+d)^2*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 5.13733, size = 3314, normalized size = 10.26 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/1376256*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2 - 32*(b^7*c - 12*a*b^5*c^2 + 48
*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768*a^3*b^2*c^3 + 256*a^4*c^4)*e^2
)*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*(43008*c^8
*e^2*x^7 + 3072*(32*c^8*d*e + 33*b*c^7*e^2)*x^6 + 256*(224*c^8*d^2 + 928*b*c^7*d*e + (243*b^2*c^6 + 476*a*c^7)
*e^2)*x^5 + 128*(1120*b*c^7*d^2 + 32*(37*b^2*c^6 + 72*a*c^7)*d*e + (3*b^3*c^5 + 1228*a*b*c^6)*e^2)*x^4 + 16*(2
24*(27*b^2*c^6 + 52*a*c^7)*d^2 + 32*(3*b^3*c^5 + 788*a*b*c^6)*d*e - (27*b^4*c^4 - 216*a*b^2*c^5 - 6608*a^2*c^6
)*e^2)*x^3 + 224*(15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^5)*d^2 - 32*(105*b^6*c^2 - 1120*a*b^4*c^3 + 3696*a^
2*b^2*c^4 - 3072*a^3*c^5)*d*e + (945*b^7*c - 10500*a*b^5*c^2 + 37744*a^2*b^3*c^3 - 42432*a^3*b*c^4)*e^2 + 8*(2
24*(b^3*c^5 + 156*a*b*c^6)*d^2 - 32*(7*b^4*c^4 - 60*a*b^2*c^5 - 1152*a^2*c^6)*d*e + (63*b^5*c^3 - 568*a*b^3*c^
4 + 1392*a^2*b*c^5)*e^2)*x^2 - 2*(224*(5*b^4*c^4 - 48*a*b^2*c^5 - 528*a^2*c^6)*d^2 - 32*(35*b^5*c^3 - 336*a*b^
3*c^4 + 912*a^2*b*c^5)*d*e + (315*b^6*c^2 - 3164*a*b^4*c^3 + 9552*a^2*b^2*c^4 - 6720*a^3*c^5)*e^2)*x)*sqrt(c*x
^2 + b*x + a))/c^6, 1/688128*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2 - 32*(b^7*c -
12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768*a^3*b^2*c^3
+ 256*a^4*c^4)*e^2)*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*(43008*c^8*e^2*x^7 + 3072*(32*c^8*d*e + 33*b*c^7*e^2)*x^6 + 256*(224*c^8*d^2 + 928*b*c^7*d*e + (243*b^2*c^6
+ 476*a*c^7)*e^2)*x^5 + 128*(1120*b*c^7*d^2 + 32*(37*b^2*c^6 + 72*a*c^7)*d*e + (3*b^3*c^5 + 1228*a*b*c^6)*e^2)
*x^4 + 16*(224*(27*b^2*c^6 + 52*a*c^7)*d^2 + 32*(3*b^3*c^5 + 788*a*b*c^6)*d*e - (27*b^4*c^4 - 216*a*b^2*c^5 -
6608*a^2*c^6)*e^2)*x^3 + 224*(15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^5)*d^2 - 32*(105*b^6*c^2 - 1120*a*b^4*c
^3 + 3696*a^2*b^2*c^4 - 3072*a^3*c^5)*d*e + (945*b^7*c - 10500*a*b^5*c^2 + 37744*a^2*b^3*c^3 - 42432*a^3*b*c^4
)*e^2 + 8*(224*(b^3*c^5 + 156*a*b*c^6)*d^2 - 32*(7*b^4*c^4 - 60*a*b^2*c^5 - 1152*a^2*c^6)*d*e + (63*b^5*c^3 -
568*a*b^3*c^4 + 1392*a^2*b*c^5)*e^2)*x^2 - 2*(224*(5*b^4*c^4 - 48*a*b^2*c^5 - 528*a^2*c^6)*d^2 - 32*(35*b^5*c^
3 - 336*a*b^3*c^4 + 912*a^2*b*c^5)*d*e + (315*b^6*c^2 - 3164*a*b^4*c^3 + 9552*a^2*b^2*c^4 - 6720*a^3*c^5)*e^2)
*x)*sqrt(c*x^2 + b*x + a))/c^6]

________________________________________________________________________________________

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

[Out]

Integral((d + e*x)**2*(a + b*x + c*x**2)**(5/2), x)

________________________________________________________________________________________

Giac [B]  time = 1.16733, size = 1035, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/344064*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(14*c^2*x*e^2 + (32*c^9*d*e + 33*b*c^8*e^2)/c^7)*x + (224*c^
9*d^2 + 928*b*c^8*d*e + 243*b^2*c^7*e^2 + 476*a*c^8*e^2)/c^7)*x + (1120*b*c^8*d^2 + 1184*b^2*c^7*d*e + 2304*a*
c^8*d*e + 3*b^3*c^6*e^2 + 1228*a*b*c^7*e^2)/c^7)*x + (6048*b^2*c^7*d^2 + 11648*a*c^8*d^2 + 96*b^3*c^6*d*e + 25
216*a*b*c^7*d*e - 27*b^4*c^5*e^2 + 216*a*b^2*c^6*e^2 + 6608*a^2*c^7*e^2)/c^7)*x + (224*b^3*c^6*d^2 + 34944*a*b
*c^7*d^2 - 224*b^4*c^5*d*e + 1920*a*b^2*c^6*d*e + 36864*a^2*c^7*d*e + 63*b^5*c^4*e^2 - 568*a*b^3*c^5*e^2 + 139
2*a^2*b*c^6*e^2)/c^7)*x - (1120*b^4*c^5*d^2 - 10752*a*b^2*c^6*d^2 - 118272*a^2*c^7*d^2 - 1120*b^5*c^4*d*e + 10
752*a*b^3*c^5*d*e - 29184*a^2*b*c^6*d*e + 315*b^6*c^3*e^2 - 3164*a*b^4*c^4*e^2 + 9552*a^2*b^2*c^5*e^2 - 6720*a
^3*c^6*e^2)/c^7)*x + (3360*b^5*c^4*d^2 - 35840*a*b^3*c^5*d^2 + 118272*a^2*b*c^6*d^2 - 3360*b^6*c^3*d*e + 35840
*a*b^4*c^4*d*e - 118272*a^2*b^2*c^5*d*e + 98304*a^3*c^6*d*e + 945*b^7*c^2*e^2 - 10500*a*b^5*c^3*e^2 + 37744*a^
2*b^3*c^4*e^2 - 42432*a^3*b*c^5*e^2)/c^7) + 5/32768*(32*b^6*c^2*d^2 - 384*a*b^4*c^3*d^2 + 1536*a^2*b^2*c^4*d^2
- 2048*a^3*c^5*d^2 - 32*b^7*c*d*e + 384*a*b^5*c^2*d*e - 1536*a^2*b^3*c^3*d*e + 2048*a^3*b*c^4*d*e + 9*b^8*e^2
- 112*a*b^6*c*e^2 + 480*a^2*b^4*c^2*e^2 - 768*a^3*b^2*c^3*e^2 + 256*a^4*c^4*e^2)*log(abs(-2*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)