\(\int \frac {(a+b x+c x^2)^2}{\sqrt {d+e x}} \, dx\) [2278]

Optimal result

Integrand size = 22, antiderivative size = 164 \[ \int \frac {\left (a+b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}{e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{3 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{5/2}}{5 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{7/2}}{7 e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 e^5} \]

[Out]

-4/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(e*x+d)^(3/2)/e^5+2/5*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(e*x+d)^(5/
2)/e^5-4/7*c*(-b*e+2*c*d)*(e*x+d)^(7/2)/e^5+2/9*c^2*(e*x+d)^(9/2)/e^5+2*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(1/2)/e^
5

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 164, 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} \[ \int \frac {\left (a+b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 (d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac {4 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac {2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac {4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 e^5} \]

[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)

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 \sqrt {d+e x}}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{5/2}}{e^4}+\frac {c^2 (d+e x)^{7/2}}{e^4}\right ) \, dx \\ & = \frac {2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}{e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{3 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{5/2}}{5 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{7/2}}{7 e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+21 e^2 \left (15 a^2 e^2+10 a b e (-2 d+e x)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-6 c e \left (-7 a e \left (8 d^2-4 d e x+3 e^2 x^2\right )+3 b \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )\right )}{315 e^5} \]

[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] (verified)

Time = 0.25 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (b e -2 c d \right ) c \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{5}}\)	\(135\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (b e -2 c d \right ) c \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{5}}\)	\(135\)
pseudoelliptic	\(\frac {2 \sqrt {e x +d}\, \left (\left (\frac {c^{2} x^{4}}{9}+\frac {2 \left (\frac {5 b x}{7}+a \right ) x^{2} c}{5}+\frac {b^{2} x^{2}}{5}+\frac {2 a b x}{3}+a^{2}\right ) e^{4}-\frac {4 \left (\frac {2 c^{2} x^{3}}{21}+\frac {2 x \left (\frac {9 b x}{14}+a \right ) c}{5}+b \left (\frac {b x}{5}+a \right )\right ) d \,e^{3}}{3}+\frac {16 \left (\frac {c^{2} x^{2}}{7}+\left (\frac {3 b x}{7}+a \right ) c +\frac {b^{2}}{2}\right ) d^{2} e^{2}}{15}-\frac {32 \left (\frac {2 c x}{9}+b \right ) c \,d^{3} e}{35}+\frac {128 c^{2} d^{4}}{315}\right )}{e^{5}}\)	\(139\)
gosper	\(\frac {2 \sqrt {e x +d}\, \left (35 c^{2} x^{4} e^{4}+90 b c \,e^{4} x^{3}-40 c^{2} d \,e^{3} x^{3}+126 a c \,e^{4} x^{2}+63 b^{2} e^{4} x^{2}-108 b c d \,e^{3} x^{2}+48 c^{2} d^{2} e^{2} x^{2}+210 a b \,e^{4} x -168 a c d \,e^{3} x -84 b^{2} d \,e^{3} x +144 b c \,d^{2} e^{2} x -64 c^{2} d^{3} e x +315 a^{2} e^{4}-420 a b d \,e^{3}+336 a c \,d^{2} e^{2}+168 b^{2} d^{2} e^{2}-288 b c \,d^{3} e +128 c^{2} d^{4}\right )}{315 e^{5}}\)	\(194\)
trager	\(\frac {2 \sqrt {e x +d}\, \left (35 c^{2} x^{4} e^{4}+90 b c \,e^{4} x^{3}-40 c^{2} d \,e^{3} x^{3}+126 a c \,e^{4} x^{2}+63 b^{2} e^{4} x^{2}-108 b c d \,e^{3} x^{2}+48 c^{2} d^{2} e^{2} x^{2}+210 a b \,e^{4} x -168 a c d \,e^{3} x -84 b^{2} d \,e^{3} x +144 b c \,d^{2} e^{2} x -64 c^{2} d^{3} e x +315 a^{2} e^{4}-420 a b d \,e^{3}+336 a c \,d^{2} e^{2}+168 b^{2} d^{2} e^{2}-288 b c \,d^{3} e +128 c^{2} d^{4}\right )}{315 e^{5}}\)	\(194\)
risch	\(\frac {2 \sqrt {e x +d}\, \left (35 c^{2} x^{4} e^{4}+90 b c \,e^{4} x^{3}-40 c^{2} d \,e^{3} x^{3}+126 a c \,e^{4} x^{2}+63 b^{2} e^{4} x^{2}-108 b c d \,e^{3} x^{2}+48 c^{2} d^{2} e^{2} x^{2}+210 a b \,e^{4} x -168 a c d \,e^{3} x -84 b^{2} d \,e^{3} x +144 b c \,d^{2} e^{2} x -64 c^{2} d^{3} e x +315 a^{2} e^{4}-420 a b d \,e^{3}+336 a c \,d^{2} e^{2}+168 b^{2} d^{2} e^{2}-288 b c \,d^{3} e +128 c^{2} d^{4}\right )}{315 e^{5}}\)	\(194\)

[In]

int((c*x^2+b*x+a)^2/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/e^5*(1/9*c^2*(e*x+d)^(9/2)+2/7*(b*e-2*c*d)*c*(e*x+d)^(7/2)+1/5*(2*(a*e^2-b*d*e+c*d^2)*c+(b*e-2*c*d)^2)*(e*x+
d)^(5/2)+2/3*(a*e^2-b*d*e+c*d^2)*(b*e-2*c*d)*(e*x+d)^(3/2)+(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (35 \, c^{2} e^{4} x^{4} + 128 \, c^{2} d^{4} - 288 \, b c d^{3} e - 420 \, a b d e^{3} + 315 \, a^{2} e^{4} + 168 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 10 \, {\left (4 \, c^{2} d e^{3} - 9 \, b c e^{4}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{2} e^{2} - 36 \, b c d e^{3} + 21 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 2 \, {\left (32 \, c^{2} d^{3} e - 72 \, b c d^{2} e^{2} - 105 \, a b e^{4} + 42 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{5}} \]

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^(1/2),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] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 a b e^{3} - 4 a c d e^{2} - 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 a c + b^{2}\right )}{3}}{\sqrt {d}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {e x + d} a^{2} + 42 \, a {\left (\frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} b}{e} + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c}{e^{2}}\right )} + \frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2}}{e^{2}} + \frac {18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b c}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )}}{315 \, e} \]

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^(1/2),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 + 3
15*sqrt(e*x + d)*d^4)*c^2/e^4)/e

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {e x + d} a^{2} + \frac {210 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b}{e} + \frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2}}{e^{2}} + \frac {42 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a c}{e^{2}} + \frac {18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b c}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )}}{315 \, e} \]

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/315*(315*sqrt(e*x + d)*a^2 + 210*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b/e + 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 + 42*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x +
d)*d^2)*a*c/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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{5\,e^5}+\frac {2\,\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,e^5} \]

[In]

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

[Out]

(2*c^2*(d + e*x)^(9/2))/(9*e^5) + ((d + e*x)^(5/2)*(2*b^2*e^2 + 12*c^2*d^2 + 4*a*c*e^2 - 12*b*c*d*e))/(5*e^5)
+ (2*(d + e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e)^2)/e^5 - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(7/2))/(7*e^5) + (4*(b*e
- 2*c*d)*(d + e*x)^(3/2)*(a*e^2 + c*d^2 - b*d*e))/(3*e^5)