Integrand size = 24, antiderivative size = 298 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{7/2}}-\frac {2 (3 b B d-10 A b e+7 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{7/2}}-\frac {16 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{21 b (b d-a e)^3 (d+e x)^{7/2}}-\frac {32 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{5/2}}-\frac {128 b e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{105 (b d-a e)^5 (d+e x)^{3/2}}-\frac {256 b^2 e (3 b B d-10 A b e+7 a B e) \sqrt {a+b x}}{105 (b d-a e)^6 \sqrt {d+e x}} \]
-2/3*(A*b-B*a)/b/(-a*e+b*d)/(b*x+a)^(3/2)/(e*x+d)^(7/2)-2/3*(-10*A*b*e+7*B *a*e+3*B*b*d)/b/(-a*e+b*d)^2/(e*x+d)^(7/2)/(b*x+a)^(1/2)-16/21*e*(-10*A*b* e+7*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/b/(-a*e+b*d)^3/(e*x+d)^(7/2)-32/35*e*(-10 *A*b*e+7*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^4/(e*x+d)^(5/2)-128/105*b *e*(-10*A*b*e+7*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^5/(e*x+d)^(3/2)-25 6/105*b^2*e*(-10*A*b*e+7*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^6/(e*x+d) ^(1/2)
Time = 0.45 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx=-\frac {2 \left (-15 B d e^4 (a+b x)^5+15 A e^5 (a+b x)^5+84 b B d e^3 (a+b x)^4 (d+e x)-105 A b e^4 (a+b x)^4 (d+e x)+21 a B e^4 (a+b x)^4 (d+e x)-210 b^2 B d e^2 (a+b x)^3 (d+e x)^2+350 A b^2 e^3 (a+b x)^3 (d+e x)^2-140 a b B e^3 (a+b x)^3 (d+e x)^2+420 b^3 B d e (a+b x)^2 (d+e x)^3-1050 A b^3 e^2 (a+b x)^2 (d+e x)^3+630 a b^2 B e^2 (a+b x)^2 (d+e x)^3+105 b^4 B d (a+b x) (d+e x)^4-525 A b^4 e (a+b x) (d+e x)^4+420 a b^3 B e (a+b x) (d+e x)^4+35 A b^5 (d+e x)^5-35 a b^4 B (d+e x)^5\right )}{105 (b d-a e)^6 (a+b x)^{3/2} (d+e x)^{7/2}} \]
(-2*(-15*B*d*e^4*(a + b*x)^5 + 15*A*e^5*(a + b*x)^5 + 84*b*B*d*e^3*(a + b* x)^4*(d + e*x) - 105*A*b*e^4*(a + b*x)^4*(d + e*x) + 21*a*B*e^4*(a + b*x)^ 4*(d + e*x) - 210*b^2*B*d*e^2*(a + b*x)^3*(d + e*x)^2 + 350*A*b^2*e^3*(a + b*x)^3*(d + e*x)^2 - 140*a*b*B*e^3*(a + b*x)^3*(d + e*x)^2 + 420*b^3*B*d* e*(a + b*x)^2*(d + e*x)^3 - 1050*A*b^3*e^2*(a + b*x)^2*(d + e*x)^3 + 630*a *b^2*B*e^2*(a + b*x)^2*(d + e*x)^3 + 105*b^4*B*d*(a + b*x)*(d + e*x)^4 - 5 25*A*b^4*e*(a + b*x)*(d + e*x)^4 + 420*a*b^3*B*e*(a + b*x)*(d + e*x)^4 + 3 5*A*b^5*(d + e*x)^5 - 35*a*b^4*B*(d + e*x)^5))/(105*(b*d - a*e)^6*(a + b*x )^(3/2)*(d + e*x)^(7/2))
Time = 0.30 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 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.
| \(\displaystyle \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(7 a B e-10 A b e+3 b B d) \int \frac {1}{(a+b x)^{3/2} (d+e x)^{9/2}}dx}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(7 a B e-10 A b e+3 b B d) \left (-\frac {8 e \int \frac {1}{\sqrt {a+b x} (d+e x)^{9/2}}dx}{b d-a e}-\frac {2}{\sqrt {a+b x} (d+e x)^{7/2} (b d-a e)}\right )}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(7 a B e-10 A b e+3 b B d) \left (-\frac {8 e \left (\frac {6 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}}dx}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{b d-a e}-\frac {2}{\sqrt {a+b x} (d+e x)^{7/2} (b d-a e)}\right )}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(7 a B e-10 A b e+3 b B d) \left (-\frac {8 e \left (\frac {6 b \left (\frac {4 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}}dx}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{b d-a e}-\frac {2}{\sqrt {a+b x} (d+e x)^{7/2} (b d-a e)}\right )}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(7 a B e-10 A b e+3 b B d) \left (-\frac {8 e \left (\frac {6 b \left (\frac {4 b \left (\frac {2 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}}dx}{3 (b d-a e)}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{b d-a e}-\frac {2}{\sqrt {a+b x} (d+e x)^{7/2} (b d-a e)}\right )}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (-\frac {8 e \left (\frac {6 b \left (\frac {4 b \left (\frac {4 b \sqrt {a+b x}}{3 \sqrt {d+e x} (b d-a e)^2}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{b d-a e}-\frac {2}{\sqrt {a+b x} (d+e x)^{7/2} (b d-a e)}\right ) (7 a B e-10 A b e+3 b B d)}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{7/2} (b d-a e)}\) |
(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(7/2)) + ((3*b *B*d - 10*A*b*e + 7*a*B*e)*(-2/((b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(7/2)) - (8*e*((2*Sqrt[a + b*x])/(7*(b*d - a*e)*(d + e*x)^(7/2)) + (6*b*((2*Sqrt [a + b*x])/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (4*b*((2*Sqrt[a + b*x])/(3*(b *d - a*e)*(d + e*x)^(3/2)) + (4*b*Sqrt[a + b*x])/(3*(b*d - a*e)^2*Sqrt[d + e*x])))/(5*(b*d - a*e))))/(7*(b*d - a*e))))/(b*d - a*e)))/(3*b*(b*d - a*e ))
3.23.61.3.1 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])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Leaf count of result is larger than twice the leaf count of optimal. \(652\) vs. \(2(262)=524\).
Time = 1.12 (sec) , antiderivative size = 653, normalized size of antiderivative = 2.19
| method | result | size |
| default | \(-\frac {2 \left (-1280 A \,b^{5} e^{5} x^{5}+896 B a \,b^{4} e^{5} x^{5}+384 B \,b^{5} d \,e^{4} x^{5}-1920 A a \,b^{4} e^{5} x^{4}-4480 A \,b^{5} d \,e^{4} x^{4}+1344 B \,a^{2} b^{3} e^{5} x^{4}+3712 B a \,b^{4} d \,e^{4} x^{4}+1344 B \,b^{5} d^{2} e^{3} x^{4}-480 A \,a^{2} b^{3} e^{5} x^{3}-6720 A a \,b^{4} d \,e^{4} x^{3}-5600 A \,b^{5} d^{2} e^{3} x^{3}+336 B \,a^{3} b^{2} e^{5} x^{3}+4848 B \,a^{2} b^{3} d \,e^{4} x^{3}+5936 B a \,b^{4} d^{2} e^{3} x^{3}+1680 B \,b^{5} d^{3} e^{2} x^{3}+80 A \,a^{3} b^{2} e^{5} x^{2}-1680 A \,a^{2} b^{3} d \,e^{4} x^{2}-8400 A a \,b^{4} d^{2} e^{3} x^{2}-2800 A \,b^{5} d^{3} e^{2} x^{2}-56 B \,a^{4} b \,e^{5} x^{2}+1152 B \,a^{3} b^{2} d \,e^{4} x^{2}+6384 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+4480 B a \,b^{4} d^{3} e^{2} x^{2}+840 B \,b^{5} d^{4} e \,x^{2}-30 A \,a^{4} b \,e^{5} x +280 A \,a^{3} b^{2} d \,e^{4} x -2100 A \,a^{2} b^{3} d^{2} e^{3} x -4200 A a \,b^{4} d^{3} e^{2} x -350 A \,b^{5} d^{4} e x +21 B \,a^{5} e^{5} x -187 B \,a^{4} b d \,e^{4} x +1386 B \,a^{3} b^{2} d^{2} e^{3} x +3570 B \,a^{2} b^{3} d^{3} e^{2} x +1505 B a \,b^{4} d^{4} e x +105 B \,b^{5} d^{5} x +15 A \,a^{5} e^{5}-105 A \,a^{4} b d \,e^{4}+350 A \,a^{3} b^{2} d^{2} e^{3}-1050 A \,a^{2} b^{3} d^{3} e^{2}-525 A a \,b^{4} d^{4} e +35 A \,b^{5} d^{5}+6 B \,a^{5} d \,e^{4}-56 B \,a^{4} b \,d^{2} e^{3}+420 B \,a^{3} b^{2} d^{3} e^{2}+840 B \,a^{2} b^{3} d^{4} e +70 B a \,b^{4} d^{5}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (b x +a \right )^{\frac {3}{2}} \left (a e -b d \right )^{6}}\) | \(653\) |
| gosper | \(-\frac {2 \left (-1280 A \,b^{5} e^{5} x^{5}+896 B a \,b^{4} e^{5} x^{5}+384 B \,b^{5} d \,e^{4} x^{5}-1920 A a \,b^{4} e^{5} x^{4}-4480 A \,b^{5} d \,e^{4} x^{4}+1344 B \,a^{2} b^{3} e^{5} x^{4}+3712 B a \,b^{4} d \,e^{4} x^{4}+1344 B \,b^{5} d^{2} e^{3} x^{4}-480 A \,a^{2} b^{3} e^{5} x^{3}-6720 A a \,b^{4} d \,e^{4} x^{3}-5600 A \,b^{5} d^{2} e^{3} x^{3}+336 B \,a^{3} b^{2} e^{5} x^{3}+4848 B \,a^{2} b^{3} d \,e^{4} x^{3}+5936 B a \,b^{4} d^{2} e^{3} x^{3}+1680 B \,b^{5} d^{3} e^{2} x^{3}+80 A \,a^{3} b^{2} e^{5} x^{2}-1680 A \,a^{2} b^{3} d \,e^{4} x^{2}-8400 A a \,b^{4} d^{2} e^{3} x^{2}-2800 A \,b^{5} d^{3} e^{2} x^{2}-56 B \,a^{4} b \,e^{5} x^{2}+1152 B \,a^{3} b^{2} d \,e^{4} x^{2}+6384 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+4480 B a \,b^{4} d^{3} e^{2} x^{2}+840 B \,b^{5} d^{4} e \,x^{2}-30 A \,a^{4} b \,e^{5} x +280 A \,a^{3} b^{2} d \,e^{4} x -2100 A \,a^{2} b^{3} d^{2} e^{3} x -4200 A a \,b^{4} d^{3} e^{2} x -350 A \,b^{5} d^{4} e x +21 B \,a^{5} e^{5} x -187 B \,a^{4} b d \,e^{4} x +1386 B \,a^{3} b^{2} d^{2} e^{3} x +3570 B \,a^{2} b^{3} d^{3} e^{2} x +1505 B a \,b^{4} d^{4} e x +105 B \,b^{5} d^{5} x +15 A \,a^{5} e^{5}-105 A \,a^{4} b d \,e^{4}+350 A \,a^{3} b^{2} d^{2} e^{3}-1050 A \,a^{2} b^{3} d^{3} e^{2}-525 A a \,b^{4} d^{4} e +35 A \,b^{5} d^{5}+6 B \,a^{5} d \,e^{4}-56 B \,a^{4} b \,d^{2} e^{3}+420 B \,a^{3} b^{2} d^{3} e^{2}+840 B \,a^{2} b^{3} d^{4} e +70 B a \,b^{4} d^{5}\right )}{105 \left (b x +a \right )^{\frac {3}{2}} \left (e x +d \right )^{\frac {7}{2}} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}\) | \(722\) |
-2/105*(-1280*A*b^5*e^5*x^5+896*B*a*b^4*e^5*x^5+384*B*b^5*d*e^4*x^5-1920*A *a*b^4*e^5*x^4-4480*A*b^5*d*e^4*x^4+1344*B*a^2*b^3*e^5*x^4+3712*B*a*b^4*d* e^4*x^4+1344*B*b^5*d^2*e^3*x^4-480*A*a^2*b^3*e^5*x^3-6720*A*a*b^4*d*e^4*x^ 3-5600*A*b^5*d^2*e^3*x^3+336*B*a^3*b^2*e^5*x^3+4848*B*a^2*b^3*d*e^4*x^3+59 36*B*a*b^4*d^2*e^3*x^3+1680*B*b^5*d^3*e^2*x^3+80*A*a^3*b^2*e^5*x^2-1680*A* a^2*b^3*d*e^4*x^2-8400*A*a*b^4*d^2*e^3*x^2-2800*A*b^5*d^3*e^2*x^2-56*B*a^4 *b*e^5*x^2+1152*B*a^3*b^2*d*e^4*x^2+6384*B*a^2*b^3*d^2*e^3*x^2+4480*B*a*b^ 4*d^3*e^2*x^2+840*B*b^5*d^4*e*x^2-30*A*a^4*b*e^5*x+280*A*a^3*b^2*d*e^4*x-2 100*A*a^2*b^3*d^2*e^3*x-4200*A*a*b^4*d^3*e^2*x-350*A*b^5*d^4*e*x+21*B*a^5* e^5*x-187*B*a^4*b*d*e^4*x+1386*B*a^3*b^2*d^2*e^3*x+3570*B*a^2*b^3*d^3*e^2* x+1505*B*a*b^4*d^4*e*x+105*B*b^5*d^5*x+15*A*a^5*e^5-105*A*a^4*b*d*e^4+350* A*a^3*b^2*d^2*e^3-1050*A*a^2*b^3*d^3*e^2-525*A*a*b^4*d^4*e+35*A*b^5*d^5+6* B*a^5*d*e^4-56*B*a^4*b*d^2*e^3+420*B*a^3*b^2*d^3*e^2+840*B*a^2*b^3*d^4*e+7 0*B*a*b^4*d^5)/(e*x+d)^(7/2)/(b*x+a)^(3/2)/(a*e-b*d)^6
Leaf count of result is larger than twice the leaf count of optimal. 1290 vs. \(2 (262) = 524\).
Time = 55.47 (sec) , antiderivative size = 1290, normalized size of antiderivative = 4.33 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx=\text {Too large to display} \]
-2/105*(15*A*a^5*e^5 + 35*(2*B*a*b^4 + A*b^5)*d^5 + 105*(8*B*a^2*b^3 - 5*A *a*b^4)*d^4*e + 210*(2*B*a^3*b^2 - 5*A*a^2*b^3)*d^3*e^2 - 14*(4*B*a^4*b - 25*A*a^3*b^2)*d^2*e^3 + 3*(2*B*a^5 - 35*A*a^4*b)*d*e^4 + 128*(3*B*b^5*d*e^ 4 + (7*B*a*b^4 - 10*A*b^5)*e^5)*x^5 + 64*(21*B*b^5*d^2*e^3 + 2*(29*B*a*b^4 - 35*A*b^5)*d*e^4 + 3*(7*B*a^2*b^3 - 10*A*a*b^4)*e^5)*x^4 + 16*(105*B*b^5 *d^3*e^2 + 7*(53*B*a*b^4 - 50*A*b^5)*d^2*e^3 + 3*(101*B*a^2*b^3 - 140*A*a* b^4)*d*e^4 + 3*(7*B*a^3*b^2 - 10*A*a^2*b^3)*e^5)*x^3 + 8*(105*B*b^5*d^4*e + 70*(8*B*a*b^4 - 5*A*b^5)*d^3*e^2 + 42*(19*B*a^2*b^3 - 25*A*a*b^4)*d^2*e^ 3 + 6*(24*B*a^3*b^2 - 35*A*a^2*b^3)*d*e^4 - (7*B*a^4*b - 10*A*a^3*b^2)*e^5 )*x^2 + (105*B*b^5*d^5 + 35*(43*B*a*b^4 - 10*A*b^5)*d^4*e + 210*(17*B*a^2* b^3 - 20*A*a*b^4)*d^3*e^2 + 42*(33*B*a^3*b^2 - 50*A*a^2*b^3)*d^2*e^3 - (18 7*B*a^4*b - 280*A*a^3*b^2)*d*e^4 + 3*(7*B*a^5 - 10*A*a^4*b)*e^5)*x)*sqrt(b *x + a)*sqrt(e*x + d)/(a^2*b^6*d^10 - 6*a^3*b^5*d^9*e + 15*a^4*b^4*d^8*e^2 - 20*a^5*b^3*d^7*e^3 + 15*a^6*b^2*d^6*e^4 - 6*a^7*b*d^5*e^5 + a^8*d^4*e^6 + (b^8*d^6*e^4 - 6*a*b^7*d^5*e^5 + 15*a^2*b^6*d^4*e^6 - 20*a^3*b^5*d^3*e^ 7 + 15*a^4*b^4*d^2*e^8 - 6*a^5*b^3*d*e^9 + a^6*b^2*e^10)*x^6 + 2*(2*b^8*d^ 7*e^3 - 11*a*b^7*d^6*e^4 + 24*a^2*b^6*d^5*e^5 - 25*a^3*b^5*d^4*e^6 + 10*a^ 4*b^4*d^3*e^7 + 3*a^5*b^3*d^2*e^8 - 4*a^6*b^2*d*e^9 + a^7*b*e^10)*x^5 + (6 *b^8*d^8*e^2 - 28*a*b^7*d^7*e^3 + 43*a^2*b^6*d^6*e^4 - 6*a^3*b^5*d^5*e^5 - 55*a^4*b^4*d^4*e^6 + 64*a^5*b^3*d^3*e^7 - 27*a^6*b^2*d^2*e^8 + 2*a^7*b...
\[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx=\int \frac {A + B x}{\left (a + b x\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {9}{2}}}\, dx \]
Exception generated. \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e-b*d)>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 3720 vs. \(2 (262) = 524\).
Time = 2.66 (sec) , antiderivative size = 3720, normalized size of antiderivative = 12.48 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx=\text {Too large to display} \]
-2/105*(((b*x + a)*((279*B*b^22*d^13*e^7*abs(b) - 2837*B*a*b^21*d^12*e^8*a bs(b) - 790*A*b^22*d^12*e^8*abs(b) + 12282*B*a^2*b^20*d^11*e^9*abs(b) + 94 80*A*a*b^21*d^11*e^9*abs(b) - 27654*B*a^3*b^19*d^10*e^10*abs(b) - 52140*A* a^2*b^20*d^10*e^10*abs(b) + 25685*B*a^4*b^18*d^9*e^11*abs(b) + 173800*A*a^ 3*b^19*d^9*e^11*abs(b) + 31977*B*a^5*b^17*d^8*e^12*abs(b) - 391050*A*a^4*b ^18*d^8*e^12*abs(b) - 146916*B*a^6*b^16*d^7*e^13*abs(b) + 625680*A*a^5*b^1 7*d^7*e^13*abs(b) + 251196*B*a^7*b^15*d^6*e^14*abs(b) - 729960*A*a^6*b^16* d^6*e^14*abs(b) - 266607*B*a^8*b^14*d^5*e^15*abs(b) + 625680*A*a^7*b^15*d^ 5*e^15*abs(b) + 191565*B*a^9*b^13*d^4*e^16*abs(b) - 391050*A*a^8*b^14*d^4* e^16*abs(b) - 94006*B*a^10*b^12*d^3*e^17*abs(b) + 173800*A*a^9*b^13*d^3*e^ 17*abs(b) + 30378*B*a^11*b^11*d^2*e^18*abs(b) - 52140*A*a^10*b^12*d^2*e^18 *abs(b) - 5853*B*a^12*b^10*d*e^19*abs(b) + 9480*A*a^11*b^11*d*e^19*abs(b) + 511*B*a^13*b^9*e^20*abs(b) - 790*A*a^12*b^10*e^20*abs(b))*(b*x + a)/(b^2 2*d^18*e^3 - 18*a*b^21*d^17*e^4 + 153*a^2*b^20*d^16*e^5 - 816*a^3*b^19*d^1 5*e^6 + 3060*a^4*b^18*d^14*e^7 - 8568*a^5*b^17*d^13*e^8 + 18564*a^6*b^16*d ^12*e^9 - 31824*a^7*b^15*d^11*e^10 + 43758*a^8*b^14*d^10*e^11 - 48620*a^9* b^13*d^9*e^12 + 43758*a^10*b^12*d^8*e^13 - 31824*a^11*b^11*d^7*e^14 + 1856 4*a^12*b^10*d^6*e^15 - 8568*a^13*b^9*d^5*e^16 + 3060*a^14*b^8*d^4*e^17 - 8 16*a^15*b^7*d^3*e^18 + 153*a^16*b^6*d^2*e^19 - 18*a^17*b^5*d*e^20 + a^18*b ^4*e^21) + 7*(132*B*b^23*d^14*e^6*abs(b) - 1483*B*a*b^22*d^13*e^7*abs(b...
Time = 3.68 (sec) , antiderivative size = 596, normalized size of antiderivative = 2.00 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{9/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {12\,B\,a^5\,d\,e^4+30\,A\,a^5\,e^5-112\,B\,a^4\,b\,d^2\,e^3-210\,A\,a^4\,b\,d\,e^4+840\,B\,a^3\,b^2\,d^3\,e^2+700\,A\,a^3\,b^2\,d^2\,e^3+1680\,B\,a^2\,b^3\,d^4\,e-2100\,A\,a^2\,b^3\,d^3\,e^2+140\,B\,a\,b^4\,d^5-1050\,A\,a\,b^4\,d^4\,e+70\,A\,b^5\,d^5}{105\,b\,e^4\,{\left (a\,e-b\,d\right )}^6}+\frac {256\,b^3\,x^5\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )}{105\,{\left (a\,e-b\,d\right )}^6}+\frac {16\,x^2\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )\,\left (-a^3\,e^3+21\,a^2\,b\,d\,e^2+105\,a\,b^2\,d^2\,e+35\,b^3\,d^3\right )}{105\,e^3\,{\left (a\,e-b\,d\right )}^6}+\frac {128\,b^2\,x^4\,\left (3\,a\,e+7\,b\,d\right )\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )}{105\,e\,{\left (a\,e-b\,d\right )}^6}+\frac {32\,b\,x^3\,\left (3\,a^2\,e^2+42\,a\,b\,d\,e+35\,b^2\,d^2\right )\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )}{105\,e^2\,{\left (a\,e-b\,d\right )}^6}+\frac {2\,x\,\left (7\,B\,a\,e-10\,A\,b\,e+3\,B\,b\,d\right )\,\left (3\,a^4\,e^4-28\,a^3\,b\,d\,e^3+210\,a^2\,b^2\,d^2\,e^2+420\,a\,b^3\,d^3\,e+35\,b^4\,d^4\right )}{105\,b\,e^4\,{\left (a\,e-b\,d\right )}^6}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a\,d^4\,\sqrt {a+b\,x}}{b\,e^4}+\frac {x^4\,\left (a\,e+4\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e}+\frac {2\,d\,x^3\,\left (2\,a\,e+3\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e^2}+\frac {d^3\,x\,\left (4\,a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e^4}+\frac {2\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e^3}} \]
-((d + e*x)^(1/2)*((30*A*a^5*e^5 + 70*A*b^5*d^5 + 140*B*a*b^4*d^5 + 12*B*a ^5*d*e^4 + 1680*B*a^2*b^3*d^4*e - 112*B*a^4*b*d^2*e^3 - 2100*A*a^2*b^3*d^3 *e^2 + 700*A*a^3*b^2*d^2*e^3 + 840*B*a^3*b^2*d^3*e^2 - 1050*A*a*b^4*d^4*e - 210*A*a^4*b*d*e^4)/(105*b*e^4*(a*e - b*d)^6) + (256*b^3*x^5*(7*B*a*e - 1 0*A*b*e + 3*B*b*d))/(105*(a*e - b*d)^6) + (16*x^2*(7*B*a*e - 10*A*b*e + 3* B*b*d)*(35*b^3*d^3 - a^3*e^3 + 105*a*b^2*d^2*e + 21*a^2*b*d*e^2))/(105*e^3 *(a*e - b*d)^6) + (128*b^2*x^4*(3*a*e + 7*b*d)*(7*B*a*e - 10*A*b*e + 3*B*b *d))/(105*e*(a*e - b*d)^6) + (32*b*x^3*(3*a^2*e^2 + 35*b^2*d^2 + 42*a*b*d* e)*(7*B*a*e - 10*A*b*e + 3*B*b*d))/(105*e^2*(a*e - b*d)^6) + (2*x*(7*B*a*e - 10*A*b*e + 3*B*b*d)*(3*a^4*e^4 + 35*b^4*d^4 + 210*a^2*b^2*d^2*e^2 + 420 *a*b^3*d^3*e - 28*a^3*b*d*e^3))/(105*b*e^4*(a*e - b*d)^6)))/(x^5*(a + b*x) ^(1/2) + (a*d^4*(a + b*x)^(1/2))/(b*e^4) + (x^4*(a*e + 4*b*d)*(a + b*x)^(1 /2))/(b*e) + (2*d*x^3*(2*a*e + 3*b*d)*(a + b*x)^(1/2))/(b*e^2) + (d^3*x*(4 *a*e + b*d)*(a + b*x)^(1/2))/(b*e^4) + (2*d^2*x^2*(3*a*e + 2*b*d)*(a + b*x )^(1/2))/(b*e^3))