\(\int (d+e x)^3 (b x+c x^2)^{3/2} \, dx\) [295]

Optimal result

Integrand size = 21, antiderivative size = 271 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=-\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}} \]

[Out]

1/128*(-b*e+2*c*d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^4+1/7*e*(e*x+d)^2*(c*x^2+b*x)
^(5/2)/c+1/280*e*(128*c^2*d^2-98*b*c*d*e+21*b^2*e^2+30*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x)^(5/2)/c^3+3/1024*b^4*(-
b*e+2*c*d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(11/2)-3/1024*b^2*(-b*e+2*c*
d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c^5

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 271, 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, 793, 626, 634, 212} \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\frac {3 b^4 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{1024 c^{11/2}}-\frac {3 b^2 (b+2 c x) \sqrt {b x+c x^2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{1024 c^5}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{128 c^4}+\frac {e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{280 c^3}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]

[In]

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

[Out]

(-3*b^2*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^5) + ((2*c*d
- b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(128*c^4) + (e*(d + e*x)^2*(b*x +
c*x^2)^(5/2))/(7*c) + (e*(128*c^2*d^2 - 98*b*c*d*e + 21*b^2*e^2 + 30*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(5/2))
/(280*c^3) + (3*b^4*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*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 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*} \text {integral}& = \frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\int (d+e x) \left (\frac {1}{2} d (14 c d-5 b e)+\frac {9}{2} e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2} \, dx}{7 c} \\ & = \frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{16 c^3} \\ & = \frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}-\frac {\left (3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \sqrt {b x+c x^2} \, dx}{256 c^4} \\ & = -\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left (3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2048 c^5} \\ & = -\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left (3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 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 )}{1024 c^5} \\ & = -\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.49 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.37 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (315 b^6 e^3-210 b^5 c e^2 (7 d+e x)+28 b^4 c^2 e \left (90 d^2+35 d e x+6 e^2 x^2\right )+32 b^2 c^4 x \left (35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right )-16 b^3 c^3 \left (105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right )+256 c^6 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+128 b c^5 x^2 \left (105 d^3+231 d^2 e x+182 d e^2 x^2+50 e^3 x^3\right )\right )+630 b^5 e \left (8 c^2 d^2+b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+420 b^4 c d \left (8 c^2 d^2+7 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{35840 c^{11/2} \sqrt {x (b+c x)}} \]

[In]

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

[Out]

(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(315*b^6*e^3 - 210*b^5*c*e^2*(7*d + e*x) + 28*b^4*c^2*e*
(90*d^2 + 35*d*e*x + 6*e^2*x^2) + 32*b^2*c^4*x*(35*d^3 + 42*d^2*e*x + 21*d*e^2*x^2 + 4*e^3*x^3) - 16*b^3*c^3*(
105*d^3 + 105*d^2*e*x + 49*d*e^2*x^2 + 9*e^3*x^3) + 256*c^6*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x
^3) + 128*b*c^5*x^2*(105*d^3 + 231*d^2*e*x + 182*d*e^2*x^2 + 50*e^3*x^3)) + 630*b^5*e*(8*c^2*d^2 + b^2*e^2)*Ar
cTanh[(Sqrt[c]*Sqrt[x])/(Sqrt[b] - Sqrt[b + c*x])] + 420*b^4*c*d*(8*c^2*d^2 + 7*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt
[x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(35840*c^(11/2)*Sqrt[x*(b + c*x)])

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(-\frac {9 \left (\left (b e -2 c d \right ) \left (b^{2} e^{2}-\frac {8}{3} b c d e +\frac {8}{3} c^{2} d^{2}\right ) b^{4} \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )-\sqrt {x \left (c x +b \right )}\, \left (\frac {128 x^{2} \left (\frac {10}{21} e^{3} x^{3}+\frac {26}{15} d \,e^{2} x^{2}+\frac {11}{5} d^{2} e x +d^{3}\right ) b \,c^{\frac {11}{2}}}{3}+\frac {256 x^{3} \left (\frac {4}{7} e^{3} x^{3}+2 d \,e^{2} x^{2}+\frac {12}{5} d^{2} e x +d^{3}\right ) c^{\frac {13}{2}}}{9}+b^{2} \left (-\frac {16 \left (\frac {3}{35} e^{3} x^{3}+\frac {7}{15} d \,e^{2} x^{2}+d^{2} e x +d^{3}\right ) b \,c^{\frac {7}{2}}}{3}+\frac {32 x \left (\frac {4}{35} e^{3} x^{3}+\frac {3}{5} d \,e^{2} x^{2}+\frac {6}{5} d^{2} e x +d^{3}\right ) c^{\frac {9}{2}}}{9}+\left (\left (\frac {8}{15} x^{2} e^{2}+\frac {28}{9} d e x +8 d^{2}\right ) c^{\frac {5}{2}}+\left (\left (-\frac {2 e x}{3}-\frac {14 d}{3}\right ) c^{\frac {3}{2}}+\sqrt {c}\, b e \right ) e b \right ) e \,b^{2}\right )\right )\right )}{1024 c^{\frac {11}{2}}}\)	\(266\)
risch	\(\frac {\left (5120 c^{6} e^{3} x^{6}+6400 b \,c^{5} e^{3} x^{5}+17920 c^{6} d \,e^{2} x^{5}+128 b^{2} c^{4} e^{3} x^{4}+23296 b \,c^{5} d \,e^{2} x^{4}+21504 c^{6} d^{2} e \,x^{4}-144 b^{3} c^{3} e^{3} x^{3}+672 b^{2} c^{4} d \,e^{2} x^{3}+29568 b \,c^{5} d^{2} e \,x^{3}+8960 c^{6} d^{3} x^{3}+168 b^{4} c^{2} e^{3} x^{2}-784 b^{3} c^{3} d \,e^{2} x^{2}+1344 b^{2} c^{4} d^{2} e \,x^{2}+13440 b \,c^{5} d^{3} x^{2}-210 b^{5} c \,e^{3} x +980 b^{4} c^{2} d \,e^{2} x -1680 b^{3} c^{3} d^{2} e x +1120 b^{2} c^{4} d^{3} x +315 b^{6} e^{3}-1470 b^{5} c d \,e^{2}+2520 b^{4} c^{2} d^{2} e -1680 b^{3} c^{3} d^{3}\right ) x \left (c x +b \right )}{35840 c^{5} \sqrt {x \left (c x +b \right )}}-\frac {3 b^{4} \left (3 b^{3} e^{3}-14 b^{2} d \,e^{2} c +24 b \,c^{2} d^{2} e -16 c^{3} d^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {11}{2}}}\)	\(364\)
default	\(d^{3} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+e^{3} \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 c}-\frac {9 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )}{14 c}\right )+3 d \,e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )+3 d^{2} e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\)	\(509\)

[In]

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

[Out]

-9/1024/c^(11/2)*((b*e-2*c*d)*(b^2*e^2-8/3*b*c*d*e+8/3*c^2*d^2)*b^4*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))-(x*(c
*x+b))^(1/2)*(128/3*x^2*(10/21*e^3*x^3+26/15*d*e^2*x^2+11/5*d^2*e*x+d^3)*b*c^(11/2)+256/9*x^3*(4/7*e^3*x^3+2*d
*e^2*x^2+12/5*d^2*e*x+d^3)*c^(13/2)+b^2*(-16/3*(3/35*e^3*x^3+7/15*d*e^2*x^2+d^2*e*x+d^3)*b*c^(7/2)+32/9*x*(4/3
5*e^3*x^3+3/5*d*e^2*x^2+6/5*d^2*e*x+d^3)*c^(9/2)+((8/15*x^2*e^2+28/9*d*e*x+8*d^2)*c^(5/2)+((-2/3*e*x-14/3*d)*c
^(3/2)+c^(1/2)*b*e)*e*b)*e*b^2)))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.61 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\left [-\frac {105 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (5120 \, c^{7} e^{3} x^{6} - 1680 \, b^{3} c^{4} d^{3} + 2520 \, b^{4} c^{3} d^{2} e - 1470 \, b^{5} c^{2} d e^{2} + 315 \, b^{6} c e^{3} + 1280 \, {\left (14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}\right )} x^{5} + 128 \, {\left (168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}\right )} x^{4} + 16 \, {\left (560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}\right )} x^{3} + 56 \, {\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )} x^{2} + 70 \, {\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{71680 \, c^{6}}, -\frac {105 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (5120 \, c^{7} e^{3} x^{6} - 1680 \, b^{3} c^{4} d^{3} + 2520 \, b^{4} c^{3} d^{2} e - 1470 \, b^{5} c^{2} d e^{2} + 315 \, b^{6} c e^{3} + 1280 \, {\left (14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}\right )} x^{5} + 128 \, {\left (168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}\right )} x^{4} + 16 \, {\left (560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}\right )} x^{3} + 56 \, {\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )} x^{2} + 70 \, {\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{35840 \, c^{6}}\right ] \]

[In]

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

[Out]

[-1/71680*(105*(16*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b^7*e^3)*sqrt(c)*log(2*c*x + b - 2*sqrt
(c*x^2 + b*x)*sqrt(c)) - 2*(5120*c^7*e^3*x^6 - 1680*b^3*c^4*d^3 + 2520*b^4*c^3*d^2*e - 1470*b^5*c^2*d*e^2 + 31
5*b^6*c*e^3 + 1280*(14*c^7*d*e^2 + 5*b*c^6*e^3)*x^5 + 128*(168*c^7*d^2*e + 182*b*c^6*d*e^2 + b^2*c^5*e^3)*x^4
+ 16*(560*c^7*d^3 + 1848*b*c^6*d^2*e + 42*b^2*c^5*d*e^2 - 9*b^3*c^4*e^3)*x^3 + 56*(240*b*c^6*d^3 + 24*b^2*c^5*
d^2*e - 14*b^3*c^4*d*e^2 + 3*b^4*c^3*e^3)*x^2 + 70*(16*b^2*c^5*d^3 - 24*b^3*c^4*d^2*e + 14*b^4*c^3*d*e^2 - 3*b
^5*c^2*e^3)*x)*sqrt(c*x^2 + b*x))/c^6, -1/35840*(105*(16*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b
^7*e^3)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (5120*c^7*e^3*x^6 - 1680*b^3*c^4*d^3 + 2520*b^4*c^
3*d^2*e - 1470*b^5*c^2*d*e^2 + 315*b^6*c*e^3 + 1280*(14*c^7*d*e^2 + 5*b*c^6*e^3)*x^5 + 128*(168*c^7*d^2*e + 18
2*b*c^6*d*e^2 + b^2*c^5*e^3)*x^4 + 16*(560*c^7*d^3 + 1848*b*c^6*d^2*e + 42*b^2*c^5*d*e^2 - 9*b^3*c^4*e^3)*x^3
+ 56*(240*b*c^6*d^3 + 24*b^2*c^5*d^2*e - 14*b^3*c^4*d*e^2 + 3*b^4*c^3*e^3)*x^2 + 70*(16*b^2*c^5*d^3 - 24*b^3*c
^4*d^2*e + 14*b^4*c^3*d*e^2 - 3*b^5*c^2*e^3)*x)*sqrt(c*x^2 + b*x))/c^6]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 864 vs. \(2 (269) = 538\).

Time = 0.52 (sec) , antiderivative size = 864, normalized size of antiderivative = 3.19 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 b^{2} \left (b^{2} d^{3} - \frac {5 b \left (3 b^{2} d^{2} e + 2 b c d^{3} - \frac {7 b \left (3 b^{2} d e^{2} + 6 b c d^{2} e - \frac {9 b \left (b^{2} e^{3} + 6 b c d e^{2} - \frac {11 b \left (\frac {15 b c e^{3}}{14} + 3 c^{2} d e^{2}\right )}{12 c} + 3 c^{2} d^{2} e\right )}{10 c} + c^{2} d^{3}\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^{3} - \frac {5 b \left (3 b^{2} d^{2} e + 2 b c d^{3} - \frac {7 b \left (3 b^{2} d e^{2} + 6 b c d^{2} e - \frac {9 b \left (b^{2} e^{3} + 6 b c d e^{2} - \frac {11 b \left (\frac {15 b c e^{3}}{14} + 3 c^{2} d e^{2}\right )}{12 c} + 3 c^{2} d^{2} e\right )}{10 c} + c^{2} d^{3}\right )}{8 c}\right )}{6 c}\right )}{4 c^{2}} + \frac {c e^{3} x^{6}}{7} + \frac {x^{5} \cdot \left (\frac {15 b c e^{3}}{14} + 3 c^{2} d e^{2}\right )}{6 c} + \frac {x^{4} \left (b^{2} e^{3} + 6 b c d e^{2} - \frac {11 b \left (\frac {15 b c e^{3}}{14} + 3 c^{2} d e^{2}\right )}{12 c} + 3 c^{2} d^{2} e\right )}{5 c} + \frac {x^{3} \cdot \left (3 b^{2} d e^{2} + 6 b c d^{2} e - \frac {9 b \left (b^{2} e^{3} + 6 b c d e^{2} - \frac {11 b \left (\frac {15 b c e^{3}}{14} + 3 c^{2} d e^{2}\right )}{12 c} + 3 c^{2} d^{2} e\right )}{10 c} + c^{2} d^{3}\right )}{4 c} + \frac {x^{2} \cdot \left (3 b^{2} d^{2} e + 2 b c d^{3} - \frac {7 b \left (3 b^{2} d e^{2} + 6 b c d^{2} e - \frac {9 b \left (b^{2} e^{3} + 6 b c d e^{2} - \frac {11 b \left (\frac {15 b c e^{3}}{14} + 3 c^{2} d e^{2}\right )}{12 c} + 3 c^{2} d^{2} e\right )}{10 c} + c^{2} d^{3}\right )}{8 c}\right )}{3 c} + \frac {x \left (b^{2} d^{3} - \frac {5 b \left (3 b^{2} d^{2} e + 2 b c d^{3} - \frac {7 b \left (3 b^{2} d e^{2} + 6 b c d^{2} e - \frac {9 b \left (b^{2} e^{3} + 6 b c d e^{2} - \frac {11 b \left (\frac {15 b c e^{3}}{14} + 3 c^{2} d e^{2}\right )}{12 c} + 3 c^{2} d^{2} e\right )}{10 c} + c^{2} d^{3}\right )}{8 c}\right )}{6 c}\right )}{2 c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {d^{3} \left (b x\right )^{\frac {5}{2}}}{5} + \frac {3 d^{2} e \left (b x\right )^{\frac {7}{2}}}{7 b} + \frac {d e^{2} \left (b x\right )^{\frac {9}{2}}}{3 b^{2}} + \frac {e^{3} \left (b x\right )^{\frac {11}{2}}}{11 b^{3}}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((3*b**2*(b**2*d**3 - 5*b*(3*b**2*d**2*e + 2*b*c*d**3 - 7*b*(3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(b**2
*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(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**3 - 5*b*(3*b**2*d**
2*e + 2*b*c*d**3 - 7*b*(3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 +
3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(8*c))/(6*c))/(4*c**2) + c*e**3*x**6/7 + x**5*(15*b
*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + x**4*(b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12
*c) + 3*c**2*d**2*e)/(5*c) + x**3*(3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c
*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(4*c) + x**2*(3*b**2*d**2*e + 2*b*c*d**3
- 7*b*(3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(
12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(8*c))/(3*c) + x*(b**2*d**3 - 5*b*(3*b**2*d**2*e + 2*b*c*d**3 - 7*b
*(3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c)
+ 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(8*c))/(6*c))/(2*c)), Ne(c, 0)), (2*(d**3*(b*x)**(5/2)/5 + 3*d**2*e*(b*x)
**(7/2)/(7*b) + d*e**2*(b*x)**(9/2)/(3*b**2) + e**3*(b*x)**(11/2)/(11*b**3))/b, Ne(b, 0)), (0, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (247) = 494\).

Time = 0.20 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.30 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} e^{3} x^{2}}{7 \, c} + \frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} d^{3} x - \frac {3 \, \sqrt {c x^{2} + b x} b^{2} d^{3} x}{32 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} b^{3} d^{2} e x}{64 \, c^{2}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d^{2} e x}{8 \, c} - \frac {21 \, \sqrt {c x^{2} + b x} b^{4} d e^{2} x}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d e^{2} x}{32 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} d e^{2} x}{2 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} b^{5} e^{3} x}{512 \, c^{4}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} e^{3} x}{64 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b e^{3} x}{28 \, c^{2}} + \frac {3 \, b^{4} d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} - \frac {9 \, b^{5} d^{2} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} + \frac {21 \, b^{6} d e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {9 \, b^{7} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} b^{3} d^{3}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d^{3}}{8 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} b^{4} d^{2} e}{128 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d^{2} e}{16 \, c^{2}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d^{2} e}{5 \, c} - \frac {21 \, \sqrt {c x^{2} + b x} b^{5} d e^{2}}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d e^{2}}{64 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d e^{2}}{20 \, c^{2}} + \frac {9 \, \sqrt {c x^{2} + b x} b^{6} e^{3}}{1024 \, c^{5}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} e^{3}}{128 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} e^{3}}{40 \, c^{3}} \]

[In]

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

[Out]

1/7*(c*x^2 + b*x)^(5/2)*e^3*x^2/c + 1/4*(c*x^2 + b*x)^(3/2)*d^3*x - 3/32*sqrt(c*x^2 + b*x)*b^2*d^3*x/c + 9/64*
sqrt(c*x^2 + b*x)*b^3*d^2*e*x/c^2 - 3/8*(c*x^2 + b*x)^(3/2)*b*d^2*e*x/c - 21/256*sqrt(c*x^2 + b*x)*b^4*d*e^2*x
/c^3 + 7/32*(c*x^2 + b*x)^(3/2)*b^2*d*e^2*x/c^2 + 1/2*(c*x^2 + b*x)^(5/2)*d*e^2*x/c + 9/512*sqrt(c*x^2 + b*x)*
b^5*e^3*x/c^4 - 3/64*(c*x^2 + b*x)^(3/2)*b^3*e^3*x/c^3 - 3/28*(c*x^2 + b*x)^(5/2)*b*e^3*x/c^2 + 3/128*b^4*d^3*
log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 9/256*b^5*d^2*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqr
t(c))/c^(7/2) + 21/1024*b^6*d*e^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) - 9/2048*b^7*e^3*log(2*
c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) - 3/64*sqrt(c*x^2 + b*x)*b^3*d^3/c^2 + 1/8*(c*x^2 + b*x)^(3/2)
*b*d^3/c + 9/128*sqrt(c*x^2 + b*x)*b^4*d^2*e/c^3 - 3/16*(c*x^2 + b*x)^(3/2)*b^2*d^2*e/c^2 + 3/5*(c*x^2 + b*x)^
(5/2)*d^2*e/c - 21/512*sqrt(c*x^2 + b*x)*b^5*d*e^2/c^4 + 7/64*(c*x^2 + b*x)^(3/2)*b^3*d*e^2/c^3 - 7/20*(c*x^2
+ b*x)^(5/2)*b*d*e^2/c^2 + 9/1024*sqrt(c*x^2 + b*x)*b^6*e^3/c^5 - 3/128*(c*x^2 + b*x)^(3/2)*b^4*e^3/c^4 + 3/40
*(c*x^2 + b*x)^(5/2)*b^2*e^3/c^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.37 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{35840} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, c e^{3} x + \frac {14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}}{c^{6}}\right )} x + \frac {168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}}{c^{6}}\right )} x + \frac {560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}}{c^{6}}\right )} x + \frac {7 \, {\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )}}{c^{6}}\right )} x + \frac {35 \, {\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )}}{c^{6}}\right )} x - \frac {105 \, {\left (16 \, b^{3} c^{4} d^{3} - 24 \, b^{4} c^{3} d^{2} e + 14 \, b^{5} c^{2} d e^{2} - 3 \, b^{6} c e^{3}\right )}}{c^{6}}\right )} - \frac {3 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \]

[In]

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

[Out]

1/35840*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(4*c*e^3*x + (14*c^7*d*e^2 + 5*b*c^6*e^3)/c^6)*x + (168*c^7*d^2*e +
182*b*c^6*d*e^2 + b^2*c^5*e^3)/c^6)*x + (560*c^7*d^3 + 1848*b*c^6*d^2*e + 42*b^2*c^5*d*e^2 - 9*b^3*c^4*e^3)/c^
6)*x + 7*(240*b*c^6*d^3 + 24*b^2*c^5*d^2*e - 14*b^3*c^4*d*e^2 + 3*b^4*c^3*e^3)/c^6)*x + 35*(16*b^2*c^5*d^3 - 2
4*b^3*c^4*d^2*e + 14*b^4*c^3*d*e^2 - 3*b^5*c^2*e^3)/c^6)*x - 105*(16*b^3*c^4*d^3 - 24*b^4*c^3*d^2*e + 14*b^5*c
^2*d*e^2 - 3*b^6*c*e^3)/c^6) - 3/2048*(16*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b^7*e^3)*log(abs
(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(11/2)

Mupad [F(-1)]

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

[In]

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

[Out]

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