Integrand size = 27, antiderivative size = 140 \[ \int \left (a+b \sqrt {x}+c x\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=a^2 d x+\frac {2}{3} a (2 b d+a e) x^{3/2}+\frac {1}{2} \left (b^2 d+2 a b e+a (2 c d+a f)\right ) x^2+\frac {2}{5} \left (b^2 e+2 a c e+2 b (c d+a f)\right ) x^{5/2}+\frac {1}{3} \left (c^2 d+b^2 f+2 c (b e+a f)\right ) x^3+\frac {2}{7} c (c e+2 b f) x^{7/2}+\frac {1}{4} c^2 f x^4 \] Output:
a^2*d*x+2/3*a*(a*e+2*b*d)*x^(3/2)+1/2*(b^2*d+2*a*b*e+a*(a*f+2*c*d))*x^2+2/ 5*(b^2*e+2*a*c*e+2*b*(a*f+c*d))*x^(5/2)+1/3*(c^2*d+b^2*f+2*c*(a*f+b*e))*x^ 3+2/7*c*(2*b*f+c*e)*x^(7/2)+1/4*c^2*f*x^4
Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.13 \[ \int \left (a+b \sqrt {x}+c x\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{6} a^2 x \left (6 d+4 e \sqrt {x}+3 f x\right )+\frac {1}{420} x^2 \left (14 b^2 \left (15 d+12 e \sqrt {x}+10 f x\right )+5 c^2 x \left (28 d+24 e \sqrt {x}+21 f x\right )+8 b c \sqrt {x} \left (42 d+35 e \sqrt {x}+30 f x\right )\right )+a \left (c d x^2+\frac {4}{5} c e x^{5/2}+\frac {2}{3} c f x^3+b \left (\frac {4}{3} d x^{3/2}+e x^2+\frac {4}{5} f x^{5/2}\right )\right ) \] Input:
Integrate[(a + b*Sqrt[x] + c*x)^2*(d + e*Sqrt[x] + f*x),x]
Output:
(a^2*x*(6*d + 4*e*Sqrt[x] + 3*f*x))/6 + (x^2*(14*b^2*(15*d + 12*e*Sqrt[x] + 10*f*x) + 5*c^2*x*(28*d + 24*e*Sqrt[x] + 21*f*x) + 8*b*c*Sqrt[x]*(42*d + 35*e*Sqrt[x] + 30*f*x)))/420 + a*(c*d*x^2 + (4*c*e*x^(5/2))/5 + (2*c*f*x^ 3)/3 + b*((4*d*x^(3/2))/3 + e*x^2 + (4*f*x^(5/2))/5))
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.
\(\displaystyle \int \left (a+b \sqrt {x}+c x\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx\) |
\(\Big \downarrow \) 2329 |
\(\displaystyle \int \left (a+b \sqrt {x}+c x\right )^2 \left (d+e \sqrt {x}+f x\right )dx\) |
Input:
Int[(a + b*Sqrt[x] + c*x)^2*(d + e*Sqrt[x] + f*x),x]
Output:
$Aborted
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Unintegrable[Pq*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])
Time = 2.35 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {c^{2} f \,x^{4}}{4}+\frac {2 \left (2 b c f +c^{2} e \right ) x^{\frac {7}{2}}}{7}+\frac {\left (\left (2 a c +b^{2}\right ) f +2 b c e +c^{2} d \right ) x^{3}}{3}+\frac {2 \left (2 a b f +\left (2 a c +b^{2}\right ) e +2 c b d \right ) x^{\frac {5}{2}}}{5}+\frac {\left (a^{2} f +2 a b e +\left (2 a c +b^{2}\right ) d \right ) x^{2}}{2}+\frac {2 \left (a^{2} e +2 a b d \right ) x^{\frac {3}{2}}}{3}+a^{2} d x\) | \(127\) |
default | \(\frac {2 x^{\frac {5}{2}} b^{2} e}{5}+\frac {\left (b^{2} f +2 b c e \right ) x^{3}}{3}+\frac {\left (2 a b e +d \,b^{2}\right ) x^{2}}{2}+\frac {2 c \left (2 b f +c e \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (a \left (2 b f +c e \right )+c \left (a e +2 b d \right )\right ) x^{\frac {5}{2}}}{5}+\frac {2 a \left (a e +2 b d \right ) x^{\frac {3}{2}}}{3}+\frac {c^{2} f \,x^{4}}{4}+\frac {\left (2 a c f +c^{2} d \right ) x^{3}}{3}+\frac {\left (a^{2} f +2 a c d \right ) x^{2}}{2}+a^{2} d x\) | \(144\) |
trager | \(\frac {\left (3 c^{2} f \,x^{3}+8 a c f \,x^{2}+4 b^{2} f \,x^{2}+8 x^{2} b c e +4 c^{2} d \,x^{2}+3 c^{2} f \,x^{2}+6 a^{2} f x +12 a b e x +12 a c d x +8 a c f x +6 b^{2} d x +4 b^{2} f x +8 b c e x +4 c^{2} x d +3 c^{2} f x +12 a^{2} d +6 a^{2} f +12 a b e +12 a c d +8 a c f +6 d \,b^{2}+4 b^{2} f +8 b c e +4 c^{2} d +3 c^{2} f \right ) \left (x -1\right )}{12}+\frac {2 x^{\frac {3}{2}} \left (30 b c f \,x^{2}+15 c^{2} e \,x^{2}+42 b a f x +42 a c e x +21 b^{2} e x +42 b c d x +35 a^{2} e +70 a b d \right )}{105}\) | \(234\) |
orering | \(\text {Expression too large to display}\) | \(2897\) |
Input:
int((a+b*x^(1/2)+c*x)^2*(d+e*x^(1/2)+f*x),x,method=_RETURNVERBOSE)
Output:
1/4*c^2*f*x^4+2/7*(2*b*c*f+c^2*e)*x^(7/2)+1/3*((2*a*c+b^2)*f+2*b*c*e+c^2*d )*x^3+2/5*(2*a*b*f+(2*a*c+b^2)*e+2*c*b*d)*x^(5/2)+1/2*(a^2*f+2*a*b*e+(2*a* c+b^2)*d)*x^2+2/3*(a^2*e+2*a*b*d)*x^(3/2)+a^2*d*x
Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.93 \[ \int \left (a+b \sqrt {x}+c x\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{4} \, c^{2} f x^{4} + a^{2} d x + \frac {1}{3} \, {\left (c^{2} d + 2 \, b c e + {\left (b^{2} + 2 \, a c\right )} f\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b e + a^{2} f + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{2} + \frac {2}{105} \, {\left (15 \, {\left (c^{2} e + 2 \, b c f\right )} x^{3} + 21 \, {\left (2 \, b c d + 2 \, a b f + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{2} + 35 \, {\left (2 \, a b d + a^{2} e\right )} x\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2)+c*x)^2*(d+e*x^(1/2)+f*x),x, algorithm="fricas")
Output:
1/4*c^2*f*x^4 + a^2*d*x + 1/3*(c^2*d + 2*b*c*e + (b^2 + 2*a*c)*f)*x^3 + 1/ 2*(2*a*b*e + a^2*f + (b^2 + 2*a*c)*d)*x^2 + 2/105*(15*(c^2*e + 2*b*c*f)*x^ 3 + 21*(2*b*c*d + 2*a*b*f + (b^2 + 2*a*c)*e)*x^2 + 35*(2*a*b*d + a^2*e)*x) *sqrt(x)
Time = 0.16 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.47 \[ \int \left (a+b \sqrt {x}+c x\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=a^{2} d x + \frac {2 a^{2} e x^{\frac {3}{2}}}{3} + \frac {a^{2} f x^{2}}{2} + \frac {4 a b d x^{\frac {3}{2}}}{3} + a b e x^{2} + \frac {4 a b f x^{\frac {5}{2}}}{5} + a c d x^{2} + \frac {4 a c e x^{\frac {5}{2}}}{5} + \frac {2 a c f x^{3}}{3} + \frac {b^{2} d x^{2}}{2} + \frac {2 b^{2} e x^{\frac {5}{2}}}{5} + \frac {b^{2} f x^{3}}{3} + \frac {4 b c d x^{\frac {5}{2}}}{5} + \frac {2 b c e x^{3}}{3} + \frac {4 b c f x^{\frac {7}{2}}}{7} + \frac {c^{2} d x^{3}}{3} + \frac {2 c^{2} e x^{\frac {7}{2}}}{7} + \frac {c^{2} f x^{4}}{4} \] Input:
integrate((a+b*x**(1/2)+c*x)**2*(d+e*x**(1/2)+f*x),x)
Output:
a**2*d*x + 2*a**2*e*x**(3/2)/3 + a**2*f*x**2/2 + 4*a*b*d*x**(3/2)/3 + a*b* e*x**2 + 4*a*b*f*x**(5/2)/5 + a*c*d*x**2 + 4*a*c*e*x**(5/2)/5 + 2*a*c*f*x* *3/3 + b**2*d*x**2/2 + 2*b**2*e*x**(5/2)/5 + b**2*f*x**3/3 + 4*b*c*d*x**(5 /2)/5 + 2*b*c*e*x**3/3 + 4*b*c*f*x**(7/2)/7 + c**2*d*x**3/3 + 2*c**2*e*x** (7/2)/7 + c**2*f*x**4/4
Time = 0.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.90 \[ \int \left (a+b \sqrt {x}+c x\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{4} \, c^{2} f x^{4} + \frac {2}{7} \, {\left (c^{2} e + 2 \, b c f\right )} x^{\frac {7}{2}} + a^{2} d x + \frac {1}{3} \, {\left (c^{2} d + 2 \, b c e + {\left (b^{2} + 2 \, a c\right )} f\right )} x^{3} + \frac {2}{5} \, {\left (2 \, b c d + 2 \, a b f + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{\frac {5}{2}} + \frac {1}{2} \, {\left (2 \, a b e + a^{2} f + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{2} + \frac {2}{3} \, {\left (2 \, a b d + a^{2} e\right )} x^{\frac {3}{2}} \] Input:
integrate((a+b*x^(1/2)+c*x)^2*(d+e*x^(1/2)+f*x),x, algorithm="maxima")
Output:
1/4*c^2*f*x^4 + 2/7*(c^2*e + 2*b*c*f)*x^(7/2) + a^2*d*x + 1/3*(c^2*d + 2*b *c*e + (b^2 + 2*a*c)*f)*x^3 + 2/5*(2*b*c*d + 2*a*b*f + (b^2 + 2*a*c)*e)*x^ (5/2) + 1/2*(2*a*b*e + a^2*f + (b^2 + 2*a*c)*d)*x^2 + 2/3*(2*a*b*d + a^2*e )*x^(3/2)
Time = 0.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.06 \[ \int \left (a+b \sqrt {x}+c x\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{4} \, c^{2} f x^{4} + \frac {2}{7} \, c^{2} e x^{\frac {7}{2}} + \frac {4}{7} \, b c f x^{\frac {7}{2}} + \frac {1}{3} \, c^{2} d x^{3} + \frac {2}{3} \, b c e x^{3} + \frac {1}{3} \, b^{2} f x^{3} + \frac {2}{3} \, a c f x^{3} + \frac {4}{5} \, b c d x^{\frac {5}{2}} + \frac {2}{5} \, b^{2} e x^{\frac {5}{2}} + \frac {4}{5} \, a c e x^{\frac {5}{2}} + \frac {4}{5} \, a b f x^{\frac {5}{2}} + \frac {1}{2} \, b^{2} d x^{2} + a c d x^{2} + a b e x^{2} + \frac {1}{2} \, a^{2} f x^{2} + \frac {4}{3} \, a b d x^{\frac {3}{2}} + \frac {2}{3} \, a^{2} e x^{\frac {3}{2}} + a^{2} d x \] Input:
integrate((a+b*x^(1/2)+c*x)^2*(d+e*x^(1/2)+f*x),x, algorithm="giac")
Output:
1/4*c^2*f*x^4 + 2/7*c^2*e*x^(7/2) + 4/7*b*c*f*x^(7/2) + 1/3*c^2*d*x^3 + 2/ 3*b*c*e*x^3 + 1/3*b^2*f*x^3 + 2/3*a*c*f*x^3 + 4/5*b*c*d*x^(5/2) + 2/5*b^2* e*x^(5/2) + 4/5*a*c*e*x^(5/2) + 4/5*a*b*f*x^(5/2) + 1/2*b^2*d*x^2 + a*c*d* x^2 + a*b*e*x^2 + 1/2*a^2*f*x^2 + 4/3*a*b*d*x^(3/2) + 2/3*a^2*e*x^(3/2) + a^2*d*x
Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.90 \[ \int \left (a+b \sqrt {x}+c x\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=x^{3/2}\,\left (\frac {2\,e\,a^2}{3}+\frac {4\,b\,d\,a}{3}\right )+x^{7/2}\,\left (\frac {2\,e\,c^2}{7}+\frac {4\,b\,f\,c}{7}\right )+x^2\,\left (\frac {f\,a^2}{2}+e\,a\,b+c\,d\,a+\frac {d\,b^2}{2}\right )+x^{5/2}\,\left (\frac {2\,b^2\,e}{5}+\frac {4\,a\,b\,f}{5}+\frac {4\,a\,c\,e}{5}+\frac {4\,b\,c\,d}{5}\right )+x^3\,\left (\frac {f\,b^2}{3}+\frac {2\,e\,b\,c}{3}+\frac {d\,c^2}{3}+\frac {2\,a\,f\,c}{3}\right )+\frac {c^2\,f\,x^4}{4}+a^2\,d\,x \] Input:
int((a + c*x + b*x^(1/2))^2*(d + f*x + e*x^(1/2)),x)
Output:
x^(3/2)*((2*a^2*e)/3 + (4*a*b*d)/3) + x^(7/2)*((2*c^2*e)/7 + (4*b*c*f)/7) + x^2*((b^2*d)/2 + (a^2*f)/2 + a*b*e + a*c*d) + x^(5/2)*((2*b^2*e)/5 + (4* a*b*f)/5 + (4*a*c*e)/5 + (4*b*c*d)/5) + x^3*((c^2*d)/3 + (b^2*f)/3 + (2*a* c*f)/3 + (2*b*c*e)/3) + (c^2*f*x^4)/4 + a^2*d*x
Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06 \[ \int \left (a+b \sqrt {x}+c x\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {x \left (280 \sqrt {x}\, a^{2} e +560 \sqrt {x}\, a b d +336 \sqrt {x}\, a b f x +336 \sqrt {x}\, a c e x +168 \sqrt {x}\, b^{2} e x +336 \sqrt {x}\, b c d x +240 \sqrt {x}\, b c f \,x^{2}+120 \sqrt {x}\, c^{2} e \,x^{2}+420 a^{2} d +210 a^{2} f x +420 a b e x +420 a c d x +280 a c f \,x^{2}+210 b^{2} d x +140 b^{2} f \,x^{2}+280 b c e \,x^{2}+140 c^{2} d \,x^{2}+105 c^{2} f \,x^{3}\right )}{420} \] Input:
int((a+b*x^(1/2)+c*x)^2*(d+e*x^(1/2)+f*x),x)
Output:
(x*(280*sqrt(x)*a**2*e + 560*sqrt(x)*a*b*d + 336*sqrt(x)*a*b*f*x + 336*sqr t(x)*a*c*e*x + 168*sqrt(x)*b**2*e*x + 336*sqrt(x)*b*c*d*x + 240*sqrt(x)*b* c*f*x**2 + 120*sqrt(x)*c**2*e*x**2 + 420*a**2*d + 210*a**2*f*x + 420*a*b*e *x + 420*a*c*d*x + 280*a*c*f*x**2 + 210*b**2*d*x + 140*b**2*f*x**2 + 280*b *c*e*x**2 + 140*c**2*d*x**2 + 105*c**2*f*x**3))/420