Integrand size = 17, antiderivative size = 154 \[ \int \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx=-\frac {2 (b c-a d)^5 \sqrt {c+d x}}{d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{3/2}}{3 d^6}-\frac {4 b^2 (b c-a d)^3 (c+d x)^{5/2}}{d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{7/2}}{7 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{9/2}}{9 d^6}+\frac {2 b^5 (c+d x)^{11/2}}{11 d^6} \]
10/3*b*(-a*d+b*c)^4*(d*x+c)^(3/2)/d^6-4*b^2*(-a*d+b*c)^3*(d*x+c)^(5/2)/d^6 +20/7*b^3*(-a*d+b*c)^2*(d*x+c)^(7/2)/d^6-10/9*b^4*(-a*d+b*c)*(d*x+c)^(9/2) /d^6+2/11*b^5*(d*x+c)^(11/2)/d^6-2*(-a*d+b*c)^5*(d*x+c)^(1/2)/d^6
Time = 0.10 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {c+d x} \left (693 a^5 d^5+1155 a^4 b d^4 (-2 c+d x)+462 a^3 b^2 d^3 \left (8 c^2-4 c d x+3 d^2 x^2\right )+198 a^2 b^3 d^2 \left (-16 c^3+8 c^2 d x-6 c d^2 x^2+5 d^3 x^3\right )+11 a b^4 d \left (128 c^4-64 c^3 d x+48 c^2 d^2 x^2-40 c d^3 x^3+35 d^4 x^4\right )+b^5 \left (-256 c^5+128 c^4 d x-96 c^3 d^2 x^2+80 c^2 d^3 x^3-70 c d^4 x^4+63 d^5 x^5\right )\right )}{693 d^6} \]
(2*Sqrt[c + d*x]*(693*a^5*d^5 + 1155*a^4*b*d^4*(-2*c + d*x) + 462*a^3*b^2* d^3*(8*c^2 - 4*c*d*x + 3*d^2*x^2) + 198*a^2*b^3*d^2*(-16*c^3 + 8*c^2*d*x - 6*c*d^2*x^2 + 5*d^3*x^3) + 11*a*b^4*d*(128*c^4 - 64*c^3*d*x + 48*c^2*d^2* x^2 - 40*c*d^3*x^3 + 35*d^4*x^4) + b^5*(-256*c^5 + 128*c^4*d*x - 96*c^3*d^ 2*x^2 + 80*c^2*d^3*x^3 - 70*c*d^4*x^4 + 63*d^5*x^5)))/(693*d^6)
Time = 0.26 (sec) , antiderivative size = 154, 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.
| \(\displaystyle \int \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (c+d x)^{7/2} (b c-a d)}{d^5}+\frac {10 b^3 (c+d x)^{5/2} (b c-a d)^2}{d^5}-\frac {10 b^2 (c+d x)^{3/2} (b c-a d)^3}{d^5}+\frac {5 b \sqrt {c+d x} (b c-a d)^4}{d^5}+\frac {(a d-b c)^5}{d^5 \sqrt {c+d x}}+\frac {b^5 (c+d x)^{9/2}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b^4 (c+d x)^{9/2} (b c-a d)}{9 d^6}+\frac {20 b^3 (c+d x)^{7/2} (b c-a d)^2}{7 d^6}-\frac {4 b^2 (c+d x)^{5/2} (b c-a d)^3}{d^6}+\frac {10 b (c+d x)^{3/2} (b c-a d)^4}{3 d^6}-\frac {2 \sqrt {c+d x} (b c-a d)^5}{d^6}+\frac {2 b^5 (c+d x)^{11/2}}{11 d^6}\) |
(-2*(b*c - a*d)^5*Sqrt[c + d*x])/d^6 + (10*b*(b*c - a*d)^4*(c + d*x)^(3/2) )/(3*d^6) - (4*b^2*(b*c - a*d)^3*(c + d*x)^(5/2))/d^6 + (20*b^3*(b*c - a*d )^2*(c + d*x)^(7/2))/(7*d^6) - (10*b^4*(b*c - a*d)*(c + d*x)^(9/2))/(9*d^6 ) + (2*b^5*(c + d*x)^(11/2))/(11*d^6)
3.15.13.3.1 Defintions of rubi rules used
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 = 0.75 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {7}{2}}}{7}+4 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {5}{2}}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \left (a d -b c \right )^{5} \sqrt {d x +c}}{d^{6}}\) | \(121\) |
| default | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {7}{2}}}{7}+4 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {5}{2}}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {3}{2}}}{3}+2 \left (a d -b c \right )^{5} \sqrt {d x +c}}{d^{6}}\) | \(121\) |
| pseudoelliptic | \(\frac {2 \sqrt {d x +c}\, \left (\left (\frac {1}{11} b^{5} x^{5}+\frac {5}{9} a \,b^{4} x^{4}+\frac {10}{7} a^{2} b^{3} x^{3}+2 a^{3} b^{2} x^{2}+\frac {5}{3} a^{4} b x +a^{5}\right ) d^{5}-\frac {10 b c \left (\frac {1}{33} b^{4} x^{4}+\frac {4}{21} a \,b^{3} x^{3}+\frac {18}{35} a^{2} b^{2} x^{2}+\frac {4}{5} a^{3} b x +a^{4}\right ) d^{4}}{3}+\frac {16 b^{2} \left (\frac {5}{231} b^{3} x^{3}+\frac {1}{7} a \,b^{2} x^{2}+\frac {3}{7} a^{2} b x +a^{3}\right ) c^{2} d^{3}}{3}-\frac {32 b^{3} c^{3} \left (\frac {1}{33} b^{2} x^{2}+\frac {2}{9} a b x +a^{2}\right ) d^{2}}{7}+\frac {128 \left (\frac {b x}{11}+a \right ) b^{4} c^{4} d}{63}-\frac {256 b^{5} c^{5}}{693}\right )}{d^{6}}\) | \(204\) |
| gosper | \(\frac {2 \sqrt {d x +c}\, \left (63 x^{5} b^{5} d^{5}+385 x^{4} a \,b^{4} d^{5}-70 x^{4} b^{5} c \,d^{4}+990 x^{3} a^{2} b^{3} d^{5}-440 x^{3} a \,b^{4} c \,d^{4}+80 x^{3} b^{5} c^{2} d^{3}+1386 x^{2} a^{3} b^{2} d^{5}-1188 x^{2} a^{2} b^{3} c \,d^{4}+528 x^{2} a \,b^{4} c^{2} d^{3}-96 x^{2} b^{5} c^{3} d^{2}+1155 x \,a^{4} b \,d^{5}-1848 x \,a^{3} b^{2} c \,d^{4}+1584 x \,a^{2} b^{3} c^{2} d^{3}-704 x a \,b^{4} c^{3} d^{2}+128 x \,b^{5} c^{4} d +693 a^{5} d^{5}-2310 a^{4} b c \,d^{4}+3696 a^{3} b^{2} c^{2} d^{3}-3168 a^{2} b^{3} c^{3} d^{2}+1408 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{693 d^{6}}\) | \(273\) |
| trager | \(\frac {2 \sqrt {d x +c}\, \left (63 x^{5} b^{5} d^{5}+385 x^{4} a \,b^{4} d^{5}-70 x^{4} b^{5} c \,d^{4}+990 x^{3} a^{2} b^{3} d^{5}-440 x^{3} a \,b^{4} c \,d^{4}+80 x^{3} b^{5} c^{2} d^{3}+1386 x^{2} a^{3} b^{2} d^{5}-1188 x^{2} a^{2} b^{3} c \,d^{4}+528 x^{2} a \,b^{4} c^{2} d^{3}-96 x^{2} b^{5} c^{3} d^{2}+1155 x \,a^{4} b \,d^{5}-1848 x \,a^{3} b^{2} c \,d^{4}+1584 x \,a^{2} b^{3} c^{2} d^{3}-704 x a \,b^{4} c^{3} d^{2}+128 x \,b^{5} c^{4} d +693 a^{5} d^{5}-2310 a^{4} b c \,d^{4}+3696 a^{3} b^{2} c^{2} d^{3}-3168 a^{2} b^{3} c^{3} d^{2}+1408 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{693 d^{6}}\) | \(273\) |
| risch | \(\frac {2 \sqrt {d x +c}\, \left (63 x^{5} b^{5} d^{5}+385 x^{4} a \,b^{4} d^{5}-70 x^{4} b^{5} c \,d^{4}+990 x^{3} a^{2} b^{3} d^{5}-440 x^{3} a \,b^{4} c \,d^{4}+80 x^{3} b^{5} c^{2} d^{3}+1386 x^{2} a^{3} b^{2} d^{5}-1188 x^{2} a^{2} b^{3} c \,d^{4}+528 x^{2} a \,b^{4} c^{2} d^{3}-96 x^{2} b^{5} c^{3} d^{2}+1155 x \,a^{4} b \,d^{5}-1848 x \,a^{3} b^{2} c \,d^{4}+1584 x \,a^{2} b^{3} c^{2} d^{3}-704 x a \,b^{4} c^{3} d^{2}+128 x \,b^{5} c^{4} d +693 a^{5} d^{5}-2310 a^{4} b c \,d^{4}+3696 a^{3} b^{2} c^{2} d^{3}-3168 a^{2} b^{3} c^{3} d^{2}+1408 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{693 d^{6}}\) | \(273\) |
2/d^6*(1/11*b^5*(d*x+c)^(11/2)+5/9*(a*d-b*c)*b^4*(d*x+c)^(9/2)+10/7*(a*d-b *c)^2*b^3*(d*x+c)^(7/2)+2*(a*d-b*c)^3*b^2*(d*x+c)^(5/2)+5/3*(a*d-b*c)^4*b* (d*x+c)^(3/2)+(a*d-b*c)^5*(d*x+c)^(1/2))
Time = 0.24 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (63 \, b^{5} d^{5} x^{5} - 256 \, b^{5} c^{5} + 1408 \, a b^{4} c^{4} d - 3168 \, a^{2} b^{3} c^{3} d^{2} + 3696 \, a^{3} b^{2} c^{2} d^{3} - 2310 \, a^{4} b c d^{4} + 693 \, a^{5} d^{5} - 35 \, {\left (2 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 10 \, {\left (8 \, b^{5} c^{2} d^{3} - 44 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} c^{3} d^{2} - 88 \, a b^{4} c^{2} d^{3} + 198 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} + {\left (128 \, b^{5} c^{4} d - 704 \, a b^{4} c^{3} d^{2} + 1584 \, a^{2} b^{3} c^{2} d^{3} - 1848 \, a^{3} b^{2} c d^{4} + 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}}{693 \, d^{6}} \]
2/693*(63*b^5*d^5*x^5 - 256*b^5*c^5 + 1408*a*b^4*c^4*d - 3168*a^2*b^3*c^3* d^2 + 3696*a^3*b^2*c^2*d^3 - 2310*a^4*b*c*d^4 + 693*a^5*d^5 - 35*(2*b^5*c* d^4 - 11*a*b^4*d^5)*x^4 + 10*(8*b^5*c^2*d^3 - 44*a*b^4*c*d^4 + 99*a^2*b^3* d^5)*x^3 - 6*(16*b^5*c^3*d^2 - 88*a*b^4*c^2*d^3 + 198*a^2*b^3*c*d^4 - 231* a^3*b^2*d^5)*x^2 + (128*b^5*c^4*d - 704*a*b^4*c^3*d^2 + 1584*a^2*b^3*c^2*d ^3 - 1848*a^3*b^2*c*d^4 + 1155*a^4*b*d^5)*x)*sqrt(d*x + c)/d^6
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (143) = 286\).
Time = 1.02 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.16 \[ \int \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (c + d x\right )^{\frac {11}{2}}}{11 d^{5}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (5 a b^{4} d - 5 b^{5} c\right )}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (10 a^{2} b^{3} d^{2} - 20 a b^{4} c d + 10 b^{5} c^{2}\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (10 a^{3} b^{2} d^{3} - 30 a^{2} b^{3} c d^{2} + 30 a b^{4} c^{2} d - 10 b^{5} c^{3}\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (5 a^{4} b d^{4} - 20 a^{3} b^{2} c d^{3} + 30 a^{2} b^{3} c^{2} d^{2} - 20 a b^{4} c^{3} d + 5 b^{5} c^{4}\right )}{3 d^{5}} + \frac {\sqrt {c + d x} \left (a^{5} d^{5} - 5 a^{4} b c d^{4} + 10 a^{3} b^{2} c^{2} d^{3} - 10 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d - b^{5} c^{5}\right )}{d^{5}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{6}}{6 b} & \text {otherwise} \end {cases}}{\sqrt {c}} & \text {otherwise} \end {cases} \]
Piecewise((2*(b**5*(c + d*x)**(11/2)/(11*d**5) + (c + d*x)**(9/2)*(5*a*b** 4*d - 5*b**5*c)/(9*d**5) + (c + d*x)**(7/2)*(10*a**2*b**3*d**2 - 20*a*b**4 *c*d + 10*b**5*c**2)/(7*d**5) + (c + d*x)**(5/2)*(10*a**3*b**2*d**3 - 30*a **2*b**3*c*d**2 + 30*a*b**4*c**2*d - 10*b**5*c**3)/(5*d**5) + (c + d*x)**( 3/2)*(5*a**4*b*d**4 - 20*a**3*b**2*c*d**3 + 30*a**2*b**3*c**2*d**2 - 20*a* b**4*c**3*d + 5*b**5*c**4)/(3*d**5) + sqrt(c + d*x)*(a**5*d**5 - 5*a**4*b* c*d**4 + 10*a**3*b**2*c**2*d**3 - 10*a**2*b**3*c**3*d**2 + 5*a*b**4*c**4*d - b**5*c**5)/d**5)/d, Ne(d, 0)), (Piecewise((a**5*x, Eq(b, 0)), ((a + b*x )**6/(6*b), True))/sqrt(c), True))
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (134) = 268\).
Time = 0.21 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (693 \, \sqrt {d x + c} a^{5} + \frac {1155 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{4} b}{d} + \frac {462 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{3} b^{2}}{d^{2}} + \frac {198 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a^{2} b^{3}}{d^{3}} + \frac {11 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a b^{4}}{d^{4}} + \frac {{\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b^{5}}{d^{5}}\right )}}{693 \, d} \]
2/693*(693*sqrt(d*x + c)*a^5 + 1155*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)* a^4*b/d + 462*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c) *c^2)*a^3*b^2/d^2 + 198*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d* x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^2*b^3/d^3 + 11*(35*(d*x + c)^(9 /2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2 )*c^3 + 315*sqrt(d*x + c)*c^4)*a*b^4/d^4 + (63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155* (d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*b^5/d^5)/d
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (134) = 268\).
Time = 0.33 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (693 \, \sqrt {d x + c} a^{5} + \frac {1155 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{4} b}{d} + \frac {462 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{3} b^{2}}{d^{2}} + \frac {198 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a^{2} b^{3}}{d^{3}} + \frac {11 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a b^{4}}{d^{4}} + \frac {{\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b^{5}}{d^{5}}\right )}}{693 \, d} \]
2/693*(693*sqrt(d*x + c)*a^5 + 1155*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)* a^4*b/d + 462*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c) *c^2)*a^3*b^2/d^2 + 198*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d* x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^2*b^3/d^3 + 11*(35*(d*x + c)^(9 /2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2 )*c^3 + 315*sqrt(d*x + c)*c^4)*a*b^4/d^4 + (63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155* (d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*b^5/d^5)/d
Time = 0.06 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^5}{\sqrt {c+d x}} \, dx=\frac {2\,b^5\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {2\,{\left (a\,d-b\,c\right )}^5\,\sqrt {c+d\,x}}{d^6}+\frac {4\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{5/2}}{d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}+\frac {10\,b\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6} \]