3.2267 \(\int \frac{\left (a+b x+c x^2\right )^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=164 \[ \frac{2 (d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac{4 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac{2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac{4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5} \]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x])/e^5 - (4*(2*c*d - b*e)*(c*d^2 - b*d*
e + a*e^2)*(d + e*x)^(3/2))/(3*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a
*e))*(d + e*x)^(5/2))/(5*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^5) + (2
*c^2*(d + e*x)^(9/2))/(9*e^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.200553, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2 (d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac{4 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac{2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac{4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x])/e^5 - (4*(2*c*d - b*e)*(c*d^2 - b*d*
e + a*e^2)*(d + e*x)^(3/2))/(3*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a
*e))*(d + e*x)^(5/2))/(5*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^5) + (2
*c^2*(d + e*x)^(9/2))/(9*e^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 40.6301, size = 160, normalized size = 0.98 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )}{7 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{5 e^{5}} + \frac{4 \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{5}} + \frac{2 \sqrt{d + e x} \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**2/(e*x+d)**(1/2),x)

[Out]

2*c**2*(d + e*x)**(9/2)/(9*e**5) + 4*c*(d + e*x)**(7/2)*(b*e - 2*c*d)/(7*e**5) +
2*(d + e*x)**(5/2)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)/(5*e**5)
+ 4*(d + e*x)**(3/2)*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)/(3*e**5) + 2*sqrt(d
+ e*x)*(a*e**2 - b*d*e + c*d**2)**2/e**5

_______________________________________________________________________________________

Mathematica [A]  time = 0.160187, size = 172, normalized size = 1.05 \[ \frac{2 \sqrt{d+e x} \left (21 e^2 \left (15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-6 c e \left (3 b \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )-7 a e \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(c^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35
*e^4*x^4) + 21*e^2*(15*a^2*e^2 + 10*a*b*e*(-2*d + e*x) + b^2*(8*d^2 - 4*d*e*x +
3*e^2*x^2)) - 6*c*e*(-7*a*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 3*b*(16*d^3 - 8*d^2*
e*x + 6*d*e^2*x^2 - 5*e^3*x^3))))/(315*e^5)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 194, normalized size = 1.2 \[{\frac{70\,{x}^{4}{c}^{2}{e}^{4}+180\,bc{e}^{4}{x}^{3}-80\,{x}^{3}{c}^{2}d{e}^{3}+252\,{x}^{2}ac{e}^{4}+126\,{b}^{2}{e}^{4}{x}^{2}-216\,bcd{e}^{3}{x}^{2}+96\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+420\,ab{e}^{4}x-336\,xacd{e}^{3}-168\,{b}^{2}d{e}^{3}x+288\,bc{d}^{2}{e}^{2}x-128\,x{c}^{2}{d}^{3}e+630\,{a}^{2}{e}^{4}-840\,abd{e}^{3}+672\,ac{d}^{2}{e}^{2}+336\,{b}^{2}{d}^{2}{e}^{2}-576\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^2/(e*x+d)^(1/2),x)

[Out]

2/315*(e*x+d)^(1/2)*(35*c^2*e^4*x^4+90*b*c*e^4*x^3-40*c^2*d*e^3*x^3+126*a*c*e^4*
x^2+63*b^2*e^4*x^2-108*b*c*d*e^3*x^2+48*c^2*d^2*e^2*x^2+210*a*b*e^4*x-168*a*c*d*
e^3*x-84*b^2*d*e^3*x+144*b*c*d^2*e^2*x-64*c^2*d^3*e*x+315*a^2*e^4-420*a*b*d*e^3+
336*a*c*d^2*e^2+168*b^2*d^2*e^2-288*b*c*d^3*e+128*c^2*d^4)/e^5

_______________________________________________________________________________________

Maxima [A]  time = 0.70578, size = 320, normalized size = 1.95 \[ \frac{2 \,{\left (315 \, \sqrt{e x + d} a^{2} + 42 \, a{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )} b}{e} + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} c}{e^{2}}\right )} + \frac{21 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} b^{2}}{e^{2}} + \frac{18 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )} b c}{e^{3}} + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )}}{315 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/315*(315*sqrt(e*x + d)*a^2 + 42*a*(5*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*b/e
+ (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c/e^2) + 21
*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2/e^2 + 18*
(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x
+ d)*d^3)*b*c/e^3 + (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)
^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2/e^4)/e

_______________________________________________________________________________________

Fricas [A]  time = 0.213025, size = 238, normalized size = 1.45 \[ \frac{2 \,{\left (35 \, c^{2} e^{4} x^{4} + 128 \, c^{2} d^{4} - 288 \, b c d^{3} e - 420 \, a b d e^{3} + 315 \, a^{2} e^{4} + 168 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 10 \,{\left (4 \, c^{2} d e^{3} - 9 \, b c e^{4}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{2} e^{2} - 36 \, b c d e^{3} + 21 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 2 \,{\left (32 \, c^{2} d^{3} e - 72 \, b c d^{2} e^{2} - 105 \, a b e^{4} + 42 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/315*(35*c^2*e^4*x^4 + 128*c^2*d^4 - 288*b*c*d^3*e - 420*a*b*d*e^3 + 315*a^2*e^
4 + 168*(b^2 + 2*a*c)*d^2*e^2 - 10*(4*c^2*d*e^3 - 9*b*c*e^4)*x^3 + 3*(16*c^2*d^2
*e^2 - 36*b*c*d*e^3 + 21*(b^2 + 2*a*c)*e^4)*x^2 - 2*(32*c^2*d^3*e - 72*b*c*d^2*e
^2 - 105*a*b*e^4 + 42*(b^2 + 2*a*c)*d*e^3)*x)*sqrt(e*x + d)/e^5

_______________________________________________________________________________________

Sympy [A]  time = 41.9127, size = 644, normalized size = 3.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**2/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*a**2*d/sqrt(d + e*x) + 2*a**2*(-d/sqrt(d + e*x) - sqrt(d + e*x))
+ 4*a*b*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 4*a*b*(d**2/sqrt(d + e*x) + 2*d
*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 4*a*c*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(
d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 4*a*c*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d
+ e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 2*b**2*d*(d**2/sqrt(d
+ e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 2*b**2*(-d**3/sqrt(d + e
*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 4*b
*c*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x
)**(5/2)/5)/e**3 + 4*b*c*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d
+ e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 + 2*c**2*d*(d*
*4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x
)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 + 2*c**2*(-d**5/sqrt(d + e*x) - 5*d**4*sqr
t(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x
)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4)/e, Ne(e, 0)), ((a**2*x + a*b*x**2 + b*c*x
**4/2 + c**2*x**5/5 + x**3*(2*a*c + b**2)/3)/sqrt(d), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.206972, size = 375, normalized size = 2.29 \[ \frac{2}{315} \,{\left (210 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a b e^{\left (-1\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} b^{2} e^{\left (-10\right )} + 42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a c e^{\left (-10\right )} + 18 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} b c e^{\left (-21\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} c^{2} e^{\left (-36\right )} + 315 \, \sqrt{x e + d} a^{2}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/sqrt(e*x + d),x, algorithm="giac")

[Out]

2/315*(210*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*e^(-1) + 21*(3*(x*e + d)^(5
/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*b^2*e^(-10) + 42*
(3*(x*e + d)^(5/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*a*
c*e^(-10) + 18*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d
)^(3/2)*d^2*e^18 - 35*sqrt(x*e + d)*d^3*e^18)*b*c*e^(-21) + (35*(x*e + d)^(9/2)*
e^32 - 180*(x*e + d)^(7/2)*d*e^32 + 378*(x*e + d)^(5/2)*d^2*e^32 - 420*(x*e + d)
^(3/2)*d^3*e^32 + 315*sqrt(x*e + d)*d^4*e^32)*c^2*e^(-36) + 315*sqrt(x*e + d)*a^
2)*e^(-1)