Integrand size = 17, antiderivative size = 152 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\frac {2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}-\frac {10 b (b c-a d)^4}{d^6 \sqrt {c+d x}}-\frac {20 b^2 (b c-a d)^3 \sqrt {c+d x}}{d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{3/2}}{3 d^6}-\frac {2 b^4 (b c-a d) (c+d x)^{5/2}}{d^6}+\frac {2 b^5 (c+d x)^{7/2}}{7 d^6} \] Output:
2/3*(-a*d+b*c)^5/d^6/(d*x+c)^(3/2)-10*b*(-a*d+b*c)^4/d^6/(d*x+c)^(1/2)-20* b^2*(-a*d+b*c)^3*(d*x+c)^(1/2)/d^6+20/3*b^3*(-a*d+b*c)^2*(d*x+c)^(3/2)/d^6 -2*b^4*(-a*d+b*c)*(d*x+c)^(5/2)/d^6+2/7*b^5*(d*x+c)^(7/2)/d^6
Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=-\frac {2 \left (7 a^5 d^5+35 a^4 b d^4 (2 c+3 d x)-70 a^3 b^2 d^3 \left (8 c^2+12 c d x+3 d^2 x^2\right )+70 a^2 b^3 d^2 \left (16 c^3+24 c^2 d x+6 c d^2 x^2-d^3 x^3\right )-7 a b^4 d \left (128 c^4+192 c^3 d x+48 c^2 d^2 x^2-8 c d^3 x^3+3 d^4 x^4\right )+b^5 \left (256 c^5+384 c^4 d x+96 c^3 d^2 x^2-16 c^2 d^3 x^3+6 c d^4 x^4-3 d^5 x^5\right )\right )}{21 d^6 (c+d x)^{3/2}} \] Input:
Integrate[(a + b*x)^5/(c + d*x)^(5/2),x]
Output:
(-2*(7*a^5*d^5 + 35*a^4*b*d^4*(2*c + 3*d*x) - 70*a^3*b^2*d^3*(8*c^2 + 12*c *d*x + 3*d^2*x^2) + 70*a^2*b^3*d^2*(16*c^3 + 24*c^2*d*x + 6*c*d^2*x^2 - d^ 3*x^3) - 7*a*b^4*d*(128*c^4 + 192*c^3*d*x + 48*c^2*d^2*x^2 - 8*c*d^3*x^3 + 3*d^4*x^4) + b^5*(256*c^5 + 384*c^4*d*x + 96*c^3*d^2*x^2 - 16*c^2*d^3*x^3 + 6*c*d^4*x^4 - 3*d^5*x^5)))/(21*d^6*(c + d*x)^(3/2))
Time = 0.24 (sec) , antiderivative size = 152, 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.
| \(\displaystyle \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (c+d x)^{3/2} (b c-a d)}{d^5}+\frac {10 b^3 \sqrt {c+d x} (b c-a d)^2}{d^5}-\frac {10 b^2 (b c-a d)^3}{d^5 \sqrt {c+d x}}+\frac {5 b (b c-a d)^4}{d^5 (c+d x)^{3/2}}+\frac {(a d-b c)^5}{d^5 (c+d x)^{5/2}}+\frac {b^5 (c+d x)^{5/2}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^4 (c+d x)^{5/2} (b c-a d)}{d^6}+\frac {20 b^3 (c+d x)^{3/2} (b c-a d)^2}{3 d^6}-\frac {20 b^2 \sqrt {c+d x} (b c-a d)^3}{d^6}-\frac {10 b (b c-a d)^4}{d^6 \sqrt {c+d x}}+\frac {2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}+\frac {2 b^5 (c+d x)^{7/2}}{7 d^6}\) |
Input:
Int[(a + b*x)^5/(c + d*x)^(5/2),x]
Output:
(2*(b*c - a*d)^5)/(3*d^6*(c + d*x)^(3/2)) - (10*b*(b*c - a*d)^4)/(d^6*Sqrt [c + d*x]) - (20*b^2*(b*c - a*d)^3*Sqrt[c + d*x])/d^6 + (20*b^3*(b*c - a*d )^2*(c + d*x)^(3/2))/(3*d^6) - (2*b^4*(b*c - a*d)*(c + d*x)^(5/2))/d^6 + ( 2*b^5*(c + d*x)^(7/2))/(7*d^6)
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 = 0.24 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.28
| method | result | size |
| risch | \(\frac {2 b^{2} \left (3 d^{3} x^{3} b^{3}+21 x^{2} a \,b^{2} d^{3}-12 x^{2} b^{3} c \,d^{2}+70 x \,a^{2} b \,d^{3}-98 x a \,b^{2} c \,d^{2}+37 x \,b^{3} c^{2} d +210 a^{3} d^{3}-560 a^{2} b c \,d^{2}+511 a \,b^{2} c^{2} d -158 b^{3} c^{3}\right ) \sqrt {x d +c}}{21 d^{6}}-\frac {2 \left (15 b d x +a d +14 b c \right ) \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{3 d^{6} \left (x d +c \right )^{\frac {3}{2}}}\) | \(194\) |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {3}{7} b^{5} x^{5}-3 a \,b^{4} x^{4}-10 a^{2} b^{3} x^{3}-30 a^{3} b^{2} x^{2}+15 a^{4} b x +a^{5}\right ) d^{5}+10 c \left (\frac {3}{35} b^{4} x^{4}+\frac {4}{5} a \,x^{3} b^{3}+6 a^{2} b^{2} x^{2}-12 a^{3} b x +a^{4}\right ) b \,d^{4}-80 c^{2} \left (\frac {1}{35} b^{3} x^{3}+\frac {3}{5} a \,b^{2} x^{2}-3 a^{2} b x +a^{3}\right ) b^{2} d^{3}+160 c^{3} b^{3} \left (\frac {3}{35} b^{2} x^{2}-\frac {6}{5} a b x +a^{2}\right ) d^{2}-128 c^{4} \left (-\frac {3 b x}{7}+a \right ) b^{4} d +\frac {256 c^{5} b^{5}}{7}\right )}{3 \left (x d +c \right )^{\frac {3}{2}} d^{6}}\) | \(204\) |
| gosper | \(-\frac {2 \left (-3 x^{5} b^{5} d^{5}-21 x^{4} a \,b^{4} d^{5}+6 x^{4} b^{5} c \,d^{4}-70 x^{3} a^{2} b^{3} d^{5}+56 x^{3} a \,b^{4} c \,d^{4}-16 x^{3} b^{5} c^{2} d^{3}-210 x^{2} a^{3} b^{2} d^{5}+420 x^{2} a^{2} b^{3} c \,d^{4}-336 x^{2} a \,b^{4} c^{2} d^{3}+96 x^{2} b^{5} c^{3} d^{2}+105 x \,a^{4} b \,d^{5}-840 x \,a^{3} b^{2} c \,d^{4}+1680 x \,a^{2} b^{3} c^{2} d^{3}-1344 x a \,b^{4} c^{3} d^{2}+384 x \,b^{5} c^{4} d +7 a^{5} d^{5}+70 a^{4} b c \,d^{4}-560 a^{3} b^{2} c^{2} d^{3}+1120 a^{2} b^{3} c^{3} d^{2}-896 a \,b^{4} c^{4} d +256 c^{5} b^{5}\right )}{21 \left (x d +c \right )^{\frac {3}{2}} d^{6}}\) | \(273\) |
| trager | \(-\frac {2 \left (-3 x^{5} b^{5} d^{5}-21 x^{4} a \,b^{4} d^{5}+6 x^{4} b^{5} c \,d^{4}-70 x^{3} a^{2} b^{3} d^{5}+56 x^{3} a \,b^{4} c \,d^{4}-16 x^{3} b^{5} c^{2} d^{3}-210 x^{2} a^{3} b^{2} d^{5}+420 x^{2} a^{2} b^{3} c \,d^{4}-336 x^{2} a \,b^{4} c^{2} d^{3}+96 x^{2} b^{5} c^{3} d^{2}+105 x \,a^{4} b \,d^{5}-840 x \,a^{3} b^{2} c \,d^{4}+1680 x \,a^{2} b^{3} c^{2} d^{3}-1344 x a \,b^{4} c^{3} d^{2}+384 x \,b^{5} c^{4} d +7 a^{5} d^{5}+70 a^{4} b c \,d^{4}-560 a^{3} b^{2} c^{2} d^{3}+1120 a^{2} b^{3} c^{3} d^{2}-896 a \,b^{4} c^{4} d +256 c^{5} b^{5}\right )}{21 \left (x d +c \right )^{\frac {3}{2}} d^{6}}\) | \(273\) |
| orering | \(-\frac {2 \left (-3 x^{5} b^{5} d^{5}-21 x^{4} a \,b^{4} d^{5}+6 x^{4} b^{5} c \,d^{4}-70 x^{3} a^{2} b^{3} d^{5}+56 x^{3} a \,b^{4} c \,d^{4}-16 x^{3} b^{5} c^{2} d^{3}-210 x^{2} a^{3} b^{2} d^{5}+420 x^{2} a^{2} b^{3} c \,d^{4}-336 x^{2} a \,b^{4} c^{2} d^{3}+96 x^{2} b^{5} c^{3} d^{2}+105 x \,a^{4} b \,d^{5}-840 x \,a^{3} b^{2} c \,d^{4}+1680 x \,a^{2} b^{3} c^{2} d^{3}-1344 x a \,b^{4} c^{3} d^{2}+384 x \,b^{5} c^{4} d +7 a^{5} d^{5}+70 a^{4} b c \,d^{4}-560 a^{3} b^{2} c^{2} d^{3}+1120 a^{2} b^{3} c^{3} d^{2}-896 a \,b^{4} c^{4} d +256 c^{5} b^{5}\right )}{21 \left (x d +c \right )^{\frac {3}{2}} d^{6}}\) | \(273\) |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (x d +c \right )^{\frac {7}{2}}}{7}+2 a \,b^{4} d \left (x d +c \right )^{\frac {5}{2}}-2 b^{5} c \left (x d +c \right )^{\frac {5}{2}}+\frac {20 a^{2} b^{3} d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {40 a \,b^{4} c d \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {20 b^{5} c^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}+20 a^{3} b^{2} d^{3} \sqrt {x d +c}-60 a^{2} b^{3} c \,d^{2} \sqrt {x d +c}+60 a \,b^{4} c^{2} d \sqrt {x d +c}-20 b^{5} c^{3} \sqrt {x d +c}-\frac {10 b \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{\sqrt {x d +c}}-\frac {2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{3 \left (x d +c \right )^{\frac {3}{2}}}}{d^{6}}\) | \(294\) |
| default | \(\frac {\frac {2 b^{5} \left (x d +c \right )^{\frac {7}{2}}}{7}+2 a \,b^{4} d \left (x d +c \right )^{\frac {5}{2}}-2 b^{5} c \left (x d +c \right )^{\frac {5}{2}}+\frac {20 a^{2} b^{3} d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {40 a \,b^{4} c d \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {20 b^{5} c^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}+20 a^{3} b^{2} d^{3} \sqrt {x d +c}-60 a^{2} b^{3} c \,d^{2} \sqrt {x d +c}+60 a \,b^{4} c^{2} d \sqrt {x d +c}-20 b^{5} c^{3} \sqrt {x d +c}-\frac {10 b \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{\sqrt {x d +c}}-\frac {2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{3 \left (x d +c \right )^{\frac {3}{2}}}}{d^{6}}\) | \(294\) |
Input:
int((b*x+a)^5/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/21*b^2*(3*b^3*d^3*x^3+21*a*b^2*d^3*x^2-12*b^3*c*d^2*x^2+70*a^2*b*d^3*x-9 8*a*b^2*c*d^2*x+37*b^3*c^2*d*x+210*a^3*d^3-560*a^2*b*c*d^2+511*a*b^2*c^2*d -158*b^3*c^3)*(d*x+c)^(1/2)/d^6-2/3*(15*b*d*x+a*d+14*b*c)*(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)/d^6/(d*x+c)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (134) = 268\).
Time = 0.10 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.86 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, b^{5} d^{5} x^{5} - 256 \, b^{5} c^{5} + 896 \, a b^{4} c^{4} d - 1120 \, a^{2} b^{3} c^{3} d^{2} + 560 \, a^{3} b^{2} c^{2} d^{3} - 70 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5} - 3 \, {\left (2 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} c^{2} d^{3} - 28 \, a b^{4} c d^{4} + 35 \, a^{2} b^{3} d^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} c^{3} d^{2} - 56 \, a b^{4} c^{2} d^{3} + 70 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} c^{4} d - 448 \, a b^{4} c^{3} d^{2} + 560 \, a^{2} b^{3} c^{2} d^{3} - 280 \, a^{3} b^{2} c d^{4} + 35 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}}{21 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \] Input:
integrate((b*x+a)^5/(d*x+c)^(5/2),x, algorithm="fricas")
Output:
2/21*(3*b^5*d^5*x^5 - 256*b^5*c^5 + 896*a*b^4*c^4*d - 1120*a^2*b^3*c^3*d^2 + 560*a^3*b^2*c^2*d^3 - 70*a^4*b*c*d^4 - 7*a^5*d^5 - 3*(2*b^5*c*d^4 - 7*a *b^4*d^5)*x^4 + 2*(8*b^5*c^2*d^3 - 28*a*b^4*c*d^4 + 35*a^2*b^3*d^5)*x^3 - 6*(16*b^5*c^3*d^2 - 56*a*b^4*c^2*d^3 + 70*a^2*b^3*c*d^4 - 35*a^3*b^2*d^5)* x^2 - 3*(128*b^5*c^4*d - 448*a*b^4*c^3*d^2 + 560*a^2*b^3*c^2*d^3 - 280*a^3 *b^2*c*d^4 + 35*a^4*b*d^5)*x)*sqrt(d*x + c)/(d^8*x^2 + 2*c*d^7*x + c^2*d^6 )
Time = 5.61 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (c + d x\right )^{\frac {7}{2}}}{7 d^{5}} - \frac {5 b \left (a d - b c\right )^{4}}{d^{5} \sqrt {c + d x}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (5 a b^{4} d - 5 b^{5} c\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (10 a^{2} b^{3} d^{2} - 20 a b^{4} c d + 10 b^{5} c^{2}\right )}{3 d^{5}} + \frac {\sqrt {c + d x} \left (10 a^{3} b^{2} d^{3} - 30 a^{2} b^{3} c d^{2} + 30 a b^{4} c^{2} d - 10 b^{5} c^{3}\right )}{d^{5}} - \frac {\left (a d - b c\right )^{5}}{3 d^{5} \left (c + d x\right )^{\frac {3}{2}}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{6}}{6 b} & \text {otherwise} \end {cases}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**5/(d*x+c)**(5/2),x)
Output:
Piecewise((2*(b**5*(c + d*x)**(7/2)/(7*d**5) - 5*b*(a*d - b*c)**4/(d**5*sq rt(c + d*x)) + (c + d*x)**(5/2)*(5*a*b**4*d - 5*b**5*c)/(5*d**5) + (c + d* x)**(3/2)*(10*a**2*b**3*d**2 - 20*a*b**4*c*d + 10*b**5*c**2)/(3*d**5) + sq rt(c + d*x)*(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)/d**5 - (a*d - b*c)**5/(3*d**5*(c + d*x)**(3/2)))/d, Ne(d, 0) ), (Piecewise((a**5*x, Eq(b, 0)), ((a + b*x)**6/(6*b), True))/c**(5/2), Tr ue))
Time = 0.04 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{5} - 21 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 70 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 210 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \sqrt {d x + c}}{d^{5}} + \frac {7 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5} - 15 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac {3}{2}} d^{5}}\right )}}{21 \, d} \] Input:
integrate((b*x+a)^5/(d*x+c)^(5/2),x, algorithm="maxima")
Output:
2/21*((3*(d*x + c)^(7/2)*b^5 - 21*(b^5*c - a*b^4*d)*(d*x + c)^(5/2) + 70*( b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*(d*x + c)^(3/2) - 210*(b^5*c^3 - 3*a* b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*sqrt(d*x + c))/d^5 + 7*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5 - 15*(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c *d^3 + a^4*b*d^4)*(d*x + c))/((d*x + c)^(3/2)*d^5))/d
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (134) = 268\).
Time = 0.14 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.20 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (d x + c\right )} b^{5} c^{4} - b^{5} c^{5} - 60 \, {\left (d x + c\right )} a b^{4} c^{3} d + 5 \, a b^{4} c^{4} d + 90 \, {\left (d x + c\right )} a^{2} b^{3} c^{2} d^{2} - 10 \, a^{2} b^{3} c^{3} d^{2} - 60 \, {\left (d x + c\right )} a^{3} b^{2} c d^{3} + 10 \, a^{3} b^{2} c^{2} d^{3} + 15 \, {\left (d x + c\right )} a^{4} b d^{4} - 5 \, a^{4} b c d^{4} + a^{5} d^{5}\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{6}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{5} d^{36} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} c d^{36} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{5} c^{2} d^{36} - 210 \, \sqrt {d x + c} b^{5} c^{3} d^{36} + 21 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{4} d^{37} - 140 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{4} c d^{37} + 630 \, \sqrt {d x + c} a b^{4} c^{2} d^{37} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{3} d^{38} - 630 \, \sqrt {d x + c} a^{2} b^{3} c d^{38} + 210 \, \sqrt {d x + c} a^{3} b^{2} d^{39}\right )}}{21 \, d^{42}} \] Input:
integrate((b*x+a)^5/(d*x+c)^(5/2),x, algorithm="giac")
Output:
-2/3*(15*(d*x + c)*b^5*c^4 - b^5*c^5 - 60*(d*x + c)*a*b^4*c^3*d + 5*a*b^4* c^4*d + 90*(d*x + c)*a^2*b^3*c^2*d^2 - 10*a^2*b^3*c^3*d^2 - 60*(d*x + c)*a ^3*b^2*c*d^3 + 10*a^3*b^2*c^2*d^3 + 15*(d*x + c)*a^4*b*d^4 - 5*a^4*b*c*d^4 + a^5*d^5)/((d*x + c)^(3/2)*d^6) + 2/21*(3*(d*x + c)^(7/2)*b^5*d^36 - 21* (d*x + c)^(5/2)*b^5*c*d^36 + 70*(d*x + c)^(3/2)*b^5*c^2*d^36 - 210*sqrt(d* x + c)*b^5*c^3*d^36 + 21*(d*x + c)^(5/2)*a*b^4*d^37 - 140*(d*x + c)^(3/2)* a*b^4*c*d^37 + 630*sqrt(d*x + c)*a*b^4*c^2*d^37 + 70*(d*x + c)^(3/2)*a^2*b ^3*d^38 - 630*sqrt(d*x + c)*a^2*b^3*c*d^38 + 210*sqrt(d*x + c)*a^3*b^2*d^3 9)/d^42
Time = 0.16 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.51 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\frac {2\,b^5\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^6}-\frac {\frac {2\,a^5\,d^5}{3}-\frac {2\,b^5\,c^5}{3}+\left (c+d\,x\right )\,\left (10\,a^4\,b\,d^4-40\,a^3\,b^2\,c\,d^3+60\,a^2\,b^3\,c^2\,d^2-40\,a\,b^4\,c^3\,d+10\,b^5\,c^4\right )-\frac {20\,a^2\,b^3\,c^3\,d^2}{3}+\frac {20\,a^3\,b^2\,c^2\,d^3}{3}+\frac {10\,a\,b^4\,c^4\,d}{3}-\frac {10\,a^4\,b\,c\,d^4}{3}}{d^6\,{\left (c+d\,x\right )}^{3/2}}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6} \] Input:
int((a + b*x)^5/(c + d*x)^(5/2),x)
Output:
(2*b^5*(c + d*x)^(7/2))/(7*d^6) - ((10*b^5*c - 10*a*b^4*d)*(c + d*x)^(5/2) )/(5*d^6) - ((2*a^5*d^5)/3 - (2*b^5*c^5)/3 + (c + d*x)*(10*b^5*c^4 + 10*a^ 4*b*d^4 - 40*a^3*b^2*c*d^3 + 60*a^2*b^3*c^2*d^2 - 40*a*b^4*c^3*d) - (20*a^ 2*b^3*c^3*d^2)/3 + (20*a^3*b^2*c^2*d^3)/3 + (10*a*b^4*c^4*d)/3 - (10*a^4*b *c*d^4)/3)/(d^6*(c + d*x)^(3/2)) + (20*b^2*(a*d - b*c)^3*(c + d*x)^(1/2))/ d^6 + (20*b^3*(a*d - b*c)^2*(c + d*x)^(3/2))/(3*d^6)
Time = 0.16 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\frac {\frac {2}{7} b^{5} d^{5} x^{5}+2 a \,b^{4} d^{5} x^{4}-\frac {4}{7} b^{5} c \,d^{4} x^{4}+\frac {20}{3} a^{2} b^{3} d^{5} x^{3}-\frac {16}{3} a \,b^{4} c \,d^{4} x^{3}+\frac {32}{21} b^{5} c^{2} d^{3} x^{3}+20 a^{3} b^{2} d^{5} x^{2}-40 a^{2} b^{3} c \,d^{4} x^{2}+32 a \,b^{4} c^{2} d^{3} x^{2}-\frac {64}{7} b^{5} c^{3} d^{2} x^{2}-10 a^{4} b \,d^{5} x +80 a^{3} b^{2} c \,d^{4} x -160 a^{2} b^{3} c^{2} d^{3} x +128 a \,b^{4} c^{3} d^{2} x -\frac {256}{7} b^{5} c^{4} d x -\frac {2}{3} a^{5} d^{5}-\frac {20}{3} a^{4} b c \,d^{4}+\frac {160}{3} a^{3} b^{2} c^{2} d^{3}-\frac {320}{3} a^{2} b^{3} c^{3} d^{2}+\frac {256}{3} a \,b^{4} c^{4} d -\frac {512}{21} b^{5} c^{5}}{\sqrt {d x +c}\, d^{6} \left (d x +c \right )} \] Input:
int((b*x+a)^5/(d*x+c)^(5/2),x)
Output:
(2*( - 7*a**5*d**5 - 70*a**4*b*c*d**4 - 105*a**4*b*d**5*x + 560*a**3*b**2* c**2*d**3 + 840*a**3*b**2*c*d**4*x + 210*a**3*b**2*d**5*x**2 - 1120*a**2*b **3*c**3*d**2 - 1680*a**2*b**3*c**2*d**3*x - 420*a**2*b**3*c*d**4*x**2 + 7 0*a**2*b**3*d**5*x**3 + 896*a*b**4*c**4*d + 1344*a*b**4*c**3*d**2*x + 336* a*b**4*c**2*d**3*x**2 - 56*a*b**4*c*d**4*x**3 + 21*a*b**4*d**5*x**4 - 256* b**5*c**5 - 384*b**5*c**4*d*x - 96*b**5*c**3*d**2*x**2 + 16*b**5*c**2*d**3 *x**3 - 6*b**5*c*d**4*x**4 + 3*b**5*d**5*x**5))/(21*sqrt(c + d*x)*d**6*(c + d*x))