\(\int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx\) [523]

Optimal result

Integrand size = 19, antiderivative size = 136 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx=-\frac {3 (c+d x)^{4/3}}{13 (b c-a d) (a+b x)^{13/3}}+\frac {27 d (c+d x)^{4/3}}{130 (b c-a d)^2 (a+b x)^{10/3}}-\frac {81 d^2 (c+d x)^{4/3}}{455 (b c-a d)^3 (a+b x)^{7/3}}+\frac {243 d^3 (c+d x)^{4/3}}{1820 (b c-a d)^4 (a+b x)^{4/3}} \] Output:

-3/13*(d*x+c)^(4/3)/(-a*d+b*c)/(b*x+a)^(13/3)+27/130*d*(d*x+c)^(4/3)/(-a*d 
+b*c)^2/(b*x+a)^(10/3)-81/455*d^2*(d*x+c)^(4/3)/(-a*d+b*c)^3/(b*x+a)^(7/3) 
+243/1820*d^3*(d*x+c)^(4/3)/(-a*d+b*c)^4/(b*x+a)^(4/3)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx=\frac {3 (c+d x)^{4/3} \left (455 a^3 d^3+195 a^2 b d^2 (-4 c+3 d x)+39 a b^2 d \left (14 c^2-12 c d x+9 d^2 x^2\right )+b^3 \left (-140 c^3+126 c^2 d x-108 c d^2 x^2+81 d^3 x^3\right )\right )}{1820 (b c-a d)^4 (a+b x)^{13/3}} \] Input:

Integrate[(c + d*x)^(1/3)/(a + b*x)^(16/3),x]
 

Output:

(3*(c + d*x)^(4/3)*(455*a^3*d^3 + 195*a^2*b*d^2*(-4*c + 3*d*x) + 39*a*b^2* 
d*(14*c^2 - 12*c*d*x + 9*d^2*x^2) + b^3*(-140*c^3 + 126*c^2*d*x - 108*c*d^ 
2*x^2 + 81*d^3*x^3)))/(1820*(b*c - a*d)^4*(a + b*x)^(13/3))
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {55, 55, 55, 48}

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.

\(\displaystyle \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {9 d \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{13/3}}dx}{13 (b c-a d)}-\frac {3 (c+d x)^{4/3}}{13 (a+b x)^{13/3} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {9 d \left (-\frac {3 d \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{10/3}}dx}{5 (b c-a d)}-\frac {3 (c+d x)^{4/3}}{10 (a+b x)^{10/3} (b c-a d)}\right )}{13 (b c-a d)}-\frac {3 (c+d x)^{4/3}}{13 (a+b x)^{13/3} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {9 d \left (-\frac {3 d \left (-\frac {3 d \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{7/3}}dx}{7 (b c-a d)}-\frac {3 (c+d x)^{4/3}}{7 (a+b x)^{7/3} (b c-a d)}\right )}{5 (b c-a d)}-\frac {3 (c+d x)^{4/3}}{10 (a+b x)^{10/3} (b c-a d)}\right )}{13 (b c-a d)}-\frac {3 (c+d x)^{4/3}}{13 (a+b x)^{13/3} (b c-a d)}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {3 (c+d x)^{4/3}}{13 (a+b x)^{13/3} (b c-a d)}-\frac {9 d \left (-\frac {3 (c+d x)^{4/3}}{10 (a+b x)^{10/3} (b c-a d)}-\frac {3 d \left (\frac {9 d (c+d x)^{4/3}}{28 (a+b x)^{4/3} (b c-a d)^2}-\frac {3 (c+d x)^{4/3}}{7 (a+b x)^{7/3} (b c-a d)}\right )}{5 (b c-a d)}\right )}{13 (b c-a d)}\)

Input:

Int[(c + d*x)^(1/3)/(a + b*x)^(16/3),x]
 

Output:

(-3*(c + d*x)^(4/3))/(13*(b*c - a*d)*(a + b*x)^(13/3)) - (9*d*((-3*(c + d* 
x)^(4/3))/(10*(b*c - a*d)*(a + b*x)^(10/3)) - (3*d*((-3*(c + d*x)^(4/3))/( 
7*(b*c - a*d)*(a + b*x)^(7/3)) + (9*d*(c + d*x)^(4/3))/(28*(b*c - a*d)^2*( 
a + b*x)^(4/3))))/(5*(b*c - a*d))))/(13*(b*c - a*d))
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26

method	result	size
gosper	\(\frac {3 \left (x d +c \right )^{\frac {4}{3}} \left (81 d^{3} x^{3} b^{3}+351 x^{2} a \,b^{2} d^{3}-108 x^{2} b^{3} c \,d^{2}+585 x \,a^{2} b \,d^{3}-468 x a \,b^{2} c \,d^{2}+126 x \,b^{3} c^{2} d +455 a^{3} d^{3}-780 a^{2} b c \,d^{2}+546 a \,b^{2} c^{2} d -140 b^{3} c^{3}\right )}{1820 \left (b x +a \right )^{\frac {13}{3}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(171\)
orering	\(\frac {3 \left (x d +c \right )^{\frac {4}{3}} \left (81 d^{3} x^{3} b^{3}+351 x^{2} a \,b^{2} d^{3}-108 x^{2} b^{3} c \,d^{2}+585 x \,a^{2} b \,d^{3}-468 x a \,b^{2} c \,d^{2}+126 x \,b^{3} c^{2} d +455 a^{3} d^{3}-780 a^{2} b c \,d^{2}+546 a \,b^{2} c^{2} d -140 b^{3} c^{3}\right )}{1820 \left (b x +a \right )^{\frac {13}{3}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(171\)

Input:

int((d*x+c)^(1/3)/(b*x+a)^(16/3),x,method=_RETURNVERBOSE)
 

Output:

3/1820*(d*x+c)^(4/3)*(81*b^3*d^3*x^3+351*a*b^2*d^3*x^2-108*b^3*c*d^2*x^2+5 
85*a^2*b*d^3*x-468*a*b^2*c*d^2*x+126*b^3*c^2*d*x+455*a^3*d^3-780*a^2*b*c*d 
^2+546*a*b^2*c^2*d-140*b^3*c^3)/(b*x+a)^(13/3)/(a^4*d^4-4*a^3*b*c*d^3+6*a^ 
2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (112) = 224\).

Time = 0.09 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.92 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx=\frac {3 \, {\left (81 \, b^{3} d^{4} x^{4} - 140 \, b^{3} c^{4} + 546 \, a b^{2} c^{3} d - 780 \, a^{2} b c^{2} d^{2} + 455 \, a^{3} c d^{3} - 27 \, {\left (b^{3} c d^{3} - 13 \, a b^{2} d^{4}\right )} x^{3} + 9 \, {\left (2 \, b^{3} c^{2} d^{2} - 13 \, a b^{2} c d^{3} + 65 \, a^{2} b d^{4}\right )} x^{2} - {\left (14 \, b^{3} c^{3} d - 78 \, a b^{2} c^{2} d^{2} + 195 \, a^{2} b c d^{3} - 455 \, a^{3} d^{4}\right )} x\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{1820 \, {\left (a^{5} b^{4} c^{4} - 4 \, a^{6} b^{3} c^{3} d + 6 \, a^{7} b^{2} c^{2} d^{2} - 4 \, a^{8} b c d^{3} + a^{9} d^{4} + {\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} x^{5} + 5 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} x^{4} + 10 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} x^{3} + 10 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} x^{2} + 5 \, {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} x\right )}} \] Input:

integrate((d*x+c)^(1/3)/(b*x+a)^(16/3),x, algorithm="fricas")
 

Output:

3/1820*(81*b^3*d^4*x^4 - 140*b^3*c^4 + 546*a*b^2*c^3*d - 780*a^2*b*c^2*d^2 
 + 455*a^3*c*d^3 - 27*(b^3*c*d^3 - 13*a*b^2*d^4)*x^3 + 9*(2*b^3*c^2*d^2 - 
13*a*b^2*c*d^3 + 65*a^2*b*d^4)*x^2 - (14*b^3*c^3*d - 78*a*b^2*c^2*d^2 + 19 
5*a^2*b*c*d^3 - 455*a^3*d^4)*x)*(b*x + a)^(2/3)*(d*x + c)^(1/3)/(a^5*b^4*c 
^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4 + (b^9* 
c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x 
^5 + 5*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 
+ a^5*b^4*d^4)*x^4 + 10*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 
 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*x^3 + 10*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d 
+ 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*x^2 + 5*(a^4*b^5*c^4 
- 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/3)/(b*x+a)**(16/3),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {16}{3}}} \,d x } \] Input:

integrate((d*x+c)^(1/3)/(b*x+a)^(16/3),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^(1/3)/(b*x + a)^(16/3), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {16}{3}}} \,d x } \] Input:

integrate((d*x+c)^(1/3)/(b*x+a)^(16/3),x, algorithm="giac")
 

Output:

integrate((d*x + c)^(1/3)/(b*x + a)^(16/3), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx=\frac {{\left (c+d\,x\right )}^{1/3}\,\left (\frac {243\,d^4\,x^4}{1820\,b\,{\left (a\,d-b\,c\right )}^4}-\frac {-1365\,a^3\,c\,d^3+2340\,a^2\,b\,c^2\,d^2-1638\,a\,b^2\,c^3\,d+420\,b^3\,c^4}{1820\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (1365\,a^3\,d^4-585\,a^2\,b\,c\,d^3+234\,a\,b^2\,c^2\,d^2-42\,b^3\,c^3\,d\right )}{1820\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {81\,d^3\,x^3\,\left (13\,a\,d-b\,c\right )}{1820\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {27\,d^2\,x^2\,\left (65\,a^2\,d^2-13\,a\,b\,c\,d+2\,b^2\,c^2\right )}{1820\,b^3\,{\left (a\,d-b\,c\right )}^4}\right )}{x^4\,{\left (a+b\,x\right )}^{1/3}+\frac {a^4\,{\left (a+b\,x\right )}^{1/3}}{b^4}+\frac {6\,a^2\,x^2\,{\left (a+b\,x\right )}^{1/3}}{b^2}+\frac {4\,a\,x^3\,{\left (a+b\,x\right )}^{1/3}}{b}+\frac {4\,a^3\,x\,{\left (a+b\,x\right )}^{1/3}}{b^3}} \] Input:

int((c + d*x)^(1/3)/(a + b*x)^(16/3),x)
 

Output:

((c + d*x)^(1/3)*((243*d^4*x^4)/(1820*b*(a*d - b*c)^4) - (420*b^3*c^4 - 13 
65*a^3*c*d^3 + 2340*a^2*b*c^2*d^2 - 1638*a*b^2*c^3*d)/(1820*b^4*(a*d - b*c 
)^4) + (x*(1365*a^3*d^4 - 42*b^3*c^3*d + 234*a*b^2*c^2*d^2 - 585*a^2*b*c*d 
^3))/(1820*b^4*(a*d - b*c)^4) + (81*d^3*x^3*(13*a*d - b*c))/(1820*b^2*(a*d 
 - b*c)^4) + (27*d^2*x^2*(65*a^2*d^2 + 2*b^2*c^2 - 13*a*b*c*d))/(1820*b^3* 
(a*d - b*c)^4)))/(x^4*(a + b*x)^(1/3) + (a^4*(a + b*x)^(1/3))/b^4 + (6*a^2 
*x^2*(a + b*x)^(1/3))/b^2 + (4*a*x^3*(a + b*x)^(1/3))/b + (4*a^3*x*(a + b* 
x)^(1/3))/b^3)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 508, normalized size of antiderivative = 3.74 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx=\frac {3 \left (d x +c \right )^{\frac {1}{3}} \left (81 b^{3} d^{4} x^{4}+351 a \,b^{2} d^{4} x^{3}-27 b^{3} c \,d^{3} x^{3}+585 a^{2} b \,d^{4} x^{2}-117 a \,b^{2} c \,d^{3} x^{2}+18 b^{3} c^{2} d^{2} x^{2}+455 a^{3} d^{4} x -195 a^{2} b c \,d^{3} x +78 a \,b^{2} c^{2} d^{2} x -14 b^{3} c^{3} d x +455 a^{3} c \,d^{3}-780 a^{2} b \,c^{2} d^{2}+546 a \,b^{2} c^{3} d -140 b^{3} c^{4}\right )}{1820 \left (b x +a \right )^{\frac {1}{3}} \left (a^{4} b^{4} d^{4} x^{4}-4 a^{3} b^{5} c \,d^{3} x^{4}+6 a^{2} b^{6} c^{2} d^{2} x^{4}-4 a \,b^{7} c^{3} d \,x^{4}+b^{8} c^{4} x^{4}+4 a^{5} b^{3} d^{4} x^{3}-16 a^{4} b^{4} c \,d^{3} x^{3}+24 a^{3} b^{5} c^{2} d^{2} x^{3}-16 a^{2} b^{6} c^{3} d \,x^{3}+4 a \,b^{7} c^{4} x^{3}+6 a^{6} b^{2} d^{4} x^{2}-24 a^{5} b^{3} c \,d^{3} x^{2}+36 a^{4} b^{4} c^{2} d^{2} x^{2}-24 a^{3} b^{5} c^{3} d \,x^{2}+6 a^{2} b^{6} c^{4} x^{2}+4 a^{7} b \,d^{4} x -16 a^{6} b^{2} c \,d^{3} x +24 a^{5} b^{3} c^{2} d^{2} x -16 a^{4} b^{4} c^{3} d x +4 a^{3} b^{5} c^{4} x +a^{8} d^{4}-4 a^{7} b c \,d^{3}+6 a^{6} b^{2} c^{2} d^{2}-4 a^{5} b^{3} c^{3} d +a^{4} b^{4} c^{4}\right )} \] Input:

int((d*x+c)^(1/3)/(b*x+a)^(16/3),x)
 

Output:

(3*(c + d*x)**(1/3)*(455*a**3*c*d**3 + 455*a**3*d**4*x - 780*a**2*b*c**2*d 
**2 - 195*a**2*b*c*d**3*x + 585*a**2*b*d**4*x**2 + 546*a*b**2*c**3*d + 78* 
a*b**2*c**2*d**2*x - 117*a*b**2*c*d**3*x**2 + 351*a*b**2*d**4*x**3 - 140*b 
**3*c**4 - 14*b**3*c**3*d*x + 18*b**3*c**2*d**2*x**2 - 27*b**3*c*d**3*x**3 
 + 81*b**3*d**4*x**4))/(1820*(a + b*x)**(1/3)*(a**8*d**4 - 4*a**7*b*c*d**3 
 + 4*a**7*b*d**4*x + 6*a**6*b**2*c**2*d**2 - 16*a**6*b**2*c*d**3*x + 6*a** 
6*b**2*d**4*x**2 - 4*a**5*b**3*c**3*d + 24*a**5*b**3*c**2*d**2*x - 24*a**5 
*b**3*c*d**3*x**2 + 4*a**5*b**3*d**4*x**3 + a**4*b**4*c**4 - 16*a**4*b**4* 
c**3*d*x + 36*a**4*b**4*c**2*d**2*x**2 - 16*a**4*b**4*c*d**3*x**3 + a**4*b 
**4*d**4*x**4 + 4*a**3*b**5*c**4*x - 24*a**3*b**5*c**3*d*x**2 + 24*a**3*b* 
*5*c**2*d**2*x**3 - 4*a**3*b**5*c*d**3*x**4 + 6*a**2*b**6*c**4*x**2 - 16*a 
**2*b**6*c**3*d*x**3 + 6*a**2*b**6*c**2*d**2*x**4 + 4*a*b**7*c**4*x**3 - 4 
*a*b**7*c**3*d*x**4 + b**8*c**4*x**4))