### 3.302 $$\int (d+e x)^3 (b x+c x^2)^{5/2} \, dx$$

Optimal. Leaf size=332 $\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{32768 c^6}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{12288 c^5}+\frac{e \left (b x+c x^2\right )^{7/2} \left (99 b^2 e^2+154 c e x (2 c d-b e)-486 b c d e+640 c^2 d^2\right )}{2016 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{768 c^4}-\frac{5 b^6 (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{32768 c^{13/2}}+\frac{e \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c}$

[Out]

(5*b^4*(2*c*d - b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(32768*c^6) - (5*b^
2*(2*c*d - b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(12288*c^5) + ((2*c*d
- b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(768*c^4) + (e*(d + e*x)^2*(b*x
+ c*x^2)^(7/2))/(9*c) + (e*(640*c^2*d^2 - 486*b*c*d*e + 99*b^2*e^2 + 154*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(
7/2))/(2016*c^3) - (5*b^6*(2*c*d - b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x +
c*x^2]])/(32768*c^(13/2))

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Rubi [A]  time = 0.454318, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.238, Rules used = {742, 779, 612, 620, 206} $\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{32768 c^6}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{12288 c^5}+\frac{e \left (b x+c x^2\right )^{7/2} \left (99 b^2 e^2+154 c e x (2 c d-b e)-486 b c d e+640 c^2 d^2\right )}{2016 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{768 c^4}-\frac{5 b^6 (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{32768 c^{13/2}}+\frac{e \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*(b*x + c*x^2)^(5/2),x]

[Out]

(5*b^4*(2*c*d - b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(32768*c^6) - (5*b^
2*(2*c*d - b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(12288*c^5) + ((2*c*d
- b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(768*c^4) + (e*(d + e*x)^2*(b*x
+ c*x^2)^(7/2))/(9*c) + (e*(640*c^2*d^2 - 486*b*c*d*e + 99*b^2*e^2 + 154*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(
7/2))/(2016*c^3) - (5*b^6*(2*c*d - b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x +
c*x^2]])/(32768*c^(13/2))

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 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 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

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)^3 \left (b x+c x^2\right )^{5/2} \, dx &=\frac{e (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{\int (d+e x) \left (\frac{1}{2} d (18 c d-7 b e)+\frac{11}{2} e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac{e (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2-486 b c d e+99 b^2 e^2+154 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}+\frac{\left ((2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right )\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{64 c^3}\\ &=\frac{(2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac{e (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2-486 b c d e+99 b^2 e^2+154 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (5 b^2 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{1536 c^4}\\ &=-\frac{5 b^2 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{(2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac{e (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2-486 b c d e+99 b^2 e^2+154 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}+\frac{\left (5 b^4 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right )\right ) \int \sqrt{b x+c x^2} \, dx}{8192 c^5}\\ &=\frac{5 b^4 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{32768 c^6}-\frac{5 b^2 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{(2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac{e (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2-486 b c d e+99 b^2 e^2+154 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (5 b^6 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{65536 c^6}\\ &=\frac{5 b^4 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{32768 c^6}-\frac{5 b^2 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{(2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac{e (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2-486 b c d e+99 b^2 e^2+154 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (5 b^6 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{32768 c^6}\\ &=\frac{5 b^4 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{32768 c^6}-\frac{5 b^2 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{(2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac{e (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2-486 b c d e+99 b^2 e^2+154 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{5 b^6 (2 c d-b e) \left (32 c^2 d^2-32 b c d e+11 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{32768 c^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.893081, size = 395, normalized size = 1.19 $\frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-84 b^6 c^2 e \left (360 d^2+135 d e x+22 e^2 x^2\right )+144 b^5 c^3 \left (140 d^2 e x+140 d^3+63 d e^2 x^2+11 e^3 x^3\right )-32 b^4 c^4 x \left (504 d^2 e x+420 d^3+243 d e^2 x^2+44 e^3 x^3\right )+256 b^3 c^5 x^2 \left (54 d^2 e x+42 d^3+27 d e^2 x^2+5 e^3 x^3\right )+1536 b^2 c^6 x^3 \left (888 d^2 e x+378 d^3+729 d e^2 x^2+206 e^3 x^3\right )+210 b^7 c e^2 (81 d+11 e x)-3465 b^8 e^3+2048 b c^7 x^4 \left (1044 d^2 e x+420 d^3+891 d e^2 x^2+259 e^3 x^3\right )+4096 c^8 x^5 \left (216 d^2 e x+84 d^3+189 d e^2 x^2+56 e^3 x^3\right )\right )+\frac{315 b^{11/2} \left (-54 b^2 c d e^2+11 b^3 e^3+96 b c^2 d^2 e-64 c^3 d^3\right ) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{2064384 c^{13/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^3*(b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-3465*b^8*e^3 + 210*b^7*c*e^2*(81*d + 11*e*x) - 84*b^6*c^2*e*(360*d^2 + 135*d*e*x
+ 22*e^2*x^2) + 256*b^3*c^5*x^2*(42*d^3 + 54*d^2*e*x + 27*d*e^2*x^2 + 5*e^3*x^3) + 144*b^5*c^3*(140*d^3 + 140
*d^2*e*x + 63*d*e^2*x^2 + 11*e^3*x^3) - 32*b^4*c^4*x*(420*d^3 + 504*d^2*e*x + 243*d*e^2*x^2 + 44*e^3*x^3) + 40
96*c^8*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 1536*b^2*c^6*x^3*(378*d^3 + 888*d^2*e*x + 729
*d*e^2*x^2 + 206*e^3*x^3) + 2048*b*c^7*x^4*(420*d^3 + 1044*d^2*e*x + 891*d*e^2*x^2 + 259*e^3*x^3)) + (315*b^(1
1/2)*(-64*c^3*d^3 + 96*b*c^2*d^2*e - 54*b^2*c*d*e^2 + 11*b^3*e^3)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]
*Sqrt[1 + (c*x)/b])))/(2064384*c^(13/2))

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Maple [B]  time = 0.056, size = 813, normalized size = 2.5 \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)^3*(c*x^2+b*x)^(5/2),x)

[Out]

9/64*d*e^2*b^2/c^2*x*(c*x^2+b*x)^(5/2)+1/12*d^3/c*(c*x^2+b*x)^(5/2)*b-5/192*d^3*b^3/c^2*(c*x^2+b*x)^(3/2)+5/51
2*d^3*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*d^3*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/9*e^3*x^2*(
c*x^2+b*x)^(7/2)/c+11/224*e^3*b^2/c^3*(c*x^2+b*x)^(7/2)-11/768*e^3*b^4/c^4*(c*x^2+b*x)^(5/2)+55/12288*e^3*b^6/
c^5*(c*x^2+b*x)^(3/2)-55/32768*e^3*b^8/c^6*(c*x^2+b*x)^(1/2)-45/1024*d*e^2*b^4/c^3*(c*x^2+b*x)^(3/2)*x+135/819
2*d*e^2*b^6/c^4*(c*x^2+b*x)^(1/2)*x-1/4*d^2*e*b/c*x*(c*x^2+b*x)^(5/2)+5/64*d^2*e*b^3/c^2*(c*x^2+b*x)^(3/2)*x-1
5/512*d^2*e*b^5/c^3*(c*x^2+b*x)^(1/2)*x-15/1024*d^2*e*b^6/c^4*(c*x^2+b*x)^(1/2)+15/2048*d^2*e*b^7/c^(9/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))-5/96*d^3*b^2/c*(c*x^2+b*x)^(3/2)*x+5/256*d^3*b^4/c^2*(c*x^2+b*x)^(1/2)*x
+3/8*d*e^2*x*(c*x^2+b*x)^(7/2)/c-27/112*d*e^2*b/c^2*(c*x^2+b*x)^(7/2)+9/128*d*e^2*b^3/c^3*(c*x^2+b*x)^(5/2)-45
/2048*d*e^2*b^5/c^4*(c*x^2+b*x)^(3/2)+135/16384*d*e^2*b^7/c^5*(c*x^2+b*x)^(1/2)-135/32768*d*e^2*b^8/c^(11/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))-1/8*d^2*e*b^2/c^2*(c*x^2+b*x)^(5/2)+5/128*d^2*e*b^4/c^3*(c*x^2+b*x)^(
3/2)-11/144*e^3*b/c^2*x*(c*x^2+b*x)^(7/2)-11/384*e^3*b^3/c^3*x*(c*x^2+b*x)^(5/2)+55/65536*e^3*b^9/c^(13/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+3/7*d^2*e*(c*x^2+b*x)^(7/2)/c+55/6144*e^3*b^5/c^4*(c*x^2+b*x)^(3/2)*x-5
5/16384*e^3*b^7/c^5*(c*x^2+b*x)^(1/2)*x+1/6*d^3*x*(c*x^2+b*x)^(5/2)

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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((e*x+d)^3*(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.2217, size = 2109, normalized size = 6.35 \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)^3*(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

[-1/4128768*(315*(64*b^6*c^3*d^3 - 96*b^7*c^2*d^2*e + 54*b^8*c*d*e^2 - 11*b^9*e^3)*sqrt(c)*log(2*c*x + b + 2*s
qrt(c*x^2 + b*x)*sqrt(c)) - 2*(229376*c^9*e^3*x^8 + 20160*b^5*c^4*d^3 - 30240*b^6*c^3*d^2*e + 17010*b^7*c^2*d*
e^2 - 3465*b^8*c*e^3 + 14336*(54*c^9*d*e^2 + 37*b*c^8*e^3)*x^7 + 3072*(288*c^9*d^2*e + 594*b*c^8*d*e^2 + 103*b
^2*c^7*e^3)*x^6 + 256*(1344*c^9*d^3 + 8352*b*c^8*d^2*e + 4374*b^2*c^7*d*e^2 + 5*b^3*c^6*e^3)*x^5 + 128*(6720*b
*c^8*d^3 + 10656*b^2*c^7*d^2*e + 54*b^3*c^6*d*e^2 - 11*b^4*c^5*e^3)*x^4 + 144*(4032*b^2*c^7*d^3 + 96*b^3*c^6*d
^2*e - 54*b^4*c^5*d*e^2 + 11*b^5*c^4*e^3)*x^3 + 168*(64*b^3*c^6*d^3 - 96*b^4*c^5*d^2*e + 54*b^5*c^4*d*e^2 - 11
*b^6*c^3*e^3)*x^2 - 210*(64*b^4*c^5*d^3 - 96*b^5*c^4*d^2*e + 54*b^6*c^3*d*e^2 - 11*b^7*c^2*e^3)*x)*sqrt(c*x^2
+ b*x))/c^7, 1/2064384*(315*(64*b^6*c^3*d^3 - 96*b^7*c^2*d^2*e + 54*b^8*c*d*e^2 - 11*b^9*e^3)*sqrt(-c)*arctan(
sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (229376*c^9*e^3*x^8 + 20160*b^5*c^4*d^3 - 30240*b^6*c^3*d^2*e + 17010*b^7*
c^2*d*e^2 - 3465*b^8*c*e^3 + 14336*(54*c^9*d*e^2 + 37*b*c^8*e^3)*x^7 + 3072*(288*c^9*d^2*e + 594*b*c^8*d*e^2 +
103*b^2*c^7*e^3)*x^6 + 256*(1344*c^9*d^3 + 8352*b*c^8*d^2*e + 4374*b^2*c^7*d*e^2 + 5*b^3*c^6*e^3)*x^5 + 128*(
6720*b*c^8*d^3 + 10656*b^2*c^7*d^2*e + 54*b^3*c^6*d*e^2 - 11*b^4*c^5*e^3)*x^4 + 144*(4032*b^2*c^7*d^3 + 96*b^3
*c^6*d^2*e - 54*b^4*c^5*d*e^2 + 11*b^5*c^4*e^3)*x^3 + 168*(64*b^3*c^6*d^3 - 96*b^4*c^5*d^2*e + 54*b^5*c^4*d*e^
2 - 11*b^6*c^3*e^3)*x^2 - 210*(64*b^4*c^5*d^3 - 96*b^5*c^4*d^2*e + 54*b^6*c^3*d*e^2 - 11*b^7*c^2*e^3)*x)*sqrt(
c*x^2 + b*x))/c^7]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{3}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(c*x**2+b*x)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)*(d + e*x)**3, x)

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Giac [A]  time = 1.73549, size = 648, normalized size = 1.95 \begin{align*} \frac{1}{2064384} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (14 \,{\left (16 \, c^{2} x e^{3} + \frac{54 \, c^{10} d e^{2} + 37 \, b c^{9} e^{3}}{c^{8}}\right )} x + \frac{3 \,{\left (288 \, c^{10} d^{2} e + 594 \, b c^{9} d e^{2} + 103 \, b^{2} c^{8} e^{3}\right )}}{c^{8}}\right )} x + \frac{1344 \, c^{10} d^{3} + 8352 \, b c^{9} d^{2} e + 4374 \, b^{2} c^{8} d e^{2} + 5 \, b^{3} c^{7} e^{3}}{c^{8}}\right )} x + \frac{6720 \, b c^{9} d^{3} + 10656 \, b^{2} c^{8} d^{2} e + 54 \, b^{3} c^{7} d e^{2} - 11 \, b^{4} c^{6} e^{3}}{c^{8}}\right )} x + \frac{9 \,{\left (4032 \, b^{2} c^{8} d^{3} + 96 \, b^{3} c^{7} d^{2} e - 54 \, b^{4} c^{6} d e^{2} + 11 \, b^{5} c^{5} e^{3}\right )}}{c^{8}}\right )} x + \frac{21 \,{\left (64 \, b^{3} c^{7} d^{3} - 96 \, b^{4} c^{6} d^{2} e + 54 \, b^{5} c^{5} d e^{2} - 11 \, b^{6} c^{4} e^{3}\right )}}{c^{8}}\right )} x - \frac{105 \,{\left (64 \, b^{4} c^{6} d^{3} - 96 \, b^{5} c^{5} d^{2} e + 54 \, b^{6} c^{4} d e^{2} - 11 \, b^{7} c^{3} e^{3}\right )}}{c^{8}}\right )} x + \frac{315 \,{\left (64 \, b^{5} c^{5} d^{3} - 96 \, b^{6} c^{4} d^{2} e + 54 \, b^{7} c^{3} d e^{2} - 11 \, b^{8} c^{2} e^{3}\right )}}{c^{8}}\right )} + \frac{5 \,{\left (64 \, b^{6} c^{3} d^{3} - 96 \, b^{7} c^{2} d^{2} e + 54 \, b^{8} c d e^{2} - 11 \, b^{9} e^{3}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{65536 \, c^{\frac{13}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

1/2064384*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(4*(14*(16*c^2*x*e^3 + (54*c^10*d*e^2 + 37*b*c^9*e^3)/c^8)*x + 3*(2
88*c^10*d^2*e + 594*b*c^9*d*e^2 + 103*b^2*c^8*e^3)/c^8)*x + (1344*c^10*d^3 + 8352*b*c^9*d^2*e + 4374*b^2*c^8*d
*e^2 + 5*b^3*c^7*e^3)/c^8)*x + (6720*b*c^9*d^3 + 10656*b^2*c^8*d^2*e + 54*b^3*c^7*d*e^2 - 11*b^4*c^6*e^3)/c^8)
*x + 9*(4032*b^2*c^8*d^3 + 96*b^3*c^7*d^2*e - 54*b^4*c^6*d*e^2 + 11*b^5*c^5*e^3)/c^8)*x + 21*(64*b^3*c^7*d^3 -
96*b^4*c^6*d^2*e + 54*b^5*c^5*d*e^2 - 11*b^6*c^4*e^3)/c^8)*x - 105*(64*b^4*c^6*d^3 - 96*b^5*c^5*d^2*e + 54*b^
6*c^4*d*e^2 - 11*b^7*c^3*e^3)/c^8)*x + 315*(64*b^5*c^5*d^3 - 96*b^6*c^4*d^2*e + 54*b^7*c^3*d*e^2 - 11*b^8*c^2*
e^3)/c^8) + 5/65536*(64*b^6*c^3*d^3 - 96*b^7*c^2*d^2*e + 54*b^8*c*d*e^2 - 11*b^9*e^3)*log(abs(-2*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(13/2)