Integrand size = 17, antiderivative size = 100 \[ \int (a+b x)^3 (c+d x)^{4/5} \, dx=-\frac {5 (b c-a d)^3 (c+d x)^{9/5}}{9 d^4}+\frac {15 b (b c-a d)^2 (c+d x)^{14/5}}{14 d^4}-\frac {15 b^2 (b c-a d) (c+d x)^{19/5}}{19 d^4}+\frac {5 b^3 (c+d x)^{24/5}}{24 d^4} \] Output:
-5/9*(-a*d+b*c)^3*(d*x+c)^(9/5)/d^4+15/14*b*(-a*d+b*c)^2*(d*x+c)^(14/5)/d^ 4-15/19*b^2*(-a*d+b*c)*(d*x+c)^(19/5)/d^4+5/24*b^3*(d*x+c)^(24/5)/d^4
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int (a+b x)^3 (c+d x)^{4/5} \, dx=\frac {5 (c+d x)^{9/5} \left (1064 a^3 d^3+228 a^2 b d^2 (-5 c+9 d x)+24 a b^2 d \left (25 c^2-45 c d x+63 d^2 x^2\right )+b^3 \left (-125 c^3+225 c^2 d x-315 c d^2 x^2+399 d^3 x^3\right )\right )}{9576 d^4} \] Input:
Integrate[(a + b*x)^3*(c + d*x)^(4/5),x]
Output:
(5*(c + d*x)^(9/5)*(1064*a^3*d^3 + 228*a^2*b*d^2*(-5*c + 9*d*x) + 24*a*b^2 *d*(25*c^2 - 45*c*d*x + 63*d^2*x^2) + b^3*(-125*c^3 + 225*c^2*d*x - 315*c* d^2*x^2 + 399*d^3*x^3)))/(9576*d^4)
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {53, 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.
\(\displaystyle \int (a+b x)^3 (c+d x)^{4/5} \, dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (-\frac {3 b^2 (c+d x)^{14/5} (b c-a d)}{d^3}+\frac {3 b (c+d x)^{9/5} (b c-a d)^2}{d^3}+\frac {(c+d x)^{4/5} (a d-b c)^3}{d^3}+\frac {b^3 (c+d x)^{19/5}}{d^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {15 b^2 (c+d x)^{19/5} (b c-a d)}{19 d^4}+\frac {15 b (c+d x)^{14/5} (b c-a d)^2}{14 d^4}-\frac {5 (c+d x)^{9/5} (b c-a d)^3}{9 d^4}+\frac {5 b^3 (c+d x)^{24/5}}{24 d^4}\) |
Input:
Int[(a + b*x)^3*(c + d*x)^(4/5),x]
Output:
(-5*(b*c - a*d)^3*(c + d*x)^(9/5))/(9*d^4) + (15*b*(b*c - a*d)^2*(c + d*x) ^(14/5))/(14*d^4) - (15*b^2*(b*c - a*d)*(c + d*x)^(19/5))/(19*d^4) + (5*b^ 3*(c + d*x)^(24/5))/(24*d^4)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {5 b^{3} \left (x d +c \right )^{\frac {24}{5}}}{24}+\frac {15 \left (a d -b c \right ) b^{2} \left (x d +c \right )^{\frac {19}{5}}}{19}+\frac {15 \left (a d -b c \right )^{2} b \left (x d +c \right )^{\frac {14}{5}}}{14}+\frac {5 \left (a d -b c \right )^{3} \left (x d +c \right )^{\frac {9}{5}}}{9}}{d^{4}}\) | \(78\) |
default | \(\frac {\frac {5 b^{3} \left (x d +c \right )^{\frac {24}{5}}}{24}+\frac {15 \left (a d -b c \right ) b^{2} \left (x d +c \right )^{\frac {19}{5}}}{19}+\frac {15 \left (a d -b c \right )^{2} b \left (x d +c \right )^{\frac {14}{5}}}{14}+\frac {5 \left (a d -b c \right )^{3} \left (x d +c \right )^{\frac {9}{5}}}{9}}{d^{4}}\) | \(78\) |
pseudoelliptic | \(\frac {5 \left (\left (\frac {3}{8} b^{3} x^{3}+\frac {27}{19} a \,b^{2} x^{2}+\frac {27}{14} a^{2} b x +a^{3}\right ) d^{3}-\frac {15 c b \left (\frac {21}{76} b^{2} x^{2}+\frac {18}{19} a b x +a^{2}\right ) d^{2}}{14}+\frac {75 c^{2} \left (\frac {3 b x}{8}+a \right ) b^{2} d}{133}-\frac {125 b^{3} c^{3}}{1064}\right ) \left (x d +c \right )^{\frac {9}{5}}}{9 d^{4}}\) | \(93\) |
gosper | \(\frac {5 \left (x d +c \right )^{\frac {9}{5}} \left (399 d^{3} x^{3} b^{3}+1512 x^{2} a \,b^{2} d^{3}-315 x^{2} b^{3} c \,d^{2}+2052 x \,a^{2} b \,d^{3}-1080 x a \,b^{2} c \,d^{2}+225 x \,b^{3} c^{2} d +1064 a^{3} d^{3}-1140 a^{2} b c \,d^{2}+600 a \,b^{2} c^{2} d -125 b^{3} c^{3}\right )}{9576 d^{4}}\) | \(116\) |
orering | \(\frac {5 \left (x d +c \right )^{\frac {9}{5}} \left (399 d^{3} x^{3} b^{3}+1512 x^{2} a \,b^{2} d^{3}-315 x^{2} b^{3} c \,d^{2}+2052 x \,a^{2} b \,d^{3}-1080 x a \,b^{2} c \,d^{2}+225 x \,b^{3} c^{2} d +1064 a^{3} d^{3}-1140 a^{2} b c \,d^{2}+600 a \,b^{2} c^{2} d -125 b^{3} c^{3}\right )}{9576 d^{4}}\) | \(116\) |
trager | \(\frac {5 \left (399 b^{3} d^{4} x^{4}+1512 a \,b^{2} d^{4} x^{3}+84 b^{3} c \,d^{3} x^{3}+2052 a^{2} b \,d^{4} x^{2}+432 a \,b^{2} c \,d^{3} x^{2}-90 b^{3} c^{2} d^{2} x^{2}+1064 a^{3} d^{4} x +912 a^{2} b c \,d^{3} x -480 a \,b^{2} c^{2} d^{2} x +100 b^{3} c^{3} d x +1064 a^{3} c \,d^{3}-1140 a^{2} b \,c^{2} d^{2}+600 a \,b^{2} c^{3} d -125 b^{3} c^{4}\right ) \left (x d +c \right )^{\frac {4}{5}}}{9576 d^{4}}\) | \(170\) |
risch | \(\frac {5 \left (399 b^{3} d^{4} x^{4}+1512 a \,b^{2} d^{4} x^{3}+84 b^{3} c \,d^{3} x^{3}+2052 a^{2} b \,d^{4} x^{2}+432 a \,b^{2} c \,d^{3} x^{2}-90 b^{3} c^{2} d^{2} x^{2}+1064 a^{3} d^{4} x +912 a^{2} b c \,d^{3} x -480 a \,b^{2} c^{2} d^{2} x +100 b^{3} c^{3} d x +1064 a^{3} c \,d^{3}-1140 a^{2} b \,c^{2} d^{2}+600 a \,b^{2} c^{3} d -125 b^{3} c^{4}\right ) \left (x d +c \right )^{\frac {4}{5}}}{9576 d^{4}}\) | \(170\) |
Input:
int((b*x+a)^3*(d*x+c)^(4/5),x,method=_RETURNVERBOSE)
Output:
5/d^4*(1/24*b^3*(d*x+c)^(24/5)+3/19*(a*d-b*c)*b^2*(d*x+c)^(19/5)+3/14*(a*d -b*c)^2*b*(d*x+c)^(14/5)+1/9*(a*d-b*c)^3*(d*x+c)^(9/5))
Time = 0.06 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.65 \[ \int (a+b x)^3 (c+d x)^{4/5} \, dx=\frac {5 \, {\left (399 \, b^{3} d^{4} x^{4} - 125 \, b^{3} c^{4} + 600 \, a b^{2} c^{3} d - 1140 \, a^{2} b c^{2} d^{2} + 1064 \, a^{3} c d^{3} + 84 \, {\left (b^{3} c d^{3} + 18 \, a b^{2} d^{4}\right )} x^{3} - 18 \, {\left (5 \, b^{3} c^{2} d^{2} - 24 \, a b^{2} c d^{3} - 114 \, a^{2} b d^{4}\right )} x^{2} + 4 \, {\left (25 \, b^{3} c^{3} d - 120 \, a b^{2} c^{2} d^{2} + 228 \, a^{2} b c d^{3} + 266 \, a^{3} d^{4}\right )} x\right )} {\left (d x + c\right )}^{\frac {4}{5}}}{9576 \, d^{4}} \] Input:
integrate((b*x+a)^3*(d*x+c)^(4/5),x, algorithm="fricas")
Output:
5/9576*(399*b^3*d^4*x^4 - 125*b^3*c^4 + 600*a*b^2*c^3*d - 1140*a^2*b*c^2*d ^2 + 1064*a^3*c*d^3 + 84*(b^3*c*d^3 + 18*a*b^2*d^4)*x^3 - 18*(5*b^3*c^2*d^ 2 - 24*a*b^2*c*d^3 - 114*a^2*b*d^4)*x^2 + 4*(25*b^3*c^3*d - 120*a*b^2*c^2* d^2 + 228*a^2*b*c*d^3 + 266*a^3*d^4)*x)*(d*x + c)^(4/5)/d^4
Time = 0.82 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.67 \[ \int (a+b x)^3 (c+d x)^{4/5} \, dx=\begin {cases} \frac {5 \left (\frac {b^{3} \left (c + d x\right )^{\frac {24}{5}}}{24 d^{3}} + \frac {\left (c + d x\right )^{\frac {19}{5}} \cdot \left (3 a b^{2} d - 3 b^{3} c\right )}{19 d^{3}} + \frac {\left (c + d x\right )^{\frac {14}{5}} \cdot \left (3 a^{2} b d^{2} - 6 a b^{2} c d + 3 b^{3} c^{2}\right )}{14 d^{3}} + \frac {\left (c + d x\right )^{\frac {9}{5}} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{9 d^{3}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {4}{5}} \left (\begin {cases} a^{3} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{4}}{4 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**3*(d*x+c)**(4/5),x)
Output:
Piecewise((5*(b**3*(c + d*x)**(24/5)/(24*d**3) + (c + d*x)**(19/5)*(3*a*b* *2*d - 3*b**3*c)/(19*d**3) + (c + d*x)**(14/5)*(3*a**2*b*d**2 - 6*a*b**2*c *d + 3*b**3*c**2)/(14*d**3) + (c + d*x)**(9/5)*(a**3*d**3 - 3*a**2*b*c*d** 2 + 3*a*b**2*c**2*d - b**3*c**3)/(9*d**3))/d, Ne(d, 0)), (c**(4/5)*Piecewi se((a**3*x, Eq(b, 0)), ((a + b*x)**4/(4*b), True)), True))
Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.18 \[ \int (a+b x)^3 (c+d x)^{4/5} \, dx=\frac {5 \, {\left (399 \, {\left (d x + c\right )}^{\frac {24}{5}} b^{3} - 1512 \, {\left (b^{3} c - a b^{2} d\right )} {\left (d x + c\right )}^{\frac {19}{5}} + 2052 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left (d x + c\right )}^{\frac {14}{5}} - 1064 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x + c\right )}^{\frac {9}{5}}\right )}}{9576 \, d^{4}} \] Input:
integrate((b*x+a)^3*(d*x+c)^(4/5),x, algorithm="maxima")
Output:
5/9576*(399*(d*x + c)^(24/5)*b^3 - 1512*(b^3*c - a*b^2*d)*(d*x + c)^(19/5) + 2052*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*(d*x + c)^(14/5) - 1064*(b^3*c ^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(d*x + c)^(9/5))/d^4
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (84) = 168\).
Time = 0.13 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.26 \[ \int (a+b x)^3 (c+d x)^{4/5} \, dx=\frac {5 \, {\left (2394 \, {\left (d x + c\right )}^{\frac {4}{5}} a^{3} c + 266 \, {\left (4 \, {\left (d x + c\right )}^{\frac {9}{5}} - 9 \, {\left (d x + c\right )}^{\frac {4}{5}} c\right )} a^{3} + \frac {798 \, {\left (4 \, {\left (d x + c\right )}^{\frac {9}{5}} - 9 \, {\left (d x + c\right )}^{\frac {4}{5}} c\right )} a^{2} b c}{d} + \frac {114 \, {\left (18 \, {\left (d x + c\right )}^{\frac {14}{5}} - 56 \, {\left (d x + c\right )}^{\frac {9}{5}} c + 63 \, {\left (d x + c\right )}^{\frac {4}{5}} c^{2}\right )} a b^{2} c}{d^{2}} + \frac {114 \, {\left (18 \, {\left (d x + c\right )}^{\frac {14}{5}} - 56 \, {\left (d x + c\right )}^{\frac {9}{5}} c + 63 \, {\left (d x + c\right )}^{\frac {4}{5}} c^{2}\right )} a^{2} b}{d} + \frac {6 \, {\left (84 \, {\left (d x + c\right )}^{\frac {19}{5}} - 342 \, {\left (d x + c\right )}^{\frac {14}{5}} c + 532 \, {\left (d x + c\right )}^{\frac {9}{5}} c^{2} - 399 \, {\left (d x + c\right )}^{\frac {4}{5}} c^{3}\right )} b^{3} c}{d^{3}} + \frac {18 \, {\left (84 \, {\left (d x + c\right )}^{\frac {19}{5}} - 342 \, {\left (d x + c\right )}^{\frac {14}{5}} c + 532 \, {\left (d x + c\right )}^{\frac {9}{5}} c^{2} - 399 \, {\left (d x + c\right )}^{\frac {4}{5}} c^{3}\right )} a b^{2}}{d^{2}} + \frac {{\left (399 \, {\left (d x + c\right )}^{\frac {24}{5}} - 2016 \, {\left (d x + c\right )}^{\frac {19}{5}} c + 4104 \, {\left (d x + c\right )}^{\frac {14}{5}} c^{2} - 4256 \, {\left (d x + c\right )}^{\frac {9}{5}} c^{3} + 2394 \, {\left (d x + c\right )}^{\frac {4}{5}} c^{4}\right )} b^{3}}{d^{3}}\right )}}{9576 \, d} \] Input:
integrate((b*x+a)^3*(d*x+c)^(4/5),x, algorithm="giac")
Output:
5/9576*(2394*(d*x + c)^(4/5)*a^3*c + 266*(4*(d*x + c)^(9/5) - 9*(d*x + c)^ (4/5)*c)*a^3 + 798*(4*(d*x + c)^(9/5) - 9*(d*x + c)^(4/5)*c)*a^2*b*c/d + 1 14*(18*(d*x + c)^(14/5) - 56*(d*x + c)^(9/5)*c + 63*(d*x + c)^(4/5)*c^2)*a *b^2*c/d^2 + 114*(18*(d*x + c)^(14/5) - 56*(d*x + c)^(9/5)*c + 63*(d*x + c )^(4/5)*c^2)*a^2*b/d + 6*(84*(d*x + c)^(19/5) - 342*(d*x + c)^(14/5)*c + 5 32*(d*x + c)^(9/5)*c^2 - 399*(d*x + c)^(4/5)*c^3)*b^3*c/d^3 + 18*(84*(d*x + c)^(19/5) - 342*(d*x + c)^(14/5)*c + 532*(d*x + c)^(9/5)*c^2 - 399*(d*x + c)^(4/5)*c^3)*a*b^2/d^2 + (399*(d*x + c)^(24/5) - 2016*(d*x + c)^(19/5)* c + 4104*(d*x + c)^(14/5)*c^2 - 4256*(d*x + c)^(9/5)*c^3 + 2394*(d*x + c)^ (4/5)*c^4)*b^3/d^3)/d
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87 \[ \int (a+b x)^3 (c+d x)^{4/5} \, dx=\frac {5\,b^3\,{\left (c+d\,x\right )}^{24/5}}{24\,d^4}-\frac {\left (15\,b^3\,c-15\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{19/5}}{19\,d^4}+\frac {5\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{9/5}}{9\,d^4}+\frac {15\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{14/5}}{14\,d^4} \] Input:
int((a + b*x)^3*(c + d*x)^(4/5),x)
Output:
(5*b^3*(c + d*x)^(24/5))/(24*d^4) - ((15*b^3*c - 15*a*b^2*d)*(c + d*x)^(19 /5))/(19*d^4) + (5*(a*d - b*c)^3*(c + d*x)^(9/5))/(9*d^4) + (15*b*(a*d - b *c)^2*(c + d*x)^(14/5))/(14*d^4)
Time = 0.16 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.69 \[ \int (a+b x)^3 (c+d x)^{4/5} \, dx=\frac {5 \left (d x +c \right )^{\frac {4}{5}} \left (399 b^{3} d^{4} x^{4}+1512 a \,b^{2} d^{4} x^{3}+84 b^{3} c \,d^{3} x^{3}+2052 a^{2} b \,d^{4} x^{2}+432 a \,b^{2} c \,d^{3} x^{2}-90 b^{3} c^{2} d^{2} x^{2}+1064 a^{3} d^{4} x +912 a^{2} b c \,d^{3} x -480 a \,b^{2} c^{2} d^{2} x +100 b^{3} c^{3} d x +1064 a^{3} c \,d^{3}-1140 a^{2} b \,c^{2} d^{2}+600 a \,b^{2} c^{3} d -125 b^{3} c^{4}\right )}{9576 d^{4}} \] Input:
int((b*x+a)^3*(d*x+c)^(4/5),x)
Output:
(5*(c + d*x)**(4/5)*(1064*a**3*c*d**3 + 1064*a**3*d**4*x - 1140*a**2*b*c** 2*d**2 + 912*a**2*b*c*d**3*x + 2052*a**2*b*d**4*x**2 + 600*a*b**2*c**3*d - 480*a*b**2*c**2*d**2*x + 432*a*b**2*c*d**3*x**2 + 1512*a*b**2*d**4*x**3 - 125*b**3*c**4 + 100*b**3*c**3*d*x - 90*b**3*c**2*d**2*x**2 + 84*b**3*c*d* *3*x**3 + 399*b**3*d**4*x**4))/(9576*d**4)