\(\int \frac {(A+B x) (b x+c x^2)^2}{(d+e x)^{3/2}} \, dx\) [1226]

Optimal result

Integrand size = 26, antiderivative size = 263 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt {d+e x}}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \sqrt {d+e x}}{e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{7/2}}{7 e^6}+\frac {2 B c^2 (d+e x)^{9/2}}{9 e^6} \]

[Out]

2/3*(A*e*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))*(e*x+d)^(3/2)/e^6-2/5*(2*A*c*e*(
-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))*(e*x+d)^(5/2)/e^6-2/7*c*(-A*c*e-2*B*b*e+5*B*c*d)*(e*x+d)^(7/2)/e
^6+2/9*B*c^2*(e*x+d)^(9/2)/e^6+2*d^2*(-A*e+B*d)*(-b*e+c*d)^2/e^6/(e*x+d)^(1/2)+2*d*(-b*e+c*d)*(B*d*(-3*b*e+5*c
*d)-2*A*e*(-b*e+2*c*d))*(e*x+d)^(1/2)/e^6

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {785} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 (d+e x)^{5/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{5 e^6}+\frac {2 (d+e x)^{3/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt {d+e x}}-\frac {2 c (d+e x)^{7/2} (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac {2 d \sqrt {d+e x} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}+\frac {2 B c^2 (d+e x)^{9/2}}{9 e^6} \]

[In]

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

[Out]

(2*d^2*(B*d - A*e)*(c*d - b*e)^2)/(e^6*Sqrt[d + e*x]) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d -
b*e))*Sqrt[d + e*x])/e^6 + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e
^2))*(d + e*x)^(3/2))/(3*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(d + e*x)^(5
/2))/(5*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(7/2))/(7*e^6) + (2*B*c^2*(d + e*x)^(9/2))/(9*e^6)

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^{3/2}}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 \sqrt {d+e x}}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \sqrt {d+e x}}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{3/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{5/2}}{e^5}+\frac {B c^2 (d+e x)^{7/2}}{e^5}\right ) \, dx \\ & = \frac {2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt {d+e x}}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \sqrt {d+e x}}{e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{7/2}}{7 e^6}+\frac {2 B c^2 (d+e x)^{9/2}}{9 e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {-6 A e \left (35 b^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )-42 b c e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+3 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )+2 B \left (63 b^2 e^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+18 b c e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 c^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )}{315 e^6 \sqrt {d+e x}} \]

[In]

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

[Out]

(-6*A*e*(35*b^2*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) - 42*b*c*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 3*c^
2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)) + 2*B*(63*b^2*e^2*(16*d^3 + 8*d^2*e*x - 2
*d*e^2*x^2 + e^3*x^3) + 18*b*c*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + 5*c^2*(2
56*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)))/(315*e^6*Sqrt[d + e*x])

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(\frac {\frac {2 \left (5 \left (7 e^{5} x^{5}-10 d \,e^{4} x^{4}+16 d^{2} e^{3} x^{3}-32 d^{3} e^{2} x^{2}+128 d^{4} e x +256 d^{5}\right ) B -1152 \left (-\frac {5}{128} e^{4} x^{4}+\frac {1}{16} d \,e^{3} x^{3}-\frac {1}{8} d^{2} e^{2} x^{2}+\frac {1}{2} d^{3} e x +d^{4}\right ) A e \right ) c^{2}}{315}+\frac {64 \left (\frac {\left (\frac {5}{16} e^{4} x^{4}-\frac {1}{2} d \,e^{3} x^{3}+d^{2} e^{2} x^{2}-4 d^{3} e x -8 d^{4}\right ) B}{7}+A e \left (\frac {1}{16} e^{3} x^{3}-\frac {1}{8} d \,e^{2} x^{2}+\frac {1}{2} d^{2} e x +d^{3}\right )\right ) e b c}{5}-\frac {16 \left (\frac {3 \left (-\frac {1}{8} e^{3} x^{3}+\frac {1}{4} d \,e^{2} x^{2}-d^{2} e x -2 d^{3}\right ) B}{5}+A e \left (-\frac {1}{8} e^{2} x^{2}+\frac {1}{2} d e x +d^{2}\right )\right ) e^{2} b^{2}}{3}}{\sqrt {e x +d}\, e^{6}}\)	\(259\)
risch	\(-\frac {2 \left (-35 B \,c^{2} x^{4} e^{4}-45 A \,c^{2} e^{4} x^{3}-90 B b c \,e^{4} x^{3}+85 B \,c^{2} d \,e^{3} x^{3}-126 A b c \,e^{4} x^{2}+117 A \,c^{2} d \,e^{3} x^{2}-63 B \,b^{2} e^{4} x^{2}+234 B b c d \,e^{3} x^{2}-165 B \,c^{2} d^{2} e^{2} x^{2}-105 A \,b^{2} e^{4} x +378 A b c d \,e^{3} x -261 A \,c^{2} d^{2} e^{2} x +189 B \,b^{2} d \,e^{3} x -522 B b c \,d^{2} e^{2} x +325 B \,c^{2} d^{3} e x +525 A \,b^{2} d \,e^{3}-1386 A b c \,d^{2} e^{2}+837 A \,c^{2} d^{3} e -693 B \,b^{2} d^{2} e^{2}+1674 B b c \,d^{3} e -965 B \,c^{2} d^{4}\right ) \sqrt {e x +d}}{315 e^{6}}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{e^{6} \sqrt {e x +d}}\)	\(324\)
gosper	\(-\frac {2 \left (-35 B \,x^{5} c^{2} e^{5}-45 A \,x^{4} c^{2} e^{5}-90 B \,x^{4} b c \,e^{5}+50 B \,x^{4} c^{2} d \,e^{4}-126 A \,x^{3} b c \,e^{5}+72 A \,x^{3} c^{2} d \,e^{4}-63 B \,x^{3} b^{2} e^{5}+144 B \,x^{3} b c d \,e^{4}-80 B \,x^{3} c^{2} d^{2} e^{3}-105 A \,x^{2} b^{2} e^{5}+252 A \,x^{2} b c d \,e^{4}-144 A \,x^{2} c^{2} d^{2} e^{3}+126 B \,x^{2} b^{2} d \,e^{4}-288 B \,x^{2} b c \,d^{2} e^{3}+160 B \,x^{2} c^{2} d^{3} e^{2}+420 A x \,b^{2} d \,e^{4}-1008 A x b c \,d^{2} e^{3}+576 A x \,c^{2} d^{3} e^{2}-504 B x \,b^{2} d^{2} e^{3}+1152 B x b c \,d^{3} e^{2}-640 B x \,c^{2} d^{4} e +840 A \,b^{2} d^{2} e^{3}-2016 A b c \,d^{3} e^{2}+1152 A \,c^{2} d^{4} e -1008 B \,b^{2} d^{3} e^{2}+2304 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{315 \sqrt {e x +d}\, e^{6}}\)	\(341\)
trager	\(-\frac {2 \left (-35 B \,x^{5} c^{2} e^{5}-45 A \,x^{4} c^{2} e^{5}-90 B \,x^{4} b c \,e^{5}+50 B \,x^{4} c^{2} d \,e^{4}-126 A \,x^{3} b c \,e^{5}+72 A \,x^{3} c^{2} d \,e^{4}-63 B \,x^{3} b^{2} e^{5}+144 B \,x^{3} b c d \,e^{4}-80 B \,x^{3} c^{2} d^{2} e^{3}-105 A \,x^{2} b^{2} e^{5}+252 A \,x^{2} b c d \,e^{4}-144 A \,x^{2} c^{2} d^{2} e^{3}+126 B \,x^{2} b^{2} d \,e^{4}-288 B \,x^{2} b c \,d^{2} e^{3}+160 B \,x^{2} c^{2} d^{3} e^{2}+420 A x \,b^{2} d \,e^{4}-1008 A x b c \,d^{2} e^{3}+576 A x \,c^{2} d^{3} e^{2}-504 B x \,b^{2} d^{2} e^{3}+1152 B x b c \,d^{3} e^{2}-640 B x \,c^{2} d^{4} e +840 A \,b^{2} d^{2} e^{3}-2016 A b c \,d^{3} e^{2}+1152 A \,c^{2} d^{4} e -1008 B \,b^{2} d^{3} e^{2}+2304 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{315 \sqrt {e x +d}\, e^{6}}\)	\(341\)
derivativedivides	\(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 B b c e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {10 B \,c^{2} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 A b c \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 A \,c^{2} d e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 B \,b^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {16 B b c d e \left (e x +d \right )^{\frac {5}{2}}}{5}+4 B \,c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}+\frac {2 A \,b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 A b c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+4 A \,c^{2} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-2 B \,b^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+8 B b c \,d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {20 B \,c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 A \,b^{2} d \,e^{3} \sqrt {e x +d}+12 A b c \,d^{2} e^{2} \sqrt {e x +d}-8 A \,c^{2} d^{3} e \sqrt {e x +d}+6 B \,b^{2} d^{2} e^{2} \sqrt {e x +d}-16 B b c \,d^{3} e \sqrt {e x +d}+10 B \,c^{2} d^{4} \sqrt {e x +d}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{\sqrt {e x +d}}}{e^{6}}\)	\(405\)
default	\(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 B b c e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {10 B \,c^{2} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 A b c \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 A \,c^{2} d e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 B \,b^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {16 B b c d e \left (e x +d \right )^{\frac {5}{2}}}{5}+4 B \,c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}+\frac {2 A \,b^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 A b c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+4 A \,c^{2} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-2 B \,b^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+8 B b c \,d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {20 B \,c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 A \,b^{2} d \,e^{3} \sqrt {e x +d}+12 A b c \,d^{2} e^{2} \sqrt {e x +d}-8 A \,c^{2} d^{3} e \sqrt {e x +d}+6 B \,b^{2} d^{2} e^{2} \sqrt {e x +d}-16 B b c \,d^{3} e \sqrt {e x +d}+10 B \,c^{2} d^{4} \sqrt {e x +d}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{\sqrt {e x +d}}}{e^{6}}\)	\(405\)

[In]

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

[Out]

2/315*((5*(7*e^5*x^5-10*d*e^4*x^4+16*d^2*e^3*x^3-32*d^3*e^2*x^2+128*d^4*e*x+256*d^5)*B-1152*(-5/128*e^4*x^4+1/
16*d*e^3*x^3-1/8*d^2*e^2*x^2+1/2*d^3*e*x+d^4)*A*e)*c^2+2016*(1/7*(5/16*e^4*x^4-1/2*d*e^3*x^3+d^2*e^2*x^2-4*d^3
*e*x-8*d^4)*B+A*e*(1/16*e^3*x^3-1/8*d*e^2*x^2+1/2*d^2*e*x+d^3))*e*b*c-840*(3/5*(-1/8*e^3*x^3+1/4*d*e^2*x^2-d^2
*e*x-2*d^3)*B+A*e*(-1/8*e^2*x^2+1/2*d*e*x+d^2))*e^2*b^2)/(e*x+d)^(1/2)/e^6

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, B c^{2} e^{5} x^{5} + 1280 \, B c^{2} d^{5} - 840 \, A b^{2} d^{2} e^{3} - 1152 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 1008 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 5 \, {\left (10 \, B c^{2} d e^{4} - 9 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + {\left (80 \, B c^{2} d^{2} e^{3} - 72 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 63 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - {\left (160 \, B c^{2} d^{3} e^{2} - 105 \, A b^{2} e^{5} - 144 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 126 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{4} e - 105 \, A b^{2} d e^{4} - 144 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 126 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{7} x + d e^{6}\right )}} \]

[In]

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

[Out]

2/315*(35*B*c^2*e^5*x^5 + 1280*B*c^2*d^5 - 840*A*b^2*d^2*e^3 - 1152*(2*B*b*c + A*c^2)*d^4*e + 1008*(B*b^2 + 2*
A*b*c)*d^3*e^2 - 5*(10*B*c^2*d*e^4 - 9*(2*B*b*c + A*c^2)*e^5)*x^4 + (80*B*c^2*d^2*e^3 - 72*(2*B*b*c + A*c^2)*d
*e^4 + 63*(B*b^2 + 2*A*b*c)*e^5)*x^3 - (160*B*c^2*d^3*e^2 - 105*A*b^2*e^5 - 144*(2*B*b*c + A*c^2)*d^2*e^3 + 12
6*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 4*(160*B*c^2*d^4*e - 105*A*b^2*d*e^4 - 144*(2*B*b*c + A*c^2)*d^3*e^2 + 126*(B
*b^2 + 2*A*b*c)*d^2*e^3)*x)*sqrt(e*x + d)/(e^7*x + d*e^6)

Sympy [A] (verification not implemented)

Time = 8.92 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{5}} + \frac {d^{2} \left (- A e + B d\right ) \left (b e - c d\right )^{2}}{e^{5} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A c^{2} e + 2 B b c e - 5 B c^{2} d\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (- 2 A b^{2} d e^{3} + 6 A b c d^{2} e^{2} - 4 A c^{2} d^{3} e + 3 B b^{2} d^{2} e^{2} - 8 B b c d^{3} e + 5 B c^{2} d^{4}\right )}{e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {A b^{2} x^{3}}{3} + \frac {B c^{2} x^{6}}{6} + \frac {x^{5} \left (A c^{2} + 2 B b c\right )}{5} + \frac {x^{4} \cdot \left (2 A b c + B b^{2}\right )}{4}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}} B c^{2} - 45 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {315 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )}}{\sqrt {e x + d} e^{5}}\right )}}{315 \, e} \]

[In]

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

[Out]

2/315*((35*(e*x + d)^(9/2)*B*c^2 - 45*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(7/2) + 63*(10*B*c^2*d^2 - 4
*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e*x + d)^(5/2) - 105*(10*B*c^2*d^3 - A*b^2*e^3 - 6*(2*B*b*c +
A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(3/2) + 315*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*(2*B*b*c + A
*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*sqrt(e*x + d))/e^5 + 315*(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*B*b*c + A*
c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)/(sqrt(e*x + d)*e^5))/e

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.68 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (B c^{2} d^{5} - 2 \, B b c d^{4} e - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, A b c d^{3} e^{2} - A b^{2} d^{2} e^{3}\right )}}{\sqrt {e x + d} e^{6}} + \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B c^{2} e^{48} - 225 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{2} d e^{48} + 630 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{2} d^{2} e^{48} - 1050 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{2} d^{3} e^{48} + 1575 \, \sqrt {e x + d} B c^{2} d^{4} e^{48} + 90 \, {\left (e x + d\right )}^{\frac {7}{2}} B b c e^{49} + 45 \, {\left (e x + d\right )}^{\frac {7}{2}} A c^{2} e^{49} - 504 \, {\left (e x + d\right )}^{\frac {5}{2}} B b c d e^{49} - 252 \, {\left (e x + d\right )}^{\frac {5}{2}} A c^{2} d e^{49} + 1260 \, {\left (e x + d\right )}^{\frac {3}{2}} B b c d^{2} e^{49} + 630 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{2} d^{2} e^{49} - 2520 \, \sqrt {e x + d} B b c d^{3} e^{49} - 1260 \, \sqrt {e x + d} A c^{2} d^{3} e^{49} + 63 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{2} e^{50} + 126 \, {\left (e x + d\right )}^{\frac {5}{2}} A b c e^{50} - 315 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{2} d e^{50} - 630 \, {\left (e x + d\right )}^{\frac {3}{2}} A b c d e^{50} + 945 \, \sqrt {e x + d} B b^{2} d^{2} e^{50} + 1890 \, \sqrt {e x + d} A b c d^{2} e^{50} + 105 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{2} e^{51} - 630 \, \sqrt {e x + d} A b^{2} d e^{51}\right )}}{315 \, e^{54}} \]

[In]

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

[Out]

2*(B*c^2*d^5 - 2*B*b*c*d^4*e - A*c^2*d^4*e + B*b^2*d^3*e^2 + 2*A*b*c*d^3*e^2 - A*b^2*d^2*e^3)/(sqrt(e*x + d)*e
^6) + 2/315*(35*(e*x + d)^(9/2)*B*c^2*e^48 - 225*(e*x + d)^(7/2)*B*c^2*d*e^48 + 630*(e*x + d)^(5/2)*B*c^2*d^2*
e^48 - 1050*(e*x + d)^(3/2)*B*c^2*d^3*e^48 + 1575*sqrt(e*x + d)*B*c^2*d^4*e^48 + 90*(e*x + d)^(7/2)*B*b*c*e^49
+ 45*(e*x + d)^(7/2)*A*c^2*e^49 - 504*(e*x + d)^(5/2)*B*b*c*d*e^49 - 252*(e*x + d)^(5/2)*A*c^2*d*e^49 + 1260*
(e*x + d)^(3/2)*B*b*c*d^2*e^49 + 630*(e*x + d)^(3/2)*A*c^2*d^2*e^49 - 2520*sqrt(e*x + d)*B*b*c*d^3*e^49 - 1260
*sqrt(e*x + d)*A*c^2*d^3*e^49 + 63*(e*x + d)^(5/2)*B*b^2*e^50 + 126*(e*x + d)^(5/2)*A*b*c*e^50 - 315*(e*x + d)
^(3/2)*B*b^2*d*e^50 - 630*(e*x + d)^(3/2)*A*b*c*d*e^50 + 945*sqrt(e*x + d)*B*b^2*d^2*e^50 + 1890*sqrt(e*x + d)
*A*b*c*d^2*e^50 + 105*(e*x + d)^(3/2)*A*b^2*e^51 - 630*sqrt(e*x + d)*A*b^2*d*e^51)/e^54

Mupad [B] (verification not implemented)

Time = 10.40 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.13 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{7\,e^6}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{3\,e^6}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{5\,e^6}+\frac {2\,B\,b^2\,d^3\,e^2-2\,A\,b^2\,d^2\,e^3-4\,B\,b\,c\,d^4\,e+4\,A\,b\,c\,d^3\,e^2+2\,B\,c^2\,d^5-2\,A\,c^2\,d^4\,e}{e^6\,\sqrt {d+e\,x}}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{e^6} \]

[In]

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

[Out]

((d + e*x)^(7/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(7*e^6) + ((d + e*x)^(3/2)*(2*A*b^2*e^3 - 20*B*c^2*d^3
+ 12*A*c^2*d^2*e - 6*B*b^2*d*e^2 - 12*A*b*c*d*e^2 + 24*B*b*c*d^2*e))/(3*e^6) + ((d + e*x)^(5/2)*(2*B*b^2*e^2 +
20*B*c^2*d^2 + 4*A*b*c*e^2 - 8*A*c^2*d*e - 16*B*b*c*d*e))/(5*e^6) + (2*B*c^2*d^5 - 2*A*c^2*d^4*e - 2*A*b^2*d^
2*e^3 + 2*B*b^2*d^3*e^2 - 4*B*b*c*d^4*e + 4*A*b*c*d^3*e^2)/(e^6*(d + e*x)^(1/2)) + (2*B*c^2*(d + e*x)^(9/2))/(
9*e^6) - (2*d*(b*e - c*d)*(d + e*x)^(1/2)*(2*A*b*e^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/e^6