\(\int \frac {(a+b x+c x^2)^2}{(d+e x)^{7/2}} \, dx\) [2281]

Optimal result

Integrand size = 22, antiderivative size = 162 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 \sqrt {d+e x}}-\frac {4 c (2 c d-b e) \sqrt {d+e x}}{e^5}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5} \]

[Out]

-2/5*(a*e^2-b*d*e+c*d^2)^2/e^5/(e*x+d)^(5/2)+4/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)/e^5/(e*x+d)^(3/2)+2/3*c^2*(e
*x+d)^(3/2)/e^5-2*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))/e^5/(e*x+d)^(1/2)-4*c*(-b*e+2*c*d)*(e*x+d)^(1/2)/e^5

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 162, 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} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 \sqrt {d+e x}}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac {2 \left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}-\frac {4 c \sqrt {d+e x} (2 c d-b e)}{e^5}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5} \]

[In]

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

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^5*(d + e*x)^(5/2)) + (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(3*e^5*(d +
e*x)^(3/2)) - (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(e^5*Sqrt[d + e*x]) - (4*c*(2*c*d - b*e)*Sqrt[d
+ e*x])/e^5 + (2*c^2*(d + e*x)^(3/2))/(3*e^5)

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 )^2}{e^4 (d+e x)^{7/2}}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^{5/2}}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^{3/2}}-\frac {2 c (2 c d-b e)}{e^4 \sqrt {d+e x}}+\frac {c^2 \sqrt {d+e x}}{e^4}\right ) \, dx \\ & = -\frac {2 \left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 \sqrt {d+e x}}-\frac {4 c (2 c d-b e) \sqrt {d+e x}}{e^5}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (3 a^2 e^4+2 a b e^3 (2 d+5 e x)+b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+2 a c e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 b c e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}} \]

[In]

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

[Out]

(-2*(3*a^2*e^4 + 2*a*b*e^3*(2*d + 5*e*x) + b^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 2*a*c*e^2*(8*d^2 + 20*d*e
*x + 15*e^2*x^2) - 6*b*c*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + c^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)))/(15*e^5*(d + e*x)^(5/2))

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(\frac {\left (10 c^{2} x^{4}-60 x^{2} \left (-b x +a \right ) c -30 b^{2} x^{2}-20 a b x -6 a^{2}\right ) e^{4}-8 \left (10 c^{2} x^{3}+\left (-45 b \,x^{2}+10 a x \right ) c +b \left (5 b x +a \right )\right ) d \,e^{3}-32 \left (15 c^{2} x^{2}+\left (-15 b x +a \right ) c +\frac {b^{2}}{2}\right ) d^{2} e^{2}+192 c \left (-\frac {10 c x}{3}+b \right ) d^{3} e -256 c^{2} d^{4}}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\)	\(144\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 b c e \sqrt {e x +d}-8 c^{2} d \sqrt {e x +d}-\frac {2 \left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (2 a b \,e^{3}-4 a c d \,e^{2}-2 b^{2} d \,e^{2}+6 b c \,d^{2} e -4 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}}{e^{5}}\)	\(192\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 b c e \sqrt {e x +d}-8 c^{2} d \sqrt {e x +d}-\frac {2 \left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (2 a b \,e^{3}-4 a c d \,e^{2}-2 b^{2} d \,e^{2}+6 b c \,d^{2} e -4 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}}{e^{5}}\)	\(192\)
gosper	\(-\frac {2 \left (-5 c^{2} x^{4} e^{4}-30 b c \,e^{4} x^{3}+40 c^{2} d \,e^{3} x^{3}+30 a c \,e^{4} x^{2}+15 b^{2} e^{4} x^{2}-180 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}+10 a b \,e^{4} x +40 a c d \,e^{3} x +20 b^{2} d \,e^{3} x -240 b c \,d^{2} e^{2} x +320 c^{2} d^{3} e x +3 a^{2} e^{4}+4 a b d \,e^{3}+16 a c \,d^{2} e^{2}+8 b^{2} d^{2} e^{2}-96 b c \,d^{3} e +128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\)	\(194\)
trager	\(-\frac {2 \left (-5 c^{2} x^{4} e^{4}-30 b c \,e^{4} x^{3}+40 c^{2} d \,e^{3} x^{3}+30 a c \,e^{4} x^{2}+15 b^{2} e^{4} x^{2}-180 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}+10 a b \,e^{4} x +40 a c d \,e^{3} x +20 b^{2} d \,e^{3} x -240 b c \,d^{2} e^{2} x +320 c^{2} d^{3} e x +3 a^{2} e^{4}+4 a b d \,e^{3}+16 a c \,d^{2} e^{2}+8 b^{2} d^{2} e^{2}-96 b c \,d^{3} e +128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\)	\(194\)
risch	\(\frac {2 c \left (c x e +6 b e -11 c d \right ) \sqrt {e x +d}}{3 e^{5}}-\frac {2 \left (30 a c \,e^{4} x^{2}+15 b^{2} e^{4} x^{2}-90 b c d \,e^{3} x^{2}+90 c^{2} d^{2} e^{2} x^{2}+10 a b \,e^{4} x +40 a c d \,e^{3} x +20 b^{2} d \,e^{3} x -150 b c \,d^{2} e^{2} x +160 c^{2} d^{3} e x +3 a^{2} e^{4}+4 a b d \,e^{3}+16 a c \,d^{2} e^{2}+8 b^{2} d^{2} e^{2}-66 b c \,d^{3} e +73 c^{2} d^{4}\right )}{15 e^{5} \sqrt {e x +d}\, \left (x^{2} e^{2}+2 d x e +d^{2}\right )}\)	\(206\)

[In]

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

[Out]

1/15*((10*c^2*x^4-60*x^2*(-b*x+a)*c-30*b^2*x^2-20*a*b*x-6*a^2)*e^4-8*(10*c^2*x^3+(-45*b*x^2+10*a*x)*c+b*(5*b*x
+a))*d*e^3-32*(15*c^2*x^2+(-15*b*x+a)*c+1/2*b^2)*d^2*e^2+192*c*(-10/3*c*x+b)*d^3*e-256*c^2*d^4)/(e*x+d)^(5/2)/
e^5

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (5 \, c^{2} e^{4} x^{4} - 128 \, c^{2} d^{4} + 96 \, b c d^{3} e - 4 \, a b d e^{3} - 3 \, a^{2} e^{4} - 8 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 10 \, {\left (4 \, c^{2} d e^{3} - 3 \, b c e^{4}\right )} x^{3} - 15 \, {\left (16 \, c^{2} d^{2} e^{2} - 12 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 10 \, {\left (32 \, c^{2} d^{3} e - 24 \, b c d^{2} e^{2} + a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]

[In]

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

[Out]

2/15*(5*c^2*e^4*x^4 - 128*c^2*d^4 + 96*b*c*d^3*e - 4*a*b*d*e^3 - 3*a^2*e^4 - 8*(b^2 + 2*a*c)*d^2*e^2 - 10*(4*c
^2*d*e^3 - 3*b*c*e^4)*x^3 - 15*(16*c^2*d^2*e^2 - 12*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 - 10*(32*c^2*d^3*e - 24
*b*c*d^2*e^2 + a*b*e^4 + 2*(b^2 + 2*a*c)*d*e^3)*x)*sqrt(e*x + d)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^
5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1180 vs. \(2 (156) = 312\).

Time = 0.56 (sec) , antiderivative size = 1180, normalized size of antiderivative = 7.28 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\begin {cases} - \frac {6 a^{2} e^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {8 a b d e^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {20 a b e^{4} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {32 a c d^{2} e^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 a c d e^{3} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {60 a c e^{4} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {16 b^{2} d^{2} e^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {40 b^{2} d e^{3} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {30 b^{2} e^{4} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {192 b c d^{3} e}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {480 b c d^{2} e^{2} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {360 b c d e^{3} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {60 b c e^{4} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {256 c^{2} d^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {640 c^{2} d^{3} e x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {480 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 c^{2} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {10 c^{2} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {2 a c x^{3}}{3} + \frac {b^{2} x^{3}}{3} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-6*a**2*e**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x))
- 8*a*b*d*e**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 20*a*b*
e**4*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 32*a*c*d**2*e**
2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*a*c*d*e**3*x/(15*
d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 60*a*c*e**4*x**2/(15*d**2*
e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 16*b**2*d**2*e**2/(15*d**2*e**5
*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 40*b**2*d*e**3*x/(15*d**2*e**5*sqrt
(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 30*b**2*e**4*x**2/(15*d**2*e**5*sqrt(d +
e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 192*b*c*d**3*e/(15*d**2*e**5*sqrt(d + e*x) +
30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 480*b*c*d**2*e**2*x/(15*d**2*e**5*sqrt(d + e*x) + 3
0*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 360*b*c*d*e**3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*
d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 60*b*c*e**4*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**
6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 256*c**2*d**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt
(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 640*c**2*d**3*e*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d +
e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 480*c**2*d**2*e**2*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d
+ e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*c**2*d*e**3*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d +
e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 10*c**2*e**4*x**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x
) + 15*e**7*x**2*sqrt(d + e*x)), Ne(e, 0)), ((a**2*x + a*b*x**2 + 2*a*c*x**3/3 + b**2*x**3/3 + b*c*x**4/2 + c*
*2*x**5/5)/d**(7/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} - 6 \, {\left (2 \, c^{2} d - b c e\right )} \sqrt {e x + d}\right )}}{e^{4}} - \frac {3 \, c^{2} d^{4} - 6 \, b c d^{3} e - 6 \, a b d e^{3} + 3 \, a^{2} e^{4} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 15 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{2} - 10 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{4}}\right )}}{15 \, e} \]

[In]

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

[Out]

2/15*(5*((e*x + d)^(3/2)*c^2 - 6*(2*c^2*d - b*c*e)*sqrt(e*x + d))/e^4 - (3*c^2*d^4 - 6*b*c*d^3*e - 6*a*b*d*e^3
+ 3*a^2*e^4 + 3*(b^2 + 2*a*c)*d^2*e^2 + 15*(6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a*c)*e^2)*(e*x + d)^2 - 10*(2*c^
2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d))/((e*x + d)^(5/2)*e^4))/e

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (90 \, {\left (e x + d\right )}^{2} c^{2} d^{2} - 20 \, {\left (e x + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} - 90 \, {\left (e x + d\right )}^{2} b c d e + 30 \, {\left (e x + d\right )} b c d^{2} e - 6 \, b c d^{3} e + 15 \, {\left (e x + d\right )}^{2} b^{2} e^{2} + 30 \, {\left (e x + d\right )}^{2} a c e^{2} - 10 \, {\left (e x + d\right )} b^{2} d e^{2} - 20 \, {\left (e x + d\right )} a c d e^{2} + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} + 10 \, {\left (e x + d\right )} a b e^{3} - 6 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{5}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} e^{10} - 12 \, \sqrt {e x + d} c^{2} d e^{10} + 6 \, \sqrt {e x + d} b c e^{11}\right )}}{3 \, e^{15}} \]

[In]

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

[Out]

-2/15*(90*(e*x + d)^2*c^2*d^2 - 20*(e*x + d)*c^2*d^3 + 3*c^2*d^4 - 90*(e*x + d)^2*b*c*d*e + 30*(e*x + d)*b*c*d
^2*e - 6*b*c*d^3*e + 15*(e*x + d)^2*b^2*e^2 + 30*(e*x + d)^2*a*c*e^2 - 10*(e*x + d)*b^2*d*e^2 - 20*(e*x + d)*a
*c*d*e^2 + 3*b^2*d^2*e^2 + 6*a*c*d^2*e^2 + 10*(e*x + d)*a*b*e^3 - 6*a*b*d*e^3 + 3*a^2*e^4)/((e*x + d)^(5/2)*e^
5) + 2/3*((e*x + d)^(3/2)*c^2*e^10 - 12*sqrt(e*x + d)*c^2*d*e^10 + 6*sqrt(e*x + d)*b*c*e^11)/e^15

Mupad [B] (verification not implemented)

Time = 9.83 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2\,\left (3\,a^2\,e^4+4\,a\,b\,d\,e^3+10\,a\,b\,e^4\,x+16\,a\,c\,d^2\,e^2+40\,a\,c\,d\,e^3\,x+30\,a\,c\,e^4\,x^2+8\,b^2\,d^2\,e^2+20\,b^2\,d\,e^3\,x+15\,b^2\,e^4\,x^2-96\,b\,c\,d^3\,e-240\,b\,c\,d^2\,e^2\,x-180\,b\,c\,d\,e^3\,x^2-30\,b\,c\,e^4\,x^3+128\,c^2\,d^4+320\,c^2\,d^3\,e\,x+240\,c^2\,d^2\,e^2\,x^2+40\,c^2\,d\,e^3\,x^3-5\,c^2\,e^4\,x^4\right )}{15\,e^5\,{\left (d+e\,x\right )}^{5/2}} \]

[In]

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

[Out]

-(2*(3*a^2*e^4 + 128*c^2*d^4 + 8*b^2*d^2*e^2 + 15*b^2*e^4*x^2 - 5*c^2*e^4*x^4 + 40*c^2*d*e^3*x^3 + 4*a*b*d*e^3
- 96*b*c*d^3*e + 10*a*b*e^4*x + 240*c^2*d^2*e^2*x^2 + 16*a*c*d^2*e^2 + 30*a*c*e^4*x^2 - 30*b*c*e^4*x^3 + 20*b
^2*d*e^3*x + 320*c^2*d^3*e*x - 240*b*c*d^2*e^2*x - 180*b*c*d*e^3*x^2 + 40*a*c*d*e^3*x))/(15*e^5*(d + e*x)^(5/2
))