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

Optimal result

Integrand size = 22, antiderivative size = 161 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \left (b^2 c-a b d+a^2 e\right )^2 \sqrt {a+b x}}{b^5}+\frac {4 (b d-2 a e) \left (b^2 c-a b d+a^2 e\right ) (a+b x)^{3/2}}{3 b^5}-\frac {2 \left (6 a b d e-6 a^2 e^2-b^2 \left (d^2+2 c e\right )\right ) (a+b x)^{5/2}}{5 b^5}+\frac {4 e (b d-2 a e) (a+b x)^{7/2}}{7 b^5}+\frac {2 e^2 (a+b x)^{9/2}}{9 b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 161, 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} \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=-\frac {2 (a+b x)^{5/2} \left (-6 a^2 e^2+6 a b d e-\left (b^2 \left (2 c e+d^2\right )\right )\right )}{5 b^5}+\frac {4 (a+b x)^{3/2} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )}{3 b^5}+\frac {2 \sqrt {a+b x} \left (a^2 e-a b d+b^2 c\right )^2}{b^5}+\frac {4 e (a+b x)^{7/2} (b d-2 a e)}{7 b^5}+\frac {2 e^2 (a+b x)^{9/2}}{9 b^5} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \left (128 a^4 e^2-32 a^3 b e (9 d+2 e x)+24 a^2 b^2 \left (7 d^2+6 d e x+2 e \left (7 c+e x^2\right )\right )-4 a b^3 \left (21 c (5 d+2 e x)+x \left (21 d^2+27 d e x+10 e^2 x^2\right )\right )+b^4 \left (315 c^2+42 c x (5 d+3 e x)+x^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )\right )\right )}{315 b^5} \]

[In]

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

[Out]

(2*Sqrt[a + b*x]*(128*a^4*e^2 - 32*a^3*b*e*(9*d + 2*e*x) + 24*a^2*b^2*(7*d^2 + 6*d*e*x + 2*e*(7*c + e*x^2)) -
4*a*b^3*(21*c*(5*d + 2*e*x) + x*(21*d^2 + 27*d*e*x + 10*e^2*x^2)) + b^4*(315*c^2 + 42*c*x*(5*d + 3*e*x) + x^2*
(63*d^2 + 90*d*e*x + 35*e^2*x^2))))/(315*b^5)

Maple [A] (verified)

Time = 5.07 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {\frac {2 e^{2} \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {4 \left (-2 a e +b d \right ) e \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a^{2} e -a b d +b^{2} c \right ) e +\left (-2 a e +b d \right )^{2}\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a^{2} e -a b d +b^{2} c \right ) \left (-2 a e +b d \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}+2 \left (a^{2} e -a b d +b^{2} c \right )^{2} \sqrt {b x +a}}{b^{5}}\)	\(135\)
default	\(\frac {\frac {2 e^{2} \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {4 \left (-2 a e +b d \right ) e \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a^{2} e -a b d +b^{2} c \right ) e +\left (-2 a e +b d \right )^{2}\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a^{2} e -a b d +b^{2} c \right ) \left (-2 a e +b d \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}+2 \left (a^{2} e -a b d +b^{2} c \right )^{2} \sqrt {b x +a}}{b^{5}}\)	\(135\)
pseudoelliptic	\(\frac {256 \sqrt {b x +a}\, \left (\frac {7 \left (\frac {5 e^{2} x^{4}}{2}+9 \left (\frac {5 d x}{7}+c \right ) x^{2} e +\frac {9 d^{2} x^{2}}{2}+15 c d x +\frac {45 c^{2}}{2}\right ) b^{4}}{64}-\frac {105 \left (\frac {2 e^{2} x^{3}}{21}+\frac {2 \left (\frac {9 d x}{14}+c \right ) x e}{5}+d \left (\frac {d x}{5}+c \right )\right ) a \,b^{3}}{32}+\frac {21 a^{2} \left (\frac {x^{2} e^{2}}{7}+\left (\frac {3 d x}{7}+c \right ) e +\frac {d^{2}}{2}\right ) b^{2}}{8}-\frac {9 e \left (\frac {2 e x}{9}+d \right ) a^{3} b}{4}+a^{4} e^{2}\right )}{315 b^{5}}\)	\(141\)
gosper	\(\frac {2 \sqrt {b x +a}\, \left (35 e^{2} x^{4} b^{4}-40 a \,b^{3} e^{2} x^{3}+90 b^{4} d e \,x^{3}+48 a^{2} b^{2} e^{2} x^{2}-108 a \,b^{3} d e \,x^{2}+126 b^{4} c e \,x^{2}+63 b^{4} d^{2} x^{2}-64 a^{3} b \,e^{2} x +144 a^{2} b^{2} d e x -168 a \,b^{3} c e x -84 a \,b^{3} d^{2} x +210 b^{4} c d x +128 a^{4} e^{2}-288 a^{3} b d e +336 a^{2} b^{2} c e +168 a^{2} b^{2} d^{2}-420 a \,b^{3} c d +315 c^{2} b^{4}\right )}{315 b^{5}}\)	\(194\)
trager	\(\frac {2 \sqrt {b x +a}\, \left (35 e^{2} x^{4} b^{4}-40 a \,b^{3} e^{2} x^{3}+90 b^{4} d e \,x^{3}+48 a^{2} b^{2} e^{2} x^{2}-108 a \,b^{3} d e \,x^{2}+126 b^{4} c e \,x^{2}+63 b^{4} d^{2} x^{2}-64 a^{3} b \,e^{2} x +144 a^{2} b^{2} d e x -168 a \,b^{3} c e x -84 a \,b^{3} d^{2} x +210 b^{4} c d x +128 a^{4} e^{2}-288 a^{3} b d e +336 a^{2} b^{2} c e +168 a^{2} b^{2} d^{2}-420 a \,b^{3} c d +315 c^{2} b^{4}\right )}{315 b^{5}}\)	\(194\)
risch	\(\frac {2 \sqrt {b x +a}\, \left (35 e^{2} x^{4} b^{4}-40 a \,b^{3} e^{2} x^{3}+90 b^{4} d e \,x^{3}+48 a^{2} b^{2} e^{2} x^{2}-108 a \,b^{3} d e \,x^{2}+126 b^{4} c e \,x^{2}+63 b^{4} d^{2} x^{2}-64 a^{3} b \,e^{2} x +144 a^{2} b^{2} d e x -168 a \,b^{3} c e x -84 a \,b^{3} d^{2} x +210 b^{4} c d x +128 a^{4} e^{2}-288 a^{3} b d e +336 a^{2} b^{2} c e +168 a^{2} b^{2} d^{2}-420 a \,b^{3} c d +315 c^{2} b^{4}\right )}{315 b^{5}}\)	\(194\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (35 \, b^{4} e^{2} x^{4} + 315 \, b^{4} c^{2} - 420 \, a b^{3} c d + 168 \, a^{2} b^{2} d^{2} + 128 \, a^{4} e^{2} + 10 \, {\left (9 \, b^{4} d e - 4 \, a b^{3} e^{2}\right )} x^{3} + 3 \, {\left (21 \, b^{4} d^{2} + 16 \, a^{2} b^{2} e^{2} + 6 \, {\left (7 \, b^{4} c - 6 \, a b^{3} d\right )} e\right )} x^{2} + 48 \, {\left (7 \, a^{2} b^{2} c - 6 \, a^{3} b d\right )} e + 2 \, {\left (105 \, b^{4} c d - 42 \, a b^{3} d^{2} - 32 \, a^{3} b e^{2} - 12 \, {\left (7 \, a b^{3} c - 6 \, a^{2} b^{2} d\right )} e\right )} x\right )} \sqrt {b x + a}}{315 \, b^{5}} \]

[In]

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

[Out]

2/315*(35*b^4*e^2*x^4 + 315*b^4*c^2 - 420*a*b^3*c*d + 168*a^2*b^2*d^2 + 128*a^4*e^2 + 10*(9*b^4*d*e - 4*a*b^3*
e^2)*x^3 + 3*(21*b^4*d^2 + 16*a^2*b^2*e^2 + 6*(7*b^4*c - 6*a*b^3*d)*e)*x^2 + 48*(7*a^2*b^2*c - 6*a^3*b*d)*e +
2*(105*b^4*c*d - 42*a*b^3*d^2 - 32*a^3*b*e^2 - 12*(7*a*b^3*c - 6*a^2*b^2*d)*e)*x)*sqrt(b*x + a)/b^5

Sympy [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.71 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 \left (\frac {e^{2} \left (a + b x\right )^{\frac {9}{2}}}{9 b^{4}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 4 a e^{2} + 2 b d e\right )}{7 b^{4}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \cdot \left (6 a^{2} e^{2} - 6 a b d e + 2 b^{2} c e + b^{2} d^{2}\right )}{5 b^{4}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- 4 a^{3} e^{2} + 6 a^{2} b d e - 4 a b^{2} c e - 2 a b^{2} d^{2} + 2 b^{3} c d\right )}{3 b^{4}} + \frac {\sqrt {a + b x} \left (a^{4} e^{2} - 2 a^{3} b d e + 2 a^{2} b^{2} c e + a^{2} b^{2} d^{2} - 2 a b^{3} c d + b^{4} c^{2}\right )}{b^{4}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {c^{2} x + c d x^{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 c e + d^{2}\right )}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.47 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {b x + a} c^{2} + 42 \, c {\left (\frac {5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} d}{b} + \frac {{\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} e}{b^{2}}\right )} + \frac {21 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} d^{2}}{b^{2}} + \frac {18 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} d e}{b^{3}} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} e^{2}}{b^{4}}\right )}}{315 \, b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.47 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {b x + a} c^{2} + \frac {210 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} c d}{b} + \frac {21 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} d^{2}}{b^{2}} + \frac {42 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} c e}{b^{2}} + \frac {18 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} d e}{b^{3}} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} e^{2}}{b^{4}}\right )}}{315 \, b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x+e x^2\right )^2}{\sqrt {a+b x}} \, dx=\frac {2\,e^2\,{\left (a+b\,x\right )}^{9/2}}{9\,b^5}+\frac {{\left (a+b\,x\right )}^{5/2}\,\left (12\,a^2\,e^2-12\,a\,b\,d\,e+2\,b^2\,d^2+4\,c\,b^2\,e\right )}{5\,b^5}+\frac {2\,\sqrt {a+b\,x}\,{\left (e\,a^2-d\,a\,b+c\,b^2\right )}^2}{b^5}-\frac {\left (8\,a\,e^2-4\,b\,d\,e\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5}-\frac {4\,\left (2\,a\,e-b\,d\right )\,{\left (a+b\,x\right )}^{3/2}\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{3\,b^5} \]

[In]

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

[Out]

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