3.12.72 \(\int (A+B x) (d+e x)^2 (b x+c x^2)^{3/2} \, dx\) [1172]

Optimal. Leaf size=345 \[ -\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}} \]

[Out]

1/384*(48*A*c^3*d^2-9*b^3*B*e^2+14*b^2*c*e*(A*e+2*B*d)-24*b*c^2*d*(2*A*e+B*d))*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^4
+1/7*B*(e*x+d)^2*(c*x^2+b*x)^(5/2)/c+1/840*(14*A*c*e*(-7*b*e+24*c*d)+B*(63*b^2*e^2-196*b*c*d*e+48*c^2*d^2)+10*
c*e*(14*A*c*e-9*B*b*e+4*B*c*d)*x)*(c*x^2+b*x)^(5/2)/c^3+1/1024*b^4*(48*A*c^3*d^2-9*b^3*B*e^2+14*b^2*c*e*(A*e+2
*B*d)-24*b*c^2*d*(2*A*e+B*d))*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(11/2)-1/1024*b^2*(48*A*c^3*d^2-9*b^3*B*e
^2+14*b^2*c*e*(A*e+2*B*d)-24*b*c^2*d*(2*A*e+B*d))*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c^5

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Rubi [A]
time = 0.22, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {846, 793, 626, 634, 212} \begin {gather*} \frac {\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+B \left (63 b^2 e^2-196 b c d e+48 c^2 d^2\right )\right )}{840 c^3}-\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^5}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{384 c^4}+\frac {b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^{11/2}}+\frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1024*(b^2*(48*A*c^3*d^2 - 9*b^3*B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*(b + 2*c*x)*Sq
rt[b*x + c*x^2])/c^5 + ((48*A*c^3*d^2 - 9*b^3*B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*(b
+ 2*c*x)*(b*x + c*x^2)^(3/2))/(384*c^4) + (B*(d + e*x)^2*(b*x + c*x^2)^(5/2))/(7*c) + ((14*A*c*e*(24*c*d - 7*b
*e) + B*(48*c^2*d^2 - 196*b*c*d*e + 63*b^2*e^2) + 10*c*e*(4*B*c*d - 9*b*B*e + 14*A*c*e)*x)*(b*x + c*x^2)^(5/2)
)/(840*c^3) + (b^4*(48*A*c^3*d^2 - 9*b^3*B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*ArcTanh[
(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(1024*c^(11/2))

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 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]

Rule 846

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])

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\int (d+e x) \left (-\frac {1}{2} (5 b B-14 A c) d+\frac {1}{2} (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{48 c^3}\\ &=\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \int \sqrt {b x+c x^2} \, dx}{256 c^4}\\ &=-\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2048 c^5}\\ &=-\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{1024 c^5}\\ &=-\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 1.16, size = 407, normalized size = 1.18 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (945 b^6 B e^2-210 b^5 c e (14 B d+7 A e+3 B e x)+96 b^2 c^4 x \left (7 A \left (5 d^2+4 d e x+e^2 x^2\right )+2 B x \left (7 d^2+7 d e x+2 e^2 x^2\right )\right )+28 b^4 c^2 \left (5 A e (36 d+7 e x)+2 B \left (45 d^2+35 d e x+9 e^2 x^2\right )\right )+256 c^6 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )-16 b^3 c^3 \left (7 A \left (45 d^2+30 d e x+7 e^2 x^2\right )+B x \left (105 d^2+98 d e x+27 e^2 x^2\right )\right )+128 b c^5 x^2 \left (7 A \left (45 d^2+66 d e x+26 e^2 x^2\right )+B x \left (231 d^2+364 d e x+150 e^2 x^2\right )\right )\right )+\frac {105 b^4 \left (-48 A c^3 d^2+9 b^3 B e^2-14 b^2 c e (2 B d+A e)+24 b c^2 d (B d+2 A e)\right ) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{107520 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(945*b^6*B*e^2 - 210*b^5*c*e*(14*B*d + 7*A*e + 3*B*e*x) + 96*b^2*c^4*x*(7*A*(5*d^2
+ 4*d*e*x + e^2*x^2) + 2*B*x*(7*d^2 + 7*d*e*x + 2*e^2*x^2)) + 28*b^4*c^2*(5*A*e*(36*d + 7*e*x) + 2*B*(45*d^2
+ 35*d*e*x + 9*e^2*x^2)) + 256*c^6*x^3*(7*A*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 4*B*x*(21*d^2 + 35*d*e*x + 15*e
^2*x^2)) - 16*b^3*c^3*(7*A*(45*d^2 + 30*d*e*x + 7*e^2*x^2) + B*x*(105*d^2 + 98*d*e*x + 27*e^2*x^2)) + 128*b*c^
5*x^2*(7*A*(45*d^2 + 66*d*e*x + 26*e^2*x^2) + B*x*(231*d^2 + 364*d*e*x + 150*e^2*x^2))) + (105*b^4*(-48*A*c^3*
d^2 + 9*b^3*B*e^2 - 14*b^2*c*e*(2*B*d + A*e) + 24*b*c^2*d*(B*d + 2*A*e))*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x
]])/(Sqrt[x]*Sqrt[b + c*x])))/(107520*c^(11/2))

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Maple [A]
time = 0.60, size = 523, normalized size = 1.52

method	result	size
default	\(B \,e^{2} \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 c}-\frac {9 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )}{14 c}\right )+\left (A \,e^{2}+2 B d e \right ) \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )+\left (2 A d e +B \,d^{2}\right ) \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )+A \,d^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )\)	\(523\)
risch	\(-\frac {\left (-15360 B \,c^{6} e^{2} x^{6}-17920 A \,c^{6} e^{2} x^{5}-19200 B b \,c^{5} e^{2} x^{5}-35840 B \,c^{6} d e \,x^{5}-23296 A b \,c^{5} e^{2} x^{4}-43008 A \,c^{6} d e \,x^{4}-384 B \,b^{2} c^{4} e^{2} x^{4}-46592 B b \,c^{5} d e \,x^{4}-21504 B \,c^{6} d^{2} x^{4}-672 A \,b^{2} c^{4} e^{2} x^{3}-59136 A b \,c^{5} d e \,x^{3}-26880 A \,c^{6} d^{2} x^{3}+432 B \,b^{3} c^{3} e^{2} x^{3}-1344 B \,b^{2} c^{4} d e \,x^{3}-29568 B b \,c^{5} d^{2} x^{3}+784 A \,b^{3} c^{3} e^{2} x^{2}-2688 A \,b^{2} c^{4} d e \,x^{2}-40320 A b \,c^{5} d^{2} x^{2}-504 B \,b^{4} c^{2} e^{2} x^{2}+1568 B \,b^{3} c^{3} d e \,x^{2}-1344 B \,b^{2} c^{4} d^{2} x^{2}-980 A \,b^{4} c^{2} e^{2} x +3360 A \,b^{3} c^{3} d e x -3360 A \,b^{2} c^{4} d^{2} x +630 B \,b^{5} c \,e^{2} x -1960 B \,b^{4} c^{2} d e x +1680 B \,b^{3} c^{3} d^{2} x +1470 A \,b^{5} c \,e^{2}-5040 A \,b^{4} c^{2} d e +5040 A \,b^{3} c^{3} d^{2}-945 B \,b^{6} e^{2}+2940 B \,b^{5} c d e -2520 B \,b^{4} c^{2} d^{2}\right ) x \left (c x +b \right )}{107520 c^{5} \sqrt {x \left (c x +b \right )}}+\frac {7 b^{6} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) A \,e^{2}}{1024 c^{\frac {9}{2}}}-\frac {3 b^{5} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) A d e}{128 c^{\frac {7}{2}}}+\frac {3 b^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) A \,d^{2}}{128 c^{\frac {5}{2}}}-\frac {9 b^{7} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) B \,e^{2}}{2048 c^{\frac {11}{2}}}+\frac {7 b^{6} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) B d e}{512 c^{\frac {9}{2}}}-\frac {3 b^{5} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) B \,d^{2}}{256 c^{\frac {7}{2}}}\)	\(652\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*e^2*(1/7*x^2*(c*x^2+b*x)^(5/2)/c-9/14*b/c*(1/6*x*(c*x^2+b*x)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x)^(5/2)/c-1/2*b
/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)))))))+(A*e^2+2*B*d*e)*(1/6*x*(c*x^2+b*x)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x)^(5/2
)/c-1/2*b/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)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))))))+(2*A*d*e+B*d^2)*(1/5*(c*x^2+b*x)^(5/2)/c-1/2*b/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)))))+A*d^2*(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))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (333) = 666\).
time = 0.28, size = 726, normalized size = 2.10 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A d^{2} x - \frac {3 \, \sqrt {c x^{2} + b x} A b^{2} d^{2} x}{32 \, c} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B x^{2} e^{2}}{7 \, c} + \frac {3 \, A b^{4} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} A b^{3} d^{2}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b d^{2}}{8 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} B b^{5} x e^{2}}{512 \, c^{4}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} x e^{2}}{64 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b x e^{2}}{28 \, c^{2}} - \frac {9 \, B b^{7} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {7 \, \sqrt {c x^{2} + b x} {\left (2 \, B d e + A e^{2}\right )} b^{4} x}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (2 \, B d e + A e^{2}\right )} b^{2} x}{96 \, c^{2}} + \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b^{3} x}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (2 \, B d e + A e^{2}\right )} x}{6 \, c} - \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b x}{8 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} B b^{6} e^{2}}{1024 \, c^{5}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4} e^{2}}{128 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2} e^{2}}{40 \, c^{3}} + \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} b^{6} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} - \frac {7 \, \sqrt {c x^{2} + b x} {\left (2 \, B d e + A e^{2}\right )} b^{5}}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (2 \, B d e + A e^{2}\right )} b^{3}}{192 \, c^{3}} + \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b^{4}}{128 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (2 \, B d e + A e^{2}\right )} b}{60 \, c^{2}} - \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}}{16 \, c^{2}} + \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{5 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c*x^2 + b*x)^(3/2)*A*d^2*x - 3/32*sqrt(c*x^2 + b*x)*A*b^2*d^2*x/c + 1/7*(c*x^2 + b*x)^(5/2)*B*x^2*e^2/c +
3/128*A*b^4*d^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 3/64*sqrt(c*x^2 + b*x)*A*b^3*d^2/c^2 +
1/8*(c*x^2 + b*x)^(3/2)*A*b*d^2/c + 9/512*sqrt(c*x^2 + b*x)*B*b^5*x*e^2/c^4 - 3/64*(c*x^2 + b*x)^(3/2)*B*b^3*
x*e^2/c^3 - 3/28*(c*x^2 + b*x)^(5/2)*B*b*x*e^2/c^2 - 9/2048*B*b^7*e^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt
(c))/c^(11/2) - 7/256*sqrt(c*x^2 + b*x)*(2*B*d*e + A*e^2)*b^4*x/c^3 + 7/96*(c*x^2 + b*x)^(3/2)*(2*B*d*e + A*e^
2)*b^2*x/c^2 + 3/64*(B*d^2 + 2*A*d*e)*sqrt(c*x^2 + b*x)*b^3*x/c^2 + 1/6*(c*x^2 + b*x)^(5/2)*(2*B*d*e + A*e^2)*
x/c - 1/8*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(3/2)*b*x/c + 9/1024*sqrt(c*x^2 + b*x)*B*b^6*e^2/c^5 - 3/128*(c*x^2
+ b*x)^(3/2)*B*b^4*e^2/c^4 + 3/40*(c*x^2 + b*x)^(5/2)*B*b^2*e^2/c^3 + 7/1024*(2*B*d*e + A*e^2)*b^6*log(2*c*x +
b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) - 3/256*(B*d^2 + 2*A*d*e)*b^5*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sq
rt(c))/c^(7/2) - 7/512*sqrt(c*x^2 + b*x)*(2*B*d*e + A*e^2)*b^5/c^4 + 7/192*(c*x^2 + b*x)^(3/2)*(2*B*d*e + A*e^
2)*b^3/c^3 + 3/128*(B*d^2 + 2*A*d*e)*sqrt(c*x^2 + b*x)*b^4/c^3 - 7/60*(c*x^2 + b*x)^(5/2)*(2*B*d*e + A*e^2)*b/
c^2 - 1/16*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(3/2)*b^2/c^2 + 1/5*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(5/2)/c

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Fricas [A]
time = 4.67, size = 969, normalized size = 2.81 \begin {gather*} \left [-\frac {105 \, {\left (24 \, {\left (B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} d^{2} - 4 \, {\left (7 \, B b^{6} c - 12 \, A b^{5} c^{2}\right )} d e + {\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (21504 \, B c^{7} d^{2} x^{4} + 2688 \, {\left (11 \, B b c^{6} + 10 \, A c^{7}\right )} d^{2} x^{3} + 1344 \, {\left (B b^{2} c^{5} + 30 \, A b c^{6}\right )} d^{2} x^{2} - 1680 \, {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{2} x + 2520 \, {\left (B b^{4} c^{3} - 2 \, A b^{3} c^{4}\right )} d^{2} + {\left (15360 \, B c^{7} x^{6} + 945 \, B b^{6} c - 1470 \, A b^{5} c^{2} + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 128 \, {\left (3 \, B b^{2} c^{5} + 182 \, A b c^{6}\right )} x^{4} - 48 \, {\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )} x^{3} + 56 \, {\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )} x^{2} - 70 \, {\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )} x\right )} e^{2} + 28 \, {\left (1280 \, B c^{7} d x^{5} + 128 \, {\left (13 \, B b c^{6} + 12 \, A c^{7}\right )} d x^{4} + 48 \, {\left (B b^{2} c^{5} + 44 \, A b c^{6}\right )} d x^{3} - 8 \, {\left (7 \, B b^{3} c^{4} - 12 \, A b^{2} c^{5}\right )} d x^{2} + 10 \, {\left (7 \, B b^{4} c^{3} - 12 \, A b^{3} c^{4}\right )} d x - 15 \, {\left (7 \, B b^{5} c^{2} - 12 \, A b^{4} c^{3}\right )} d\right )} e\right )} \sqrt {c x^{2} + b x}}{215040 \, c^{6}}, \frac {105 \, {\left (24 \, {\left (B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} d^{2} - 4 \, {\left (7 \, B b^{6} c - 12 \, A b^{5} c^{2}\right )} d e + {\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (21504 \, B c^{7} d^{2} x^{4} + 2688 \, {\left (11 \, B b c^{6} + 10 \, A c^{7}\right )} d^{2} x^{3} + 1344 \, {\left (B b^{2} c^{5} + 30 \, A b c^{6}\right )} d^{2} x^{2} - 1680 \, {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{2} x + 2520 \, {\left (B b^{4} c^{3} - 2 \, A b^{3} c^{4}\right )} d^{2} + {\left (15360 \, B c^{7} x^{6} + 945 \, B b^{6} c - 1470 \, A b^{5} c^{2} + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 128 \, {\left (3 \, B b^{2} c^{5} + 182 \, A b c^{6}\right )} x^{4} - 48 \, {\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )} x^{3} + 56 \, {\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )} x^{2} - 70 \, {\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )} x\right )} e^{2} + 28 \, {\left (1280 \, B c^{7} d x^{5} + 128 \, {\left (13 \, B b c^{6} + 12 \, A c^{7}\right )} d x^{4} + 48 \, {\left (B b^{2} c^{5} + 44 \, A b c^{6}\right )} d x^{3} - 8 \, {\left (7 \, B b^{3} c^{4} - 12 \, A b^{2} c^{5}\right )} d x^{2} + 10 \, {\left (7 \, B b^{4} c^{3} - 12 \, A b^{3} c^{4}\right )} d x - 15 \, {\left (7 \, B b^{5} c^{2} - 12 \, A b^{4} c^{3}\right )} d\right )} e\right )} \sqrt {c x^{2} + b x}}{107520 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2*(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

[-1/215040*(105*(24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^2 - 4*(7*B*b^6*c - 12*A*b^5*c^2)*d*e + (9*B*b^7 - 14*A*b^6*c)*
e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(21504*B*c^7*d^2*x^4 + 2688*(11*B*b*c^6 + 10*A*c
^7)*d^2*x^3 + 1344*(B*b^2*c^5 + 30*A*b*c^6)*d^2*x^2 - 1680*(B*b^3*c^4 - 2*A*b^2*c^5)*d^2*x + 2520*(B*b^4*c^3 -
2*A*b^3*c^4)*d^2 + (15360*B*c^7*x^6 + 945*B*b^6*c - 1470*A*b^5*c^2 + 1280*(15*B*b*c^6 + 14*A*c^7)*x^5 + 128*(
3*B*b^2*c^5 + 182*A*b*c^6)*x^4 - 48*(9*B*b^3*c^4 - 14*A*b^2*c^5)*x^3 + 56*(9*B*b^4*c^3 - 14*A*b^3*c^4)*x^2 - 7
0*(9*B*b^5*c^2 - 14*A*b^4*c^3)*x)*e^2 + 28*(1280*B*c^7*d*x^5 + 128*(13*B*b*c^6 + 12*A*c^7)*d*x^4 + 48*(B*b^2*c
^5 + 44*A*b*c^6)*d*x^3 - 8*(7*B*b^3*c^4 - 12*A*b^2*c^5)*d*x^2 + 10*(7*B*b^4*c^3 - 12*A*b^3*c^4)*d*x - 15*(7*B*
b^5*c^2 - 12*A*b^4*c^3)*d)*e)*sqrt(c*x^2 + b*x))/c^6, 1/107520*(105*(24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^2 - 4*(7*B
*b^6*c - 12*A*b^5*c^2)*d*e + (9*B*b^7 - 14*A*b^6*c)*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (
21504*B*c^7*d^2*x^4 + 2688*(11*B*b*c^6 + 10*A*c^7)*d^2*x^3 + 1344*(B*b^2*c^5 + 30*A*b*c^6)*d^2*x^2 - 1680*(B*b
^3*c^4 - 2*A*b^2*c^5)*d^2*x + 2520*(B*b^4*c^3 - 2*A*b^3*c^4)*d^2 + (15360*B*c^7*x^6 + 945*B*b^6*c - 1470*A*b^5
*c^2 + 1280*(15*B*b*c^6 + 14*A*c^7)*x^5 + 128*(3*B*b^2*c^5 + 182*A*b*c^6)*x^4 - 48*(9*B*b^3*c^4 - 14*A*b^2*c^5
)*x^3 + 56*(9*B*b^4*c^3 - 14*A*b^3*c^4)*x^2 - 70*(9*B*b^5*c^2 - 14*A*b^4*c^3)*x)*e^2 + 28*(1280*B*c^7*d*x^5 +
128*(13*B*b*c^6 + 12*A*c^7)*d*x^4 + 48*(B*b^2*c^5 + 44*A*b*c^6)*d*x^3 - 8*(7*B*b^3*c^4 - 12*A*b^2*c^5)*d*x^2 +
10*(7*B*b^4*c^3 - 12*A*b^3*c^4)*d*x - 15*(7*B*b^5*c^2 - 12*A*b^4*c^3)*d)*e)*sqrt(c*x^2 + b*x))/c^6]

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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 {3}{2}} \left (A + B x\right ) \left (d + e x\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.78, size = 518, normalized size = 1.50 \begin {gather*} \frac {1}{107520} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B c x e^{2} + \frac {28 \, B c^{7} d e + 15 \, B b c^{6} e^{2} + 14 \, A c^{7} e^{2}}{c^{6}}\right )} x + \frac {168 \, B c^{7} d^{2} + 364 \, B b c^{6} d e + 336 \, A c^{7} d e + 3 \, B b^{2} c^{5} e^{2} + 182 \, A b c^{6} e^{2}}{c^{6}}\right )} x + \frac {3 \, {\left (616 \, B b c^{6} d^{2} + 560 \, A c^{7} d^{2} + 28 \, B b^{2} c^{5} d e + 1232 \, A b c^{6} d e - 9 \, B b^{3} c^{4} e^{2} + 14 \, A b^{2} c^{5} e^{2}\right )}}{c^{6}}\right )} x + \frac {7 \, {\left (24 \, B b^{2} c^{5} d^{2} + 720 \, A b c^{6} d^{2} - 28 \, B b^{3} c^{4} d e + 48 \, A b^{2} c^{5} d e + 9 \, B b^{4} c^{3} e^{2} - 14 \, A b^{3} c^{4} e^{2}\right )}}{c^{6}}\right )} x - \frac {35 \, {\left (24 \, B b^{3} c^{4} d^{2} - 48 \, A b^{2} c^{5} d^{2} - 28 \, B b^{4} c^{3} d e + 48 \, A b^{3} c^{4} d e + 9 \, B b^{5} c^{2} e^{2} - 14 \, A b^{4} c^{3} e^{2}\right )}}{c^{6}}\right )} x + \frac {105 \, {\left (24 \, B b^{4} c^{3} d^{2} - 48 \, A b^{3} c^{4} d^{2} - 28 \, B b^{5} c^{2} d e + 48 \, A b^{4} c^{3} d e + 9 \, B b^{6} c e^{2} - 14 \, A b^{5} c^{2} e^{2}\right )}}{c^{6}}\right )} + \frac {{\left (24 \, B b^{5} c^{2} d^{2} - 48 \, A b^{4} c^{3} d^{2} - 28 \, B b^{6} c d e + 48 \, A b^{5} c^{2} d e + 9 \, B b^{7} e^{2} - 14 \, A b^{6} c e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/107520*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(12*B*c*x*e^2 + (28*B*c^7*d*e + 15*B*b*c^6*e^2 + 14*A*c^7*e^2)/c^6)
*x + (168*B*c^7*d^2 + 364*B*b*c^6*d*e + 336*A*c^7*d*e + 3*B*b^2*c^5*e^2 + 182*A*b*c^6*e^2)/c^6)*x + 3*(616*B*b
*c^6*d^2 + 560*A*c^7*d^2 + 28*B*b^2*c^5*d*e + 1232*A*b*c^6*d*e - 9*B*b^3*c^4*e^2 + 14*A*b^2*c^5*e^2)/c^6)*x +
7*(24*B*b^2*c^5*d^2 + 720*A*b*c^6*d^2 - 28*B*b^3*c^4*d*e + 48*A*b^2*c^5*d*e + 9*B*b^4*c^3*e^2 - 14*A*b^3*c^4*e
^2)/c^6)*x - 35*(24*B*b^3*c^4*d^2 - 48*A*b^2*c^5*d^2 - 28*B*b^4*c^3*d*e + 48*A*b^3*c^4*d*e + 9*B*b^5*c^2*e^2 -
14*A*b^4*c^3*e^2)/c^6)*x + 105*(24*B*b^4*c^3*d^2 - 48*A*b^3*c^4*d^2 - 28*B*b^5*c^2*d*e + 48*A*b^4*c^3*d*e + 9
*B*b^6*c*e^2 - 14*A*b^5*c^2*e^2)/c^6) + 1/2048*(24*B*b^5*c^2*d^2 - 48*A*b^4*c^3*d^2 - 28*B*b^6*c*d*e + 48*A*b^
5*c^2*d*e + 9*B*b^7*e^2 - 14*A*b^6*c*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(11/2)

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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 )}^{3/2}\,\left (A+B\,x\right )\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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