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

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

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Rubi [A]  time = 0.45, antiderivative size = 332, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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])

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 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]

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*}

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Mathematica [A]  time = 0.83, size = 395, normalized size = 1.19 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {315 b^{11/2} \left (11 b^3 e^3-54 b^2 c d e^2+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}}+\sqrt {c} \left (-3465 b^8 e^3+210 b^7 c e^2 (81 d+11 e x)-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^3+140 d^2 e x+63 d e^2 x^2+11 e^3 x^3\right )-32 b^4 c^4 x \left (420 d^3+504 d^2 e x+243 d e^2 x^2+44 e^3 x^3\right )+256 b^3 c^5 x^2 \left (42 d^3+54 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right )+1536 b^2 c^6 x^3 \left (378 d^3+888 d^2 e x+729 d e^2 x^2+206 e^3 x^3\right )+2048 b c^7 x^4 \left (420 d^3+1044 d^2 e x+891 d e^2 x^2+259 e^3 x^3\right )+4096 c^8 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )\right )\right )}{2064384 c^{13/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 1.75, size = 488, normalized size = 1.47 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-3465 b^8 e^3+17010 b^7 c d e^2+2310 b^7 c e^3 x-30240 b^6 c^2 d^2 e-11340 b^6 c^2 d e^2 x-1848 b^6 c^2 e^3 x^2+20160 b^5 c^3 d^3+20160 b^5 c^3 d^2 e x+9072 b^5 c^3 d e^2 x^2+1584 b^5 c^3 e^3 x^3-13440 b^4 c^4 d^3 x-16128 b^4 c^4 d^2 e x^2-7776 b^4 c^4 d e^2 x^3-1408 b^4 c^4 e^3 x^4+10752 b^3 c^5 d^3 x^2+13824 b^3 c^5 d^2 e x^3+6912 b^3 c^5 d e^2 x^4+1280 b^3 c^5 e^3 x^5+580608 b^2 c^6 d^3 x^3+1363968 b^2 c^6 d^2 e x^4+1119744 b^2 c^6 d e^2 x^5+316416 b^2 c^6 e^3 x^6+860160 b c^7 d^3 x^4+2138112 b c^7 d^2 e x^5+1824768 b c^7 d e^2 x^6+530432 b c^7 e^3 x^7+344064 c^8 d^3 x^5+884736 c^8 d^2 e x^6+774144 c^8 d e^2 x^7+229376 c^8 e^3 x^8\right )}{2064384 c^6}-\frac {5 \left (11 b^9 e^3-54 b^8 c d e^2+96 b^7 c^2 d^2 e-64 b^6 c^3 d^3\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{65536 c^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(20160*b^5*c^3*d^3 - 30240*b^6*c^2*d^2*e + 17010*b^7*c*d*e^2 - 3465*b^8*e^3 - 13440*b^4*c^4
*d^3*x + 20160*b^5*c^3*d^2*e*x - 11340*b^6*c^2*d*e^2*x + 2310*b^7*c*e^3*x + 10752*b^3*c^5*d^3*x^2 - 16128*b^4*
c^4*d^2*e*x^2 + 9072*b^5*c^3*d*e^2*x^2 - 1848*b^6*c^2*e^3*x^2 + 580608*b^2*c^6*d^3*x^3 + 13824*b^3*c^5*d^2*e*x
^3 - 7776*b^4*c^4*d*e^2*x^3 + 1584*b^5*c^3*e^3*x^3 + 860160*b*c^7*d^3*x^4 + 1363968*b^2*c^6*d^2*e*x^4 + 6912*b
^3*c^5*d*e^2*x^4 - 1408*b^4*c^4*e^3*x^4 + 344064*c^8*d^3*x^5 + 2138112*b*c^7*d^2*e*x^5 + 1119744*b^2*c^6*d*e^2
*x^5 + 1280*b^3*c^5*e^3*x^5 + 884736*c^8*d^2*e*x^6 + 1824768*b*c^7*d*e^2*x^6 + 316416*b^2*c^6*e^3*x^6 + 774144
*c^8*d*e^2*x^7 + 530432*b*c^7*e^3*x^7 + 229376*c^8*e^3*x^8))/(2064384*c^6) - (5*(-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[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*x^2]])/(65536*c^(13/2))

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fricas [A]  time = 0.45, size = 915, normalized size = 2.76 \begin {gather*} \left [-\frac {315 \, {\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 )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (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 \, {\left (54 \, c^{9} d e^{2} + 37 \, b c^{8} e^{3}\right )} x^{7} + 3072 \, {\left (288 \, c^{9} d^{2} e + 594 \, b c^{8} d e^{2} + 103 \, b^{2} c^{7} e^{3}\right )} x^{6} + 256 \, {\left (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}\right )} x^{5} + 128 \, {\left (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}\right )} x^{4} + 144 \, {\left (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}\right )} x^{3} + 168 \, {\left (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}\right )} x^{2} - 210 \, {\left (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}\right )} x\right )} \sqrt {c x^{2} + b x}}{4128768 \, c^{7}}, \frac {315 \, {\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 )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (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 \, {\left (54 \, c^{9} d e^{2} + 37 \, b c^{8} e^{3}\right )} x^{7} + 3072 \, {\left (288 \, c^{9} d^{2} e + 594 \, b c^{8} d e^{2} + 103 \, b^{2} c^{7} e^{3}\right )} x^{6} + 256 \, {\left (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}\right )} x^{5} + 128 \, {\left (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}\right )} x^{4} + 144 \, {\left (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}\right )} x^{3} + 168 \, {\left (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}\right )} x^{2} - 210 \, {\left (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}\right )} x\right )} \sqrt {c x^{2} + b x}}{2064384 \, c^{7}}\right ] \end {gather*}

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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giac [A]  time = 0.30, size = 480, normalized size = 1.45 \begin {gather*} \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 {gather*}

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)

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maple [B]  time = 0.05, size = 813, normalized size = 2.45 \begin {gather*} \frac {55 b^{9} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{65536 c^{\frac {13}{2}}}-\frac {135 b^{8} d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{32768 c^{\frac {11}{2}}}+\frac {15 b^{7} d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {9}{2}}}-\frac {5 b^{6} d^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {7}{2}}}-\frac {55 \sqrt {c \,x^{2}+b x}\, b^{7} e^{3} x}{16384 c^{5}}+\frac {135 \sqrt {c \,x^{2}+b x}\, b^{6} d \,e^{2} x}{8192 c^{4}}-\frac {15 \sqrt {c \,x^{2}+b x}\, b^{5} d^{2} e x}{512 c^{3}}+\frac {5 \sqrt {c \,x^{2}+b x}\, b^{4} d^{3} x}{256 c^{2}}-\frac {55 \sqrt {c \,x^{2}+b x}\, b^{8} e^{3}}{32768 c^{6}}+\frac {135 \sqrt {c \,x^{2}+b x}\, b^{7} d \,e^{2}}{16384 c^{5}}-\frac {15 \sqrt {c \,x^{2}+b x}\, b^{6} d^{2} e}{1024 c^{4}}+\frac {5 \sqrt {c \,x^{2}+b x}\, b^{5} d^{3}}{512 c^{3}}+\frac {55 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{5} e^{3} x}{6144 c^{4}}-\frac {45 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{4} d \,e^{2} x}{1024 c^{3}}+\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{3} d^{2} e x}{64 c^{2}}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} d^{3} x}{96 c}+\frac {55 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{6} e^{3}}{12288 c^{5}}-\frac {45 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{5} d \,e^{2}}{2048 c^{4}}+\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{4} d^{2} e}{128 c^{3}}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{3} d^{3}}{192 c^{2}}-\frac {11 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{3} e^{3} x}{384 c^{3}}+\frac {9 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{2} d \,e^{2} x}{64 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} b \,d^{2} e x}{4 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}} e^{3} x^{2}}{9 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} d^{3} x}{6}-\frac {11 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{4} e^{3}}{768 c^{4}}+\frac {9 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{3} d \,e^{2}}{128 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{2} d^{2} e}{8 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} b \,d^{3}}{12 c}-\frac {11 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} b \,e^{3} x}{144 c^{2}}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} d \,e^{2} x}{8 c}+\frac {11 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} b^{2} e^{3}}{224 c^{3}}-\frac {27 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} b d \,e^{2}}{112 c^{2}}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} d^{2} e}{7 c} \end {gather*}

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]

-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)*ln((c*x+1/2*b)/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)-5/96*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+5/64*d^2*e*b^3/c^2*(c*x^2+b*x)^(3/2)*x-15/512*d^2*e*b^5/c^3*(c*x^2+b*x)^(1/2)*x-1/4*d^2*e*b/c*x*(c*x^2
+b*x)^(5/2)+9/64*d*e^2*b^2/c^2*x*(c*x^2+b*x)^(5/2)-45/1024*d*e^2*b^4/c^3*(c*x^2+b*x)^(3/2)*x+135/8192*d*e^2*b^
6/c^4*(c*x^2+b*x)^(1/2)*x+1/6*d^3*x*(c*x^2+b*x)^(5/2)+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)+55/65536*e^3*b^9/c^(13/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))+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)+5/512*d^3*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*d^3*b^6/c^(7/2)*ln((c*x+
1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))+3/7*d^2*e*(c*x^2+b*x)^(7/2)/c+5/128*d^2*e*b^4/c^3*(c*x^2+b*x)^(3/2)-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((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))-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)+55/6144*e^3*b^5/c^4*(c*x^2+b*x)^(3/2)*x-55/1
6384*e^3*b^7/c^5*(c*x^2+b*x)^(1/2)*x+3/8*d*e^2*x*(c*x^2+b*x)^(7/2)/c

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maxima [B]  time = 1.56, size = 808, normalized size = 2.43 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} e^{3} x^{2}}{9 \, c} + \frac {1}{6} \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d^{3} x + \frac {5 \, \sqrt {c x^{2} + b x} b^{4} d^{3} x}{256 \, c^{2}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d^{3} x}{96 \, c} - \frac {15 \, \sqrt {c x^{2} + b x} b^{5} d^{2} e x}{512 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d^{2} e x}{64 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d^{2} e x}{4 \, c} + \frac {135 \, \sqrt {c x^{2} + b x} b^{6} d e^{2} x}{8192 \, c^{4}} - \frac {45 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} d e^{2} x}{1024 \, c^{3}} + \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} d e^{2} x}{64 \, c^{2}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} d e^{2} x}{8 \, c} - \frac {55 \, \sqrt {c x^{2} + b x} b^{7} e^{3} x}{16384 \, c^{5}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5} e^{3} x}{6144 \, c^{4}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{3} e^{3} x}{384 \, c^{3}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b e^{3} x}{144 \, c^{2}} - \frac {5 \, b^{6} d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {7}{2}}} + \frac {15 \, b^{7} d^{2} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {9}{2}}} - \frac {135 \, b^{8} d e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {11}{2}}} + \frac {55 \, b^{9} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{65536 \, c^{\frac {13}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} b^{5} d^{3}}{512 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d^{3}}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d^{3}}{12 \, c} - \frac {15 \, \sqrt {c x^{2} + b x} b^{6} d^{2} e}{1024 \, c^{4}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} d^{2} e}{128 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} d^{2} e}{8 \, c^{2}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} d^{2} e}{7 \, c} + \frac {135 \, \sqrt {c x^{2} + b x} b^{7} d e^{2}}{16384 \, c^{5}} - \frac {45 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5} d e^{2}}{2048 \, c^{4}} + \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{3} d e^{2}}{128 \, c^{3}} - \frac {27 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b d e^{2}}{112 \, c^{2}} - \frac {55 \, \sqrt {c x^{2} + b x} b^{8} e^{3}}{32768 \, c^{6}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{6} e^{3}}{12288 \, c^{5}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{4} e^{3}}{768 \, c^{4}} + \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b^{2} e^{3}}{224 \, c^{3}} \end {gather*}

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]

1/9*(c*x^2 + b*x)^(7/2)*e^3*x^2/c + 1/6*(c*x^2 + b*x)^(5/2)*d^3*x + 5/256*sqrt(c*x^2 + b*x)*b^4*d^3*x/c^2 - 5/
96*(c*x^2 + b*x)^(3/2)*b^2*d^3*x/c - 15/512*sqrt(c*x^2 + b*x)*b^5*d^2*e*x/c^3 + 5/64*(c*x^2 + b*x)^(3/2)*b^3*d
^2*e*x/c^2 - 1/4*(c*x^2 + b*x)^(5/2)*b*d^2*e*x/c + 135/8192*sqrt(c*x^2 + b*x)*b^6*d*e^2*x/c^4 - 45/1024*(c*x^2
+ b*x)^(3/2)*b^4*d*e^2*x/c^3 + 9/64*(c*x^2 + b*x)^(5/2)*b^2*d*e^2*x/c^2 + 3/8*(c*x^2 + b*x)^(7/2)*d*e^2*x/c -
55/16384*sqrt(c*x^2 + b*x)*b^7*e^3*x/c^5 + 55/6144*(c*x^2 + b*x)^(3/2)*b^5*e^3*x/c^4 - 11/384*(c*x^2 + b*x)^(
5/2)*b^3*e^3*x/c^3 - 11/144*(c*x^2 + b*x)^(7/2)*b*e^3*x/c^2 - 5/1024*b^6*d^3*log(2*c*x + b + 2*sqrt(c*x^2 + b*
x)*sqrt(c))/c^(7/2) + 15/2048*b^7*d^2*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) - 135/32768*b^8*d
*e^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) + 55/65536*b^9*e^3*log(2*c*x + b + 2*sqrt(c*x^2 + b
*x)*sqrt(c))/c^(13/2) + 5/512*sqrt(c*x^2 + b*x)*b^5*d^3/c^3 - 5/192*(c*x^2 + b*x)^(3/2)*b^3*d^3/c^2 + 1/12*(c*
x^2 + b*x)^(5/2)*b*d^3/c - 15/1024*sqrt(c*x^2 + b*x)*b^6*d^2*e/c^4 + 5/128*(c*x^2 + b*x)^(3/2)*b^4*d^2*e/c^3 -
1/8*(c*x^2 + b*x)^(5/2)*b^2*d^2*e/c^2 + 3/7*(c*x^2 + b*x)^(7/2)*d^2*e/c + 135/16384*sqrt(c*x^2 + b*x)*b^7*d*e
^2/c^5 - 45/2048*(c*x^2 + b*x)^(3/2)*b^5*d*e^2/c^4 + 9/128*(c*x^2 + b*x)^(5/2)*b^3*d*e^2/c^3 - 27/112*(c*x^2 +
b*x)^(7/2)*b*d*e^2/c^2 - 55/32768*sqrt(c*x^2 + b*x)*b^8*e^3/c^6 + 55/12288*(c*x^2 + b*x)^(3/2)*b^6*e^3/c^5 -
11/768*(c*x^2 + b*x)^(5/2)*b^4*e^3/c^4 + 11/224*(c*x^2 + b*x)^(7/2)*b^2*e^3/c^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,x^2+b\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{3}\, dx \end {gather*}

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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