Integrand size = 20, antiderivative size = 105 \[ \int \left (a+b \sqrt {x}+c x\right )^2 (d+e x) \, dx=a^2 d x+\frac {4}{3} a b d x^{3/2}+\frac {1}{2} \left (b^2 d+a (2 c d+a e)\right ) x^2+\frac {4}{5} b (c d+a e) x^{5/2}+\frac {1}{3} \left (c^2 d+b^2 e+2 a c e\right ) x^3+\frac {4}{7} b c e x^{7/2}+\frac {1}{4} c^2 e x^4 \] Output:
a^2*d*x+4/3*a*b*d*x^(3/2)+1/2*(b^2*d+a*(a*e+2*c*d))*x^2+4/5*b*(a*e+c*d)*x^ (5/2)+1/3*(2*a*c*e+b^2*e+c^2*d)*x^3+4/7*b*c*e*x^(7/2)+1/4*c^2*e*x^4
Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92 \[ \int \left (a+b \sqrt {x}+c x\right )^2 (d+e x) \, dx=\frac {1}{420} x \left (210 a^2 (2 d+e x)+28 a \left (5 c x (3 d+2 e x)+4 b \sqrt {x} (5 d+3 e x)\right )+x \left (70 b^2 (3 d+2 e x)+35 c^2 x (4 d+3 e x)+48 b c \sqrt {x} (7 d+5 e x)\right )\right ) \] Input:
Integrate[(a + b*Sqrt[x] + c*x)^2*(d + e*x),x]
Output:
(x*(210*a^2*(2*d + e*x) + 28*a*(5*c*x*(3*d + 2*e*x) + 4*b*Sqrt[x]*(5*d + 3 *e*x)) + x*(70*b^2*(3*d + 2*e*x) + 35*c^2*x*(4*d + 3*e*x) + 48*b*c*Sqrt[x] *(7*d + 5*e*x))))/420
Time = 0.49 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2308, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (d+e x) \left (a+b \sqrt {x}+c x\right )^2 \, dx\) |
\(\Big \downarrow \) 2308 |
\(\displaystyle \int \left (d \left (a+b \sqrt {x}+c x\right )^2+e x \left (a+b \sqrt {x}+c x\right )^2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{2} d x^2 \left (2 a c+b^2\right )+\frac {1}{3} e x^3 \left (2 a c+b^2\right )+\frac {4}{3} a b d x^{3/2}+\frac {4}{5} a b e x^{5/2}+\frac {4}{5} b c d x^{5/2}+\frac {4}{7} b c e x^{7/2}+\frac {1}{3} c^2 d x^3+\frac {1}{4} c^2 e x^4\) |
Input:
Int[(a + b*Sqrt[x] + c*x)^2*(d + e*x),x]
Output:
a^2*d*x + (4*a*b*d*x^(3/2))/3 + ((b^2 + 2*a*c)*d*x^2)/2 + (a^2*e*x^2)/2 + (4*b*c*d*x^(5/2))/5 + (4*a*b*e*x^(5/2))/5 + (c^2*d*x^3)/3 + ((b^2 + 2*a*c) *e*x^3)/3 + (4*b*c*e*x^(7/2))/7 + (c^2*e*x^4)/4
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c , n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && IGtQ[p, 0]
Time = 0.44 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {c^{2} e \,x^{4}}{4}+\frac {4 b c e \,x^{\frac {7}{2}}}{7}+\frac {\left (\left (2 a c +b^{2}\right ) e +c^{2} d \right ) x^{3}}{3}+\frac {2 \left (2 a b e +2 c b d \right ) x^{\frac {5}{2}}}{5}+\frac {\left (a^{2} e +\left (2 a c +b^{2}\right ) d \right ) x^{2}}{2}+\frac {4 a b d \,x^{\frac {3}{2}}}{3}+a^{2} d x\) | \(91\) |
default | \(b^{2} \left (\frac {1}{3} e \,x^{3}+\frac {1}{2} d \,x^{2}\right )+2 b \left (\frac {2 c e \,x^{\frac {7}{2}}}{7}+\frac {2 \left (a e +c d \right ) x^{\frac {5}{2}}}{5}+\frac {2 a d \,x^{\frac {3}{2}}}{3}\right )+\frac {c^{2} e \,x^{4}}{4}+\frac {\left (2 a c e +c^{2} d \right ) x^{3}}{3}+\frac {\left (a^{2} e +2 a c d \right ) x^{2}}{2}+a^{2} d x\) | \(96\) |
trager | \(\frac {\left (3 c^{2} e \,x^{3}+8 a c e \,x^{2}+4 b^{2} e \,x^{2}+4 c^{2} d \,x^{2}+3 c^{2} e \,x^{2}+6 a^{2} e x +12 a c d x +8 a c e x +6 b^{2} d x +4 b^{2} e x +4 c^{2} x d +3 c^{2} e x +12 a^{2} d +6 a^{2} e +12 a c d +8 a c e +6 d \,b^{2}+4 b^{2} e +4 c^{2} d +3 c^{2} e \right ) \left (x -1\right )}{12}+\frac {4 b \,x^{\frac {3}{2}} \left (15 c e \,x^{2}+21 a e x +21 x c d +35 a d \right )}{105}\) | \(173\) |
orering | \(-\frac {\left (195 c^{4} e^{2} x^{6}+253 a \,c^{3} e^{2} x^{5}-220 b^{2} c^{2} e^{2} x^{5}+473 c^{4} d e \,x^{5}-306 a^{2} c^{2} e^{2} x^{4}+252 a \,b^{2} c \,e^{2} x^{4}+711 a \,c^{3} d e \,x^{4}-558 b^{2} c^{2} d e \,x^{4}+252 c^{4} d^{2} x^{4}-294 a^{3} c \,e^{2} x^{3}-1022 a^{2} c^{2} d e \,x^{3}+686 a \,b^{2} c d e \,x^{3}+392 a \,c^{3} d^{2} x^{3}-294 b^{2} c^{2} d^{2} x^{3}-210 a^{3} c d e \,x^{2}-560 a^{2} c^{2} d^{2} x^{2}+350 a \,b^{2} c \,d^{2} x^{2}+420 a^{4} d e x +140 a^{4} d^{2}\right ) \left (a +b \sqrt {x}+c x \right )^{2}}{420 \left (-c^{3} x^{3}-a \,c^{2} x^{2}+b^{2} c \,x^{2}+a^{2} c x -x a \,b^{2}+a^{3}\right ) c \left (e x +d \right )}+\frac {\left (15 c^{4} e \,x^{5}+23 a \,c^{3} e \,x^{4}-20 b^{2} c^{2} e \,x^{4}+28 c^{4} d \,x^{4}-34 a^{2} c^{2} e \,x^{3}+28 a \,b^{2} c e \,x^{3}+56 a \,c^{3} d \,x^{3}-42 b^{2} c^{2} d \,x^{3}-42 a^{3} c e \,x^{2}-112 a^{2} c^{2} d \,x^{2}+70 a \,b^{2} c d \,x^{2}+140 a^{4} d \right ) x \left (2 \left (a +b \sqrt {x}+c x \right ) \left (e x +d \right ) \left (\frac {b}{2 \sqrt {x}}+c \right )+\left (a +b \sqrt {x}+c x \right )^{2} e \right )}{210 c \left (c^{2} x^{2}+2 x a c -b^{2} x +a^{2}\right ) \left (-c x +a \right ) \left (e x +d \right )}\) | \(507\) |
Input:
int((a+b*x^(1/2)+c*x)^2*(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/4*c^2*e*x^4+4/7*b*c*e*x^(7/2)+1/3*((2*a*c+b^2)*e+c^2*d)*x^3+2/5*(2*a*b*e +2*b*c*d)*x^(5/2)+1/2*(a^2*e+(2*a*c+b^2)*d)*x^2+4/3*a*b*d*x^(3/2)+a^2*d*x
Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \left (a+b \sqrt {x}+c x\right )^2 (d+e x) \, dx=\frac {1}{4} \, c^{2} e x^{4} + a^{2} d x + \frac {1}{3} \, {\left (c^{2} d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (a^{2} e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{2} + \frac {4}{105} \, {\left (15 \, b c e x^{3} + 35 \, a b d x + 21 \, {\left (b c d + a b e\right )} x^{2}\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2)+c*x)^2*(e*x+d),x, algorithm="fricas")
Output:
1/4*c^2*e*x^4 + a^2*d*x + 1/3*(c^2*d + (b^2 + 2*a*c)*e)*x^3 + 1/2*(a^2*e + (b^2 + 2*a*c)*d)*x^2 + 4/105*(15*b*c*e*x^3 + 35*a*b*d*x + 21*(b*c*d + a*b *e)*x^2)*sqrt(x)
Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.25 \[ \int \left (a+b \sqrt {x}+c x\right )^2 (d+e x) \, dx=a^{2} d x + \frac {a^{2} e x^{2}}{2} + \frac {4 a b d x^{\frac {3}{2}}}{3} + \frac {4 a b e x^{\frac {5}{2}}}{5} + a c d x^{2} + \frac {2 a c e x^{3}}{3} + \frac {b^{2} d x^{2}}{2} + \frac {b^{2} e x^{3}}{3} + \frac {4 b c d x^{\frac {5}{2}}}{5} + \frac {4 b c e x^{\frac {7}{2}}}{7} + \frac {c^{2} d x^{3}}{3} + \frac {c^{2} e x^{4}}{4} \] Input:
integrate((a+b*x**(1/2)+c*x)**2*(e*x+d),x)
Output:
a**2*d*x + a**2*e*x**2/2 + 4*a*b*d*x**(3/2)/3 + 4*a*b*e*x**(5/2)/5 + a*c*d *x**2 + 2*a*c*e*x**3/3 + b**2*d*x**2/2 + b**2*e*x**3/3 + 4*b*c*d*x**(5/2)/ 5 + 4*b*c*e*x**(7/2)/7 + c**2*d*x**3/3 + c**2*e*x**4/4
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84 \[ \int \left (a+b \sqrt {x}+c x\right )^2 (d+e x) \, dx=\frac {1}{4} \, c^{2} e x^{4} + \frac {4}{7} \, b c e x^{\frac {7}{2}} + \frac {4}{3} \, a b d x^{\frac {3}{2}} + a^{2} d x + \frac {1}{3} \, {\left (c^{2} d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{3} + \frac {4}{5} \, {\left (b c d + a b e\right )} x^{\frac {5}{2}} + \frac {1}{2} \, {\left (a^{2} e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{2} \] Input:
integrate((a+b*x^(1/2)+c*x)^2*(e*x+d),x, algorithm="maxima")
Output:
1/4*c^2*e*x^4 + 4/7*b*c*e*x^(7/2) + 4/3*a*b*d*x^(3/2) + a^2*d*x + 1/3*(c^2 *d + (b^2 + 2*a*c)*e)*x^3 + 4/5*(b*c*d + a*b*e)*x^(5/2) + 1/2*(a^2*e + (b^ 2 + 2*a*c)*d)*x^2
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.94 \[ \int \left (a+b \sqrt {x}+c x\right )^2 (d+e x) \, dx=\frac {1}{4} \, c^{2} e x^{4} + \frac {4}{7} \, b c e x^{\frac {7}{2}} + \frac {1}{3} \, c^{2} d x^{3} + \frac {1}{3} \, b^{2} e x^{3} + \frac {2}{3} \, a c e x^{3} + \frac {4}{5} \, b c d x^{\frac {5}{2}} + \frac {4}{5} \, a b e x^{\frac {5}{2}} + \frac {1}{2} \, b^{2} d x^{2} + a c d x^{2} + \frac {1}{2} \, a^{2} e x^{2} + \frac {4}{3} \, a b d x^{\frac {3}{2}} + a^{2} d x \] Input:
integrate((a+b*x^(1/2)+c*x)^2*(e*x+d),x, algorithm="giac")
Output:
1/4*c^2*e*x^4 + 4/7*b*c*e*x^(7/2) + 1/3*c^2*d*x^3 + 1/3*b^2*e*x^3 + 2/3*a* c*e*x^3 + 4/5*b*c*d*x^(5/2) + 4/5*a*b*e*x^(5/2) + 1/2*b^2*d*x^2 + a*c*d*x^ 2 + 1/2*a^2*e*x^2 + 4/3*a*b*d*x^(3/2) + a^2*d*x
Time = 21.67 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84 \[ \int \left (a+b \sqrt {x}+c x\right )^2 (d+e x) \, dx=x^2\,\left (\frac {e\,a^2}{2}+c\,d\,a+\frac {d\,b^2}{2}\right )+x^3\,\left (\frac {e\,b^2}{3}+\frac {d\,c^2}{3}+\frac {2\,a\,e\,c}{3}\right )+\frac {4\,b\,x^{5/2}\,\left (a\,e+c\,d\right )}{5}+\frac {c^2\,e\,x^4}{4}+a^2\,d\,x+\frac {4\,a\,b\,d\,x^{3/2}}{3}+\frac {4\,b\,c\,e\,x^{7/2}}{7} \] Input:
int((d + e*x)*(a + c*x + b*x^(1/2))^2,x)
Output:
x^2*((a^2*e)/2 + (b^2*d)/2 + a*c*d) + x^3*((b^2*e)/3 + (c^2*d)/3 + (2*a*c* e)/3) + (4*b*x^(5/2)*(a*e + c*d))/5 + (c^2*e*x^4)/4 + a^2*d*x + (4*a*b*d*x ^(3/2))/3 + (4*b*c*e*x^(7/2))/7
Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int \left (a+b \sqrt {x}+c x\right )^2 (d+e x) \, dx=\frac {x \left (560 \sqrt {x}\, a b d +336 \sqrt {x}\, a b e x +336 \sqrt {x}\, b c d x +240 \sqrt {x}\, b c e \,x^{2}+420 a^{2} d +210 a^{2} e x +420 a c d x +280 a c e \,x^{2}+210 b^{2} d x +140 b^{2} e \,x^{2}+140 c^{2} d \,x^{2}+105 c^{2} e \,x^{3}\right )}{420} \] Input:
int((a+b*x^(1/2)+c*x)^2*(e*x+d),x)
Output:
(x*(560*sqrt(x)*a*b*d + 336*sqrt(x)*a*b*e*x + 336*sqrt(x)*b*c*d*x + 240*sq rt(x)*b*c*e*x**2 + 420*a**2*d + 210*a**2*e*x + 420*a*c*d*x + 280*a*c*e*x** 2 + 210*b**2*d*x + 140*b**2*e*x**2 + 140*c**2*d*x**2 + 105*c**2*e*x**3))/4 20