\(\int \frac {(a+b x+c x^2)^3}{(d+e x)^{7/2}} \, dx\) [2288]

Optimal result

Integrand size = 22, antiderivative size = 278 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 \sqrt {d+e x}}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \sqrt {d+e x}}{e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3/2}}{e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \]

[Out]

-2/5*(a*e^2-b*d*e+c*d^2)^3/e^7/(e*x+d)^(5/2)+2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2/e^7/(e*x+d)^(3/2)+2*c*(5*c^2
*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(3/2)/e^7-6/5*c^2*(-b*e+2*c*d)*(e*x+d)^(5/2)/e^7+2/7*c^3*(e*x+d)^(7/2)/
e^7-6*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))/e^7/(e*x+d)^(1/2)-2*(-b*e+2*c*d)*(10*c^2*d^2+b^
2*e^2-2*c*e*(-3*a*e+5*b*d))*(e*x+d)^(1/2)/e^7

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac {6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac {6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \]

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^3)/(5*e^7*(d + e*x)^(5/2)) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^7*(d +
e*x)^(3/2)) - (6*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^7*Sqrt[d + e*x]) - (2*
(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*Sqrt[d + e*x])/e^7 + (2*c*(5*c^2*d^2 + b^2*e^2 -
c*e*(5*b*d - a*e))*(d + e*x)^(3/2))/e^7 - (6*c^2*(2*c*d - b*e)*(d + e*x)^(5/2))/(5*e^7) + (2*c^3*(d + e*x)^(7/
2))/(7*e^7)

Rule 712

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

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )+7 e^3 \left (a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left (8 d^2+20 d e x+15 e^2 x^2\right )-b^3 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )+7 c e^2 \left (a^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a b e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+b^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )-7 c^2 e \left (a e \left (-128 d^4-320 d^3 e x-240 d^2 e^2 x^2-40 d e^3 x^3+5 e^4 x^4\right )+b \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )\right )}{35 e^7 (d+e x)^{5/2}} \]

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(-2*(c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6
*x^6) + 7*e^3*(a^3*e^3 + a^2*b*e^2*(2*d + 5*e*x) + a*b^2*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - b^3*(16*d^3 + 40*
d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)) + 7*c*e^2*(a^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*a*b*e*(16*d^3 + 40
*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + b^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4
)) - 7*c^2*e*(a*e*(-128*d^4 - 320*d^3*e*x - 240*d^2*e^2*x^2 - 40*d*e^3*x^3 + 5*e^4*x^4) + b*(256*d^5 + 640*d^4
*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5))))/(35*e^7*(d + e*x)^(5/2))

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.20

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (-\frac {5 c^{3} x^{6}}{7}+\left (-3 b \,x^{5}-5 a \,x^{4}\right ) c^{2}+\left (-5 b^{2} x^{4}-30 a b \,x^{3}+15 a^{2} x^{2}\right ) c -5 b^{3} x^{3}+15 a \,b^{2} x^{2}+5 a^{2} b x +a^{3}\right ) e^{6}+2 \left (\frac {6 c^{3} x^{5}}{7}+\left (5 b \,x^{4}+20 a \,x^{3}\right ) c^{2}+\left (20 b^{2} x^{3}-90 a b \,x^{2}+10 a^{2} x \right ) c +b \left (-15 b^{2} x^{2}+10 a b x +a^{2}\right )\right ) d \,e^{5}+8 \left (-\frac {5 c^{3} x^{4}}{7}+\left (-10 b \,x^{3}+30 x^{2} a \right ) c^{2}+\left (30 b^{2} x^{2}-30 a b x +a^{2}\right ) c +b^{2} \left (-5 b x +a \right )\right ) d^{2} e^{4}-96 \left (-\frac {10 c^{3} x^{3}}{21}+\left (5 b \,x^{2}-\frac {10}{3} a x \right ) c^{2}+b \left (-\frac {10 b x}{3}+a \right ) c +\frac {b^{3}}{6}\right ) d^{3} e^{3}+128 \left (\frac {15 c^{2} x^{2}}{7}+\left (-5 b x +a \right ) c +b^{2}\right ) c \,d^{4} e^{2}-256 c^{2} \left (-\frac {10 c x}{7}+b \right ) d^{5} e +\frac {1024 c^{3} d^{6}}{7}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} e^{7}}\)	\(333\)
risch	\(\frac {2 \left (5 c^{3} x^{3} e^{3}+21 b \,c^{2} e^{3} x^{2}-27 c^{3} d \,e^{2} x^{2}+35 a \,c^{2} e^{3} x +35 b^{2} c \,e^{3} x -133 b \,c^{2} d \,e^{2} x +106 c^{3} d^{2} e x +210 a b c \,e^{3}-385 a \,c^{2} d \,e^{2}+35 b^{3} e^{3}-385 b^{2} c d \,e^{2}+896 b \,c^{2} d^{2} e -562 c^{3} d^{3}\right ) \sqrt {e x +d}}{35 e^{7}}-\frac {2 \left (15 a c \,e^{4} x^{2}+15 b^{2} e^{4} x^{2}-75 b c d \,e^{3} x^{2}+75 c^{2} d^{2} e^{2} x^{2}+5 a b \,e^{4} x +20 a c d \,e^{3} x +25 b^{2} d \,e^{3} x -135 b c \,d^{2} e^{2} x +140 c^{2} d^{3} e x +a^{2} e^{4}+3 a b d \,e^{3}+7 a c \,d^{2} e^{2}+11 b^{2} d^{2} e^{2}-62 b c \,d^{3} e +66 c^{2} d^{4}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )}{5 e^{7} \sqrt {e x +d}\, \left (x^{2} e^{2}+2 d e x +d^{2}\right )}\)	\(338\)
gosper	\(-\frac {2 \left (-5 c^{3} x^{6} e^{6}-21 b \,c^{2} e^{6} x^{5}+12 c^{3} d \,e^{5} x^{5}-35 a \,c^{2} e^{6} x^{4}-35 b^{2} c \,e^{6} x^{4}+70 b \,c^{2} d \,e^{5} x^{4}-40 c^{3} d^{2} e^{4} x^{4}-210 a b c \,e^{6} x^{3}+280 a \,c^{2} d \,e^{5} x^{3}-35 b^{3} e^{6} x^{3}+280 b^{2} c d \,e^{5} x^{3}-560 b \,c^{2} d^{2} e^{4} x^{3}+320 c^{3} d^{3} e^{3} x^{3}+105 a^{2} c \,e^{6} x^{2}+105 a \,b^{2} e^{6} x^{2}-1260 a b c d \,e^{5} x^{2}+1680 a \,c^{2} d^{2} e^{4} x^{2}-210 b^{3} d \,e^{5} x^{2}+1680 b^{2} c \,d^{2} e^{4} x^{2}-3360 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+35 a^{2} b \,e^{6} x +140 a^{2} c d \,e^{5} x +140 a \,b^{2} d \,e^{5} x -1680 a b c \,d^{2} e^{4} x +2240 a \,c^{2} d^{3} e^{3} x -280 b^{3} d^{2} e^{4} x +2240 b^{2} c \,d^{3} e^{3} x -4480 b \,c^{2} d^{4} e^{2} x +2560 c^{3} d^{5} e x +7 a^{3} e^{6}+14 a^{2} b d \,e^{5}+56 a^{2} c \,d^{2} e^{4}+56 a \,b^{2} d^{2} e^{4}-672 a b c \,d^{3} e^{3}+896 a \,c^{2} d^{4} e^{2}-112 b^{3} d^{3} e^{3}+896 b^{2} c \,d^{4} e^{2}-1792 b \,c^{2} d^{5} e +1024 c^{3} d^{6}\right )}{35 \left (e x +d \right )^{\frac {5}{2}} e^{7}}\)	\(495\)
trager	\(-\frac {2 \left (-5 c^{3} x^{6} e^{6}-21 b \,c^{2} e^{6} x^{5}+12 c^{3} d \,e^{5} x^{5}-35 a \,c^{2} e^{6} x^{4}-35 b^{2} c \,e^{6} x^{4}+70 b \,c^{2} d \,e^{5} x^{4}-40 c^{3} d^{2} e^{4} x^{4}-210 a b c \,e^{6} x^{3}+280 a \,c^{2} d \,e^{5} x^{3}-35 b^{3} e^{6} x^{3}+280 b^{2} c d \,e^{5} x^{3}-560 b \,c^{2} d^{2} e^{4} x^{3}+320 c^{3} d^{3} e^{3} x^{3}+105 a^{2} c \,e^{6} x^{2}+105 a \,b^{2} e^{6} x^{2}-1260 a b c d \,e^{5} x^{2}+1680 a \,c^{2} d^{2} e^{4} x^{2}-210 b^{3} d \,e^{5} x^{2}+1680 b^{2} c \,d^{2} e^{4} x^{2}-3360 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+35 a^{2} b \,e^{6} x +140 a^{2} c d \,e^{5} x +140 a \,b^{2} d \,e^{5} x -1680 a b c \,d^{2} e^{4} x +2240 a \,c^{2} d^{3} e^{3} x -280 b^{3} d^{2} e^{4} x +2240 b^{2} c \,d^{3} e^{3} x -4480 b \,c^{2} d^{4} e^{2} x +2560 c^{3} d^{5} e x +7 a^{3} e^{6}+14 a^{2} b d \,e^{5}+56 a^{2} c \,d^{2} e^{4}+56 a \,b^{2} d^{2} e^{4}-672 a b c \,d^{3} e^{3}+896 a \,c^{2} d^{4} e^{2}-112 b^{3} d^{3} e^{3}+896 b^{2} c \,d^{4} e^{2}-1792 b \,c^{2} d^{5} e +1024 c^{3} d^{6}\right )}{35 \left (e x +d \right )^{\frac {5}{2}} e^{7}}\)	\(495\)
derivativedivides	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}+2 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+2 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-10 b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}+10 c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}+12 a b c \,e^{3} \sqrt {e x +d}-24 a \,c^{2} d \,e^{2} \sqrt {e x +d}+2 b^{3} e^{3} \sqrt {e x +d}-24 b^{2} c d \,e^{2} \sqrt {e x +d}+60 b \,c^{2} d^{2} e \sqrt {e x +d}-40 c^{3} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{3} e^{6}-3 a^{2} b d \,e^{5}+3 a^{2} c \,d^{2} e^{4}+3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 a \,c^{2} d^{4} e^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (3 c \,a^{2} e^{4}+3 b^{2} a \,e^{4}-18 a b c d \,e^{3}+18 a \,c^{2} d^{2} e^{2}-3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 b \,c^{2} d^{3} e +15 c^{3} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (3 a^{2} b \,e^{5}-6 a^{2} c d \,e^{4}-6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 a \,c^{2} d^{3} e^{2}+3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 c^{3} d^{5}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\)	\(506\)
default	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}+2 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+2 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-10 b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}+10 c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}+12 a b c \,e^{3} \sqrt {e x +d}-24 a \,c^{2} d \,e^{2} \sqrt {e x +d}+2 b^{3} e^{3} \sqrt {e x +d}-24 b^{2} c d \,e^{2} \sqrt {e x +d}+60 b \,c^{2} d^{2} e \sqrt {e x +d}-40 c^{3} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{3} e^{6}-3 a^{2} b d \,e^{5}+3 a^{2} c \,d^{2} e^{4}+3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 a \,c^{2} d^{4} e^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (3 c \,a^{2} e^{4}+3 b^{2} a \,e^{4}-18 a b c d \,e^{3}+18 a \,c^{2} d^{2} e^{2}-3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 b \,c^{2} d^{3} e +15 c^{3} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (3 a^{2} b \,e^{5}-6 a^{2} c d \,e^{4}-6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 a \,c^{2} d^{3} e^{2}+3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 c^{3} d^{5}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\)	\(506\)

[In]

int((c*x^2+b*x+a)^3/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/5/(e*x+d)^(5/2)*((-5/7*c^3*x^6+(-3*b*x^5-5*a*x^4)*c^2+(-5*b^2*x^4-30*a*b*x^3+15*a^2*x^2)*c-5*b^3*x^3+15*a*b
^2*x^2+5*a^2*b*x+a^3)*e^6+2*(6/7*c^3*x^5+(5*b*x^4+20*a*x^3)*c^2+(20*b^2*x^3-90*a*b*x^2+10*a^2*x)*c+b*(-15*b^2*
x^2+10*a*b*x+a^2))*d*e^5+8*(-5/7*c^3*x^4+(-10*b*x^3+30*a*x^2)*c^2+(30*b^2*x^2-30*a*b*x+a^2)*c+b^2*(-5*b*x+a))*
d^2*e^4-96*(-10/21*c^3*x^3+(5*b*x^2-10/3*a*x)*c^2+b*(-10/3*b*x+a)*c+1/6*b^3)*d^3*e^3+128*(15/7*c^2*x^2+(-5*b*x
+a)*c+b^2)*c*d^4*e^2-256*c^2*(-10/7*c*x+b)*d^5*e+1024/7*c^3*d^6)/e^7

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.58 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (5 \, c^{3} e^{6} x^{6} - 1024 \, c^{3} d^{6} + 1792 \, b c^{2} d^{5} e - 14 \, a^{2} b d e^{5} - 7 \, a^{3} e^{6} - 896 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 112 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 56 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \, {\left (4 \, c^{3} d e^{5} - 7 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (8 \, c^{3} d^{2} e^{4} - 14 \, b c^{2} d e^{5} + 7 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{3} e^{3} - 112 \, b c^{2} d^{2} e^{4} + 56 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - 7 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 15 \, {\left (128 \, c^{3} d^{4} e^{2} - 224 \, b c^{2} d^{3} e^{3} + 112 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 14 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 7 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 5 \, {\left (512 \, c^{3} d^{5} e - 896 \, b c^{2} d^{4} e^{2} + 7 \, a^{2} b e^{6} + 448 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 56 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 28 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*c^3*e^6*x^6 - 1024*c^3*d^6 + 1792*b*c^2*d^5*e - 14*a^2*b*d*e^5 - 7*a^3*e^6 - 896*(b^2*c + a*c^2)*d^4*e
^2 + 112*(b^3 + 6*a*b*c)*d^3*e^3 - 56*(a*b^2 + a^2*c)*d^2*e^4 - 3*(4*c^3*d*e^5 - 7*b*c^2*e^6)*x^5 + 5*(8*c^3*d
^2*e^4 - 14*b*c^2*d*e^5 + 7*(b^2*c + a*c^2)*e^6)*x^4 - 5*(64*c^3*d^3*e^3 - 112*b*c^2*d^2*e^4 + 56*(b^2*c + a*c
^2)*d*e^5 - 7*(b^3 + 6*a*b*c)*e^6)*x^3 - 15*(128*c^3*d^4*e^2 - 224*b*c^2*d^3*e^3 + 112*(b^2*c + a*c^2)*d^2*e^4
- 14*(b^3 + 6*a*b*c)*d*e^5 + 7*(a*b^2 + a^2*c)*e^6)*x^2 - 5*(512*c^3*d^5*e - 896*b*c^2*d^4*e^2 + 7*a^2*b*e^6
+ 448*(b^2*c + a*c^2)*d^3*e^3 - 56*(b^3 + 6*a*b*c)*d^2*e^4 + 28*(a*b^2 + a^2*c)*d*e^5)*x)*sqrt(e*x + d)/(e^10*
x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

Sympy [A] (verification not implemented)

Time = 22.52 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{e^{6}} - \frac {3 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{6} \sqrt {d + e x}} - \frac {\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{6} \left (d + e x\right )^{\frac {3}{2}}} - \frac {\left (a e^{2} - b d e + c d^{2}\right )^{3}}{5 e^{6} \left (d + e x\right )^{\frac {5}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((c*x**2+b*x+a)**3/(e*x+d)**(7/2),x)

[Out]

Piecewise((2*(c**3*(d + e*x)**(7/2)/(7*e**6) + (d + e*x)**(5/2)*(3*b*c**2*e - 6*c**3*d)/(5*e**6) + (d + e*x)**
(3/2)*(3*a*c**2*e**2 + 3*b**2*c*e**2 - 15*b*c**2*d*e + 15*c**3*d**2)/(3*e**6) + sqrt(d + e*x)*(6*a*b*c*e**3 -
12*a*c**2*d*e**2 + b**3*e**3 - 12*b**2*c*d*e**2 + 30*b*c**2*d**2*e - 20*c**3*d**3)/e**6 - 3*(a*e**2 - b*d*e +
c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(e**6*sqrt(d + e*x)) - (b*e - 2*c*d)*(a*e**2 - b*d*e
+ c*d**2)**2/(e**6*(d + e*x)**(3/2)) - (a*e**2 - b*d*e + c*d**2)**3/(5*e**6*(d + e*x)**(5/2)))/e, Ne(e, 0)), (
(a**3*x + 3*a**2*b*x**2/2 + b*c**2*x**6/2 + c**3*x**7/7 + x**5*(3*a*c**2 + 3*b**2*c)/5 + x**4*(6*a*b*c + b**3)
/4 + x**3*(3*a**2*c + 3*a*b**2)/3)/d**(7/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} - 21 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 35 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} \sqrt {e x + d}}{e^{6}} - \frac {7 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 15 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{6}}\right )}}{35 \, e} \]

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/35*((5*(e*x + d)^(7/2)*c^3 - 21*(2*c^3*d - b*c^2*e)*(e*x + d)^(5/2) + 35*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c +
a*c^2)*e^2)*(e*x + d)^(3/2) - 35*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a*b*c)*e^
3)*sqrt(e*x + d))/e^6 - 7*(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^
3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4 + 15*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2
- (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^2 - 5*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2
*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d))/((e*x + d)^(5/2)*e^6))/e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 611 vs. \(2 (258) = 516\).

Time = 0.28 (sec) , antiderivative size = 611, normalized size of antiderivative = 2.20 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (75 \, {\left (e x + d\right )}^{2} c^{3} d^{4} - 10 \, {\left (e x + d\right )} c^{3} d^{5} + c^{3} d^{6} - 150 \, {\left (e x + d\right )}^{2} b c^{2} d^{3} e + 25 \, {\left (e x + d\right )} b c^{2} d^{4} e - 3 \, b c^{2} d^{5} e + 90 \, {\left (e x + d\right )}^{2} b^{2} c d^{2} e^{2} + 90 \, {\left (e x + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \, {\left (e x + d\right )} b^{2} c d^{3} e^{2} - 20 \, {\left (e x + d\right )} a c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - 15 \, {\left (e x + d\right )}^{2} b^{3} d e^{3} - 90 \, {\left (e x + d\right )}^{2} a b c d e^{3} + 5 \, {\left (e x + d\right )} b^{3} d^{2} e^{3} + 30 \, {\left (e x + d\right )} a b c d^{2} e^{3} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 15 \, {\left (e x + d\right )}^{2} a b^{2} e^{4} + 15 \, {\left (e x + d\right )}^{2} a^{2} c e^{4} - 10 \, {\left (e x + d\right )} a b^{2} d e^{4} - 10 \, {\left (e x + d\right )} a^{2} c d e^{4} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} + 5 \, {\left (e x + d\right )} a^{2} b e^{5} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )}}{5 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{7}} + \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} e^{42} - 42 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d e^{42} + 175 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt {e x + d} c^{3} d^{3} e^{42} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} b c^{2} e^{43} - 175 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{2} d e^{43} + 1050 \, \sqrt {e x + d} b c^{2} d^{2} e^{43} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c e^{44} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{2} e^{44} - 420 \, \sqrt {e x + d} b^{2} c d e^{44} - 420 \, \sqrt {e x + d} a c^{2} d e^{44} + 35 \, \sqrt {e x + d} b^{3} e^{45} + 210 \, \sqrt {e x + d} a b c e^{45}\right )}}{35 \, e^{49}} \]

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

-2/5*(75*(e*x + d)^2*c^3*d^4 - 10*(e*x + d)*c^3*d^5 + c^3*d^6 - 150*(e*x + d)^2*b*c^2*d^3*e + 25*(e*x + d)*b*c
^2*d^4*e - 3*b*c^2*d^5*e + 90*(e*x + d)^2*b^2*c*d^2*e^2 + 90*(e*x + d)^2*a*c^2*d^2*e^2 - 20*(e*x + d)*b^2*c*d^
3*e^2 - 20*(e*x + d)*a*c^2*d^3*e^2 + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - 15*(e*x + d)^2*b^3*d*e^3 - 90*(e*x +
d)^2*a*b*c*d*e^3 + 5*(e*x + d)*b^3*d^2*e^3 + 30*(e*x + d)*a*b*c*d^2*e^3 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 15*(
e*x + d)^2*a*b^2*e^4 + 15*(e*x + d)^2*a^2*c*e^4 - 10*(e*x + d)*a*b^2*d*e^4 - 10*(e*x + d)*a^2*c*d*e^4 + 3*a*b^
2*d^2*e^4 + 3*a^2*c*d^2*e^4 + 5*(e*x + d)*a^2*b*e^5 - 3*a^2*b*d*e^5 + a^3*e^6)/((e*x + d)^(5/2)*e^7) + 2/35*(5
*(e*x + d)^(7/2)*c^3*e^42 - 42*(e*x + d)^(5/2)*c^3*d*e^42 + 175*(e*x + d)^(3/2)*c^3*d^2*e^42 - 700*sqrt(e*x +
d)*c^3*d^3*e^42 + 21*(e*x + d)^(5/2)*b*c^2*e^43 - 175*(e*x + d)^(3/2)*b*c^2*d*e^43 + 1050*sqrt(e*x + d)*b*c^2*
d^2*e^43 + 35*(e*x + d)^(3/2)*b^2*c*e^44 + 35*(e*x + d)^(3/2)*a*c^2*e^44 - 420*sqrt(e*x + d)*b^2*c*d*e^44 - 42
0*sqrt(e*x + d)*a*c^2*d*e^44 + 35*sqrt(e*x + d)*b^3*e^45 + 210*sqrt(e*x + d)*a*b*c*e^45)/e^49

Mupad [B] (verification not implemented)

Time = 9.84 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{3\,e^7}-\frac {{\left (d+e\,x\right )}^2\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )-\left (d+e\,x\right )\,\left (-2\,a^2\,b\,e^5+4\,a^2\,c\,d\,e^4+4\,a\,b^2\,d\,e^4-12\,a\,b\,c\,d^2\,e^3+8\,a\,c^2\,d^3\,e^2-2\,b^3\,d^2\,e^3+8\,b^2\,c\,d^3\,e^2-10\,b\,c^2\,d^4\,e+4\,c^3\,d^5\right )+\frac {2\,a^3\,e^6}{5}+\frac {2\,c^3\,d^6}{5}-\frac {2\,b^3\,d^3\,e^3}{5}+\frac {6\,a\,b^2\,d^2\,e^4}{5}+\frac {6\,a\,c^2\,d^4\,e^2}{5}+\frac {6\,a^2\,c\,d^2\,e^4}{5}+\frac {6\,b^2\,c\,d^4\,e^2}{5}-\frac {6\,a^2\,b\,d\,e^5}{5}-\frac {6\,b\,c^2\,d^5\,e}{5}-\frac {12\,a\,b\,c\,d^3\,e^3}{5}}{e^7\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{e^7} \]

[In]

int((a + b*x + c*x^2)^3/(d + e*x)^(7/2),x)

[Out]

(2*c^3*(d + e*x)^(7/2))/(7*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(5/2))/(5*e^7) + ((d + e*x)^(3/2)*(30*c^3*
d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(3*e^7) - ((d + e*x)^2*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^
4 - 6*b^3*d*e^3 + 36*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e - 36*a*b*c*d*e^3) - (d + e*x)*(4*c^3*d^
5 - 2*a^2*b*e^5 - 2*b^3*d^2*e^3 + 8*a*c^2*d^3*e^2 + 8*b^2*c*d^3*e^2 + 4*a*b^2*d*e^4 + 4*a^2*c*d*e^4 - 10*b*c^2
*d^4*e - 12*a*b*c*d^2*e^3) + (2*a^3*e^6)/5 + (2*c^3*d^6)/5 - (2*b^3*d^3*e^3)/5 + (6*a*b^2*d^2*e^4)/5 + (6*a*c^
2*d^4*e^2)/5 + (6*a^2*c*d^2*e^4)/5 + (6*b^2*c*d^4*e^2)/5 - (6*a^2*b*d*e^5)/5 - (6*b*c^2*d^5*e)/5 - (12*a*b*c*d
^3*e^3)/5)/(e^7*(d + e*x)^(5/2)) + (2*(b*e - 2*c*d)*(d + e*x)^(1/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c
*d*e))/e^7