3.1566 \(\int (b+2 c x) (d+e x)^2 (a+b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=289 \[ \frac{\left (a+b x+c x^2\right )^{7/2} \left (-2 c e (16 a e+9 b d)+9 b^2 e^2+14 c e x (2 c d-b e)+32 c^2 d^2\right )}{504 c^2}+\frac{5 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{8192 c^5}-\frac{5 e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{3072 c^4}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{192 c^3}-\frac{5 e \left (b^2-4 a c\right )^4 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2} \]

[Out]

(5*(b^2 - 4*a*c)^3*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*c^5) - (5*(b^2 - 4*a*c)^2*e*(2*c*d
- b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(3072*c^4) + ((b^2 - 4*a*c)*e*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x
+ c*x^2)^(5/2))/(192*c^3) + (2*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/9 + ((32*c^2*d^2 + 9*b^2*e^2 - 2*c*e*(9*b*
d + 16*a*e) + 14*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/2))/(504*c^2) - (5*(b^2 - 4*a*c)^4*e*(2*c*d - b*e)*
ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16384*c^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.547529, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {832, 779, 612, 621, 206} \[ \frac{\left (a+b x+c x^2\right )^{7/2} \left (-2 c e (16 a e+9 b d)+9 b^2 e^2+14 c e x (2 c d-b e)+32 c^2 d^2\right )}{504 c^2}+\frac{5 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{8192 c^5}-\frac{5 e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{3072 c^4}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{192 c^3}-\frac{5 e \left (b^2-4 a c\right )^4 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b^2 - 4*a*c)^3*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*c^5) - (5*(b^2 - 4*a*c)^2*e*(2*c*d
- b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(3072*c^4) + ((b^2 - 4*a*c)*e*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x
+ c*x^2)^(5/2))/(192*c^3) + (2*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/9 + ((32*c^2*d^2 + 9*b^2*e^2 - 2*c*e*(9*b*
d + 16*a*e) + 14*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/2))/(504*c^2) - (5*(b^2 - 4*a*c)^4*e*(2*c*d - b*e)*
ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16384*c^(11/2))

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 (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac{\int (d+e x) (2 c (b d-2 a e)+2 c (2 c d-b e) x) \left (a+b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}+\frac{\left (\left (b^2-4 a c\right ) e (2 c d-b e)\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^3}+\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}-\frac{\left (5 \left (b^2-4 a c\right )^2 e (2 c d-b e)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{384 c^3}\\ &=-\frac{5 \left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^4}+\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^3}+\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}+\frac{\left (5 \left (b^2-4 a c\right )^3 e (2 c d-b e)\right ) \int \sqrt{a+b x+c x^2} \, dx}{2048 c^4}\\ &=\frac{5 \left (b^2-4 a c\right )^3 e (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^5}-\frac{5 \left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^4}+\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^3}+\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}-\frac{\left (5 \left (b^2-4 a c\right )^4 e (2 c d-b e)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16384 c^5}\\ &=\frac{5 \left (b^2-4 a c\right )^3 e (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^5}-\frac{5 \left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^4}+\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^3}+\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}-\frac{\left (5 \left (b^2-4 a c\right )^4 e (2 c d-b e)\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 )}{8192 c^5}\\ &=\frac{5 \left (b^2-4 a c\right )^3 e (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^5}-\frac{5 \left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^4}+\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^3}+\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}-\frac{5 \left (b^2-4 a c\right )^4 e (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.67517, size = 252, normalized size = 0.87 \[ \frac{(a+x (b+c x))^{7/2} \left (-2 c e (16 a e+9 b d+7 b e x)+9 b^2 e^2+4 c^2 d (8 d+7 e x)\right )}{504 c^2}-\frac{e \left (b^2-4 a c\right ) (b e-2 c d) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{49152 c^{11/2}}+\frac{2}{9} (d+e x)^2 (a+x (b+c x))^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^2*(a + x*(b + c*x))^(7/2))/9 + ((a + x*(b + c*x))^(7/2)*(9*b^2*e^2 + 4*c^2*d*(8*d + 7*e*x) - 2*c*
e*(9*b*d + 16*a*e + 7*b*e*x)))/(504*c^2) - ((b^2 - 4*a*c)*e*(-2*c*d + b*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)])]))))/(4
9152*c^(11/2))

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Maple [B]  time = 0.012, size = 1358, 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((2*c*x+b)*(e*x+d)^2*(c*x^2+b*x+a)^(5/2),x)

[Out]

2/7*(c*x^2+b*x+a)^(7/2)*d^2+15/128*b^2/c*(c*x^2+b*x+a)^(1/2)*x*a^2*d*e-15/512*b^4/c^2*(c*x^2+b*x+a)^(1/2)*x*a*
d*e+5/96*b^2/c*(c*x^2+b*x+a)^(3/2)*x*a*d*e-5/8192*b^8/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e-
5/32*a^4/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e-15/256/c^2*e^2*b^3*(c*x^2+b*x+a)^(1/2)*x*a^2+
5/64/c*e^2*b*a^3*(c*x^2+b*x+a)^(1/2)*x+5/96/c*e^2*b*a^2*(c*x^2+b*x+a)^(3/2)*x+5/32*b^2/c^(3/2)*ln((1/2*b+c*x)/
c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3*d*e-15/256*b^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d*e+5/
192*b^3/c^2*(c*x^2+b*x+a)^(3/2)*a*d*e+5/2048*b^6/c^3*(c*x^2+b*x+a)^(1/2)*x*d*e+1/48*b^2/c*(c*x^2+b*x+a)^(5/2)*
x*d*e-5/768*b^4/c^2*(c*x^2+b*x+a)^(3/2)*x*d*e-5/64*a^3/c*(c*x^2+b*x+a)^(1/2)*b*d*e-1/24*a/c*(c*x^2+b*x+a)^(5/2
)*b*d*e-5/96*a^2/c*(c*x^2+b*x+a)^(3/2)*b*d*e+15/256*b^3/c^2*(c*x^2+b*x+a)^(1/2)*a^2*d*e-15/1024*b^5/c^3*(c*x^2
+b*x+a)^(1/2)*a*d*e-5/192/c^2*e^2*b^3*(c*x^2+b*x+a)^(3/2)*x*a+1/24/c*e^2*b*a*(c*x^2+b*x+a)^(5/2)*x+15/1024/c^3
*e^2*b^5*(c*x^2+b*x+a)^(1/2)*x*a+5/512*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*e-5/8192/c^
5*e^2*b^8*(c*x^2+b*x+a)^(1/2)-1/192/c^3*e^2*b^4*(c*x^2+b*x+a)^(5/2)+5/16384/c^(11/2)*e^2*b^9*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))+1/2*x*(c*x^2+b*x+a)^(7/2)*d*e+1/56/c^2*e^2*b^2*(c*x^2+b*x+a)^(7/2)-4/63/c*e^2*a*(c*
x^2+b*x+a)^(7/2)+5/3072/c^4*e^2*b^6*(c*x^2+b*x+a)^(3/2)+2/9*e^2*x^2*(c*x^2+b*x+a)^(7/2)+1/96*b^3/c^2*(c*x^2+b*
x+a)^(5/2)*d*e-1/28*b/c*(c*x^2+b*x+a)^(7/2)*d*e-5/32*a^3*(c*x^2+b*x+a)^(1/2)*x*d*e-1/12*a*(c*x^2+b*x+a)^(5/2)*
x*d*e-5/48*a^2*(c*x^2+b*x+a)^(3/2)*x*d*e-5/1536*b^5/c^3*(c*x^2+b*x+a)^(3/2)*d*e-1/96/c^2*e^2*b^3*(c*x^2+b*x+a)
^(5/2)*x+5/1536/c^3*e^2*b^5*(c*x^2+b*x+a)^(3/2)*x+5/128/c^2*e^2*b^2*a^3*(c*x^2+b*x+a)^(1/2)+1/48/c^2*e^2*b^2*a
*(c*x^2+b*x+a)^(5/2)-15/512/c^3*e^2*b^4*(c*x^2+b*x+a)^(1/2)*a^2+15/2048/c^4*e^2*b^6*(c*x^2+b*x+a)^(1/2)*a+5/19
2/c^2*e^2*b^2*a^2*(c*x^2+b*x+a)^(3/2)-5/384/c^3*e^2*b^4*(c*x^2+b*x+a)^(3/2)*a-5/4096/c^4*e^2*b^7*(c*x^2+b*x+a)
^(1/2)*x-1/36/c*e^2*b*x*(c*x^2+b*x+a)^(7/2)-5/1024/c^(9/2)*e^2*b^7*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
*a-5/64/c^(5/2)*e^2*b^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+15/512/c^(7/2)*e^2*b^5*ln((1/2*b+c*x)/
c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+5/64/c^(3/2)*e^2*b*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/4096*b^7
/c^4*(c*x^2+b*x+a)^(1/2)*d*e

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.96551, size = 3571, normalized size = 12.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2064384*(315*(2*(b^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5)*d*e - (b^9 - 16*a*
b^7*c + 96*a^2*b^5*c^2 - 256*a^3*b^3*c^3 + 256*a^4*b*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*(114688*c^9*e^2*x^8 + 147456*a^3*c^6*d^2 + 14336*(18*c^9*d*
e + 23*b*c^8*e^2)*x^7 + 1024*(144*c^9*d^2 + 738*b*c^8*d*e + (303*b^2*c^7 + 304*a*c^8)*e^2)*x^6 + 256*(1728*b*c
^8*d^2 + 6*(475*b^2*c^7 + 476*a*c^8)*d*e + (367*b^3*c^6 + 2220*a*b*c^7)*e^2)*x^5 + 128*(3456*(b^2*c^7 + a*c^8)
*d^2 + 6*(299*b^3*c^6 + 1804*a*b*c^7)*d*e - (b^4*c^5 - 1884*a*b^2*c^6 - 1920*a^2*c^7)*e^2)*x^4 + 16*(9216*(b^3
*c^6 + 6*a*b*c^7)*d^2 - 6*(3*b^4*c^5 - 6520*a*b^2*c^6 - 6608*a^2*c^7)*d*e + (9*b^5*c^4 - 104*a*b^3*c^5 + 10896
*a^2*b*c^6)*e^2)*x^3 + 6*(105*b^7*c^2 - 1540*a*b^5*c^3 + 8176*a^2*b^3*c^4 - 17856*a^3*b*c^5)*d*e - (315*b^8*c
- 4620*a*b^6*c^2 + 24528*a^2*b^4*c^3 - 53568*a^3*b^2*c^4 + 32768*a^4*c^5)*e^2 + 8*(55296*(a*b^2*c^6 + a^2*c^7)
*d^2 + 6*(7*b^5*c^4 - 88*a*b^3*c^5 + 10608*a^2*b*c^6)*d*e - (21*b^6*c^3 - 264*a*b^4*c^4 + 1104*a^2*b^2*c^5 - 2
048*a^3*c^6)*e^2)*x^2 + 2*(221184*a^2*b*c^6*d^2 - 6*(35*b^6*c^3 - 476*a*b^4*c^4 + 2256*a^2*b^2*c^5 - 6720*a^3*
c^6)*d*e + (105*b^7*c^2 - 1428*a*b^5*c^3 + 6768*a^2*b^3*c^4 - 11968*a^3*b*c^5)*e^2)*x)*sqrt(c*x^2 + b*x + a))/
c^6, 1/1032192*(315*(2*(b^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5)*d*e - (b^9 - 16
*a*b^7*c + 96*a^2*b^5*c^2 - 256*a^3*b^3*c^3 + 256*a^4*b*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*(114688*c^9*e^2*x^8 + 147456*a^3*c^6*d^2 + 14336*(18*c^9*d*e +
23*b*c^8*e^2)*x^7 + 1024*(144*c^9*d^2 + 738*b*c^8*d*e + (303*b^2*c^7 + 304*a*c^8)*e^2)*x^6 + 256*(1728*b*c^8*
d^2 + 6*(475*b^2*c^7 + 476*a*c^8)*d*e + (367*b^3*c^6 + 2220*a*b*c^7)*e^2)*x^5 + 128*(3456*(b^2*c^7 + a*c^8)*d^
2 + 6*(299*b^3*c^6 + 1804*a*b*c^7)*d*e - (b^4*c^5 - 1884*a*b^2*c^6 - 1920*a^2*c^7)*e^2)*x^4 + 16*(9216*(b^3*c^
6 + 6*a*b*c^7)*d^2 - 6*(3*b^4*c^5 - 6520*a*b^2*c^6 - 6608*a^2*c^7)*d*e + (9*b^5*c^4 - 104*a*b^3*c^5 + 10896*a^
2*b*c^6)*e^2)*x^3 + 6*(105*b^7*c^2 - 1540*a*b^5*c^3 + 8176*a^2*b^3*c^4 - 17856*a^3*b*c^5)*d*e - (315*b^8*c - 4
620*a*b^6*c^2 + 24528*a^2*b^4*c^3 - 53568*a^3*b^2*c^4 + 32768*a^4*c^5)*e^2 + 8*(55296*(a*b^2*c^6 + a^2*c^7)*d^
2 + 6*(7*b^5*c^4 - 88*a*b^3*c^5 + 10608*a^2*b*c^6)*d*e - (21*b^6*c^3 - 264*a*b^4*c^4 + 1104*a^2*b^2*c^5 - 2048
*a^3*c^6)*e^2)*x^2 + 2*(221184*a^2*b*c^6*d^2 - 6*(35*b^6*c^3 - 476*a*b^4*c^4 + 2256*a^2*b^2*c^5 - 6720*a^3*c^6
)*d*e + (105*b^7*c^2 - 1428*a*b^5*c^3 + 6768*a^2*b^3*c^4 - 11968*a^3*b*c^5)*e^2)*x)*sqrt(c*x^2 + b*x + a))/c^6
]

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

[Out]

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

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Giac [B]  time = 1.34572, size = 1112, normalized size = 3.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/516096*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(8*c^3*x*e^2 + (18*c^11*d*e + 23*b*c^10*e^2)/c^8)*x + (14
4*c^11*d^2 + 738*b*c^10*d*e + 303*b^2*c^9*e^2 + 304*a*c^10*e^2)/c^8)*x + (1728*b*c^10*d^2 + 2850*b^2*c^9*d*e +
2856*a*c^10*d*e + 367*b^3*c^8*e^2 + 2220*a*b*c^9*e^2)/c^8)*x + (3456*b^2*c^9*d^2 + 3456*a*c^10*d^2 + 1794*b^3
*c^8*d*e + 10824*a*b*c^9*d*e - b^4*c^7*e^2 + 1884*a*b^2*c^8*e^2 + 1920*a^2*c^9*e^2)/c^8)*x + (9216*b^3*c^8*d^2
+ 55296*a*b*c^9*d^2 - 18*b^4*c^7*d*e + 39120*a*b^2*c^8*d*e + 39648*a^2*c^9*d*e + 9*b^5*c^6*e^2 - 104*a*b^3*c^
7*e^2 + 10896*a^2*b*c^8*e^2)/c^8)*x + (55296*a*b^2*c^8*d^2 + 55296*a^2*c^9*d^2 + 42*b^5*c^6*d*e - 528*a*b^3*c^
7*d*e + 63648*a^2*b*c^8*d*e - 21*b^6*c^5*e^2 + 264*a*b^4*c^6*e^2 - 1104*a^2*b^2*c^7*e^2 + 2048*a^3*c^8*e^2)/c^
8)*x + (221184*a^2*b*c^8*d^2 - 210*b^6*c^5*d*e + 2856*a*b^4*c^6*d*e - 13536*a^2*b^2*c^7*d*e + 40320*a^3*c^8*d*
e + 105*b^7*c^4*e^2 - 1428*a*b^5*c^5*e^2 + 6768*a^2*b^3*c^6*e^2 - 11968*a^3*b*c^7*e^2)/c^8)*x + (147456*a^3*c^
8*d^2 + 630*b^7*c^4*d*e - 9240*a*b^5*c^5*d*e + 49056*a^2*b^3*c^6*d*e - 107136*a^3*b*c^7*d*e - 315*b^8*c^3*e^2
+ 4620*a*b^6*c^4*e^2 - 24528*a^2*b^4*c^5*e^2 + 53568*a^3*b^2*c^6*e^2 - 32768*a^4*c^7*e^2)/c^8) + 5/16384*(2*b^
8*c*d*e - 32*a*b^6*c^2*d*e + 192*a^2*b^4*c^3*d*e - 512*a^3*b^2*c^4*d*e + 512*a^4*c^5*d*e - b^9*e^2 + 16*a*b^7*
c*e^2 - 96*a^2*b^5*c^2*e^2 + 256*a^3*b^3*c^3*e^2 - 256*a^4*b*c^4*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*sqrt(c) - b))/c^(11/2)