\(\int (d+e x)^2 (b x+c x^2)^{3/2} \, dx\) [296]

Optimal result

Integrand size = 21, antiderivative size = 214 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=-\frac {b^2 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {b^4 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}} \]

[Out]

1/192*(7*b^2*e^2-24*b*c*d*e+24*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^3+7/60*e*(-b*e+2*c*d)*(c*x^2+b*x)^(5/2)/
c^2+1/6*e*(e*x+d)*(c*x^2+b*x)^(5/2)/c+1/512*b^4*(7*b^2*e^2-24*b*c*d*e+24*c^2*d^2)*arctanh(x*c^(1/2)/(c*x^2+b*x
)^(1/2))/c^(9/2)-1/512*b^2*(7*b^2*e^2-24*b*c*d*e+24*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c^4

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {756, 654, 626, 634, 212} \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\frac {b^4 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{512 c^{9/2}}-\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{512 c^4}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{192 c^3}+\frac {7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c} \]

[In]

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

[Out]

-1/512*(b^2*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/c^4 + ((24*c^2*d^2 - 24*b*c*d
*e + 7*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(192*c^3) + (7*e*(2*c*d - b*e)*(b*x + c*x^2)^(5/2))/(60*c^2)
+ (e*(d + e*x)*(b*x + c*x^2)^(5/2))/(6*c) + (b^4*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqr
t[b*x + c*x^2]])/(512*c^(9/2))

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 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {\int \left (\frac {1}{2} d (12 c d-5 b e)+\frac {7}{2} e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2} \, dx}{6 c} \\ & = \frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (c d (12 c d-5 b e)-\frac {7}{2} b e (2 c d-b e)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{12 c^2} \\ & = \frac {\left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}-\frac {\left (b^2 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right )\right ) \int \sqrt {b x+c x^2} \, dx}{128 c^3} \\ & = -\frac {b^2 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^4 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{1024 c^4} \\ & = -\frac {b^2 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^4 \left (24 c^2 d^2-24 b c d e+7 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 )}{512 c^4} \\ & = -\frac {b^2 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {b^4 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.82 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.31 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (-105 b^5 e^2+10 b^4 c e (36 d+7 e x)+48 b^2 c^3 x \left (5 d^2+4 d e x+e^2 x^2\right )-8 b^3 c^2 \left (45 d^2+30 d e x+7 e^2 x^2\right )+128 c^5 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+64 b c^4 x^2 \left (45 d^2+66 d e x+26 e^2 x^2\right )\right )+720 b^5 c d e \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+30 b^4 \left (24 c^2 d^2+7 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{7680 c^{9/2} \sqrt {x (b+c x)}} \]

[In]

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

[Out]

(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(-105*b^5*e^2 + 10*b^4*c*e*(36*d + 7*e*x) + 48*b^2*c^3*x
*(5*d^2 + 4*d*e*x + e^2*x^2) - 8*b^3*c^2*(45*d^2 + 30*d*e*x + 7*e^2*x^2) + 128*c^5*x^3*(15*d^2 + 24*d*e*x + 10
*e^2*x^2) + 64*b*c^4*x^2*(45*d^2 + 66*d*e*x + 26*e^2*x^2)) + 720*b^5*c*d*e*ArcTanh[(Sqrt[c]*Sqrt[x])/(Sqrt[b]
- Sqrt[b + c*x])] + 30*b^4*(24*c^2*d^2 + 7*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(7
680*c^(9/2)*Sqrt[x*(b + c*x)])

Maple [A] (verified)

Time = 2.02 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {\frac {7 b^{4} \left (b^{2} e^{2}-\frac {24}{7} b c d e +\frac {24}{7} c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{512}-\frac {7 \left (-\frac {128 x^{3} \left (\frac {2}{3} x^{2} e^{2}+\frac {8}{5} d e x +d^{2}\right ) c^{\frac {11}{2}}}{7}+\left (\frac {24 \left (\frac {7}{45} x^{2} e^{2}+\frac {2}{3} d e x +d^{2}\right ) b^{2} c^{\frac {5}{2}}}{7}-\frac {16 x \left (\frac {1}{5} x^{2} e^{2}+\frac {4}{5} d e x +d^{2}\right ) b \,c^{\frac {7}{2}}}{7}-\frac {192 x^{2} \left (\frac {26}{45} x^{2} e^{2}+\frac {22}{15} d e x +d^{2}\right ) c^{\frac {9}{2}}}{7}+\left (\left (-\frac {2 e x}{3}-\frac {24 d}{7}\right ) c^{\frac {3}{2}}+\sqrt {c}\, b e \right ) e \,b^{3}\right ) b \right ) \sqrt {x \left (c x +b \right )}}{512}}{c^{\frac {9}{2}}}\)	\(188\)
risch	\(-\frac {\left (-1280 c^{5} e^{2} x^{5}-1664 b \,c^{4} e^{2} x^{4}-3072 c^{5} d e \,x^{4}-48 b^{2} c^{3} e^{2} x^{3}-4224 b \,c^{4} d e \,x^{3}-1920 c^{5} d^{2} x^{3}+56 b^{3} c^{2} e^{2} x^{2}-192 b^{2} c^{3} d e \,x^{2}-2880 b \,c^{4} d^{2} x^{2}-70 b^{4} c \,e^{2} x +240 b^{3} c^{2} d e x -240 b^{2} c^{3} d^{2} x +105 b^{5} e^{2}-360 b^{4} c d e +360 b^{3} c^{2} d^{2}\right ) x \left (c x +b \right )}{7680 c^{4} \sqrt {x \left (c x +b \right )}}+\frac {b^{4} \left (7 b^{2} e^{2}-24 b c d e +24 c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}\)	\(246\)
default	\(d^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )+2 d e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\)	\(342\)

[In]

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

[Out]

7/512/c^(9/2)*(b^4*(b^2*e^2-24/7*b*c*d*e+24/7*c^2*d^2)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))-(-128/7*x^3*(2/3*x
^2*e^2+8/5*d*e*x+d^2)*c^(11/2)+(24/7*(7/45*x^2*e^2+2/3*d*e*x+d^2)*b^2*c^(5/2)-16/7*x*(1/5*x^2*e^2+4/5*d*e*x+d^
2)*b*c^(7/2)-192/7*x^2*(26/45*x^2*e^2+22/15*d*e*x+d^2)*c^(9/2)+((-2/3*e*x-24/7*d)*c^(3/2)+c^(1/2)*b*e)*e*b^3)*
b)*(x*(c*x+b))^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.29 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\left [\frac {15 \, {\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (1280 \, c^{6} e^{2} x^{5} - 360 \, b^{3} c^{3} d^{2} + 360 \, b^{4} c^{2} d e - 105 \, b^{5} c e^{2} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 48 \, {\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )} x^{3} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}\right )} x^{2} + 10 \, {\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{15360 \, c^{5}}, -\frac {15 \, {\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (1280 \, c^{6} e^{2} x^{5} - 360 \, b^{3} c^{3} d^{2} + 360 \, b^{4} c^{2} d e - 105 \, b^{5} c e^{2} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 48 \, {\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )} x^{3} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}\right )} x^{2} + 10 \, {\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{7680 \, c^{5}}\right ] \]

[In]

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

[Out]

[1/15360*(15*(24*b^4*c^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))
+ 2*(1280*c^6*e^2*x^5 - 360*b^3*c^3*d^2 + 360*b^4*c^2*d*e - 105*b^5*c*e^2 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^
4 + 48*(40*c^6*d^2 + 88*b*c^5*d*e + b^2*c^4*e^2)*x^3 + 8*(360*b*c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)*x^2
+ 10*(24*b^2*c^4*d^2 - 24*b^3*c^3*d*e + 7*b^4*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^5, -1/7680*(15*(24*b^4*c^2*d^2
- 24*b^5*c*d*e + 7*b^6*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (1280*c^6*e^2*x^5 - 360*b^3*c^
3*d^2 + 360*b^4*c^2*d*e - 105*b^5*c*e^2 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^4 + 48*(40*c^6*d^2 + 88*b*c^5*d*e
+ b^2*c^4*e^2)*x^3 + 8*(360*b*c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)*x^2 + 10*(24*b^2*c^4*d^2 - 24*b^3*c^3*
d*e + 7*b^4*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^5]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (207) = 414\).

Time = 0.49 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.66 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 b^{2} \left (b^{2} d^{2} - \frac {5 b \left (2 b^{2} d e + 2 b c d^{2} - \frac {7 b \left (b^{2} e^{2} + 4 b c d e - \frac {9 b \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{10 c} + c^{2} d^{2}\right )}{8 c}\right )}{6 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{8 c^{2}} + \sqrt {b x + c x^{2}} \left (- \frac {3 b \left (b^{2} d^{2} - \frac {5 b \left (2 b^{2} d e + 2 b c d^{2} - \frac {7 b \left (b^{2} e^{2} + 4 b c d e - \frac {9 b \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{10 c} + c^{2} d^{2}\right )}{8 c}\right )}{6 c}\right )}{4 c^{2}} + \frac {c e^{2} x^{5}}{6} + \frac {x^{4} \cdot \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{5 c} + \frac {x^{3} \left (b^{2} e^{2} + 4 b c d e - \frac {9 b \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{10 c} + c^{2} d^{2}\right )}{4 c} + \frac {x^{2} \cdot \left (2 b^{2} d e + 2 b c d^{2} - \frac {7 b \left (b^{2} e^{2} + 4 b c d e - \frac {9 b \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{10 c} + c^{2} d^{2}\right )}{8 c}\right )}{3 c} + \frac {x \left (b^{2} d^{2} - \frac {5 b \left (2 b^{2} d e + 2 b c d^{2} - \frac {7 b \left (b^{2} e^{2} + 4 b c d e - \frac {9 b \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{10 c} + c^{2} d^{2}\right )}{8 c}\right )}{6 c}\right )}{2 c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {d^{2} \left (b x\right )^{\frac {5}{2}}}{5} + \frac {2 d e \left (b x\right )^{\frac {7}{2}}}{7 b} + \frac {e^{2} \left (b x\right )^{\frac {9}{2}}}{9 b^{2}}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((3*b**2*(b**2*d**2 - 5*b*(2*b**2*d*e + 2*b*c*d**2 - 7*b*(b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12
+ 2*c**2*d*e)/(10*c) + c**2*d**2)/(8*c))/(6*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x + c*x**2) + 2*c*x)/sqrt
(c), Ne(b**2/c, 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True))/(8*c**2) + sqrt(b*x + c*
x**2)*(-3*b*(b**2*d**2 - 5*b*(2*b**2*d*e + 2*b*c*d**2 - 7*b*(b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c
**2*d*e)/(10*c) + c**2*d**2)/(8*c))/(6*c))/(4*c**2) + c*e**2*x**5/6 + x**4*(13*b*c*e**2/12 + 2*c**2*d*e)/(5*c)
+ x**3*(b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(4*c) + x**2*(2*b**2*d*
e + 2*b*c*d**2 - 7*b*(b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(8*c))/(3*
c) + x*(b**2*d**2 - 5*b*(2*b**2*d*e + 2*b*c*d**2 - 7*b*(b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d
*e)/(10*c) + c**2*d**2)/(8*c))/(6*c))/(2*c)), Ne(c, 0)), (2*(d**2*(b*x)**(5/2)/5 + 2*d*e*(b*x)**(7/2)/(7*b) +
e**2*(b*x)**(9/2)/(9*b**2))/b, Ne(b, 0)), (0, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (190) = 380\).

Time = 0.20 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.94 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} d^{2} x - \frac {3 \, \sqrt {c x^{2} + b x} b^{2} d^{2} x}{32 \, c} + \frac {3 \, \sqrt {c x^{2} + b x} b^{3} d e x}{32 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d e x}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} b^{4} e^{2} x}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} e^{2} x}{96 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} e^{2} x}{6 \, c} + \frac {3 \, b^{4} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} - \frac {3 \, b^{5} d e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {7}{2}}} + \frac {7 \, b^{6} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} b^{3} d^{2}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d^{2}}{8 \, c} + \frac {3 \, \sqrt {c x^{2} + b x} b^{4} d e}{64 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d e}{8 \, c^{2}} + \frac {2 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d e}{5 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} b^{5} e^{2}}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} e^{2}}{192 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b e^{2}}{60 \, c^{2}} \]

[In]

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

[Out]

1/4*(c*x^2 + b*x)^(3/2)*d^2*x - 3/32*sqrt(c*x^2 + b*x)*b^2*d^2*x/c + 3/32*sqrt(c*x^2 + b*x)*b^3*d*e*x/c^2 - 1/
4*(c*x^2 + b*x)^(3/2)*b*d*e*x/c - 7/256*sqrt(c*x^2 + b*x)*b^4*e^2*x/c^3 + 7/96*(c*x^2 + b*x)^(3/2)*b^2*e^2*x/c
^2 + 1/6*(c*x^2 + b*x)^(5/2)*e^2*x/c + 3/128*b^4*d^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 3/
128*b^5*d*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) + 7/1024*b^6*e^2*log(2*c*x + b + 2*sqrt(c*x^2
+ b*x)*sqrt(c))/c^(9/2) - 3/64*sqrt(c*x^2 + b*x)*b^3*d^2/c^2 + 1/8*(c*x^2 + b*x)^(3/2)*b*d^2/c + 3/64*sqrt(c*
x^2 + b*x)*b^4*d*e/c^3 - 1/8*(c*x^2 + b*x)^(3/2)*b^2*d*e/c^2 + 2/5*(c*x^2 + b*x)^(5/2)*d*e/c - 7/512*sqrt(c*x^
2 + b*x)*b^5*e^2/c^4 + 7/192*(c*x^2 + b*x)^(3/2)*b^3*e^2/c^3 - 7/60*(c*x^2 + b*x)^(5/2)*b*e^2/c^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.22 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c e^{2} x + \frac {24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac {3 \, {\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )}}{c^{5}}\right )} x + \frac {360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}}{c^{5}}\right )} x + \frac {5 \, {\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )}}{c^{5}}\right )} x - \frac {15 \, {\left (24 \, b^{3} c^{3} d^{2} - 24 \, b^{4} c^{2} d e + 7 \, b^{5} c e^{2}\right )}}{c^{5}}\right )} - \frac {{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \]

[In]

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

[Out]

1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*c*e^2*x + (24*c^6*d*e + 13*b*c^5*e^2)/c^5)*x + 3*(40*c^6*d^2 + 88*b*c
^5*d*e + b^2*c^4*e^2)/c^5)*x + (360*b*c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)/c^5)*x + 5*(24*b^2*c^4*d^2 - 2
4*b^3*c^3*d*e + 7*b^4*c^2*e^2)/c^5)*x - 15*(24*b^3*c^3*d^2 - 24*b^4*c^2*d*e + 7*b^5*c*e^2)/c^5) - 1/1024*(24*b
^4*c^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(9/2)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^2 \,d x \]

[In]

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

[Out]

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