3.4.2 \(\int (d+e x)^3 (b x+c x^2)^{5/2} \, dx\) [302]

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

[Out]

-5/12288*b^2*(-b*e+2*c*d)*(11*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^5+1/768*(-b*e+2*c*d
)*(11*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c^4+1/9*e*(e*x+d)^2*(c*x^2+b*x)^(7/2)/c+1/201
6*e*(640*c^2*d^2-486*b*c*d*e+99*b^2*e^2+154*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x)^(7/2)/c^3-5/32768*b^6*(-b*e+2*c*d)
*(11*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(13/2)+5/32768*b^4*(-b*e+2*c*d)*(11
*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c^6

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Rubi [A]
time = 0.28, 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 = {756, 793, 626, 634, 212} \begin {gather*} -\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} \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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

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 634

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 756

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 793

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 ) \text {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 = 0.87, size = 396, normalized size = 1.19 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\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 )+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 )+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 )+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 )+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 )\right )-\frac {315 b^6 \left (-64 c^3 d^3+96 b c^2 d^2 e-54 b^2 c d e^2+11 b^3 e^3\right ) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {x} \sqrt {b+c x}}\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^6*
(-64*c^3*d^3 + 96*b*c^2*d^2*e - 54*b^2*c*d*e^2 + 11*b^3*e^3)*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]])/(Sqrt[x]
*Sqrt[b + c*x])))/(2064384*c^(13/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(632\) vs. \(2(304)=608\).
time = 0.44, size = 633, normalized size = 1.91

method	result	size
risch	\(-\frac {\left (-229376 c^{8} e^{3} x^{8}-530432 b \,c^{7} e^{3} x^{7}-774144 c^{8} d \,e^{2} x^{7}-316416 b^{2} c^{6} e^{3} x^{6}-1824768 b \,c^{7} d \,e^{2} x^{6}-884736 c^{8} d^{2} e \,x^{6}-1280 b^{3} c^{5} e^{3} x^{5}-1119744 b^{2} c^{6} d \,e^{2} x^{5}-2138112 b \,c^{7} d^{2} e \,x^{5}-344064 c^{8} d^{3} x^{5}+1408 b^{4} c^{4} e^{3} x^{4}-6912 b^{3} c^{5} d \,e^{2} x^{4}-1363968 b^{2} c^{6} d^{2} e \,x^{4}-860160 b \,c^{7} d^{3} x^{4}-1584 b^{5} c^{3} e^{3} x^{3}+7776 b^{4} c^{4} d \,e^{2} x^{3}-13824 b^{3} c^{5} d^{2} e \,x^{3}-580608 b^{2} c^{6} d^{3} x^{3}+1848 b^{6} c^{2} e^{3} x^{2}-9072 b^{5} c^{3} d \,e^{2} x^{2}+16128 b^{4} c^{4} d^{2} e \,x^{2}-10752 b^{3} c^{5} d^{3} x^{2}-2310 b^{7} c \,e^{3} x +11340 b^{6} c^{2} d \,e^{2} x -20160 b^{5} c^{3} d^{2} e x +13440 b^{4} c^{4} d^{3} x +3465 b^{8} e^{3}-17010 b^{7} c d \,e^{2}+30240 b^{6} c^{2} d^{2} e -20160 b^{5} c^{3} d^{3}\right ) x \left (c x +b \right )}{2064384 c^{6} \sqrt {x \left (c x +b \right )}}+\frac {55 b^{9} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) e^{3}}{65536 c^{\frac {13}{2}}}-\frac {135 b^{8} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) d \,e^{2}}{32768 c^{\frac {11}{2}}}+\frac {15 b^{7} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) d^{2} e}{2048 c^{\frac {9}{2}}}-\frac {5 b^{6} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) d^{3}}{1024 c^{\frac {7}{2}}}\)	\(553\)
default	\(e^{3} \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 c}-\frac {11 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )+3 d \,e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )+3 d^{2} e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )+d^{3} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )\)	\(633\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^3*(1/9*x^2*(c*x^2+b*x)^(7/2)/c-11/18*b/c*(1/8*x*(c*x^2+b*x)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x)^(7/2)/c-1/2*b/
c*(1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5/24*b^2/c*(1/8*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c-3/16*b^2/c*(1/4*(2*c*x+b)*
(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))))))))+3*d*e^2*(1/8*x*(c*x^2+b*x)
^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5/24*b^2/c*(1/8*(2*c*x+
b)*(c*x^2+b*x)^(3/2)/c-3/16*b^2/c*(1/4*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x)^(1/2)))))))+3*d^2*e*(1/7*(c*x^2+b*x)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5/24*b^2/c*
(1/8*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c-3/16*b^2/c*(1/4*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x)^(1/2))))))+d^3*(1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5/24*b^2/c*(1/8*(2*c*x+b)*(c*x^2+b*x)
^(3/2)/c-3/16*b^2/c*(1/4*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2
)))))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (313) = 626\).
time = 0.30, size = 797, normalized size = 2.40 \begin {gather*} \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} x e}{512 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d^{2} x e}{64 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d^{2} x e}{4 \, c} - \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 {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 {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} x^{2} e^{3}}{9 \, c} + \frac {135 \, \sqrt {c x^{2} + b x} b^{6} d x e^{2}}{8192 \, c^{4}} - \frac {45 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} d x e^{2}}{1024 \, c^{3}} + \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} d x e^{2}}{64 \, c^{2}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} d x e^{2}}{8 \, 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 \, 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 \, \sqrt {c x^{2} + b x} b^{7} x e^{3}}{16384 \, c^{5}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5} x e^{3}}{6144 \, c^{4}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{3} x e^{3}}{384 \, c^{3}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b x e^{3}}{144 \, c^{2}} + \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 \, 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 {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]

1/6*(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*x*e/c^3 + 5/64*(c*x^2 + b*x)^(3/2)*b^3*d^2*x*e/c^2 - 1/4*(c*x^2 + b*x)^(5/2)
*b*d^2*x*e/c - 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) + 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 + 1/9*(c*x^2 + b*x)^(7/2)*x^2*e^3/c + 135/8192*sqrt(c*x^2 + b
*x)*b^6*d*x*e^2/c^4 - 45/1024*(c*x^2 + b*x)^(3/2)*b^4*d*x*e^2/c^3 + 9/64*(c*x^2 + b*x)^(5/2)*b^2*d*x*e^2/c^2 +
3/8*(c*x^2 + b*x)^(7/2)*d*x*e^2/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 - 135/32768*b^8*d*e^2*log(2
*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) - 55/16384*sqrt(c*x^2 + b*x)*b^7*x*e^3/c^5 + 55/6144*(c*x^2 +
b*x)^(3/2)*b^5*x*e^3/c^4 - 11/384*(c*x^2 + b*x)^(5/2)*b^3*x*e^3/c^3 - 11/144*(c*x^2 + b*x)^(7/2)*b*x*e^3/c^2
+ 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 + 55/65536*b^9*e^3*log(2*c*x + b + 2*sqrt(c*x^2 +
b*x)*sqrt(c))/c^(13/2) - 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

________________________________________________________________________________________

Fricas [A]
time = 2.47, size = 879, normalized size = 2.65 \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 (344064 \, c^{9} d^{3} x^{5} + 860160 \, b c^{8} d^{3} x^{4} + 580608 \, b^{2} c^{7} d^{3} x^{3} + 10752 \, b^{3} c^{6} d^{3} x^{2} - 13440 \, b^{4} c^{5} d^{3} x + 20160 \, b^{5} c^{4} d^{3} + {\left (229376 \, c^{9} x^{8} + 530432 \, b c^{8} x^{7} + 316416 \, b^{2} c^{7} x^{6} + 1280 \, b^{3} c^{6} x^{5} - 1408 \, b^{4} c^{5} x^{4} + 1584 \, b^{5} c^{4} x^{3} - 1848 \, b^{6} c^{3} x^{2} + 2310 \, b^{7} c^{2} x - 3465 \, b^{8} c\right )} e^{3} + 54 \, {\left (14336 \, c^{9} d x^{7} + 33792 \, b c^{8} d x^{6} + 20736 \, b^{2} c^{7} d x^{5} + 128 \, b^{3} c^{6} d x^{4} - 144 \, b^{4} c^{5} d x^{3} + 168 \, b^{5} c^{4} d x^{2} - 210 \, b^{6} c^{3} d x + 315 \, b^{7} c^{2} d\right )} e^{2} + 288 \, {\left (3072 \, c^{9} d^{2} x^{6} + 7424 \, b c^{8} d^{2} x^{5} + 4736 \, b^{2} c^{7} d^{2} x^{4} + 48 \, b^{3} c^{6} d^{2} x^{3} - 56 \, b^{4} c^{5} d^{2} x^{2} + 70 \, b^{5} c^{4} d^{2} x - 105 \, b^{6} c^{3} d^{2}\right )} e\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 (344064 \, c^{9} d^{3} x^{5} + 860160 \, b c^{8} d^{3} x^{4} + 580608 \, b^{2} c^{7} d^{3} x^{3} + 10752 \, b^{3} c^{6} d^{3} x^{2} - 13440 \, b^{4} c^{5} d^{3} x + 20160 \, b^{5} c^{4} d^{3} + {\left (229376 \, c^{9} x^{8} + 530432 \, b c^{8} x^{7} + 316416 \, b^{2} c^{7} x^{6} + 1280 \, b^{3} c^{6} x^{5} - 1408 \, b^{4} c^{5} x^{4} + 1584 \, b^{5} c^{4} x^{3} - 1848 \, b^{6} c^{3} x^{2} + 2310 \, b^{7} c^{2} x - 3465 \, b^{8} c\right )} e^{3} + 54 \, {\left (14336 \, c^{9} d x^{7} + 33792 \, b c^{8} d x^{6} + 20736 \, b^{2} c^{7} d x^{5} + 128 \, b^{3} c^{6} d x^{4} - 144 \, b^{4} c^{5} d x^{3} + 168 \, b^{5} c^{4} d x^{2} - 210 \, b^{6} c^{3} d x + 315 \, b^{7} c^{2} d\right )} e^{2} + 288 \, {\left (3072 \, c^{9} d^{2} x^{6} + 7424 \, b c^{8} d^{2} x^{5} + 4736 \, b^{2} c^{7} d^{2} x^{4} + 48 \, b^{3} c^{6} d^{2} x^{3} - 56 \, b^{4} c^{5} d^{2} x^{2} + 70 \, b^{5} c^{4} d^{2} x - 105 \, b^{6} c^{3} d^{2}\right )} e\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*(344064*c^9*d^3*x^5 + 860160*b*c^8*d^3*x^4 + 580608*b^2*c^7*d^3*x^3 + 10752*b^3*
c^6*d^3*x^2 - 13440*b^4*c^5*d^3*x + 20160*b^5*c^4*d^3 + (229376*c^9*x^8 + 530432*b*c^8*x^7 + 316416*b^2*c^7*x^
6 + 1280*b^3*c^6*x^5 - 1408*b^4*c^5*x^4 + 1584*b^5*c^4*x^3 - 1848*b^6*c^3*x^2 + 2310*b^7*c^2*x - 3465*b^8*c)*e
^3 + 54*(14336*c^9*d*x^7 + 33792*b*c^8*d*x^6 + 20736*b^2*c^7*d*x^5 + 128*b^3*c^6*d*x^4 - 144*b^4*c^5*d*x^3 + 1
68*b^5*c^4*d*x^2 - 210*b^6*c^3*d*x + 315*b^7*c^2*d)*e^2 + 288*(3072*c^9*d^2*x^6 + 7424*b*c^8*d^2*x^5 + 4736*b^
2*c^7*d^2*x^4 + 48*b^3*c^6*d^2*x^3 - 56*b^4*c^5*d^2*x^2 + 70*b^5*c^4*d^2*x - 105*b^6*c^3*d^2)*e)*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(sq
rt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (344064*c^9*d^3*x^5 + 860160*b*c^8*d^3*x^4 + 580608*b^2*c^7*d^3*x^3 + 10752*
b^3*c^6*d^3*x^2 - 13440*b^4*c^5*d^3*x + 20160*b^5*c^4*d^3 + (229376*c^9*x^8 + 530432*b*c^8*x^7 + 316416*b^2*c^
7*x^6 + 1280*b^3*c^6*x^5 - 1408*b^4*c^5*x^4 + 1584*b^5*c^4*x^3 - 1848*b^6*c^3*x^2 + 2310*b^7*c^2*x - 3465*b^8*
c)*e^3 + 54*(14336*c^9*d*x^7 + 33792*b*c^8*d*x^6 + 20736*b^2*c^7*d*x^5 + 128*b^3*c^6*d*x^4 - 144*b^4*c^5*d*x^3
+ 168*b^5*c^4*d*x^2 - 210*b^6*c^3*d*x + 315*b^7*c^2*d)*e^2 + 288*(3072*c^9*d^2*x^6 + 7424*b*c^8*d^2*x^5 + 473
6*b^2*c^7*d^2*x^4 + 48*b^3*c^6*d^2*x^3 - 56*b^4*c^5*d^2*x^2 + 70*b^5*c^4*d^2*x - 105*b^6*c^3*d^2)*e)*sqrt(c*x^
2 + b*x))/c^7]

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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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Giac [A]
time = 1.97, 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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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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