Integrand size = 19, antiderivative size = 171 \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {32 d^2 \sqrt {c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac {128 d^3 \sqrt {c+d x}}{315 (b c-a d)^4 (a+b x)^{3/2}}-\frac {256 d^4 \sqrt {c+d x}}{315 (b c-a d)^5 \sqrt {a+b x}} \]
-2/9*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(9/2)+16/63*d*(d*x+c)^(1/2)/(-a*d+b* c)^2/(b*x+a)^(7/2)-32/105*d^2*(d*x+c)^(1/2)/(-a*d+b*c)^3/(b*x+a)^(5/2)+128 /315*d^3*(d*x+c)^(1/2)/(-a*d+b*c)^4/(b*x+a)^(3/2)-256/315*d^4*(d*x+c)^(1/2 )/(-a*d+b*c)^5/(b*x+a)^(1/2)
Time = 0.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {c+d x} \left (315 a^4 d^4-420 a^3 b d^3 (c-2 d x)+126 a^2 b^2 d^2 \left (3 c^2-4 c d x+8 d^2 x^2\right )+36 a b^3 d \left (-5 c^3+6 c^2 d x-8 c d^2 x^2+16 d^3 x^3\right )+b^4 \left (35 c^4-40 c^3 d x+48 c^2 d^2 x^2-64 c d^3 x^3+128 d^4 x^4\right )\right )}{315 (b c-a d)^5 (a+b x)^{9/2}} \]
(-2*Sqrt[c + d*x]*(315*a^4*d^4 - 420*a^3*b*d^3*(c - 2*d*x) + 126*a^2*b^2*d ^2*(3*c^2 - 4*c*d*x + 8*d^2*x^2) + 36*a*b^3*d*(-5*c^3 + 6*c^2*d*x - 8*c*d^ 2*x^2 + 16*d^3*x^3) + b^4*(35*c^4 - 40*c^3*d*x + 48*c^2*d^2*x^2 - 64*c*d^3 *x^3 + 128*d^4*x^4)))/(315*(b*c - a*d)^5*(a + b*x)^(9/2))
Time = 0.22 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.23, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {55, 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.
| \(\displaystyle \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {8 d \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}}dx}{9 (b c-a d)}-\frac {2 \sqrt {c+d x}}{9 (a+b x)^{9/2} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {8 d \left (-\frac {6 d \int \frac {1}{(a+b x)^{7/2} \sqrt {c+d x}}dx}{7 (b c-a d)}-\frac {2 \sqrt {c+d x}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 \sqrt {c+d x}}{9 (a+b x)^{9/2} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {8 d \left (-\frac {6 d \left (-\frac {4 d \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}}dx}{5 (b c-a d)}-\frac {2 \sqrt {c+d x}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2 \sqrt {c+d x}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 \sqrt {c+d x}}{9 (a+b x)^{9/2} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {8 d \left (-\frac {6 d \left (-\frac {4 d \left (-\frac {2 d \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}}dx}{3 (b c-a d)}-\frac {2 \sqrt {c+d x}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{5 (b c-a d)}-\frac {2 \sqrt {c+d x}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2 \sqrt {c+d x}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 \sqrt {c+d x}}{9 (a+b x)^{9/2} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {8 d \left (-\frac {6 d \left (-\frac {4 d \left (\frac {4 d \sqrt {c+d x}}{3 \sqrt {a+b x} (b c-a d)^2}-\frac {2 \sqrt {c+d x}}{3 (a+b x)^{3/2} (b c-a d)}\right )}{5 (b c-a d)}-\frac {2 \sqrt {c+d x}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2 \sqrt {c+d x}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 \sqrt {c+d x}}{9 (a+b x)^{9/2} (b c-a d)}\) |
(-2*Sqrt[c + d*x])/(9*(b*c - a*d)*(a + b*x)^(9/2)) - (8*d*((-2*Sqrt[c + d* x])/(7*(b*c - a*d)*(a + b*x)^(7/2)) - (6*d*((-2*Sqrt[c + d*x])/(5*(b*c - a *d)*(a + b*x)^(5/2)) - (4*d*((-2*Sqrt[c + d*x])/(3*(b*c - a*d)*(a + b*x)^( 3/2)) + (4*d*Sqrt[c + d*x])/(3*(b*c - a*d)^2*Sqrt[a + b*x])))/(5*(b*c - a* d))))/(7*(b*c - a*d))))/(9*(b*c - a*d))
3.16.1.3.1 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 = 0.52 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02
| method | result | size |
| default | \(-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\) | \(175\) |
| gosper | \(\frac {2 \sqrt {d x +c}\, \left (128 d^{4} x^{4} b^{4}+576 a \,b^{3} d^{4} x^{3}-64 b^{4} c \,d^{3} x^{3}+1008 a^{2} b^{2} d^{4} x^{2}-288 a \,b^{3} c \,d^{3} x^{2}+48 b^{4} c^{2} d^{2} x^{2}+840 a^{3} b \,d^{4} x -504 a^{2} b^{2} c \,d^{3} x +216 a \,b^{3} c^{2} d^{2} x -40 b^{4} c^{3} d x +315 a^{4} d^{4}-420 a^{3} b c \,d^{3}+378 a^{2} b^{2} c^{2} d^{2}-180 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right )}{315 \left (b x +a \right )^{\frac {9}{2}} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\) | \(256\) |
-2/9*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(9/2)-8/9*d/(-a*d+b*c)*(-2/7*(d*x+c) ^(1/2)/(-a*d+b*c)/(b*x+a)^(7/2)-6/7*d/(-a*d+b*c)*(-2/5*(d*x+c)^(1/2)/(-a*d +b*c)/(b*x+a)^(5/2)-4/5*d/(-a*d+b*c)*(-2/3*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a )^(3/2)+4/3*d*(d*x+c)^(1/2)/(-a*d+b*c)^2/(b*x+a)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (141) = 282\).
Time = 2.05 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.73 \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left (128 \, b^{4} d^{4} x^{4} + 35 \, b^{4} c^{4} - 180 \, a b^{3} c^{3} d + 378 \, a^{2} b^{2} c^{2} d^{2} - 420 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} - 64 \, {\left (b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 48 \, {\left (b^{4} c^{2} d^{2} - 6 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} - 8 \, {\left (5 \, b^{4} c^{3} d - 27 \, a b^{3} c^{2} d^{2} + 63 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{315 \, {\left (a^{5} b^{5} c^{5} - 5 \, a^{6} b^{4} c^{4} d + 10 \, a^{7} b^{3} c^{3} d^{2} - 10 \, a^{8} b^{2} c^{2} d^{3} + 5 \, a^{9} b c d^{4} - a^{10} d^{5} + {\left (b^{10} c^{5} - 5 \, a b^{9} c^{4} d + 10 \, a^{2} b^{8} c^{3} d^{2} - 10 \, a^{3} b^{7} c^{2} d^{3} + 5 \, a^{4} b^{6} c d^{4} - a^{5} b^{5} d^{5}\right )} x^{5} + 5 \, {\left (a b^{9} c^{5} - 5 \, a^{2} b^{8} c^{4} d + 10 \, a^{3} b^{7} c^{3} d^{2} - 10 \, a^{4} b^{6} c^{2} d^{3} + 5 \, a^{5} b^{5} c d^{4} - a^{6} b^{4} d^{5}\right )} x^{4} + 10 \, {\left (a^{2} b^{8} c^{5} - 5 \, a^{3} b^{7} c^{4} d + 10 \, a^{4} b^{6} c^{3} d^{2} - 10 \, a^{5} b^{5} c^{2} d^{3} + 5 \, a^{6} b^{4} c d^{4} - a^{7} b^{3} d^{5}\right )} x^{3} + 10 \, {\left (a^{3} b^{7} c^{5} - 5 \, a^{4} b^{6} c^{4} d + 10 \, a^{5} b^{5} c^{3} d^{2} - 10 \, a^{6} b^{4} c^{2} d^{3} + 5 \, a^{7} b^{3} c d^{4} - a^{8} b^{2} d^{5}\right )} x^{2} + 5 \, {\left (a^{4} b^{6} c^{5} - 5 \, a^{5} b^{5} c^{4} d + 10 \, a^{6} b^{4} c^{3} d^{2} - 10 \, a^{7} b^{3} c^{2} d^{3} + 5 \, a^{8} b^{2} c d^{4} - a^{9} b d^{5}\right )} x\right )}} \]
-2/315*(128*b^4*d^4*x^4 + 35*b^4*c^4 - 180*a*b^3*c^3*d + 378*a^2*b^2*c^2*d ^2 - 420*a^3*b*c*d^3 + 315*a^4*d^4 - 64*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 48 *(b^4*c^2*d^2 - 6*a*b^3*c*d^3 + 21*a^2*b^2*d^4)*x^2 - 8*(5*b^4*c^3*d - 27* a*b^3*c^2*d^2 + 63*a^2*b^2*c*d^3 - 105*a^3*b*d^4)*x)*sqrt(b*x + a)*sqrt(d* x + c)/(a^5*b^5*c^5 - 5*a^6*b^4*c^4*d + 10*a^7*b^3*c^3*d^2 - 10*a^8*b^2*c^ 2*d^3 + 5*a^9*b*c*d^4 - a^10*d^5 + (b^10*c^5 - 5*a*b^9*c^4*d + 10*a^2*b^8* c^3*d^2 - 10*a^3*b^7*c^2*d^3 + 5*a^4*b^6*c*d^4 - a^5*b^5*d^5)*x^5 + 5*(a*b ^9*c^5 - 5*a^2*b^8*c^4*d + 10*a^3*b^7*c^3*d^2 - 10*a^4*b^6*c^2*d^3 + 5*a^5 *b^5*c*d^4 - a^6*b^4*d^5)*x^4 + 10*(a^2*b^8*c^5 - 5*a^3*b^7*c^4*d + 10*a^4 *b^6*c^3*d^2 - 10*a^5*b^5*c^2*d^3 + 5*a^6*b^4*c*d^4 - a^7*b^3*d^5)*x^3 + 1 0*(a^3*b^7*c^5 - 5*a^4*b^6*c^4*d + 10*a^5*b^5*c^3*d^2 - 10*a^6*b^4*c^2*d^3 + 5*a^7*b^3*c*d^4 - a^8*b^2*d^5)*x^2 + 5*(a^4*b^6*c^5 - 5*a^5*b^5*c^4*d + 10*a^6*b^4*c^3*d^2 - 10*a^7*b^3*c^2*d^3 + 5*a^8*b^2*c*d^4 - a^9*b*d^5)*x)
\[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {11}{2}} \sqrt {c + d x}}\, dx \]
Exception generated. \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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. 596 vs. \(2 (141) = 282\).
Time = 0.39 (sec) , antiderivative size = 596, normalized size of antiderivative = 3.49 \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {512 \, {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4} - 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{6} c^{3} + 27 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{2} d - 27 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c d^{2} + 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} d^{3} + 36 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{4} c^{2} - 72 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c d + 36 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} d^{2} - 84 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{2} c + 84 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b d + 126 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8}\right )} \sqrt {b d} b^{5} d^{4}}{315 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{9} {\left | b \right |}} \]
-512/315*(b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4 - 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a* b*d))^2*b^6*c^3 + 27*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^5*c^2*d - 27*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^4*c*d^2 + 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^ 2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^3*d^3 + 36*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^4*c^2 - 72*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^3*c*d + 36*(sqrt(b*d)*sq rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^2*d^2 - 84*(sqr t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^2*c + 84*( sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b*d + 1 26*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8)*sqrt (b*d)*b^5*d^4/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b *x + a)*b*d - a*b*d))^2)^9*abs(b))
Time = 1.34 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.77 \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {256\,d^4\,x^4}{315\,{\left (a\,d-b\,c\right )}^5}+\frac {630\,a^4\,d^4-840\,a^3\,b\,c\,d^3+756\,a^2\,b^2\,c^2\,d^2-360\,a\,b^3\,c^3\,d+70\,b^4\,c^4}{315\,b^4\,{\left (a\,d-b\,c\right )}^5}+\frac {x\,\left (1680\,a^3\,b\,d^4-1008\,a^2\,b^2\,c\,d^3+432\,a\,b^3\,c^2\,d^2-80\,b^4\,c^3\,d\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^5}+\frac {128\,d^3\,x^3\,\left (9\,a\,d-b\,c\right )}{315\,b\,{\left (a\,d-b\,c\right )}^5}+\frac {32\,d^2\,x^2\,\left (21\,a^2\,d^2-6\,a\,b\,c\,d+b^2\,c^2\right )}{105\,b^2\,{\left (a\,d-b\,c\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a^4\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a^2\,x^2\,\sqrt {a+b\,x}}{b^2}+\frac {4\,a\,x^3\,\sqrt {a+b\,x}}{b}+\frac {4\,a^3\,x\,\sqrt {a+b\,x}}{b^3}} \]
((c + d*x)^(1/2)*((256*d^4*x^4)/(315*(a*d - b*c)^5) + (630*a^4*d^4 + 70*b^ 4*c^4 + 756*a^2*b^2*c^2*d^2 - 360*a*b^3*c^3*d - 840*a^3*b*c*d^3)/(315*b^4* (a*d - b*c)^5) + (x*(1680*a^3*b*d^4 - 80*b^4*c^3*d + 432*a*b^3*c^2*d^2 - 1 008*a^2*b^2*c*d^3))/(315*b^4*(a*d - b*c)^5) + (128*d^3*x^3*(9*a*d - b*c))/ (315*b*(a*d - b*c)^5) + (32*d^2*x^2*(21*a^2*d^2 + b^2*c^2 - 6*a*b*c*d))/(1 05*b^2*(a*d - b*c)^5)))/(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)