Integrand size = 19, antiderivative size = 136 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\frac {6 (a+b x)^{13/6}}{31 (b c-a d) (c+d x)^{31/6}}+\frac {108 b (a+b x)^{13/6}}{775 (b c-a d)^2 (c+d x)^{25/6}}+\frac {1296 b^2 (a+b x)^{13/6}}{14725 (b c-a d)^3 (c+d x)^{19/6}}+\frac {7776 b^3 (a+b x)^{13/6}}{191425 (b c-a d)^4 (c+d x)^{13/6}} \] Output:
6/31*(b*x+a)^(13/6)/(-a*d+b*c)/(d*x+c)^(31/6)+108/775*b*(b*x+a)^(13/6)/(-a *d+b*c)^2/(d*x+c)^(25/6)+1296/14725*b^2*(b*x+a)^(13/6)/(-a*d+b*c)^3/(d*x+c )^(19/6)+7776/191425*b^3*(b*x+a)^(13/6)/(-a*d+b*c)^4/(d*x+c)^(13/6)
Time = 1.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\frac {6 (a+b x)^{13/6} \left (-6175 a^3 d^3+741 a^2 b d^2 (31 c+6 d x)-39 a b^2 d \left (775 c^2+372 c d x+72 d^2 x^2\right )+b^3 \left (14725 c^3+13950 c^2 d x+6696 c d^2 x^2+1296 d^3 x^3\right )\right )}{191425 (b c-a d)^4 (c+d x)^{31/6}} \] Input:
Integrate[(a + b*x)^(7/6)/(c + d*x)^(37/6),x]
Output:
(6*(a + b*x)^(13/6)*(-6175*a^3*d^3 + 741*a^2*b*d^2*(31*c + 6*d*x) - 39*a*b ^2*d*(775*c^2 + 372*c*d*x + 72*d^2*x^2) + b^3*(14725*c^3 + 13950*c^2*d*x + 6696*c*d^2*x^2 + 1296*d^3*x^3)))/(191425*(b*c - a*d)^4*(c + d*x)^(31/6))
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.
| \(\displaystyle \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {18 b \int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}}dx}{31 (b c-a d)}+\frac {6 (a+b x)^{13/6}}{31 (c+d x)^{31/6} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {18 b \left (\frac {12 b \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}}dx}{25 (b c-a d)}+\frac {6 (a+b x)^{13/6}}{25 (c+d x)^{25/6} (b c-a d)}\right )}{31 (b c-a d)}+\frac {6 (a+b x)^{13/6}}{31 (c+d x)^{31/6} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {18 b \left (\frac {12 b \left (\frac {6 b \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}}dx}{19 (b c-a d)}+\frac {6 (a+b x)^{13/6}}{19 (c+d x)^{19/6} (b c-a d)}\right )}{25 (b c-a d)}+\frac {6 (a+b x)^{13/6}}{25 (c+d x)^{25/6} (b c-a d)}\right )}{31 (b c-a d)}+\frac {6 (a+b x)^{13/6}}{31 (c+d x)^{31/6} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {6 (a+b x)^{13/6}}{31 (c+d x)^{31/6} (b c-a d)}+\frac {18 b \left (\frac {6 (a+b x)^{13/6}}{25 (c+d x)^{25/6} (b c-a d)}+\frac {12 b \left (\frac {36 b (a+b x)^{13/6}}{247 (c+d x)^{13/6} (b c-a d)^2}+\frac {6 (a+b x)^{13/6}}{19 (c+d x)^{19/6} (b c-a d)}\right )}{25 (b c-a d)}\right )}{31 (b c-a d)}\) |
Input:
Int[(a + b*x)^(7/6)/(c + d*x)^(37/6),x]
Output:
(6*(a + b*x)^(13/6))/(31*(b*c - a*d)*(c + d*x)^(31/6)) + (18*b*((6*(a + b* x)^(13/6))/(25*(b*c - a*d)*(c + d*x)^(25/6)) + (12*b*((6*(a + b*x)^(13/6)) /(19*(b*c - a*d)*(c + d*x)^(19/6)) + (36*b*(a + b*x)^(13/6))/(247*(b*c - a *d)^2*(c + d*x)^(13/6))))/(25*(b*c - a*d))))/(31*(b*c - a*d))
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]
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])
Time = 0.19 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
| method | result | size |
| gosper | \(-\frac {6 \left (b x +a \right )^{\frac {13}{6}} \left (-1296 d^{3} x^{3} b^{3}+2808 x^{2} a \,b^{2} d^{3}-6696 x^{2} b^{3} c \,d^{2}-4446 x \,a^{2} b \,d^{3}+14508 x a \,b^{2} c \,d^{2}-13950 x \,b^{3} c^{2} d +6175 a^{3} d^{3}-22971 a^{2} b c \,d^{2}+30225 a \,b^{2} c^{2} d -14725 b^{3} c^{3}\right )}{191425 \left (x d +c \right )^{\frac {31}{6}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\) | \(171\) |
| orering | \(-\frac {6 \left (b x +a \right )^{\frac {13}{6}} \left (-1296 d^{3} x^{3} b^{3}+2808 x^{2} a \,b^{2} d^{3}-6696 x^{2} b^{3} c \,d^{2}-4446 x \,a^{2} b \,d^{3}+14508 x a \,b^{2} c \,d^{2}-13950 x \,b^{3} c^{2} d +6175 a^{3} d^{3}-22971 a^{2} b c \,d^{2}+30225 a \,b^{2} c^{2} d -14725 b^{3} c^{3}\right )}{191425 \left (x d +c \right )^{\frac {31}{6}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\) | \(171\) |
Input:
int((b*x+a)^(7/6)/(d*x+c)^(37/6),x,method=_RETURNVERBOSE)
Output:
-6/191425*(b*x+a)^(13/6)*(-1296*b^3*d^3*x^3+2808*a*b^2*d^3*x^2-6696*b^3*c* d^2*x^2-4446*a^2*b*d^3*x+14508*a*b^2*c*d^2*x-13950*b^3*c^2*d*x+6175*a^3*d^ 3-22971*a^2*b*c*d^2+30225*a*b^2*c^2*d-14725*b^3*c^3)/(d*x+c)^(31/6)/(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)
Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (112) = 224\).
Time = 0.11 (sec) , antiderivative size = 649, normalized size of antiderivative = 4.77 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\frac {6 \, {\left (1296 \, b^{5} d^{3} x^{5} + 14725 \, a^{2} b^{3} c^{3} - 30225 \, a^{3} b^{2} c^{2} d + 22971 \, a^{4} b c d^{2} - 6175 \, a^{5} d^{3} + 216 \, {\left (31 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4} + 18 \, {\left (775 \, b^{5} c^{2} d - 62 \, a b^{4} c d^{2} + 7 \, a^{2} b^{3} d^{3}\right )} x^{3} + {\left (14725 \, b^{5} c^{3} - 2325 \, a b^{4} c^{2} d + 651 \, a^{2} b^{3} c d^{2} - 91 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (14725 \, a b^{4} c^{3} - 23250 \, a^{2} b^{3} c^{2} d + 15717 \, a^{3} b^{2} c d^{2} - 3952 \, a^{4} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{191425 \, {\left (b^{4} c^{10} - 4 \, a b^{3} c^{9} d + 6 \, a^{2} b^{2} c^{8} d^{2} - 4 \, a^{3} b c^{7} d^{3} + a^{4} c^{6} d^{4} + {\left (b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} x^{6} + 6 \, {\left (b^{4} c^{5} d^{5} - 4 \, a b^{3} c^{4} d^{6} + 6 \, a^{2} b^{2} c^{3} d^{7} - 4 \, a^{3} b c^{2} d^{8} + a^{4} c d^{9}\right )} x^{5} + 15 \, {\left (b^{4} c^{6} d^{4} - 4 \, a b^{3} c^{5} d^{5} + 6 \, a^{2} b^{2} c^{4} d^{6} - 4 \, a^{3} b c^{3} d^{7} + a^{4} c^{2} d^{8}\right )} x^{4} + 20 \, {\left (b^{4} c^{7} d^{3} - 4 \, a b^{3} c^{6} d^{4} + 6 \, a^{2} b^{2} c^{5} d^{5} - 4 \, a^{3} b c^{4} d^{6} + a^{4} c^{3} d^{7}\right )} x^{3} + 15 \, {\left (b^{4} c^{8} d^{2} - 4 \, a b^{3} c^{7} d^{3} + 6 \, a^{2} b^{2} c^{6} d^{4} - 4 \, a^{3} b c^{5} d^{5} + a^{4} c^{4} d^{6}\right )} x^{2} + 6 \, {\left (b^{4} c^{9} d - 4 \, a b^{3} c^{8} d^{2} + 6 \, a^{2} b^{2} c^{7} d^{3} - 4 \, a^{3} b c^{6} d^{4} + a^{4} c^{5} d^{5}\right )} x\right )}} \] Input:
integrate((b*x+a)^(7/6)/(d*x+c)^(37/6),x, algorithm="fricas")
Output:
6/191425*(1296*b^5*d^3*x^5 + 14725*a^2*b^3*c^3 - 30225*a^3*b^2*c^2*d + 229 71*a^4*b*c*d^2 - 6175*a^5*d^3 + 216*(31*b^5*c*d^2 - a*b^4*d^3)*x^4 + 18*(7 75*b^5*c^2*d - 62*a*b^4*c*d^2 + 7*a^2*b^3*d^3)*x^3 + (14725*b^5*c^3 - 2325 *a*b^4*c^2*d + 651*a^2*b^3*c*d^2 - 91*a^3*b^2*d^3)*x^2 + 2*(14725*a*b^4*c^ 3 - 23250*a^2*b^3*c^2*d + 15717*a^3*b^2*c*d^2 - 3952*a^4*b*d^3)*x)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b^4*c^10 - 4*a*b^3*c^9*d + 6*a^2*b^2*c^8*d^2 - 4 *a^3*b*c^7*d^3 + a^4*c^6*d^4 + (b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2* c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^10)*x^6 + 6*(b^4*c^5*d^5 - 4*a*b^3*c^4*d^6 + 6*a^2*b^2*c^3*d^7 - 4*a^3*b*c^2*d^8 + a^4*c*d^9)*x^5 + 15*(b^4*c^6*d^4 - 4*a*b^3*c^5*d^5 + 6*a^2*b^2*c^4*d^6 - 4*a^3*b*c^3*d^7 + a^4*c^2*d^8)*x^4 + 20*(b^4*c^7*d^3 - 4*a*b^3*c^6*d^4 + 6*a^2*b^2*c^5*d^5 - 4*a^3*b*c^4*d^6 + a^4*c^3*d^7)*x^3 + 15*(b^4*c^8*d^2 - 4*a*b^3*c^7*d^3 + 6*a^2*b^2*c^6*d^ 4 - 4*a^3*b*c^5*d^5 + a^4*c^4*d^6)*x^2 + 6*(b^4*c^9*d - 4*a*b^3*c^8*d^2 + 6*a^2*b^2*c^7*d^3 - 4*a^3*b*c^6*d^4 + a^4*c^5*d^5)*x)
Timed out. \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**(7/6)/(d*x+c)**(37/6),x)
Output:
Timed out
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {37}{6}}} \,d x } \] Input:
integrate((b*x+a)^(7/6)/(d*x+c)^(37/6),x, algorithm="maxima")
Output:
integrate((b*x + a)^(7/6)/(d*x + c)^(37/6), x)
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {37}{6}}} \,d x } \] Input:
integrate((b*x+a)^(7/6)/(d*x+c)^(37/6),x, algorithm="giac")
Output:
integrate((b*x + a)^(7/6)/(d*x + c)^(37/6), x)
Time = 0.81 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.83 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx=\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {7776\,b^5\,x^5\,{\left (a+b\,x\right )}^{1/6}}{191425\,d^3\,{\left (a\,d-b\,c\right )}^4}-\frac {{\left (a+b\,x\right )}^{1/6}\,\left (37050\,a^5\,d^3-137826\,a^4\,b\,c\,d^2+181350\,a^3\,b^2\,c^2\,d-88350\,a^2\,b^3\,c^3\right )}{191425\,d^6\,{\left (a\,d-b\,c\right )}^4}+\frac {x^2\,{\left (a+b\,x\right )}^{1/6}\,\left (-546\,a^3\,b^2\,d^3+3906\,a^2\,b^3\,c\,d^2-13950\,a\,b^4\,c^2\,d+88350\,b^5\,c^3\right )}{191425\,d^6\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,{\left (a+b\,x\right )}^{1/6}\,\left (-47424\,a^4\,b\,d^3+188604\,a^3\,b^2\,c\,d^2-279000\,a^2\,b^3\,c^2\,d+176700\,a\,b^4\,c^3\right )}{191425\,d^6\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b^3\,x^3\,{\left (a+b\,x\right )}^{1/6}\,\left (7\,a^2\,d^2-62\,a\,b\,c\,d+775\,b^2\,c^2\right )}{191425\,d^5\,{\left (a\,d-b\,c\right )}^4}-\frac {1296\,b^4\,x^4\,\left (a\,d-31\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{191425\,d^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^6+\frac {c^6}{d^6}+\frac {6\,c\,x^5}{d}+\frac {6\,c^5\,x}{d^5}+\frac {15\,c^2\,x^4}{d^2}+\frac {20\,c^3\,x^3}{d^3}+\frac {15\,c^4\,x^2}{d^4}} \] Input:
int((a + b*x)^(7/6)/(c + d*x)^(37/6),x)
Output:
((c + d*x)^(5/6)*((7776*b^5*x^5*(a + b*x)^(1/6))/(191425*d^3*(a*d - b*c)^4 ) - ((a + b*x)^(1/6)*(37050*a^5*d^3 - 88350*a^2*b^3*c^3 + 181350*a^3*b^2*c ^2*d - 137826*a^4*b*c*d^2))/(191425*d^6*(a*d - b*c)^4) + (x^2*(a + b*x)^(1 /6)*(88350*b^5*c^3 - 546*a^3*b^2*d^3 + 3906*a^2*b^3*c*d^2 - 13950*a*b^4*c^ 2*d))/(191425*d^6*(a*d - b*c)^4) + (x*(a + b*x)^(1/6)*(176700*a*b^4*c^3 - 47424*a^4*b*d^3 - 279000*a^2*b^3*c^2*d + 188604*a^3*b^2*c*d^2))/(191425*d^ 6*(a*d - b*c)^4) + (108*b^3*x^3*(a + b*x)^(1/6)*(7*a^2*d^2 + 775*b^2*c^2 - 62*a*b*c*d))/(191425*d^5*(a*d - b*c)^4) - (1296*b^4*x^4*(a*d - 31*b*c)*(a + b*x)^(1/6))/(191425*d^4*(a*d - b*c)^4)))/(x^6 + c^6/d^6 + (6*c*x^5)/d + (6*c^5*x)/d^5 + (15*c^2*x^4)/d^2 + (20*c^3*x^3)/d^3 + (15*c^4*x^2)/d^4)
Time = 17.03 (sec) , antiderivative size = 2504, normalized size of antiderivative = 18.41 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^(7/6)/(d*x+c)^(37/6),x)
Output:
(2*( - 4823910*(c + d*x)*(a + b*x)**(3/2)*a**5*b*d**5 + 27603485*(c + d*x) *(a + b*x)**(3/2)*a**4*b**2*c*d**4 + 3483935*(c + d*x)*(a + b*x)**(3/2)*a* *4*b**2*d**5*x - 67151890*(c + d*x)*(a + b*x)**(3/2)*a**3*b**3*c**2*d**3 - 23889840*(c + d*x)*(a + b*x)**(3/2)*a**3*b**3*c*d**4*x - 4977050*(c + d*x )*(a + b*x)**(3/2)*a**3*b**3*d**5*x**2 + 92037140*(c + d*x)*(a + b*x)**(3/ 2)*a**2*b**4*c**3*d**2 + 74655750*(c + d*x)*(a + b*x)**(3/2)*a**2*b**4*c** 2*d**3*x + 38820990*(c + d*x)*(a + b*x)**(3/2)*a**2*b**4*c*d**4*x**2 + 796 3280*(c + d*x)*(a + b*x)**(3/2)*a**2*b**4*d**5*x**3 - 84839560*(c + d*x)*( a + b*x)**(3/2)*a*b**5*c**4*d - 155283960*(c + d*x)*(a + b*x)**(3/2)*a*b** 5*c**3*d**2*x - 158270190*(c + d*x)*(a + b*x)**(3/2)*a*b**5*c**2*d**3*x**2 - 79632800*(c + d*x)*(a + b*x)**(3/2)*a*b**5*c*d**4*x**3 - 15926560*(c + d*x)*(a + b*x)**(3/2)*a*b**5*d**5*x**4 + 37174735*(c + d*x)*(a + b*x)**(3/ 2)*b**6*c**5 + 101034115*(c + d*x)*(a + b*x)**(3/2)*b**6*c**4*d*x + 124426 250*(c + d*x)*(a + b*x)**(3/2)*b**6*c**3*d**2*x**2 + 71669520*(c + d*x)*(a + b*x)**(3/2)*b**6*c**2*d**3*x**3 + 15926560*(c + d*x)*(a + b*x)**(3/2)*b **6*c*d**4*x**4 - 2567565*sqrt(a + b*x)*a**7*d**6 + 18486468*sqrt(a + b*x) *a**6*b*c*d**5 + 513513*sqrt(a + b*x)*a**6*b*d**6*x - 57810753*sqrt(a + b* x)*a**5*b**2*c**2*d**4 - 4702698*sqrt(a + b*x)*a**5*b**2*c*d**5*x - 810810 *sqrt(a + b*x)*a**5*b**2*d**6*x**2 + 103118400*sqrt(a + b*x)*a**4*b**3*c** 3*d**3 + 20301435*sqrt(a + b*x)*a**4*b**3*c**2*d**4*x + 8544690*sqrt(a ...