Integrand size = 19, antiderivative size = 101 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx=-\frac {2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {8 d (c+d x)^{5/2}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {16 d^2 (c+d x)^{5/2}}{315 (b c-a d)^3 (a+b x)^{5/2}} \]
-2/9*(d*x+c)^(5/2)/(-a*d+b*c)/(b*x+a)^(9/2)+8/63*d*(d*x+c)^(5/2)/(-a*d+b*c )^2/(b*x+a)^(7/2)-16/315*d^2*(d*x+c)^(5/2)/(-a*d+b*c)^3/(b*x+a)^(5/2)
Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx=-\frac {2 (c+d x)^{5/2} \left (63 a^2 d^2+18 a b d (-5 c+2 d x)+b^2 \left (35 c^2-20 c d x+8 d^2 x^2\right )\right )}{315 (b c-a d)^3 (a+b x)^{9/2}} \]
(-2*(c + d*x)^(5/2)*(63*a^2*d^2 + 18*a*b*d*(-5*c + 2*d*x) + b^2*(35*c^2 - 20*c*d*x + 8*d^2*x^2)))/(315*(b*c - a*d)^3*(a + b*x)^(9/2))
Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {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.
\(\displaystyle \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {4 d \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}}dx}{9 (b c-a d)}-\frac {2 (c+d x)^{5/2}}{9 (a+b x)^{9/2} (b c-a d)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {4 d \left (-\frac {2 d \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2}}dx}{7 (b c-a d)}-\frac {2 (c+d x)^{5/2}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{5/2}}{9 (a+b x)^{9/2} (b c-a d)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {2 (c+d x)^{5/2}}{9 (a+b x)^{9/2} (b c-a d)}-\frac {4 d \left (\frac {4 d (c+d x)^{5/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{9 (b c-a d)}\) |
(-2*(c + d*x)^(5/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) - (4*d*((-2*(c + d*x) ^(5/2))/(7*(b*c - a*d)*(a + b*x)^(7/2)) + (4*d*(c + d*x)^(5/2))/(35*(b*c - a*d)^2*(a + b*x)^(5/2))))/(9*(b*c - a*d))
3.15.79.3.1 Defintions of rubi rules used
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 = 0.77 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (8 d^{2} x^{2} b^{2}+36 x a b \,d^{2}-20 x \,b^{2} c d +63 a^{2} d^{2}-90 a b c d +35 b^{2} c^{2}\right )}{315 \left (b x +a \right )^{\frac {9}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
default | \(-\frac {\left (d x +c \right )^{\frac {3}{2}}}{3 b \left (b x +a \right )^{\frac {9}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{4 b \left (b x +a \right )^{\frac {9}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{8 b}\right )}{2 b}\) | \(241\) |
2/315*(d*x+c)^(5/2)*(8*b^2*d^2*x^2+36*a*b*d^2*x-20*b^2*c*d*x+63*a^2*d^2-90 *a*b*c*d+35*b^2*c^2)/(b*x+a)^(9/2)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^ 3*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (83) = 166\).
Time = 2.30 (sec) , antiderivative size = 426, normalized size of antiderivative = 4.22 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx=-\frac {2 \, {\left (8 \, b^{2} d^{4} x^{4} + 35 \, b^{2} c^{4} - 90 \, a b c^{3} d + 63 \, a^{2} c^{2} d^{2} - 4 \, {\left (b^{2} c d^{3} - 9 \, a b d^{4}\right )} x^{3} + 3 \, {\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (25 \, b^{2} c^{3} d - 72 \, a b c^{2} d^{2} + 63 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{315 \, {\left (a^{5} b^{3} c^{3} - 3 \, a^{6} b^{2} c^{2} d + 3 \, a^{7} b c d^{2} - a^{8} d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{5} + 5 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{4} + 10 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{3} + 10 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x^{2} + 5 \, {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} x\right )}} \]
-2/315*(8*b^2*d^4*x^4 + 35*b^2*c^4 - 90*a*b*c^3*d + 63*a^2*c^2*d^2 - 4*(b^ 2*c*d^3 - 9*a*b*d^4)*x^3 + 3*(b^2*c^2*d^2 - 6*a*b*c*d^3 + 21*a^2*d^4)*x^2 + 2*(25*b^2*c^3*d - 72*a*b*c^2*d^2 + 63*a^2*c*d^3)*x)*sqrt(b*x + a)*sqrt(d *x + c)/(a^5*b^3*c^3 - 3*a^6*b^2*c^2*d + 3*a^7*b*c*d^2 - a^8*d^3 + (b^8*c^ 3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*x^5 + 5*(a*b^7*c^3 - 3* a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^4 + 10*(a^2*b^6*c^3 - 3*a ^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*x^3 + 10*(a^3*b^5*c^3 - 3*a^ 4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*x^2 + 5*(a^4*b^4*c^3 - 3*a^5* b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*x)
\[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\left (a + b x\right )^{\frac {11}{2}}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1394 vs. \(2 (83) = 166\).
Time = 0.54 (sec) , antiderivative size = 1394, normalized size of antiderivative = 13.80 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx=\text {Too large to display} \]
-32/315*(sqrt(b*d)*b^12*c^6*d^4*abs(b) - 6*sqrt(b*d)*a*b^11*c^5*d^5*abs(b) + 15*sqrt(b*d)*a^2*b^10*c^4*d^6*abs(b) - 20*sqrt(b*d)*a^3*b^9*c^3*d^7*abs (b) + 15*sqrt(b*d)*a^4*b^8*c^2*d^8*abs(b) - 6*sqrt(b*d)*a^5*b^7*c*d^9*abs( b) + sqrt(b*d)*a^6*b^6*d^10*abs(b) - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^10*c^5*d^4*abs(b) + 45*sqrt(b*d )*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^9* c^4*d^5*abs(b) - 90*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^8*c^3*d^6*abs(b) + 90*sqrt(b*d)*(sqrt(b*d)*sqr t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^7*c^2*d^7*abs(b) - 45*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a* b*d))^2*a^4*b^6*c*d^8*abs(b) + 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt (b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^5*d^9*abs(b) + 36*sqrt(b*d)*(sqrt (b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^8*c^4*d^4*a bs(b) - 144*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b* d - a*b*d))^4*a*b^7*c^3*d^5*abs(b) + 216*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a ) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^6*c^2*d^6*abs(b) - 144*sq rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4* a^3*b^5*c*d^7*abs(b) + 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b^4*d^8*abs(b) + 126*sqrt(b*d)*(sqrt(b*d)* sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^6*c^3*d^4*abs(...
Time = 1.34 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.65 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {126\,a^2\,c^2\,d^2-180\,a\,b\,c^3\,d+70\,b^2\,c^4}{315\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {x^2\,\left (126\,a^2\,d^4-36\,a\,b\,c\,d^3+6\,b^2\,c^2\,d^2\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {16\,d^4\,x^4}{315\,b^2\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,d^3\,x^3\,\left (9\,a\,d-b\,c\right )}{315\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {4\,c\,d\,x\,\left (63\,a^2\,d^2-72\,a\,b\,c\,d+25\,b^2\,c^2\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^3}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a^4\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a^2\,x^2\,\sqrt {a+b\,x}}{b^2}+\frac {4\,a\,x^3\,\sqrt {a+b\,x}}{b}+\frac {4\,a^3\,x\,\sqrt {a+b\,x}}{b^3}} \]
((c + d*x)^(1/2)*((70*b^2*c^4 + 126*a^2*c^2*d^2 - 180*a*b*c^3*d)/(315*b^4* (a*d - b*c)^3) + (x^2*(126*a^2*d^4 + 6*b^2*c^2*d^2 - 36*a*b*c*d^3))/(315*b ^4*(a*d - b*c)^3) + (16*d^4*x^4)/(315*b^2*(a*d - b*c)^3) + (8*d^3*x^3*(9*a *d - b*c))/(315*b^3*(a*d - b*c)^3) + (4*c*d*x*(63*a^2*d^2 + 25*b^2*c^2 - 7 2*a*b*c*d))/(315*b^4*(a*d - b*c)^3)))/(x^4*(a + b*x)^(1/2) + (a^4*(a + b*x )^(1/2))/b^4 + (6*a^2*x^2*(a + b*x)^(1/2))/b^2 + (4*a*x^3*(a + b*x)^(1/2)) /b + (4*a^3*x*(a + b*x)^(1/2))/b^3)
Time = 0.03 (sec) , antiderivative size = 592, normalized size of antiderivative = 5.86 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx=\frac {-\frac {16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{4} d^{4}}{315}-\frac {64 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{3} b \,d^{4} x}{315}-\frac {32 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} b^{2} d^{4} x^{2}}{105}-\frac {64 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a \,b^{3} d^{4} x^{3}}{315}-\frac {16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{4} d^{4} x^{4}}{315}+\frac {2 \sqrt {d x +c}\, a^{2} b^{3} c^{2} d^{2}}{5}+\frac {4 \sqrt {d x +c}\, a^{2} b^{3} c \,d^{3} x}{5}+\frac {2 \sqrt {d x +c}\, a^{2} b^{3} d^{4} x^{2}}{5}-\frac {4 \sqrt {d x +c}\, a \,b^{4} c^{3} d}{7}-\frac {32 \sqrt {d x +c}\, a \,b^{4} c^{2} d^{2} x}{35}-\frac {4 \sqrt {d x +c}\, a \,b^{4} c \,d^{3} x^{2}}{35}+\frac {8 \sqrt {d x +c}\, a \,b^{4} d^{4} x^{3}}{35}+\frac {2 \sqrt {d x +c}\, b^{5} c^{4}}{9}+\frac {20 \sqrt {d x +c}\, b^{5} c^{3} d x}{63}+\frac {2 \sqrt {d x +c}\, b^{5} c^{2} d^{2} x^{2}}{105}-\frac {8 \sqrt {d x +c}\, b^{5} c \,d^{3} x^{3}}{315}+\frac {16 \sqrt {d x +c}\, b^{5} d^{4} x^{4}}{315}}{\sqrt {b x +a}\, b^{3} \left (a^{3} b^{4} d^{3} x^{4}-3 a^{2} b^{5} c \,d^{2} x^{4}+3 a \,b^{6} c^{2} d \,x^{4}-b^{7} c^{3} x^{4}+4 a^{4} b^{3} d^{3} x^{3}-12 a^{3} b^{4} c \,d^{2} x^{3}+12 a^{2} b^{5} c^{2} d \,x^{3}-4 a \,b^{6} c^{3} x^{3}+6 a^{5} b^{2} d^{3} x^{2}-18 a^{4} b^{3} c \,d^{2} x^{2}+18 a^{3} b^{4} c^{2} d \,x^{2}-6 a^{2} b^{5} c^{3} x^{2}+4 a^{6} b \,d^{3} x -12 a^{5} b^{2} c \,d^{2} x +12 a^{4} b^{3} c^{2} d x -4 a^{3} b^{4} c^{3} x +a^{7} d^{3}-3 a^{6} b c \,d^{2}+3 a^{5} b^{2} c^{2} d -a^{4} b^{3} c^{3}\right )} \]
int((sqrt(c + d*x)*(c + d*x))/(sqrt(a + b*x)*(a**5 + 5*a**4*b*x + 10*a**3* b**2*x**2 + 10*a**2*b**3*x**3 + 5*a*b**4*x**4 + b**5*x**5)),x)
(2*( - 8*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**4*d**4 - 32*sqrt(d)*sqrt(b)*sqrt (a + b*x)*a**3*b*d**4*x - 48*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*b**2*d**4* x**2 - 32*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b**3*d**4*x**3 - 8*sqrt(d)*sqrt( b)*sqrt(a + b*x)*b**4*d**4*x**4 + 63*sqrt(c + d*x)*a**2*b**3*c**2*d**2 + 1 26*sqrt(c + d*x)*a**2*b**3*c*d**3*x + 63*sqrt(c + d*x)*a**2*b**3*d**4*x**2 - 90*sqrt(c + d*x)*a*b**4*c**3*d - 144*sqrt(c + d*x)*a*b**4*c**2*d**2*x - 18*sqrt(c + d*x)*a*b**4*c*d**3*x**2 + 36*sqrt(c + d*x)*a*b**4*d**4*x**3 + 35*sqrt(c + d*x)*b**5*c**4 + 50*sqrt(c + d*x)*b**5*c**3*d*x + 3*sqrt(c + d*x)*b**5*c**2*d**2*x**2 - 4*sqrt(c + d*x)*b**5*c*d**3*x**3 + 8*sqrt(c + d *x)*b**5*d**4*x**4))/(315*sqrt(a + b*x)*b**3*(a**7*d**3 - 3*a**6*b*c*d**2 + 4*a**6*b*d**3*x + 3*a**5*b**2*c**2*d - 12*a**5*b**2*c*d**2*x + 6*a**5*b* *2*d**3*x**2 - a**4*b**3*c**3 + 12*a**4*b**3*c**2*d*x - 18*a**4*b**3*c*d** 2*x**2 + 4*a**4*b**3*d**3*x**3 - 4*a**3*b**4*c**3*x + 18*a**3*b**4*c**2*d* x**2 - 12*a**3*b**4*c*d**2*x**3 + a**3*b**4*d**3*x**4 - 6*a**2*b**5*c**3*x **2 + 12*a**2*b**5*c**2*d*x**3 - 3*a**2*b**5*c*d**2*x**4 - 4*a*b**6*c**3*x **3 + 3*a*b**6*c**2*d*x**4 - b**7*c**3*x**4))