### 3.1220 $$\int (b d+2 c d x)^2 (a+b x+c x^2)^{5/2} \, dx$$

Optimal. Leaf size=207 $-\frac{5 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}-\frac{5 d^2 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8192 c^{7/2}}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}$

[Out]

(-5*(b^2 - 4*a*c)^3*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4096*c^3) + (5*(b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^3*S
qrt[a + b*x + c*x^2])/(2048*c^3) - (5*(b^2 - 4*a*c)*d^2*(b + 2*c*x)^3*(a + b*x + c*x^2)^(3/2))/(384*c^2) + (d^
2*(b + 2*c*x)^3*(a + b*x + c*x^2)^(5/2))/(16*c) - (5*(b^2 - 4*a*c)^4*d^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(8192*c^(7/2))

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Rubi [A]  time = 0.124275, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {685, 692, 621, 206} $-\frac{5 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}-\frac{5 d^2 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8192 c^{7/2}}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-5*(b^2 - 4*a*c)^3*d^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4096*c^3) + (5*(b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^3*S
qrt[a + b*x + c*x^2])/(2048*c^3) - (5*(b^2 - 4*a*c)*d^2*(b + 2*c*x)^3*(a + b*x + c*x^2)^(3/2))/(384*c^2) + (d^
2*(b + 2*c*x)^3*(a + b*x + c*x^2)^(5/2))/(16*c) - (5*(b^2 - 4*a*c)^4*d^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(8192*c^(7/2))

Rule 685

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(d*p*(b^2 - 4*a*c))/(b*e*(m + 2*p + 1)), Int[(d + e*x)^m*(a +
b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] &&
NeQ[m + 2*p + 3, 0] && GtQ[p, 0] &&  !LtQ[m, -1] &&  !(IGtQ[(m - 1)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p]))
&& RationalQ[m] && IntegerQ[2*p]

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

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 d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac{\left (5 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx}{32 c}\\ &=-\frac{5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}+\frac{\left (5 \left (b^2-4 a c\right )^2\right ) \int (b d+2 c d x)^2 \sqrt{a+b x+c x^2} \, dx}{256 c^2}\\ &=\frac{5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac{\left (5 \left (b^2-4 a c\right )^3\right ) \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx}{4096 c^3}\\ &=-\frac{5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac{\left (5 \left (b^2-4 a c\right )^4 d^2\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8192 c^3}\\ &=-\frac{5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac{\left (5 \left (b^2-4 a c\right )^4 d^2\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 )}{4096 c^3}\\ &=-\frac{5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac{5 \left (b^2-4 a c\right )^4 d^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8192 c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.989312, size = 225, normalized size = 1.09 $\frac{1}{4} d^2 (b+2 c x) \sqrt{a+x (b+c x)} \left (\frac{\left (b^2-4 a c\right ) \left (16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (11 c x^2-20 a\right )+32 b c^2 x \left (13 a+8 c x^2\right )-40 b^3 c x+15 b^4\right )}{3072 c^3}-\frac{5 \sqrt{c} \sqrt{4 a-\frac{b^2}{c}} (a+x (b+c x))^3 \sinh ^{-1}\left (\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right )}{2048 (b+2 c x) \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{7/2}}+(a+x (b+c x))^3\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d^2*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*((a + x*(b + c*x))^3 + ((b^2 - 4*a*c)*(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)))/(3072*c^3) - (5*Sq
rt[4*a - b^2/c]*Sqrt[c]*(a + x*(b + c*x))^3*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])])/(2048*(b + 2*c*x
)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(7/2))))/4

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Maple [B]  time = 0.049, size = 634, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/128*d^2*b^2*(c*x^2+b*x+a)^(1/2)*x*a^2+5/96*d^2*b^2*(c*x^2+b*x+a)^(3/2)*x*a+15/256*d^2/c*b^3*(c*x^2+b*x+a)^(
1/2)*a^2-15/1024*d^2/c^2*b^5*(c*x^2+b*x+a)^(1/2)*a+5/2048*d^2/c^2*b^6*(c*x^2+b*x+a)^(1/2)*x-1/12*d^2*c*a*x*(c*
x^2+b*x+a)^(5/2)-5/48*d^2*c*a^2*(c*x^2+b*x+a)^(3/2)*x-5/32*d^2*c*a^3*(c*x^2+b*x+a)^(1/2)*x-5/768*d^2/c*b^4*(c*
x^2+b*x+a)^(3/2)*x+5/192*d^2/c*b^3*(c*x^2+b*x+a)^(3/2)*a-15/512*d^2/c*b^4*(c*x^2+b*x+a)^(1/2)*x*a-5/1536*d^2/c
^2*b^5*(c*x^2+b*x+a)^(3/2)+5/4096*d^2/c^3*b^7*(c*x^2+b*x+a)^(1/2)+1/96*d^2/c*b^3*(c*x^2+b*x+a)^(5/2)+1/2*d^2*c
*x*(c*x^2+b*x+a)^(7/2)-1/24*d^2*a*(c*x^2+b*x+a)^(5/2)*b-5/96*d^2*a^2*(c*x^2+b*x+a)^(3/2)*b-5/64*d^2*a^3*(c*x^2
+b*x+a)^(1/2)*b-5/32*d^2*c^(1/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/8192*d^2*b^8/c^(7/2)*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/48*d^2*b^2*x*(c*x^2+b*x+a)^(5/2)+1/4*d^2*b*(c*x^2+b*x+a)^(7/2)+5/32*d^
2*b^2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-15/256*d^2*b^4/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*a^2+5/512*d^2*b^6/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.12294, size = 1569, normalized size = 7.58 \begin{align*} \left [\frac{15 \,{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt{c} d^{2} \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 (6144 \, c^{8} d^{2} x^{7} + 21504 \, b c^{7} d^{2} x^{6} + 256 \,{\left (109 \, b^{2} c^{6} + 68 \, a c^{7}\right )} d^{2} x^{5} + 640 \,{\left (25 \, b^{3} c^{5} + 68 \, a b c^{6}\right )} d^{2} x^{4} + 16 \,{\left (219 \, b^{4} c^{4} + 2248 \, a b^{2} c^{5} + 944 \, a^{2} c^{6}\right )} d^{2} x^{3} + 8 \,{\left (b^{5} c^{3} + 1304 \, a b^{3} c^{4} + 2832 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 2 \,{\left (5 \, b^{6} c^{2} - 68 \, a b^{4} c^{3} - 4944 \, a^{2} b^{2} c^{4} - 960 \, a^{3} c^{5}\right )} d^{2} x +{\left (15 \, b^{7} c - 220 \, a b^{5} c^{2} + 1168 \, a^{2} b^{3} c^{3} + 960 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a}}{49152 \, c^{4}}, \frac{15 \,{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt{-c} d^{2} \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 (6144 \, c^{8} d^{2} x^{7} + 21504 \, b c^{7} d^{2} x^{6} + 256 \,{\left (109 \, b^{2} c^{6} + 68 \, a c^{7}\right )} d^{2} x^{5} + 640 \,{\left (25 \, b^{3} c^{5} + 68 \, a b c^{6}\right )} d^{2} x^{4} + 16 \,{\left (219 \, b^{4} c^{4} + 2248 \, a b^{2} c^{5} + 944 \, a^{2} c^{6}\right )} d^{2} x^{3} + 8 \,{\left (b^{5} c^{3} + 1304 \, a b^{3} c^{4} + 2832 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 2 \,{\left (5 \, b^{6} c^{2} - 68 \, a b^{4} c^{3} - 4944 \, a^{2} b^{2} c^{4} - 960 \, a^{3} c^{5}\right )} d^{2} x +{\left (15 \, b^{7} c - 220 \, a b^{5} c^{2} + 1168 \, a^{2} b^{3} c^{3} + 960 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a}}{24576 \, c^{4}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/49152*(15*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(c)*d^2*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*(6144*c^8*d^2*x^7 + 21504*b*c^7*d^2*x
^6 + 256*(109*b^2*c^6 + 68*a*c^7)*d^2*x^5 + 640*(25*b^3*c^5 + 68*a*b*c^6)*d^2*x^4 + 16*(219*b^4*c^4 + 2248*a*b
^2*c^5 + 944*a^2*c^6)*d^2*x^3 + 8*(b^5*c^3 + 1304*a*b^3*c^4 + 2832*a^2*b*c^5)*d^2*x^2 - 2*(5*b^6*c^2 - 68*a*b^
4*c^3 - 4944*a^2*b^2*c^4 - 960*a^3*c^5)*d^2*x + (15*b^7*c - 220*a*b^5*c^2 + 1168*a^2*b^3*c^3 + 960*a^3*b*c^4)*
d^2)*sqrt(c*x^2 + b*x + a))/c^4, 1/24576*(15*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^
4)*sqrt(-c)*d^2*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*(6144*c^8*d
^2*x^7 + 21504*b*c^7*d^2*x^6 + 256*(109*b^2*c^6 + 68*a*c^7)*d^2*x^5 + 640*(25*b^3*c^5 + 68*a*b*c^6)*d^2*x^4 +
16*(219*b^4*c^4 + 2248*a*b^2*c^5 + 944*a^2*c^6)*d^2*x^3 + 8*(b^5*c^3 + 1304*a*b^3*c^4 + 2832*a^2*b*c^5)*d^2*x^
2 - 2*(5*b^6*c^2 - 68*a*b^4*c^3 - 4944*a^2*b^2*c^4 - 960*a^3*c^5)*d^2*x + (15*b^7*c - 220*a*b^5*c^2 + 1168*a^2
*b^3*c^3 + 960*a^3*b*c^4)*d^2)*sqrt(c*x^2 + b*x + a))/c^4]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{2} \left (\int a^{2} b^{2} \sqrt{a + b x + c x^{2}}\, dx + \int b^{4} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 c^{4} x^{6} \sqrt{a + b x + c x^{2}}\, dx + \int 2 a b^{3} x \sqrt{a + b x + c x^{2}}\, dx + \int 8 a c^{3} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 4 a^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 12 b c^{3} x^{5} \sqrt{a + b x + c x^{2}}\, dx + \int 13 b^{2} c^{2} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 6 b^{3} c x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 16 a b c^{2} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 10 a b^{2} c x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 a^{2} b c x \sqrt{a + b x + c x^{2}}\, dx\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**2*(Integral(a**2*b**2*sqrt(a + b*x + c*x**2), x) + Integral(b**4*x**2*sqrt(a + b*x + c*x**2), x) + Integral
(4*c**4*x**6*sqrt(a + b*x + c*x**2), x) + Integral(2*a*b**3*x*sqrt(a + b*x + c*x**2), x) + Integral(8*a*c**3*x
**4*sqrt(a + b*x + c*x**2), x) + Integral(4*a**2*c**2*x**2*sqrt(a + b*x + c*x**2), x) + Integral(12*b*c**3*x**
5*sqrt(a + b*x + c*x**2), x) + Integral(13*b**2*c**2*x**4*sqrt(a + b*x + c*x**2), x) + Integral(6*b**3*c*x**3*
sqrt(a + b*x + c*x**2), x) + Integral(16*a*b*c**2*x**3*sqrt(a + b*x + c*x**2), x) + Integral(10*a*b**2*c*x**2*
sqrt(a + b*x + c*x**2), x) + Integral(4*a**2*b*c*x*sqrt(a + b*x + c*x**2), x))

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Giac [B]  time = 1.21522, size = 525, normalized size = 2.54 \begin{align*} \frac{1}{12288} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (2 \, c^{4} d^{2} x + 7 \, b c^{3} d^{2}\right )} x + \frac{109 \, b^{2} c^{9} d^{2} + 68 \, a c^{10} d^{2}}{c^{7}}\right )} x + \frac{5 \,{\left (25 \, b^{3} c^{8} d^{2} + 68 \, a b c^{9} d^{2}\right )}}{c^{7}}\right )} x + \frac{219 \, b^{4} c^{7} d^{2} + 2248 \, a b^{2} c^{8} d^{2} + 944 \, a^{2} c^{9} d^{2}}{c^{7}}\right )} x + \frac{b^{5} c^{6} d^{2} + 1304 \, a b^{3} c^{7} d^{2} + 2832 \, a^{2} b c^{8} d^{2}}{c^{7}}\right )} x - \frac{5 \, b^{6} c^{5} d^{2} - 68 \, a b^{4} c^{6} d^{2} - 4944 \, a^{2} b^{2} c^{7} d^{2} - 960 \, a^{3} c^{8} d^{2}}{c^{7}}\right )} x + \frac{15 \, b^{7} c^{4} d^{2} - 220 \, a b^{5} c^{5} d^{2} + 1168 \, a^{2} b^{3} c^{6} d^{2} + 960 \, a^{3} b c^{7} d^{2}}{c^{7}}\right )} + \frac{5 \,{\left (b^{8} d^{2} - 16 \, a b^{6} c d^{2} + 96 \, a^{2} b^{4} c^{2} d^{2} - 256 \, a^{3} b^{2} c^{3} d^{2} + 256 \, a^{4} c^{4} d^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8192 \, c^{\frac{7}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12288*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(2*c^4*d^2*x + 7*b*c^3*d^2)*x + (109*b^2*c^9*d^2 + 68*a*c^10*
d^2)/c^7)*x + 5*(25*b^3*c^8*d^2 + 68*a*b*c^9*d^2)/c^7)*x + (219*b^4*c^7*d^2 + 2248*a*b^2*c^8*d^2 + 944*a^2*c^9
*d^2)/c^7)*x + (b^5*c^6*d^2 + 1304*a*b^3*c^7*d^2 + 2832*a^2*b*c^8*d^2)/c^7)*x - (5*b^6*c^5*d^2 - 68*a*b^4*c^6*
d^2 - 4944*a^2*b^2*c^7*d^2 - 960*a^3*c^8*d^2)/c^7)*x + (15*b^7*c^4*d^2 - 220*a*b^5*c^5*d^2 + 1168*a^2*b^3*c^6*
d^2 + 960*a^3*b*c^7*d^2)/c^7) + 5/8192*(b^8*d^2 - 16*a*b^6*c*d^2 + 96*a^2*b^4*c^2*d^2 - 256*a^3*b^2*c^3*d^2 +
256*a^4*c^4*d^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)