Integrand size = 20, antiderivative size = 166 \[ \int \left (a+b \sqrt {x}+c x\right )^3 (d+e x) \, dx=a^3 d x+2 a^2 b d x^{3/2}+\frac {1}{2} a \left (3 b^2 d+a (3 c d+a e)\right ) x^2+\frac {2}{5} b \left (b^2 d+3 a (2 c d+a e)\right ) x^{5/2}+\left (b^2+a c\right ) (c d+a e) x^3+\frac {2}{7} b \left (3 c^2 d+b^2 e+6 a c e\right ) x^{7/2}+\frac {1}{4} c \left (c^2 d+3 b^2 e+3 a c e\right ) x^4+\frac {2}{3} b c^2 e x^{9/2}+\frac {1}{5} c^3 e x^5 \] Output:
a^3*d*x+2*a^2*b*d*x^(3/2)+1/2*a*(3*b^2*d+a*(a*e+3*c*d))*x^2+2/5*b*(b^2*d+3 *a*(a*e+2*c*d))*x^(5/2)+(a*c+b^2)*(a*e+c*d)*x^3+2/7*b*(6*a*c*e+b^2*e+3*c^2 *d)*x^(7/2)+1/4*c*(3*a*c*e+3*b^2*e+c^2*d)*x^4+2/3*b*c^2*e*x^(9/2)+1/5*c^3* e*x^5
Time = 0.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.03 \[ \int \left (a+b \sqrt {x}+c x\right )^3 (d+e x) \, dx=\frac {1}{420} x \left (210 a^3 (2 d+e x)+105 b^2 c x^2 (4 d+3 e x)+21 c^3 x^3 (5 d+4 e x)+24 b^3 x^{3/2} (7 d+5 e x)+40 b c^2 x^{5/2} (9 d+7 e x)+42 a^2 \left (5 c x (3 d+2 e x)+4 b \sqrt {x} (5 d+3 e x)\right )+3 a x \left (70 b^2 (3 d+2 e x)+35 c^2 x (4 d+3 e x)+48 b c \sqrt {x} (7 d+5 e x)\right )\right ) \] Input:
Integrate[(a + b*Sqrt[x] + c*x)^3*(d + e*x),x]
Output:
(x*(210*a^3*(2*d + e*x) + 105*b^2*c*x^2*(4*d + 3*e*x) + 21*c^3*x^3*(5*d + 4*e*x) + 24*b^3*x^(3/2)*(7*d + 5*e*x) + 40*b*c^2*x^(5/2)*(9*d + 7*e*x) + 4 2*a^2*(5*c*x*(3*d + 2*e*x) + 4*b*Sqrt[x]*(5*d + 3*e*x)) + 3*a*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 = 0.64 (sec) , antiderivative size = 190, 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.
| \(\displaystyle \int (d+e x) \left (a+b \sqrt {x}+c x\right )^3 \, dx\) |
| \(\Big \downarrow \) 2308 |
| \(\displaystyle \int \left (d \left (a+b \sqrt {x}+c x\right )^3+e x \left (a+b \sqrt {x}+c x\right )^3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 d x+\frac {1}{2} a^3 e x^2+2 a^2 b d x^{3/2}+\frac {6}{5} a^2 b e x^{5/2}+\frac {2}{5} b d x^{5/2} \left (6 a c+b^2\right )+c d x^3 \left (a c+b^2\right )+\frac {3}{2} a d x^2 \left (a c+b^2\right )+\frac {2}{7} b e x^{7/2} \left (6 a c+b^2\right )+\frac {3}{4} c e x^4 \left (a c+b^2\right )+a e x^3 \left (a c+b^2\right )+\frac {6}{7} b c^2 d x^{7/2}+\frac {2}{3} b c^2 e x^{9/2}+\frac {1}{4} c^3 d x^4+\frac {1}{5} c^3 e x^5\) |
Input:
Int[(a + b*Sqrt[x] + c*x)^3*(d + e*x),x]
Output:
a^3*d*x + 2*a^2*b*d*x^(3/2) + (3*a*(b^2 + a*c)*d*x^2)/2 + (a^3*e*x^2)/2 + (2*b*(b^2 + 6*a*c)*d*x^(5/2))/5 + (6*a^2*b*e*x^(5/2))/5 + c*(b^2 + a*c)*d* x^3 + a*(b^2 + a*c)*e*x^3 + (6*b*c^2*d*x^(7/2))/7 + (2*b*(b^2 + 6*a*c)*e*x ^(7/2))/7 + (c^3*d*x^4)/4 + (3*c*(b^2 + a*c)*e*x^4)/4 + (2*b*c^2*e*x^(9/2) )/3 + (c^3*e*x^5)/5
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 = 1.16 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(b^{3} \left (\frac {2 e \,x^{\frac {7}{2}}}{7}+\frac {2 d \,x^{\frac {5}{2}}}{5}\right )+3 b^{2} \left (\frac {c e \,x^{4}}{4}+\frac {\left (a e +c d \right ) x^{3}}{3}+\frac {a d \,x^{2}}{2}\right )+3 b \left (\frac {2 c^{2} e \,x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a c e +c^{2} d \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (a^{2} e +2 a c d \right ) x^{\frac {5}{2}}}{5}+\frac {2 a^{2} d \,x^{\frac {3}{2}}}{3}\right )+\frac {c^{3} e \,x^{5}}{5}+\frac {\left (3 a \,c^{2} e +c^{3} d \right ) x^{4}}{4}+\frac {\left (3 a^{2} c e +3 a \,c^{2} d \right ) x^{3}}{3}+\frac {\left (a^{3} e +3 a^{2} c d \right ) x^{2}}{2}+a^{3} d x\) | \(176\) |
| derivativedivides | \(\frac {c^{3} e \,x^{5}}{5}+\frac {2 b \,c^{2} e \,x^{\frac {9}{2}}}{3}+\frac {\left (\left (a \,c^{2}+2 c \,b^{2}+c \left (2 a c +b^{2}\right )\right ) e +c^{3} d \right ) x^{4}}{4}+\frac {2 \left (\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) e +3 b \,c^{2} d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +a^{2} c \right ) e +\left (a \,c^{2}+2 c \,b^{2}+c \left (2 a c +b^{2}\right )\right ) d \right ) x^{3}}{3}+\frac {2 \left (3 b \,a^{2} e +\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) d \right ) x^{\frac {5}{2}}}{5}+\frac {\left (a^{3} e +\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +a^{2} c \right ) d \right ) x^{2}}{2}+2 a^{2} b d \,x^{\frac {3}{2}}+a^{3} d x\) | \(223\) |
| trager | \(\frac {\left (4 e \,c^{3} x^{4}+15 a \,c^{2} e \,x^{3}+15 b^{2} c e \,x^{3}+5 c^{3} d \,x^{3}+4 c^{3} e \,x^{3}+20 a^{2} c e \,x^{2}+20 a \,b^{2} e \,x^{2}+20 a \,c^{2} d \,x^{2}+15 a \,c^{2} e \,x^{2}+20 b^{2} c d \,x^{2}+15 b^{2} c e \,x^{2}+5 c^{3} d \,x^{2}+4 c^{3} e \,x^{2}+10 a^{3} e x +30 a^{2} c d x +20 a^{2} c e x +30 a \,b^{2} d x +20 a \,b^{2} e x +20 a \,c^{2} d x +15 a \,c^{2} e x +20 b^{2} c d x +15 b^{2} c e x +5 c^{3} d x +4 e \,c^{3} x +20 a^{3} d +10 a^{3} e +30 a^{2} c d +20 a^{2} c e +30 a \,b^{2} d +20 a \,b^{2} e +20 a \,c^{2} d +15 a \,c^{2} e +20 b^{2} c d +15 b^{2} c e +5 c^{3} d +4 e \,c^{3}\right ) \left (x -1\right )}{20}+\frac {2 b \,x^{\frac {3}{2}} \left (35 c^{2} e \,x^{3}+90 a c e \,x^{2}+15 b^{2} e \,x^{2}+45 c^{2} d \,x^{2}+63 a^{2} e x +126 a c d x +21 b^{2} d x +105 a^{2} d \right )}{105}\) | \(366\) |
| orering | \(\text {Expression too large to display}\) | \(1085\) |
Input:
int((a+b*x^(1/2)+c*x)^3*(e*x+d),x,method=_RETURNVERBOSE)
Output:
b^3*(2/7*e*x^(7/2)+2/5*d*x^(5/2))+3*b^2*(1/4*c*e*x^4+1/3*(a*e+c*d)*x^3+1/2 *a*d*x^2)+3*b*(2/9*c^2*e*x^(9/2)+2/7*(2*a*c*e+c^2*d)*x^(7/2)+2/5*(a^2*e+2* a*c*d)*x^(5/2)+2/3*a^2*d*x^(3/2))+1/5*c^3*e*x^5+1/4*(3*a*c^2*e+c^3*d)*x^4+ 1/3*(3*a^2*c*e+3*a*c^2*d)*x^3+1/2*(a^3*e+3*a^2*c*d)*x^2+a^3*d*x
Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.02 \[ \int \left (a+b \sqrt {x}+c x\right )^3 (d+e x) \, dx=\frac {1}{5} \, c^{3} e x^{5} + a^{3} d x + \frac {1}{4} \, {\left (c^{3} d + 3 \, {\left (b^{2} c + a c^{2}\right )} e\right )} x^{4} + {\left ({\left (b^{2} c + a c^{2}\right )} d + {\left (a b^{2} + a^{2} c\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (a^{3} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d\right )} x^{2} + \frac {2}{105} \, {\left (35 \, b c^{2} e x^{4} + 105 \, a^{2} b d x + 15 \, {\left (3 \, b c^{2} d + {\left (b^{3} + 6 \, a b c\right )} e\right )} x^{3} + 21 \, {\left (3 \, a^{2} b e + {\left (b^{3} + 6 \, a b c\right )} d\right )} x^{2}\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2)+c*x)^3*(e*x+d),x, algorithm="fricas")
Output:
1/5*c^3*e*x^5 + a^3*d*x + 1/4*(c^3*d + 3*(b^2*c + a*c^2)*e)*x^4 + ((b^2*c + a*c^2)*d + (a*b^2 + a^2*c)*e)*x^3 + 1/2*(a^3*e + 3*(a*b^2 + a^2*c)*d)*x^ 2 + 2/105*(35*b*c^2*e*x^4 + 105*a^2*b*d*x + 15*(3*b*c^2*d + (b^3 + 6*a*b*c )*e)*x^3 + 21*(3*a^2*b*e + (b^3 + 6*a*b*c)*d)*x^2)*sqrt(x)
Time = 0.21 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.49 \[ \int \left (a+b \sqrt {x}+c x\right )^3 (d+e x) \, dx=a^{3} d x + \frac {a^{3} e x^{2}}{2} + 2 a^{2} b d x^{\frac {3}{2}} + \frac {6 a^{2} b e x^{\frac {5}{2}}}{5} + \frac {3 a^{2} c d x^{2}}{2} + a^{2} c e x^{3} + \frac {3 a b^{2} d x^{2}}{2} + a b^{2} e x^{3} + \frac {12 a b c d x^{\frac {5}{2}}}{5} + \frac {12 a b c e x^{\frac {7}{2}}}{7} + a c^{2} d x^{3} + \frac {3 a c^{2} e x^{4}}{4} + \frac {2 b^{3} d x^{\frac {5}{2}}}{5} + \frac {2 b^{3} e x^{\frac {7}{2}}}{7} + b^{2} c d x^{3} + \frac {3 b^{2} c e x^{4}}{4} + \frac {6 b c^{2} d x^{\frac {7}{2}}}{7} + \frac {2 b c^{2} e x^{\frac {9}{2}}}{3} + \frac {c^{3} d x^{4}}{4} + \frac {c^{3} e x^{5}}{5} \] Input:
integrate((a+b*x**(1/2)+c*x)**3*(e*x+d),x)
Output:
a**3*d*x + a**3*e*x**2/2 + 2*a**2*b*d*x**(3/2) + 6*a**2*b*e*x**(5/2)/5 + 3 *a**2*c*d*x**2/2 + a**2*c*e*x**3 + 3*a*b**2*d*x**2/2 + a*b**2*e*x**3 + 12* a*b*c*d*x**(5/2)/5 + 12*a*b*c*e*x**(7/2)/7 + a*c**2*d*x**3 + 3*a*c**2*e*x* *4/4 + 2*b**3*d*x**(5/2)/5 + 2*b**3*e*x**(7/2)/7 + b**2*c*d*x**3 + 3*b**2* c*e*x**4/4 + 6*b*c**2*d*x**(7/2)/7 + 2*b*c**2*e*x**(9/2)/3 + c**3*d*x**4/4 + c**3*e*x**5/5
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.99 \[ \int \left (a+b \sqrt {x}+c x\right )^3 (d+e x) \, dx=\frac {1}{5} \, c^{3} e x^{5} + \frac {2}{3} \, b c^{2} e x^{\frac {9}{2}} + 2 \, a^{2} b d x^{\frac {3}{2}} + a^{3} d x + \frac {1}{4} \, {\left (c^{3} d + 3 \, {\left (b^{2} c + a c^{2}\right )} e\right )} x^{4} + \frac {2}{7} \, {\left (3 \, b c^{2} d + {\left (b^{3} + 6 \, a b c\right )} e\right )} x^{\frac {7}{2}} + {\left ({\left (b^{2} c + a c^{2}\right )} d + {\left (a b^{2} + a^{2} c\right )} e\right )} x^{3} + \frac {2}{5} \, {\left (3 \, a^{2} b e + {\left (b^{3} + 6 \, a b c\right )} d\right )} x^{\frac {5}{2}} + \frac {1}{2} \, {\left (a^{3} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d\right )} x^{2} \] Input:
integrate((a+b*x^(1/2)+c*x)^3*(e*x+d),x, algorithm="maxima")
Output:
1/5*c^3*e*x^5 + 2/3*b*c^2*e*x^(9/2) + 2*a^2*b*d*x^(3/2) + a^3*d*x + 1/4*(c ^3*d + 3*(b^2*c + a*c^2)*e)*x^4 + 2/7*(3*b*c^2*d + (b^3 + 6*a*b*c)*e)*x^(7 /2) + ((b^2*c + a*c^2)*d + (a*b^2 + a^2*c)*e)*x^3 + 2/5*(3*a^2*b*e + (b^3 + 6*a*b*c)*d)*x^(5/2) + 1/2*(a^3*e + 3*(a*b^2 + a^2*c)*d)*x^2
Time = 0.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.12 \[ \int \left (a+b \sqrt {x}+c x\right )^3 (d+e x) \, dx=\frac {1}{5} \, c^{3} e x^{5} + \frac {2}{3} \, b c^{2} e x^{\frac {9}{2}} + \frac {1}{4} \, c^{3} d x^{4} + \frac {3}{4} \, b^{2} c e x^{4} + \frac {3}{4} \, a c^{2} e x^{4} + \frac {6}{7} \, b c^{2} d x^{\frac {7}{2}} + \frac {2}{7} \, b^{3} e x^{\frac {7}{2}} + \frac {12}{7} \, a b c e x^{\frac {7}{2}} + b^{2} c d x^{3} + a c^{2} d x^{3} + a b^{2} e x^{3} + a^{2} c e x^{3} + \frac {2}{5} \, b^{3} d x^{\frac {5}{2}} + \frac {12}{5} \, a b c d x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} b e x^{\frac {5}{2}} + \frac {3}{2} \, a b^{2} d x^{2} + \frac {3}{2} \, a^{2} c d x^{2} + \frac {1}{2} \, a^{3} e x^{2} + 2 \, a^{2} b d x^{\frac {3}{2}} + a^{3} d x \] Input:
integrate((a+b*x^(1/2)+c*x)^3*(e*x+d),x, algorithm="giac")
Output:
1/5*c^3*e*x^5 + 2/3*b*c^2*e*x^(9/2) + 1/4*c^3*d*x^4 + 3/4*b^2*c*e*x^4 + 3/ 4*a*c^2*e*x^4 + 6/7*b*c^2*d*x^(7/2) + 2/7*b^3*e*x^(7/2) + 12/7*a*b*c*e*x^( 7/2) + b^2*c*d*x^3 + a*c^2*d*x^3 + a*b^2*e*x^3 + a^2*c*e*x^3 + 2/5*b^3*d*x ^(5/2) + 12/5*a*b*c*d*x^(5/2) + 6/5*a^2*b*e*x^(5/2) + 3/2*a*b^2*d*x^2 + 3/ 2*a^2*c*d*x^2 + 1/2*a^3*e*x^2 + 2*a^2*b*d*x^(3/2) + a^3*d*x
Time = 22.06 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.90 \[ \int \left (a+b \sqrt {x}+c x\right )^3 (d+e x) \, dx=x^2\,\left (\frac {e\,a^3}{2}+\frac {3\,c\,d\,a^2}{2}+\frac {3\,d\,a\,b^2}{2}\right )+x^4\,\left (\frac {3\,e\,b^2\,c}{4}+\frac {d\,c^3}{4}+\frac {3\,a\,e\,c^2}{4}\right )+x^3\,\left (b^2+a\,c\right )\,\left (a\,e+c\,d\right )+\frac {c^3\,e\,x^5}{5}+\frac {2\,b\,x^{5/2}\,\left (3\,e\,a^2+6\,c\,d\,a+d\,b^2\right )}{5}+\frac {2\,b\,x^{7/2}\,\left (e\,b^2+3\,d\,c^2+6\,a\,e\,c\right )}{7}+a^3\,d\,x+2\,a^2\,b\,d\,x^{3/2}+\frac {2\,b\,c^2\,e\,x^{9/2}}{3} \] Input:
int((d + e*x)*(a + c*x + b*x^(1/2))^3,x)
Output:
x^2*((a^3*e)/2 + (3*a*b^2*d)/2 + (3*a^2*c*d)/2) + x^4*((c^3*d)/4 + (3*a*c^ 2*e)/4 + (3*b^2*c*e)/4) + x^3*(a*c + b^2)*(a*e + c*d) + (c^3*e*x^5)/5 + (2 *b*x^(5/2)*(3*a^2*e + b^2*d + 6*a*c*d))/5 + (2*b*x^(7/2)*(b^2*e + 3*c^2*d + 6*a*c*e))/7 + a^3*d*x + 2*a^2*b*d*x^(3/2) + (2*b*c^2*e*x^(9/2))/3
Time = 0.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.17 \[ \int \left (a+b \sqrt {x}+c x\right )^3 (d+e x) \, dx=\frac {x \left (840 \sqrt {x}\, a^{2} b d +504 \sqrt {x}\, a^{2} b e x +1008 \sqrt {x}\, a b c d x +720 \sqrt {x}\, a b c e \,x^{2}+168 \sqrt {x}\, b^{3} d x +120 \sqrt {x}\, b^{3} e \,x^{2}+360 \sqrt {x}\, b \,c^{2} d \,x^{2}+280 \sqrt {x}\, b \,c^{2} e \,x^{3}+420 a^{3} d +210 a^{3} e x +630 a^{2} c d x +420 a^{2} c e \,x^{2}+630 a \,b^{2} d x +420 a \,b^{2} e \,x^{2}+420 a \,c^{2} d \,x^{2}+315 a \,c^{2} e \,x^{3}+420 b^{2} c d \,x^{2}+315 b^{2} c e \,x^{3}+105 c^{3} d \,x^{3}+84 c^{3} e \,x^{4}\right )}{420} \] Input:
int((a+b*x^(1/2)+c*x)^3*(e*x+d),x)
Output:
(x*(840*sqrt(x)*a**2*b*d + 504*sqrt(x)*a**2*b*e*x + 1008*sqrt(x)*a*b*c*d*x + 720*sqrt(x)*a*b*c*e*x**2 + 168*sqrt(x)*b**3*d*x + 120*sqrt(x)*b**3*e*x* *2 + 360*sqrt(x)*b*c**2*d*x**2 + 280*sqrt(x)*b*c**2*e*x**3 + 420*a**3*d + 210*a**3*e*x + 630*a**2*c*d*x + 420*a**2*c*e*x**2 + 630*a*b**2*d*x + 420*a *b**2*e*x**2 + 420*a*c**2*d*x**2 + 315*a*c**2*e*x**3 + 420*b**2*c*d*x**2 + 315*b**2*c*e*x**3 + 105*c**3*d*x**3 + 84*c**3*e*x**4))/420