Integrand size = 25, antiderivative size = 112 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{7/2}} \, dx=-\frac {2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right )}{5 b^4 (a+b x)^{5/2}}-\frac {2 \left (b^2 B-2 a b C+3 a^2 D\right )}{3 b^4 (a+b x)^{3/2}}-\frac {2 (b C-3 a D)}{b^4 \sqrt {a+b x}}+\frac {2 D \sqrt {a+b x}}{b^4} \] Output:
1/5*(-2*A*b^3+2*a*(B*b^2-C*a*b+D*a^2))/b^4/(b*x+a)^(5/2)-2/3*(B*b^2-2*C*a* b+3*D*a^2)/b^4/(b*x+a)^(3/2)-2*(C*b-3*D*a)/b^4/(b*x+a)^(1/2)+2*D*(b*x+a)^( 1/2)/b^4
Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{7/2}} \, dx=\frac {2 \left (-3 A b^3+48 a^3 D-8 a^2 b (C-15 D x)-2 a b^2 (B+5 x (2 C-9 D x))-5 b^3 x (B+3 x (C-D x))\right )}{15 b^4 (a+b x)^{5/2}} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/(a + b*x)^(7/2),x]
Output:
(2*(-3*A*b^3 + 48*a^3*D - 8*a^2*b*(C - 15*D*x) - 2*a*b^2*(B + 5*x*(2*C - 9 *D*x)) - 5*b^3*x*(B + 3*x*(C - D*x))))/(15*b^4*(a + b*x)^(5/2))
Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2389, 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.
\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{7/2}} \, dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{b^3 (a+b x)^{7/2}}+\frac {3 a^2 D-2 a b C+b^2 B}{b^3 (a+b x)^{5/2}}+\frac {b C-3 a D}{b^3 (a+b x)^{3/2}}+\frac {D}{b^3 \sqrt {a+b x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{5 b^4 (a+b x)^{5/2}}-\frac {2 \left (3 a^2 D-2 a b C+b^2 B\right )}{3 b^4 (a+b x)^{3/2}}-\frac {2 (b C-3 a D)}{b^4 \sqrt {a+b x}}+\frac {2 D \sqrt {a+b x}}{b^4}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/(a + b*x)^(7/2),x]
Output:
(-2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D)))/(5*b^4*(a + b*x)^(5/2)) - (2*(b^2 *B - 2*a*b*C + 3*a^2*D))/(3*b^4*(a + b*x)^(3/2)) - (2*(b*C - 3*a*D))/(b^4* Sqrt[a + b*x]) + (2*D*Sqrt[a + b*x])/b^4
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.38 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.66
method | result | size |
pseudoelliptic | \(\frac {\left (30 D x^{3}-30 C \,x^{2}-10 B x -6 A \right ) b^{3}-4 a \left (-45 D x^{2}+10 C x +B \right ) b^{2}-16 a^{2} \left (-15 D x +C \right ) b +96 a^{3} D}{15 \left (b x +a \right )^{\frac {5}{2}} b^{4}}\) | \(74\) |
gosper | \(-\frac {2 \left (-15 D x^{3} b^{3}+15 C \,x^{2} b^{3}-90 D x^{2} a \,b^{2}+5 b^{3} B x +20 C x a \,b^{2}-120 D x \,a^{2} b +3 b^{3} A +2 a \,b^{2} B +8 a^{2} b C -48 a^{3} D\right )}{15 \left (b x +a \right )^{\frac {5}{2}} b^{4}}\) | \(91\) |
trager | \(-\frac {2 \left (-15 D x^{3} b^{3}+15 C \,x^{2} b^{3}-90 D x^{2} a \,b^{2}+5 b^{3} B x +20 C x a \,b^{2}-120 D x \,a^{2} b +3 b^{3} A +2 a \,b^{2} B +8 a^{2} b C -48 a^{3} D\right )}{15 \left (b x +a \right )^{\frac {5}{2}} b^{4}}\) | \(91\) |
orering | \(-\frac {2 \left (-15 D x^{3} b^{3}+15 C \,x^{2} b^{3}-90 D x^{2} a \,b^{2}+5 b^{3} B x +20 C x a \,b^{2}-120 D x \,a^{2} b +3 b^{3} A +2 a \,b^{2} B +8 a^{2} b C -48 a^{3} D\right )}{15 \left (b x +a \right )^{\frac {5}{2}} b^{4}}\) | \(91\) |
derivativedivides | \(\frac {2 D \sqrt {b x +a}-\frac {2 \left (C b -3 D a \right )}{\sqrt {b x +a}}-\frac {2 \left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right )}{5 \left (b x +a \right )^{\frac {5}{2}}}-\frac {2 \left (B \,b^{2}-2 C a b +3 D a^{2}\right )}{3 \left (b x +a \right )^{\frac {3}{2}}}}{b^{4}}\) | \(93\) |
default | \(\frac {2 D \sqrt {b x +a}-\frac {2 \left (C b -3 D a \right )}{\sqrt {b x +a}}-\frac {2 \left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right )}{5 \left (b x +a \right )^{\frac {5}{2}}}-\frac {2 \left (B \,b^{2}-2 C a b +3 D a^{2}\right )}{3 \left (b x +a \right )^{\frac {3}{2}}}}{b^{4}}\) | \(93\) |
Input:
int((D*x^3+C*x^2+B*x+A)/(b*x+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/15*((30*D*x^3-30*C*x^2-10*B*x-6*A)*b^3-4*a*(-45*D*x^2+10*C*x+B)*b^2-16*a ^2*(-15*D*x+C)*b+96*a^3*D)/(b*x+a)^(5/2)/b^4
Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{7/2}} \, dx=\frac {2 \, {\left (15 \, D b^{3} x^{3} + 48 \, D a^{3} - 8 \, C a^{2} b - 2 \, B a b^{2} - 3 \, A b^{3} + 15 \, {\left (6 \, D a b^{2} - C b^{3}\right )} x^{2} + 5 \, {\left (24 \, D a^{2} b - 4 \, C a b^{2} - B b^{3}\right )} x\right )} \sqrt {b x + a}}{15 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^(7/2),x, algorithm="fricas")
Output:
2/15*(15*D*b^3*x^3 + 48*D*a^3 - 8*C*a^2*b - 2*B*a*b^2 - 3*A*b^3 + 15*(6*D* a*b^2 - C*b^3)*x^2 + 5*(24*D*a^2*b - 4*C*a*b^2 - B*b^3)*x)*sqrt(b*x + a)/( b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (112) = 224\).
Time = 0.47 (sec) , antiderivative size = 629, normalized size of antiderivative = 5.62 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{7/2}} \, dx=\begin {cases} - \frac {6 A b^{3}}{15 a^{2} b^{4} \sqrt {a + b x} + 30 a b^{5} x \sqrt {a + b x} + 15 b^{6} x^{2} \sqrt {a + b x}} - \frac {4 B a b^{2}}{15 a^{2} b^{4} \sqrt {a + b x} + 30 a b^{5} x \sqrt {a + b x} + 15 b^{6} x^{2} \sqrt {a + b x}} - \frac {10 B b^{3} x}{15 a^{2} b^{4} \sqrt {a + b x} + 30 a b^{5} x \sqrt {a + b x} + 15 b^{6} x^{2} \sqrt {a + b x}} - \frac {16 C a^{2} b}{15 a^{2} b^{4} \sqrt {a + b x} + 30 a b^{5} x \sqrt {a + b x} + 15 b^{6} x^{2} \sqrt {a + b x}} - \frac {40 C a b^{2} x}{15 a^{2} b^{4} \sqrt {a + b x} + 30 a b^{5} x \sqrt {a + b x} + 15 b^{6} x^{2} \sqrt {a + b x}} - \frac {30 C b^{3} x^{2}}{15 a^{2} b^{4} \sqrt {a + b x} + 30 a b^{5} x \sqrt {a + b x} + 15 b^{6} x^{2} \sqrt {a + b x}} + \frac {96 D a^{3}}{15 a^{2} b^{4} \sqrt {a + b x} + 30 a b^{5} x \sqrt {a + b x} + 15 b^{6} x^{2} \sqrt {a + b x}} + \frac {240 D a^{2} b x}{15 a^{2} b^{4} \sqrt {a + b x} + 30 a b^{5} x \sqrt {a + b x} + 15 b^{6} x^{2} \sqrt {a + b x}} + \frac {180 D a b^{2} x^{2}}{15 a^{2} b^{4} \sqrt {a + b x} + 30 a b^{5} x \sqrt {a + b x} + 15 b^{6} x^{2} \sqrt {a + b x}} + \frac {30 D b^{3} x^{3}}{15 a^{2} b^{4} \sqrt {a + b x} + 30 a b^{5} x \sqrt {a + b x} + 15 b^{6} x^{2} \sqrt {a + b x}} & \text {for}\: b \neq 0 \\\frac {A x + \frac {B x^{2}}{2} + \frac {C x^{3}}{3} + \frac {D x^{4}}{4}}{a^{\frac {7}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)**(7/2),x)
Output:
Piecewise((-6*A*b**3/(15*a**2*b**4*sqrt(a + b*x) + 30*a*b**5*x*sqrt(a + b* x) + 15*b**6*x**2*sqrt(a + b*x)) - 4*B*a*b**2/(15*a**2*b**4*sqrt(a + b*x) + 30*a*b**5*x*sqrt(a + b*x) + 15*b**6*x**2*sqrt(a + b*x)) - 10*B*b**3*x/(1 5*a**2*b**4*sqrt(a + b*x) + 30*a*b**5*x*sqrt(a + b*x) + 15*b**6*x**2*sqrt( a + b*x)) - 16*C*a**2*b/(15*a**2*b**4*sqrt(a + b*x) + 30*a*b**5*x*sqrt(a + b*x) + 15*b**6*x**2*sqrt(a + b*x)) - 40*C*a*b**2*x/(15*a**2*b**4*sqrt(a + b*x) + 30*a*b**5*x*sqrt(a + b*x) + 15*b**6*x**2*sqrt(a + b*x)) - 30*C*b** 3*x**2/(15*a**2*b**4*sqrt(a + b*x) + 30*a*b**5*x*sqrt(a + b*x) + 15*b**6*x **2*sqrt(a + b*x)) + 96*D*a**3/(15*a**2*b**4*sqrt(a + b*x) + 30*a*b**5*x*s qrt(a + b*x) + 15*b**6*x**2*sqrt(a + b*x)) + 240*D*a**2*b*x/(15*a**2*b**4* sqrt(a + b*x) + 30*a*b**5*x*sqrt(a + b*x) + 15*b**6*x**2*sqrt(a + b*x)) + 180*D*a*b**2*x**2/(15*a**2*b**4*sqrt(a + b*x) + 30*a*b**5*x*sqrt(a + b*x) + 15*b**6*x**2*sqrt(a + b*x)) + 30*D*b**3*x**3/(15*a**2*b**4*sqrt(a + b*x) + 30*a*b**5*x*sqrt(a + b*x) + 15*b**6*x**2*sqrt(a + b*x)), Ne(b, 0)), ((A *x + B*x**2/2 + C*x**3/3 + D*x**4/4)/a**(7/2), True))
Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {15 \, \sqrt {b x + a} D}{b^{3}} + \frac {3 \, D a^{3} - 3 \, C a^{2} b + 3 \, B a b^{2} - 3 \, A b^{3} + 15 \, {\left (3 \, D a - C b\right )} {\left (b x + a\right )}^{2} - 5 \, {\left (3 \, D a^{2} - 2 \, C a b + B b^{2}\right )} {\left (b x + a\right )}}{{\left (b x + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{15 \, b} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^(7/2),x, algorithm="maxima")
Output:
2/15*(15*sqrt(b*x + a)*D/b^3 + (3*D*a^3 - 3*C*a^2*b + 3*B*a*b^2 - 3*A*b^3 + 15*(3*D*a - C*b)*(b*x + a)^2 - 5*(3*D*a^2 - 2*C*a*b + B*b^2)*(b*x + a))/ ((b*x + a)^(5/2)*b^3))/b
Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{7/2}} \, dx=\frac {2 \, \sqrt {b x + a} D}{b^{4}} + \frac {2 \, {\left (45 \, {\left (b x + a\right )}^{2} D a - 15 \, {\left (b x + a\right )} D a^{2} + 3 \, D a^{3} - 15 \, {\left (b x + a\right )}^{2} C b + 10 \, {\left (b x + a\right )} C a b - 3 \, C a^{2} b - 5 \, {\left (b x + a\right )} B b^{2} + 3 \, B a b^{2} - 3 \, A b^{3}\right )}}{15 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{4}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^(7/2),x, algorithm="giac")
Output:
2*sqrt(b*x + a)*D/b^4 + 2/15*(45*(b*x + a)^2*D*a - 15*(b*x + a)*D*a^2 + 3* D*a^3 - 15*(b*x + a)^2*C*b + 10*(b*x + a)*C*a*b - 3*C*a^2*b - 5*(b*x + a)* B*b^2 + 3*B*a*b^2 - 3*A*b^3)/((b*x + a)^(5/2)*b^4)
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{7/2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^{7/2}} \,d x \] Input:
int((A + B*x + C*x^2 + x^3*D)/(a + b*x)^(7/2),x)
Output:
int((A + B*x + C*x^2 + x^3*D)/(a + b*x)^(7/2), x)
Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^{7/2}} \, dx=\frac {2 b^{3} d \,x^{3}+12 a \,b^{2} d \,x^{2}-2 b^{3} c \,x^{2}+16 a^{2} b d x -\frac {8}{3} a \,b^{2} c x -\frac {2}{3} b^{4} x +\frac {32}{5} a^{3} d -\frac {16}{15} a^{2} b c -\frac {2}{3} a \,b^{3}}{\sqrt {b x +a}\, b^{4} \left (b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:
int((D*x^3+C*x^2+B*x+A)/(b*x+a)^(7/2),x)
Output:
(2*(48*a**3*d - 8*a**2*b*c + 120*a**2*b*d*x - 5*a*b**3 - 20*a*b**2*c*x + 9 0*a*b**2*d*x**2 - 5*b**4*x - 15*b**3*c*x**2 + 15*b**3*d*x**3))/(15*sqrt(a + b*x)*b**4*(a**2 + 2*a*b*x + b**2*x**2))